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WAMIT R
USER MANUAL
Versions 6.3, 6.3PC, 6.3S, 6.3S-PC

WAMIT, Inc.
www.wamit.com

R
WAMIT

USER MANUAL
Versions 6.3, 6.3PC, 6.3S, 6.3S-PC
This User Manual has been prepared for users of WAMIT Version 6.3, including the sourcecode Version 6.3, and the PC executable Version 6.3PC, and the second-order extension
V6.3S.
WAMIT is distributed by WAMIT, Inc. under a special agreement with the Massachusetts
Institute of Technology. Further information is available from the website shown below.
The WAMIT software and this User Manual are Copyrighted by WAMIT, Inc., and by
the Massachusetts Institute of Technology. The User Manual can be downloaded from
the website without a license, and individuals are permitted to make one hardcopy of the
entire manual, or portions thereof, for use in conjunction with the WAMIT demonstration
program. A hardcopy of the User Manual is provided to licensed users.

WAMIT, Inc.
822 Boylston St. – Suite 202
Chestnut Hill, MA 02467-2504
USA
phone 617-739-4488 fax 617-739-4499
www.wamit.com

0–1

c 1998-2006 WAMIT, Incorporated and Massachusetts Institute of Technology.
Copyright
No part of the WAMIT software or documentation may be reproduced, translated, or
transmitted in any form without the express written permission of WAMIT, Inc.

WAMIT is a registered trademark of WAMIT, Inc.
Compaq is a registered trademark of Compaq Computer Corporation.
Windows is a registered trademark of Microsoft Corporation.
MultiSurf and RGKernel are registered trademarks of AeroHydro, Inc.
Tecplot is a registered trademark of Amtec Engineering, Inc.

0–2

TABLE OF CONTENTS
1. INTRODUCTION
1.1
1.2
1.3
1.4
1.5

WAMIT Version 6
Changes in WAMIT Version 6.0
Additional changes in Version 6.1
Additional changes in Version 6.2
Additional changes in Version 6.3

2. GETTING STARTED
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Installation and setup
Demonstration programs
Standard test runs
Running Test 01
Running Test 11
Other test runs
Memory and storage restrictions
Modifying the input ﬁles

3. USING WAMIT
3.1 The Potential Control File (Alternative form 1)
3.2 The Potential Control File (Alternative form 2)
3.3 The Force Control File (Alternative form 1)
3.4 The Force Control File (Alternative form 2)
3.5 Deﬁnition of ﬁxed or free modes
3.6 Filenames list FNAMES.WAM
3.7 Conﬁguration ﬁle CONFIG.WAM or ∗.CFG
3.8 File names
3.9 File format
3.10 Uniform arrays of ﬁeld points
4. DEFINITION OF QUANTITIES EVALUATED BY WAMIT
4.1 Hydrostatic data
4.2 Added-mass and damping coeﬃcients
4.3 Exciting forces
4.4 Body motions in waves
4.5 Hydrodynamic pressure
4.6 Free-surface elevation
4.7 Velocity vector on the body and in the ﬂuid domain
4.8 Mean drift force and moment
4.9 Format of numeric output ﬁles
4.10 Body pressure output for the higher-order method
4.11 Body pressure and ﬂuid velocity at speciﬁed points
4.12 Radiation and diﬀraction components of the pressure and velocity
4.13 Radiation pressure and velocity for zero and inﬁnite periods

0–3

5. THE LOW-ORDER METHOD (ILOWHI=0)
5.1
5.2
5.3
5.4

The geometric data ﬁle
Use of the source formulation (ISOR=1)
Body near vertical walls
Bodies with thin submerged elements

6. THE HIGHER-ORDER METHOD (ILOWHI=1)
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13

Subdivision of the body surface in patches and panels
B-Spline representation of the solution
Order of Gauss Quadratures
The Geometric Data File
Geometry represented by low-order panels (IGDEF=0)
Geometry represented by B-Splines (IGDEF=1)
Geometry represented by MultiSurf (IGDEF=2)
Analytic representation of the geometry
Modifying the DLL Subroutine GEOMXACT
Bodies with thin submerged elements
The Optional Spline Control File
The use of default values to simplify inputs
Advantages and disadvantages of the higher-order method

7. ANALYSIS OF MULTIPLE BODIES (NBODY > 1)
7.1
7.2
7.3
7.4
7.5
7.6
7.7

Input to POTEN (IALTPOT=±1)
Input to POTEN (IALTPOT=2)
Input to FORCE (IALTFRC=1)
Input to FORCE (IALTFRC=2)
Input to FORCE (IALTFRC=3)
Parameters in the CONFIG.WAM ﬁle
Output

8. GENERALIZED BODY MODES (NEWMDS > 0)
8.1
8.2
8.3
8.4
8.5

Input Files
Using DEFMOD with the Low-Order Method
Using the DLL Subroutine NEWMODES
Hydrostatics
NBODY Analysis

9. USE OF IRREGULAR FREQUENCY OPTION (IRR > 0)
9.1 Input parameters
9.2 Automatic free-surface discretization (IRR=3)
10. SPECIAL TOPICS
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8

Error messages and log ﬁles
Reserved ﬁle names
Scratch ﬁles
Memory requirements and number of unknowns
Modifying DLL ﬁles
Auxiliary output ﬁles for the geometry
Internal tanks
Radiated waves from wavemakers in tank walls

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11. VERSION 6S (Second-order module)
11.1
11.2
11.3
11.4
11.4.1
11.4.2
11.4.3
11.4.4
11.4.5
11.4.6
11.4.7
11.5
11.6
11.7
11.8

The Conﬁguration ﬁle (CFG)
The Force control ﬁle (FRC)
The Potential control ﬁle 2 (PT2)
The Free surface data ﬁle (FDF)
Alternative form 1 FDF
Alternative form 2 FDF
Partion circle and inner circle
Restriction on very long waves
Approximation without free surface forcing
Higher-order method (ILOWHI=1)
Visualizing FDF
Fnames.wam
Deﬁnitions of output
Numeric output ﬁles
Installation of V6.3S-PC

12. THEORY
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11

The boundary-value problem
Integral equations for the velocity potential
Integral equations for the source formulation
Discretization of the integral equations in the low-order method (ILOWHI=0)
Discretization of the integral equations in the higher-order method (ILOWHI=1)
Solution of the linear systems
Mean drift force and moment based on pressure integration
Removal of irregular frequencies
Integral equation for bodies with zero thickness
Mean drift force and moment using control surface
Internal tank eﬀects

13. THE F2T UTILITY
13.1
13.2
13.3
13.4
13.5
13.6

Deﬁnitions of radiation and diﬀraction outputs
Acquiring input data for F2T with WAMIT
How to use F2T
Output ﬁles
Options 5,6,7
Theory

14. MEAN DRIFT FORCES USING CONTROL SURFACE
14.1
14.2
14.3
14.4

Control surface ﬁle (CSF)
Low-order control surface ﬁle
Higher-order control surface ﬁle
Output

REFERENCES

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APPENDIX A – DESCRIPTION OF TEST RUNS
A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.11
A.12
A.13
A.14
A.15
A.16
A.17
A.18
A.19
A.20
A.21
A.22
A.23

Truncated vertical cylinder (TEST01)
Irregular-frequency removal (TEST02)
Local pressure drift force (TEST03)
Body near a wall (TEST04)
Multiple bodies (TEST05)
The ISSC Tension-Leg Platform (TEST06)
The ISSC Tension-Leg Platform (TEST07)
Elastic column with generalized modes (TEST08)
Spar with three strakes (TEST09)
Higher-order analysis of circular cylinder (TEST11)
Irregular-frequency removal (TEST12)
Multiple bodies (TEST13)
ISSC TLP (TEST14)
Semi-Sub (TEST15)
Barge with bending modes (TEST16)
Cylinder with moonpool (TEST17)
Elastic column (TEST18)
Catamaran barge (TEST19)
MultiSurf Barge (TEST20)
Spar with three strakes (TEST21)
FPSO with two internal tanks (TEST22)
Radiated wave ﬁeld from a bank of wavemakers (TEST23)

A.24 Motions of a hinged vessel (TEST24)
APPENDIX B – DESCRIPTION OF SECOND-ORDER TEST RUNS
B.1 Bottom mounted cylinder (TEST101)
B.2 Truncated circular cylinder (TEST102)
B.3 Multiple bodies (Approximation) (TEST103)
B.3A Multiple bodies (TEST103A)
B.11 Bottom mounted cylinder (TEST111)
B.12 Truncated circular cylinder (TEST112)
B.13 Multiple bodies (Approximation) (TEST113)
B.13A Multiple bodies (TEST113A)
APPENDIX C – USING THE WAMIT-RGKERNEL INTERFACE (By J. S. Letcher, Jr.)

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Chapter 1

INTRODUCTION
This manual describes the use of WAMIT (Version 6.3). This program exists in the standard source-code form, designated as ‘V6.3’, and in the PC-executable version ‘V6.3PC’
which is logically equivalent to ‘V6.3’. A chapter concerning the use of the secondorder extension ‘V6.3S’ is included for completeness. V6.3S includes the analysis for the
second-order solution in bichromatic and bidirectional waves, including sum- and diﬀerencefrequency components.
V6.3 is a radiation/diﬀraction panel program developed for the linear analysis of the
interaction of surface waves with oﬀshore structures. A major feature of V6.0 and higher
is the option to use a higher-order method based on B-splines to represent the velocity
potential and pressure on the body surface, and to represent the geometry of the body
surface in various manners. In addition to the conventional low-order panel discretization,
these include new options to represent the body surface more accurately and/or with less
work.
In V6.1, an option was introduced to link WAMIT with the Relational Geometry Kernel
(RGKernel) of the CAD program MultiSurf With this option MultiSurf can be used to
represent a wide variety of body geometries, and the same respresentation can be input
directly in the higher-order solution of WAMIT. With this technique there is no need to
approximate the geometry or to prepare extensive input ﬁles. (To take advantage of this
approach the user must license MultiSurf and special .DLL ﬁles in addition to the standard
WAMIT software.)
Starting in V6.2, a separate utility program, F2T (Frequency-to-Time domain), has
been provided. F2T evaluates time-domain impulse-response functions (IRF’s), from the
outputs of WAMIT. IRF’s can be evaluated for any combinations of linear outputs options
1-7 described in Chapter 4. IRFs may be useful for the time-domain simulation based on
the linear hydrodynamic loads and other nonlinear constraints or excitations.
The execution of WAMIT has been simpliﬁed relative to earlier versions where two main
programs (‘POTEN’ and ‘FORCE’) were invoked sequentially. In Version 6.0 and higher
only one main program ‘WAMIT’ is used, with options to control the separate functions
of the earlier two programs in the most eﬀective manner for each application.

1–1

The remainder of this Chapter gives a general description of WAMIT Version 6.3 and
changes made from earlier versions. Users of earlier versions should refer particularly to
Sections 1.2, 1.3, 1.4, and 1.5 concerning changes and extensions introduced in Versions
6.0, 6.1, 6.2, and 6.3, respectively. A mark in the left margin, as on this line, is used to
call attention to changes in Version 6.3.
The description and use of WAMIT for both the low-order and higher-order methods
of solution has been uniﬁed as much as possible. Most but not all of the input and output
ﬁles are ‘generic’, applicable to both methods in the same form. The principal exception is
the Geometric Data File, which speciﬁes the geometry of the body surface. To simplify the
understanding and use of this User Manual, chapters are organized separately for generic
information common to both methods, and for speciﬁc information which refers to either
the low- or higher-order method separately.
In Chapter 2 a tutorial description is given to help the user get started using WAMIT
in the PC environment. Source-code users may also beneﬁt by reading this tutorial, or
following the indicated examples on a PC. The examples described in Chapter 2 are for
the simplest context of a single body.
Chapter 3 gives more detailed information regarding the generic input ﬁles, including the
Potential Control File (POT) and Force Control File (FRC) which specify the principal
non-geometric inputs for WAMIT. Also described in this Chapter are the optional ﬁles
FNAMES.WAM and CONFIG.WAM, which are useful to specify input ﬁlenames and to
specify various options.
Chapter 4 deﬁnes the various quantities which can be evaluated by WAMIT, and which
are contained in the output ﬁles.
Chapter 5 describes special topics which are relevant only to the low-order method. This
includes the Geometric Data File (GDF), which deﬁnes the coordinates of panel vertices,
the use of the source formulation to evaluate the ﬂuid velocity and second-order mean
pressure on the body surface, the analysis of a body in the presence of vertical walls, and
the analysis of bodies with thin elements such as damping plates or strakes.
Chapter 6 describes special topics which are relevant only to the higher-order method,
including the subdivision used to represent the body surface and velocity potential on this
surface, and the representation of the potential in terms of B-splines. Alternative methods
for deﬁning the body geometry are described including the use of low-order panels, the use
of B-splines to provide a higher-order continuous deﬁnition, the use of explicit analytical
formulae, and the use of MultiSurf geometry ﬁles. Starting with V6.2 WAMIT permits
’dipole patches’ to represent thin submerged elements and their use is described in Section
6.10.
Chapters 7-9, which are generic, describe the use of extended features in WAMIT. These
include the analysis of multiple interacting bodies (Chapter 7). the use of generalized modes
of body motion which can be used to describe structural deformations, motions of hinged
bodies, etc. (Chapter 8), and the use of a method to remove the eﬀect of the irregular
frequencies (Chapter 9).
Chapter 10 describes the relationships between memory requirements, numbers of un1–2

knowns, and input parameters, to provide a qualitative basis for estimating the requirements for RAM and hard disk storage, and for estimating run times. This Chapter also
outlines the procedure for modiﬁcation and use of .dll ﬁles, which can be used in Version
6 to describe the geometry and generalized modes.
Chapter 11 describes the second-order module V6.3S, including descriptions of the additional input ﬁles and deﬁnitions of the additional outputs.
A brief outline for the theoretical basis of WAMIT is presented in Chapter 12. Reference
26 contains a more complete review of the pertinent theory.
Chapter 13 describes the utility F2T (Frequency-to-Time domain) which can be used to
transform the linear WAMIT outputs to the corresponding time-domain impulse-response
functions.
A list of pertinent references and descriptions of the test runs are appended.

1.1

WAMIT Version 6

WAMIT is a radiation/diﬀraction program developed for the analysis of the interaction
of surface waves with oﬀshore structures. WAMIT is based on a three-dimensional panel
method, following the theory which is outlined in Chapter 12. The main program consists
of two top-level sub-programs POTEN and FORCE which evaluate the velocity potentials
and desired hydrodynamic parameters, respectively. The water depth can be inﬁnite or
ﬁnite, and either one or multiple interacting bodies can be analyzed. The bodies may
be located on the free surface, submerged, or mounted on the sea bottom. A variety of
options permit the dynamic analysis of bodies which are freely ﬂoating, restrained, or ﬁxed
in position.
The ﬂow is assumed to be ideal and time-harmonic. The free-surface condition is linearized (except in Version 6.1S where the second-order free-surface condition and body
boundary conditions are imposed). We refer to this as the ‘linear’ or ‘ﬁrst-order’ analysis. Mean second-order forces are included in this analysis, since they can be computed
rigorously from the linear solution.
The radiation and diﬀraction velocity potentials on the body wetted surface are determined from the solution of an integral equation obtained by using Green’s theorem with
the free-surface source-potential as the Green function.
All earlier versions of WAMIT were based strictly on the low-order panel method, where
the geometric form of the submerged body surface is deﬁned by ﬂat quadrilateral elements
(low-order panels), and the solutions for the velocity potential and/or source strength are
assumed constant on each panel. WAMIT Version 6 has been extended to include the
complementary higher-order panel method based on a continuous B-spline representation
for the velocity potential, and several alternative schemes for deﬁning the body surface
including explicit analytic formulae. The order of the B-splines is controlled by userspeciﬁed input parameters.
The two diﬀerent uses of the word order should be noted to avoid confusion. Following
1–3

the usual conventions of marine hydrodynamics, first-order and second-order are always
used here to refer to linearization of the boundary conditions and solution, whereas loworder and higher-order are used to refer to the method for representation of the body
surface and solution.
The following quantities can be evaluated by WAMIT Version 6.3:
• Hydrostatic coeﬃcients
• Added-mass and damping coeﬃcients for all modes
• Added-mass coeﬃcients for the limiting cases of zero or inﬁnite wave periods
• Wave exciting forces and moments using the Haskind relations, or directly by
pressure-integration from the solutions of the diﬀraction or scattering problems.
• Motion amplitudes and phases for a freely-ﬂoating body
• Forces restraining a body which is freely-ﬂoating in some but not all modes
• Hydrodynamic pressure and ﬂuid velocity on the body surface
• Hydrodynamic pressure and ﬂuid velocity in the ﬂuid domain
• Free-surface elevation
• Horizontal drift forces and mean yaw moment by momentum integration
• All quantities listed above for user-speciﬁed generalized modes
• All components of the drift force and moment by pressure integration
• Drift force and moment in bidirectional waves
The following additional quantities can be evaluated by WAMIT Version 6.3S:
• Second-order forces on ﬁxed or ﬂoating bodies
• Second-order pressure on the body surface
• Second-order pressure force on the waterline
• Second-order pressure in the ﬂuid domain
• Second-order free-surface elevation
• Second-order motion amplitude on ﬂoating bodies
Two, one or no planes of geometric symmetry may be present. Part or all of the rigidbody modes can be analyzed. The program is designed to optimize the use of the available
storage and minimize the computational eﬀort for the speciﬁed planes of symmetry and
modes. Matrix elements that share evaluations of the wave source potential are evaluated
simultaneously.
1–4

Several techniques have been developed and implemented in WAMIT to improve the
accuracy and eﬃciency of the solution and exploit the capabilities of a wide range of
contemporary computing systems, ranging from personal computers to supercomputers.
Important features of WAMIT include the use of special algorithms for the evaluation of
the free-surface wave-source potential, the option to use direct, iterative, or block-iterative
solution algorithms for the complex matrix equation, and the option to use either the loworder or higher-order panel methods. In combination these result in a fast, versatile, and
robust code capable of analyzing oﬀshore structures with complicated geometry.
WAMIT is designed to be ﬂexible in its use with a variety of practical applications. It
consists of two subprograms, POTEN and FORCE, which normally are run sequentially.
POTEN solves for the radiation and diﬀraction velocity potentials (and source strengths)
on the body surface for the speciﬁed modes, frequencies and wave headings. FORCE
computes global quantities including the hydrodynamic coeﬃcients, motions, and ﬁrstand second-order forces. Velocities and pressures on the body surface are evaluated by
FORCE. Additional ﬁeld data may also be evaluated by FORCE, including velocities and
pressures at speciﬁed positions in the ﬂuid domain and wave elevations on the free surface.
Since the principal computational burden is in POTEN, the intermediate output data from
this subprogram is saved in a binary ‘P2F’ ﬁle. Thus it is possible to make several runs
with FORCE, varying the requested parameters to be output, without re-running POTEN
in each instance. (V6.3S is an exception where the computational burden in FORCE can
be greater than that in POTEN.)
In Version 6 the separate functions of POTEN and FORCE have been integrated into
a single main program WAMIT, for ease of use. This diﬀers from all prior versions where
two separate main programs were employed. The use of Version 6 is initiated in all cases
by executing the single command ‘WAMIT’. This integration simpliﬁes the use of WAMIT
in typical applications where POTEN and FORCE are required sequentially, and options
can be speciﬁed in the conﬁguration ﬁle to perform separate runs of either subprogram, as
explained in Chapter 2 and in Section 3.7.
Figure 1.1 shows the architecture of the two subprograms and the principal input/output
ﬁles. (For simpliﬁcation this ﬁgure does not include additional input ﬁles required for the
case of multiple bodies, the optional spline control ﬁle which may be used to vary B-spline
parameters in the higher-order analysis, additional input ﬁles which are required for the
second-order analysis program V6.3S, and three output ﬁles which log errors, warnings,
and other auxiliary data.)
The analysis for the generalized modes also requires an additional input ﬁle or special
subroutine, to deﬁne the user-speciﬁed modes. Figure 9.1 in Chapter 9 shows the ﬂow
chart of POTEN for this case.
WAMIT Version 6 is written in FORTRAN-90 (ANSI/ISO Standard) and has been
tested on a wide variety of contemporary computer systems. The subroutine DATTIM,
which is used to set the date and time of the run, uses a generic F-90 intrinsic function
and no longer requires modiﬁcation of the source code for use on diﬀerent systems. The
only non-standard extension which has been retained from previous versions of WAMIT is
the dollar-sign prompt ‘$’, which facilitates interactive data input. Source-code users who
1–5

gdf
pot
GDF(gdf)

OR

POTEN

POT(pot)

pot.P2F

FNAMES.WAM
gdf
pot
frc

CONFIG.WAM

FORCE
OR
FRC(frc)

pot
frc
frc.OUT

frc.*

Figure 1.1: Flow chart of WAMIT showing the subprograms POTEN and FORCE with their associated
input and output files. Filenames in italics are specified by the user. The three primary input files
described in Chapters 3, 5, and 6 are indicated in the left-hand column. The names of these files are
prescribed either by the optional file FNAMES.WAM, or by the interactive inputs represented by the top
and bottom arrows in the right-hand column. Note that the P2F file output from POTEN is given the
same filename as the input control file, with the extension P2F. The output file from FORCE is given the
same filename as the force control file, with the extension OUT. The P2F file may be saved and reused for
various applications of the FORCE module where the same velocity potentials apply. Asterisks (∗) denote
the extensions corresponding to each option in the numeric output files, as listed in the table in Section
3.3.

1–6

are not able to use this extension should remove the symbol $ by editing the source ﬁle
RDFILESC.F.
The source code of WAMIT assumes the use of a PC-based Windows/DOS operating
system. ’README.TXT’ contains information regarding system requirements and conﬁguration to run the PC-executable version of WAMIT. Appropriate modiﬁcations of the
source code may be necessary for all other compilers.

1.2

CHANGES INTRODUCED IN Version 6.0

In relation to WAMIT Version 5.4, new features which aﬀect the use of Version 6 are
outlined below. For users familiar with versions of WAMIT prior to V5.4, and earlier
versions of the higher-order code HIPAN, additional changes are also listed below. More
detailed information is provided in the subsequent chapters as indicated.
• The main program WAMIT includes POTEN and FORCE as subprograms.
• The new input parameter ILOWHI has been added to control the use of the low- or
higher-order panel methods.
• Alternative options have been introduced to specify the input ﬁlenames.
• Faster algorithms have been implemented to reduce redundant evaluations of the
Green function.
• In the low-order method, some or all of the body panels can be deﬁned to represent
thin plates or appendages of small thickness, such as damper plates or strakes.
• For generalized modes the mode shapes can be speciﬁed in a separate .dll ﬁle.
• In the higher-order method the geometry can be speciﬁed by B-splines, explicit algorithms, or ﬂat panels.
• Explicit algorithms to specify the geometry can be included in a .dll ﬁle, and related
body dimensions can be speciﬁed in the .gdf input ﬁle. Several diﬀerent generic types
of bodies can be analyzed simply by inputting the relevant dimensions, without the
eﬀort required to prepare a conventional panelization.
• New parameters IALTPOT and IALTFRC have been included to provide a more
uniﬁed and ﬂexible range of input formats.
• Version 6.0PC has been compiled with the Compaq Digital Fortran compiler, optimized for use on PC systems with the Pentium CPU.
In relation to WAMIT Version 5.3, additional new features which were incorporated in
Version 5.4 also apply to Version 6.0 as outlined below.

1–7

• All restrictions on the maximum number of inputs are removed. In V6.0 all of the
relevant arrays are dimensioned at run time. In practice there may be practical
limitations, e.g. regarding the number of panels, depending on the memory and disk
capacity of the hardware and on the run time, but these limits will generally be much
larger than in earlier versions.
• A new direct-solver subroutine has been utilized. This substantially reduces the run
time when the direct solution method (ISOLVE=1) or block-iteration (ISOLVE>1)
are used.
• The PC-executable code is compiled in a generic form for all licensed users, and speciﬁc
licensees are identiﬁed by including a special binary input ﬁle userid.wam. This ﬁle
must be available for input during the WAMIT runs. Provision has been made in the
config.wam ﬁle to permit users to access this input ﬁle from a speciﬁed sub-directory,
as explained in Section 3.9; this option avoids the need to copy userid.wam to all
directories where WAMIT is used.
• Provision has been made in the conﬁguration ﬁle (config.wam or *.cfg) to permit
users to specify a diﬀerent sub-directory or drive for scratch storage of large arrays,
as explained in Section 3.9. This option eﬀectively doubles the maximum limit on
available disk memory for systems with multiple hard drives and/or partitions.
• The warning and error messages have been extended to provide more speciﬁc information concerning errors in input ﬁles.
• A new optional input array has been added to the .frc control ﬁle, as explained in
Sections 3.3 and 3.4. This permits the user to specify that some or all of the modes
selected in the inputs to POTEN are restrained with zero amplitude, and to re-run
the FORCE subprogram alone. (Previously this required re-running POTEN without
such modes.)
• Optional output ﬁles can be used to generate low-order GDF ﬁles and to generate
data ﬁles for perspective plots of the panels and/or patches.
In relation to the higher-order code HIPAN Version 2, additional new features which
are incorporated in WAMIT Version 6.0 are listed below. These features are described in
Chapter 6.
• Options exist to deﬁne the B-spline parameters automatically, to achieve a speciﬁed
maximum panel size.
• Multiple bodies with separate planes of symmetry can be input, with automatic reﬂection of the geometry.
• In the higher-order method the geometry can be speciﬁed by B-splines, explicit algorithms, or ﬂat panels.
• Body geometry for barges and similar structures can be deﬁned using a minimum
number of ﬂat quadrilateral elements, with the same input format as for the loworder analysis.
1–8

• Explicit algorithms to specify the geometry can be speciﬁed in a .dll ﬁle and related
body dimensions can be speciﬁed in the .gdf input ﬁle.
• Filenames for the .gdf ﬁles and related data can be input either in the .pot ﬁle, as in
HIPAN, or in the same format as used in earlier versions of WAMIT.

1.3

ADDITIONAL CHANGES IN Version 6.1

New features which are included in Version 6.1 are outlined below.
• Arrays of uniformly spaced wave periods and heading angles can be input more compactly in the Potential Control File, by specifying the number, ﬁrst value, and increment (see Section 3.1).
• The conﬁguration ﬁle CONFIG.WAM can be assigned a diﬀerent ﬁlename, deﬁned by
the user, with the extension .CFG (see Section 3.7).
• In the higher-order method, the default points for hydrodynamic pressure on the body
surface are modiﬁed, as described in Section 4.10.
• The hydrodynamic pressure on the body surface can be evaluated at user-speciﬁed
points (x, y, z) which are listed in a special input ﬁle. This extension, which is available in both the low-order and higher-order solutions, facilitates the integration of
WAMIT with structural-analysis codes where the hydrodynamic pressure is required
at speciﬁed locations. This procedure is described in Section 4.11.
• The separate components of the hydrodynamic pressure on the body surface due to
each of the radiation modes and due to the diﬀraction ﬁeld can be output separately.
This extension is useful for post-processing when the dynamics of the body are modiﬁed. This procedure is described in Section 4.12.
• In the higher-order solution method geometry models developed with MultiSurf can
be used directly in WAMIT. This procedure is described in Section 6.7.
• One or two additional .dll ﬁles are required to run Version 6.1PC (see Sections 2.1
and 6.7)
• The second-order extension has been updated to include both the low-order and
higher-order methods of solution, and to include other features of the V6 ﬁrst-order
code. Chapter 11 contains a description of these features.

1–9

1.4

ADDITIONAL CHANGES IN Version 6.2

New features which are included in Version 6.2 are outlined below.
• Internal tanks with free surfaces can be analysed either separately or coupled with
the dynamics of vessels. (See Section 10.7)
• Impulse response functions (IRFs) in the time domain can be evaluated for the output
options 1-7 using a utility program F2T. F2T is provided as a separate program along
with WAMIT V6.2. F2T converts the outputs of WAMIT into IRFs using Fourier
transforms. (See Chapter 13 and TEST 14A in Appendix A14.)
• In addition to the added mass and damping, the pressure and the ﬂuid velocity on the
body surface (Option 5) and at the ﬁeld points (Options 6 and 7) can be evaluated
for zero and inﬁnite wave frequencies (Section 3.1).
• The domain of the parameters for B-Splines representing the body surface is no longer
limited to (−1, 1) and B-Splines deﬁned over an arbitrary domain can be accepted.
(Section 6.6)
• Zero and inﬁnite frequencies can be speciﬁed with IRR > 0 (irregular frequency removal option), which was not permitted in the previous version. IRR > 0 is ignored
for these two frequencies. This update enables users to run these frequencies with
other frequencies in one execution of WAMIT. (Section 3.1)
• The subroutine FGR0I1, evaluating the Rankine source in two parallel horizontal
walls, is extended to calcuate double derivatives and renamed as FGR0I2. This extension is necessary to evaluate the ﬂuid velocity at zero and inﬁnite frequencies when
IOPTION(7)=1.
• The separate components of the hydrodynamic pressure and the ﬂuid velocity at the
ﬁeld points due to each of the radiation modes and due to the diﬀraction ﬁeld can be
output separately. This extension is useful for post-processing when the dynamics of
the body are modiﬁed. This procedure is described in Section 4.12.
• In the higher-order method, some or all of the body surface can be deﬁned to represent
thin structures (see Section 6.10). Similar extension was made in conjuction with the
low-order method in V6.0.
• Mean drift forces by pressure integration on the rigid bodies can be evaluated when
the generalized modes are used on other bodies in multi-body interaction. This option
was not supported in previous versions.
• A new output ﬁle wamitlog.txt is created during the WAMIT run to provide an
archival record of the run, including copies of the principal input ﬁles, and various
auxiliary information. Further information is provided in Section 10.1.

1–10

1.5

ADDITIONAL CHANGES IN Version 6.3

New features which are included in Version 6.3 are outlined below.
• Uniform arrays of ﬁeld points can be input in a more convenient manner. (See Section
3.10)
• A special option can be used to eﬃciently analyze the wave ﬁeld generated by one or
more wavemakers situated in planes of symmetry. (See Section 10.8)
• All six components of the mean drift forces and moments can be evaluated from the
momentum ﬂux through a control surface surrounding the body.
• A symmetry plane can be used when there are ﬂat dipole elements on the plane of
symmetry. (See Section 5.4 and Section 6.10)
• A supplementary output ﬁle out.hst is created to output the hydrostatic matrix of
restoring coeﬃcients, including generalized modes and/or tanks. (See Sections 4.9,
8.4, and 10.7.)
• The DLL library ﬁle GEOMXACT has been extended to include several new analytical
geometries.
• The DLL library ﬁle NEWMODES has been extended to include generalized modes
to represent a bank of paddle-type wavemakers, and to represent the vertical modes
of a vessel with hinges.

1–11

Chapter 2

GETTING STARTED
In this Chapter instructions are provided for installing the WAMIT-PC software, and
making simple test runs.
The two principal subprograms of WAMIT are POTEN and FORCE. POTEN solves
for the velocity potential on the body surface, and optionally also for the source strength.
FORCE evaluates physical parameters including the force and motion coeﬃcients, and ﬁeld
data including the ﬂuid pressure, velocity, and free-surface elevation. The basic sequence
in a typical application of WAMIT is (1) prepare the input ﬁles; (2) run WAMIT. The principal results are then contained in output ﬁles which may be printed and post-processed.
This architecture is illustrated in Figure 1.1.
The principal input ﬁles to the subprogram POTEN are the Potential Control File
(POT) which speciﬁes parameters including the ﬂuid depth, wave periods, and wave heading angles, and the Geometric Data File (GDF), which describes the geometry of the
structure to be analyzed. These ﬁles are discussed brieﬂy below, and in more detail in
subsequent Chapters. The principal input ﬁles to the subprogram FORCE are the Force
Control File (FRC), which speciﬁes inputs regarding the body dynamics, and the P2F ﬁle
(‘Poten to Force’) which transfers data from POTEN to FORCE.
There are three additional input ﬁles which should also be considered: (1) Licensed
users of WAMIT V6PC must utilize a unique input ﬁle userid.wam, which identiﬁes the
site license, (2) the optional input ﬁle fnames.wam is recommended to specify the input
ﬁlenames, and (3) the optional conﬁguration ﬁle (config.wam or *.cfg) may be used to
specify certain options and other information used by the program. The ﬁle userid.wam is
provided by WAMIT Inc. Samples of the fnames.wam and conﬁguration ﬁles are included
with the test runs; for example the ﬁles fnames.1 and config.1 are intended for use with
TEST01. (Before running TEST01 the user must rename or copy these ﬁles, as explained
below.)

2–1

2.1

INSTALLATION AND SETUP

The WAMIT-PC software is delivered in two forms, (1) on a CD-ROM disk, or (2) as
a compressed zip ﬁle suitable for electronic transfer. This software includes the main
executable program wamit.exe and a complete set of input ﬁles for the standard test runs
TESTn (n=01,02, ...), and the labeled output ﬁles TESTn.OUT for these test runs. Also
included is the text ﬁle readme.txt with additional information.
In the standard Version 6.3PC there are four additional dynamic link library ﬁles (DLL)
which must be installed in the same directory (i.e. ‘folder’) as the executable ﬁle wamit.exe.
If these DLL ﬁles are missing the program will not run, regardless of the inputs and
options speciﬁed. The four DLL ﬁles are geomxact.dll, newmodes.dll, rg2wamit.dll,
and dforrt.dll. (The extended Version including the capability to input MultiSurf models
requires ﬁve DLL ﬁles, as explained in Section 6.7.)
Before installing the software a new main directory (folder) should be made. The
recommended name is C:\WAMITv6, but the user may prefer to use a diﬀerent drive.Copy
all of the software to this directory, preserving the same subdirectories (folders) within the
new directory as are on the CD ROM or in the zip ﬁle. If the software is received in a
zip ﬁle by electronic transfer, it must be unzipped using a local utility, and all unzipped
ﬁles should be saved in the directory C:\WAMITv6 and in subdirectories corresponding
to those in the zip ﬁle. The text ﬁle readme.txt includes an outline of the directory tree
and ﬁles. If any diﬃculties are encountered during installation and testing the user should
conﬁrm that the subdirectories and ﬁles correspond to the description in readme.txt.

2.2

DEMONSTRATION PROGRAMS

A special demonstration version of WAMIT can be downloaded in .zip format from the web
site www.wamit.com. This program, wamit demo.exe, can be used for demonstration
or educational purposes, without obtaining an end-user license, subject to the conditions
stated in the website. The installation and use of this software is the same as the licensed
version, with the following exceptions:
• The program only accepts geometry inputs from the standard input ﬁles for the test
runs described in the Appendix.
• The site license identiﬁcation ﬁle userid.wam and DLL ﬁles are not required.
• The program runs are interrupted after display of the header, and the user is prompted
to press the Enter key to continue the run.
The downloadable zip ﬁle includes all of the input ﬁles required to execute the standard
test runs.

2–2

2.3

STANDARD TEST RUNS

Various standard test runs are included with the software, to illustrate diﬀerent types of
applications and features of the program. The results of these test runs can be used to
conﬁrm that the installation and setup of the program have been performed correctly by
the user. The test runs also provide opportunities to use (and modify) existing input
ﬁles, for tutorial purposes. The remainder of this Chapter is intended to guide new users
through these procedures. Descriptions of each test run are included in the Appendix.
If the WAMIT software is installed in accordance with the instructions above, the required .EXE, .DLL, and USERID.WAM ﬁles will be installed in the directory C:\WAMITv6.
All required input ﬁles for the standard test runs will be copied to the subdirectory
C:\WAMITv6\TESTRUNS. Benchmark versions of the output ﬁles test*.out will be
copied to the subdirectory C:\WAMITv6\TESTRUNS\OUT. These benchmark output
ﬁles can be compared with results obtained by the user to ensure that the software is
installed correctly.
Before running WAMIT with the standard test runs, the user should (1) open a DOS
Command Prompt Window (in the Windows environment a Command Prompt Window is
opened by clicking on ‘Start’, ‘Programs’, ‘Command Prompt’), and (2) change the default
subdirectory by entering the command cd \WAMITv6\TESTRUNS.
Since the executable ﬁle wamit.exe is resident in the directory C:\WAMITv6, the appropriate command to execute WAMIT is ‘C:\WAMITv6\WAMIT’. There are three alternative shortcuts which may be used to simplify this command: (1) add C:\WAMITv6 to
the system PATH, (2) copy the ﬁles wamit.exe, *.dll, and userid.wam from this directory
to another directory which is included in the PATH, or (3) use the batch ﬁle wamit.bat
which is supplied in the testruns subdirectory.

2.4

RUNNING TEST01

Test Run 01 evaluates the added-mass and damping coeﬃcients, exciting forces, motions,
wave elevations, ﬁeld pressures, ﬂuid velocities and drift forces for a freely-ﬂoating truncated vertical circular cylinder of radius 1.0m and draft 0.5m, in inﬁnite water depth for
three wave periods and one wave heading angle. Further details are contained in Appendix
A.
The corresponding input ﬁles test01.gdf, test01.pot, and test01.frc are included in
the subdirectory C:\WAMITv6\TESTRUNS. In order to specify the appropriate ﬁlenames
during the run, ﬁrst copy the ﬁle fnames.01 to the ﬁle fnames.wam. (Copying is recommended, in preference to renaming the ﬁle, to preserve the original ﬁle.) The appropriate
DOS command is
copy fnames.01 fnames.wam
Next enter wamit to start the run. During execution of the subprogram POTEN the
monitor displays the starting time, and after the solutions for the velocity potentials are
2–3

obtained at each wave period the monitor will display a new line of information including
the period (in seconds), time, and the maximum number of iterations required for the
radiation and diﬀraction solutions. After the ﬁrst solutions are displayed the results for
subsequent periods run faster, since the panel integration of the Rankine components of the
source potential are only evaluated initially and saved for reuse. After the third period the
intermediate storage ﬁle test01.p2f is created on the disk, storing the velocity potentials
and other inputs to the subprogram FORCE. Output from FORCE will appear relatively
quickly on the screen, and the same output is stored in the ﬁle test01.out. The latter
ﬁle includes useful identiﬁcation information concerning the inputs, body parameters, run
times and dates. This is followed for each period by tabulations of the hydrodynamic
parameters requested in test01.frc. Assuming the standard version of test01.out has
been saved in a subdirectory, as recommended in Section 2.1, the data in the new version of test01.out can be compared with the standard ﬁle with the same name. On a
contemporary PC the total run time should be a few seconds.

2.5

RUNNING TEST11

Test Run 11 is intended to complement TEST01, but using the higher-order method of
solution (ILOWHI=1). The body dimensions and other inputs are the same, but the
surface of the cylinder and also the solution for the velocity potential are represented in a
more accurate, continuous manner by B-splines, as explained in Chapter 6.
To run TEST11 follow the same procedure outlined above for TEST01, but replace ‘01’
by ‘11’ in all references to ﬁlenames and extensions. A comparison of the results from these
complementary tests is summarized in Section A.11 of the Appendix.

2.6

OTHER TEST RUNS

The procedure for running the other test runs is the same as explained above, except for
copying the appropriate ﬁles to fnames.wam from the original versions fnames.n where
‘n’ is the test number.
A special batch ﬁle runtests.bat is included in the licensed version and a similar ﬁle
rundemo.bat in the demonstration version. Executing the corresponding batch ﬁle command will run all of the tests in succession.

2.7

MEMORY AND STORAGE RESTRICTIONS

WAMIT uses scratch ﬁles on the hard disk for temporary storage at runtime. Depending
on the run parameters, the total number of scratch ﬁles and/or their size may become quite
large. Normally these ﬁles are deleted by WAMIT after the ﬁles are no longer needed, and
before the program stops. However some PC/Windows conﬁgurations save these deleted
ﬁles in a ‘recycled’ directory, and this can cause the hard disk to become overloaded. Users
2–4

who experience this problem should delete the accumulated ﬁles in the recycled directory,
or alternatively change their system setup to avoid saving a backup of all deleted ﬁles.
Contemporary Pentium PC systems generally have suﬃcient capacity for all of the Test
Runs. Further information regarding memory requirements is given in Chapter 10.

2.8

MODIFYING THE INPUT FILES

A text editor can be used to edit the input ﬁles. Users should refer to Chapter 3 for
a complete explanation of the data in these ﬁles, but a few simple modiﬁcations will be
outlined here for tutorial purposes, in the context of Test Run 01. Before proceeding
further check that the ﬁles FNAMES.WAM and CONFIG.WAM correspond to this test
run, and re-copy these ﬁles if necessary following the instructions in Section 2.4.
As the ﬁrst modiﬁcation we might request FORCE to perform additional computations,
for the same periods and wave headings analyzed by POTEN, but with modiﬁed values of
the vertical center of gravity (VCG) and radii of gyration (XPRDCT) in the force control
ﬁle TEST01.FRC. In the standard ﬁle VCG=0.0 (center of gravity in the waterplane)
and the radii of gyration are set equal to 1.0 (the three diagonal elements of the XPRDCT
matrix). A positive VCG will move the center of gravity above the waterplane, reducing the
pitch/roll hydrostatic stability and aﬀecting these RAO’s in longer wave periods. Modifying
the radii of gyration should change the same RAO’s primarily at shorter wave periods.
It is not necessary to re-run POTEN in this case, provided the ﬁle TEST01.P2F has
been retained for the POTEN output. To avoid the extra run time of POTEN, add the
line IPOTEN=0 to the conﬁguration ﬁle conﬁg.wam (See Section 3.7, and also the ﬁle
CONFIG.17b which includes the same line). After modiﬁcations are made to the ﬁle
TEST01.FRC it is advisable to save the modiﬁed ﬁle a with diﬀerent ﬁlename, and to
edit the FNAMES.WAM ﬁle to show the corresponding new ﬁlename for the force control
ﬁle. If this procedure is followed, the output ﬁle will carry the same ﬁlename (with the
.out extension) to distinguish it from the original ﬁle TEST01.OUT. The user may then
compare the RAO’s in the diﬀerent output ﬁles to discern the eﬀect of these changes.
As the second modiﬁcation WAMIT may be re-run with a more extensive list of wave
periods. Edit the potential control ﬁle TEST01.POT with the following changes:
• on line 5 change the number of wave periods (NPER) from 3 to 10
• on line 6 replace the three original wave periods by ten new periods in decimal format
• depending on your preference, (1) save the modiﬁed ﬁle with the original name
TEST01.POT, or (2) save the modiﬁed ﬁle with a new ﬁlename and correct the
FNAMES.WAM ﬁle on line 2 with the new ﬁlename.
Since it is necessary to re-run POTEN in this case, either delete the line IPOTEN=0
if this was added to conﬁg.wam, or copy conﬁg.01 once again to conﬁg.wam. During the
run, if the original ﬁlename TEST01.POT is retained, the user will be prompted whether
or not to overwrite the old output ﬁle TEST01.P2F; overwriting is the simplest procedure
2–5

to follow, in this circumstance, otherwise the new .P2F ﬁlename must be speciﬁed before
the FORCE run is executed.
The input ﬁles for other test runs can be used to illustrate various options and modiﬁcations.

2–6

Chapter 3

USING WAMIT
A typical application of the standard WAMIT program V6 or V6PC will consist of (a)
preparing appropriate input files; (b) running WAMIT; and (c) using the resulting output
files. Most of the required input files are ‘generic’, with the same format and data irrespective of whether the low-order or high-order method is used. These files are described in
this Chapter. The principal exception is the geometric data file (GDF), which is described
separately for the two methods in Chapters 5 and 6, respectively.
To simplify the presentation this Chapter will describe the required input files and
resulting output files for a basic application involving the analysis of a single body. Further information is given in Chapters 7-9 for the appropriate modifications of the input
files for specific purposes, including the analysis of multiple bodies, the analysis including
generalized modes of body motion, and the option to remove irregular-frequency effects.
The execution of a WAMIT run is divided between two subprograms, POTEN and
FORCE, as explained in Chapter 1. In special circumstances it is useful to run WAMIT
and execute only one of the two subprograms, using the optional parameters IPOTEN=0
or IFORCE=0 to skip the corresponding subprogram execution. These parameters can
be input in the configuration file, as explained below in Section 3.7. In the default case
(IPOTEN=1, IFORCE=1) both subprograms are executed sequentially in the same run.
Some input files are used only by POTEN or FORCE, whereas other input files are
used by both. The following table lists the input files which are prepared by the user, and
indicates the relevant subprogram(s):
Filename

Usage

runidp.pot
POTEN
body.gdf
POTEN
body.spl
POTEN
FNAMES.WAM POTEN/FORCE
∗.cfg
POTEN/FORCE
runidf.frc
FORCE

Description
Potential Control File (Sections 3.1-2)
Geometric Data File (Chapters 5,6)
Spline Control File (Section 6.11)
Filenames list (Section 3.8)
Configuration file (Section 3.7)
Force Control File (Sections 3.3-4)

3–1

Here runidp is a user-specified filename assigned to the POT file. The same filename is
assigned to the P2F file, which is output by POTEN and input to FORCE. Similarly,
runidf is a user-specified filename assigned to the FRC file and this filename is assigned
to the principal output files from FORCE. body is the user-specified filename assigned to
the GDF file. If the optional Spline Control File is used it must have the same filename.
The following table lists the output files which are produced by each subprogram:
Filename
Program Description
runidp.p2f
POTEN P2F File (binary data for transfer to FORCE)
errorp.log
POTEN Error Log File (This section)
errorf.log
FORCE Error Log File (This section)
runidf.out
FORCE Formatted output file (Chapter 4)
runidf.num FORCE Numeric output files (Section 4.9)
body.pnl
FORCE Panel data file (Section 4.9)
runidf.fpt
FORCE List of field points (Section 4.9)
The structure of input and output files is illustrated in the flow chart shown in Figure
1.1. Further details are provided below.
The primary generic data files are the two ‘control files’ input to POTEN and FORCE.
These are referred to as the Potential Control File (POT), with the extension ’.pot’, and
the Force Control File (FRC), with the extension ’.frc’.
All WAMIT input files are ASCII files. The first line of most files is reserved for a userspecified header, consisting of up to 72 characters which may be used to identify the file.
If no header is specified a blank line must be inserted to avoid a run-time error reading the
file. The remaining data in each file is read by a sequence of free-format READ statements.
Thus the precise format of the input files is not important, provided at least one blank
space is used to separate data on the same line of the file. Further details regarding the
formats and names of files are contained in Section 3.10.
Additional input files may be used to assist in using WAMIT, to select various options,
and to optimize its use for specific applications. The optional file FNAMES.WAM is used
to specify the names of the GDF, POT, and FRC input files to avoid interactive input
of these names (see Section 3.6). The optional input file CONFIG.WAM may be used to
configure WAMIT and to specify various options, as described in Section 3.7. (In Version
6.2-3 this configuration file may be assigned a different filename by the user.) The third
additional input file, the optional Spline Control File (SPL), is used only in the higher-order
method, as described in Section 6.11.
In V6PC the input file ‘userid.wam’ is read by both POTEN and FORCE, to identify
the licensee name and address for output to the headers at run time, and to write this
information in the header of the .out output file. This file is prepared by WAMIT, Inc.,
and must be available for input to POTEN and FORCE at runtime. To be available for
input, the file USERID.WAM must either be copied to the default directory with other
input/output files, or else the pathname indicating the resident directory must be listed in
CONFIG,WAM, as explained in Section 3.7.
Two alternative formats for the POT and FRC files are described separately in Sections
3–2

3.1-2 and 3.3-4.
Alternative Form 1 for the POT file is identical to the POT file used in earlier versions of
WAMIT. Alternative Form 2 was developed initially for the higher-order program HIPAN,
and is retained here as an option for convenience. Either Form 1 or Form 2 can be used
irrespective of whether the low- or higher-order method is used. Form 2 is particularly
convenient for the analysis of multiple bodies.
Alternative Forms 1 and 2 for the FRC file are identical to the corresponding alternative
forms used in Version 5 of WAMIT. However a more consistent specification of these two
alternatives is now available, via the parameter IALTFRC in the configuration file. For
a rigid body which is freely floating, and not subject to external constraints, Alternative
Form 1 (Section 3.3) may be used, with the inertia matrix of the body specified in terms
of a 3 × 3 matrix of radii of gyration. Alternative Form 2 (Section 3.4) permits inputs of
up to three 6 × 6 mass, damping, and stiffness matrices to allow for a more general body
inertia matrix, and for any linear combination of external forces and moments. (A third
alternative format may be used for multiple bodies, as described in Section 7.5.)
Several output files are created by WAMIT with assigned filenames. The output from
POTEN for use by FORCE is stored in the P2F file (Poten to Force) and automatically
assigned the extension P2F. The final output from FORCE is saved in a file with the
extension OUT which includes extensive text, labels and summaries of the input data.
FORCE also writes a separate numeric output file for the data corresponding to each
requested option, in a more suitable form for post-processing; these files are distinguished
by their extensions, which correspond to the option numbers listed in the table in Section
3.3.
Two additional numeric files are generated when the FRC file specifies either Option
5 (pressure and fluid velocity on the body surface) or Options 6-7 (pressure and fluid
velocity at field points in the fluid), to assist in post-processing of these data. For Option
5 a ‘panel geometry’ file with the extension PNL is created with data to specify the area,
normal vector, coordinates of the centroid, and moment cross-product for each panel on
the body surface. For Options 6-7 a ‘field point’ file with the extension FPT specifies the
coordinates of the field points in the fluid.
The filenames assigned to the various output files are intended to correspond logically
with the pertinent inputs, and to simplify file maintenance. Thus the filename of the P2F
file is derived in the program from the filename of the POT file, and similarly for the
OUT output file from the FRC file. To make this convention more specific it is necessary
to define the input filenames specified by the user. For the present discussion these are
assumed to be, respectively pot.POT and frc.FRC where the italicized portion is userspecified. The resulting output filenames are then assigned as follows: pot.P2F is the
output file from POTEN, input to FORCE. frc.OUT is the principal output file from
FORCE. The various numeric output filenames are assigned the same name as the OUT
file, but with extensions corresponding to each option number. (The alternative name
OPTN may be specified by setting the optional switch NUMNAM=1 in the configuration
file, as described in Section 3.7.) For Option 5, the panel data file is assigned the filename
from the Geometric Data File, gdf.PNL, since this data is specific to the body identified
3–3

in the GDF file. If NUMNAM=1, the generic name ‘gdf.pnl’ is assigned to the panel data
file. For Options 6-7, the field point file is assigned the filename frc.FPT (or OPTN.FPT,
if NUMNAM=1), to correspond to the force control file and other numeric output files.
These conventions are illustrated in Figure 1.1.

3–4

3.1

THE POTENTIAL CONTROL FILE (Alternative Form 1)

The Potential Control File is used to input various parameters to the POTEN subprogram.
The name of the POT file can be any legal filename accepted by the operating system,
with a maximum length of 16 ASCII characters, followed by the extension ‘.pot’.
Two alternative forms of the POT file can be used in Version 6, as specified by the
optional parameter IALTPOT in the config.wam file. The default case (IALTPOT=1) is
described in this Section. The alternative form (IALTPOT=2) is described separately in
Section 3.2.
If IALTPOT is not specified, or if it is specified with the default value (IALTPOT=1),
the data in the Potential Control File must be in the format listed below:
header
ISOR†
IRR†
HBOT {XBODY(1) XBODY(2) XBODY(3) XBODY(4)}†
IQUAD† ILOG† IDIAG†
IRAD IDIFF
MODE(1) MODE(2) MODE(3) MODE(4) MODE(5) MODE(6)
NPER
PER(1) PER(2) PER(3) ... PER(NPER)
NBETA
BETA(1) BETA(2) BETA(3) ... BETA(NBETA)
The symbol † denotes that the corresponding parameters may be input via the configuration file and not included in the Potential Control File (See Section 3.7). If the
higher-order method is used (ILOWHI=1) the parameters ISOR, IQUAD, IDIAG are ignored, but they must be included either in the configuration file or in the Potential Control
File to correctly read the remainder of the data.
The data shown on each line above are read consecutively by corresponding read statements. Thus it is recommended to preserve the line breaks indicated above, except that
if a large number of periods (PER) and/or wave heading angles (BETA) are input, these
may be placed on an arbitrary number of consecutive lines.
The definition of each variable in the Potential Control File is as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72. This line is
available for the user to insert a brief description of the file.
ISOR is the integer used to specify whether the source strength is evaluated:
ISOR= 0: Do not evaluate the source strength.
ISOR= 1: Evaluate the source strength.
The source strength is required in the low-order method (ILOWHI=0) if FORCE evaluates the fluid velocity on the body (IOPTN(5)=2 or 3), the pressure/free-surface elevation
3–5

or velocity in the fluid domain by the source formulation (IOPTN(6)=2 or IOPTN(7)=2)
and the mean drift force and moment from pressure integration (IOPTN(9)=1 or 2). Further information on these options is given in Section 5.2. Running POTEN with ISOR=1
requires substantially longer run time and larger scratch storage. In the higher-order
method (ILOWHI=1) all of the FORCE evaluations are made directly from the solution
for the velocity potential, and ISOR=0 is required.
IRR is the integer used to specify whether the irregular frequencies are removed or not.
A detailed explanation for this parameter is provided in Chapter 9.
IRR= 0: Do not remove the effect of the irregular frequencies.
IRR= 1: Do remove the effect of the irregular frequencies. (User needs to represent the
interior free surface by panels or patches.)
IRR= 2: Do remove the effect of the irregular frequencies. (Program projects the body
panels on the interior free surface.)
IRR= 3: Do remove the effect of the irregular frequencies. (Program automatically
discretizes the interior free surface.)
IRR>1 is not permitted in the higher-order method (ILOWHI=1).
The parameters ISOR and IRR can be input either in the POT or the configuration file.
If ISOR or IRR is specified in the configuration file the corresponding line of the POT file
should be deleted, as described in Section 3.7.
HBOT is the dimensional water depth. By convention in WAMIT, a value of HBOT
less than or equal to zero is interpreted to mean that the water depth is infinite. It is
recommended to set HBOT=-1. in this case. If HBOT is positive it must be within the
range of values
10−5 < HBOT × ω 2 /GRAV < 105
where ω = 2π/PER is the radian frequency of the incident waves and GRAV is the gravitational acceleration constant. For each run the value of GRAV is input in the GDF file, as
described in Chapters 5 and 6. All dimensional inputs with the units of length, including
HBOT, must be consistent with the input GRAV.
XBODY(1), XBODY(2), XBODY(3) are the dimensional (X, Y, Z) coordinates of the
origin of the body-fixed coordinate system relative to the global coordinate system,
input in the same unit of length as GRAV. The global coordinate system is required when
walls are present (Section 5.3) and when multiple bodies are analyzed (Chapter 7). The
global coordinate system is also used in place of the body coordinate system to define
field-point data (fluid pressures, velocities, and free-surface elevation). Normally, in the
absence of walls or multiple bodies, the coordinates XBODY(1) and YBODY(1) should be
set equal to zero unless it is desired to refer the field-point data to a different coordinate
system from that of the body. (The origin of the global coordinate system is on the free
surface.) The incident-wave velocity potential is defined relative to the global coordinate
system. Consequently, the phases of the exciting forces, motions, hydrodynamic pressure
and field velocity induced by the incident wave are understood relative to the incident-wave
elevation at X = Y = 0.
3–6

XBODY(4) is the angle in degrees of the x−axis of the body coordinate system relative
to the X−axis of the global system in counterclockwise sense (see Figure 5.2).
The array XBODY may be moved from the POT file to the configuration file (See
Section 3.7).
IQUAD, ILOG, IDIAG are control indices which may be used to increase the precision of the panel integration of the Green function and its derivatives, at the expense of
computation time; in each case the default setting zero will minimize the computation
time. IQUAD and IDIAG are only used in the low-order method (ILOWHI=0). However
if the Alternative Form 1 POT file is used IQUAD and IDIAG must be input, even when
ILOWHI=1, to avoid an error reading the input data.
IQUAD= 0: The integration of the regular wavelike part of the Green function and its
derivatives is carried out by using a single node at the centroid of each panel.
IQUAD= 1: The integration is carried out by using a four-node Gauss quadrature.
(This option can be used to verify the accuracy of computations carried out with the faster
single-node quadrature. However this practice is generally not as useful as the alternatives
of increasing the number of panels or using the higher-order method.)
ILOG= 0: The logarithmic singularity is included with the wavelike component of the
Green function and is integrated by quadrature over each panel.
ILOG= 1: The logarithmic singularity in the Green function is subtracted and integrated analytically for pairs of panels for which the Rankine image singularity 1/r0 is also
integrated analytically. (This option produces more accurate results.) ILOG=1 is required
when panels are defined in the plane of the free surface, including the following two cases:
(1) bodies with horizontal physical surfaces lying in the plane of the free surface, and (2)
use of the irregular frequency option, where panels are located on the interior free surface
inside the body waterline. In these two cases, execution of the program is interrupted with
an error message if ILOG=0.
IDIAG= 0: In determining those pairs of panels where the above analytic integration is
required, the distance between their centroids is compared with the characteristic length
based on the square root of their area.
IDIAG= 1: The characteristic length of each panel is based on its maximum diagonal.
[This option is more accurate for panels with very large aspect ratios.]
Generally it is recommended to use the default values IQUAD= 0, ILOG= 0, and
IDIAG= 0. ILOG= 1 may be useful when studying local characteristics such as run up
near the waterline of the body. IDIAG= 1 may be useful when some of the panels used to
describe the body have a very large aspect ratio.
Any or all of these control indices may be moved from the POT file to the configuration
file (See Section 3.7).
IRAD, IDIFF are indices used to specify the components of the radiation and diffraction
problems to be solved. The following options are available depending on the values of
IRAD and IDIFF:

3–7

IRAD= 1: Solve for the radiation velocity potentials due to all six rigid-body modes of
motion.
IRAD= 0: Solve the radiation problem only for those modes of motion specified by
setting the elements of the array MODE(I)=1 (see below).
IRAD= −1: Do not solve any component of the radiation problem.
IDIFF= 1: Solve for all diffraction components, i.e. the complete diffraction problem.
IDIFF= 0: Solve only for the diffraction problem component(s) required to evaluate the
exciting forces in the modes specified by MODE(I)=1.
IDIFF= −1: Do not solve the diffraction problem.
MODE is a six-element array of indices, where I=1,2,3 correspond to the surge, sway and
heave translational modes along the body-fixed (x, y, z) axes, and I=4,5,6 to the roll, pitch
and yaw rotational modes around the same axes, respectively. Each of these six indices
should be set equal to 0 or 1, depending on whether the corresponding radiation mode(s)
and diffraction component(s) are required. (See the options IRAD=0 and IDIFF=0 above.)
The MODE array in the radiation solution specifies which modes of the forced motion
problem will be solved. To understand the significance of the MODE array in the diffraction solution, it should be noted that, when symmetry planes are defined, the complete
diffraction problem is decomposed into symmetric/antisymmetric components in a manner
which permits the most efficient solution, and when IDIFF=0, only those components of
the diffraction potential required to evaluate the exciting force for the specified modes are
evaluated. For example, if ISX=1, IDIFF=0, MODE(1)=1, and the remaining elements of
MODE are set equal to zero, then the only component of the diffraction potential which
is solved is that part which is antisymmetric in x. If the complete diffraction potential is
required, for example to evaluate the drift forces or field data, IDIFF should be set equal
to one. (For further discussion of these options see Section 3.3.)
NPER is the number of wave periods to be analyzed. NPER must be an integer. The
following alternatives can be used in Versions 6.2-3:
N P ER = 0: Read inputs and evaluate hydrostatic coefficients only.
NP ER > 0: Execute the hydrodynamic analysis for NPER wave periods PER
N P ER < 0: Execute the hydrodynamic analysis for |NP ER| wave periods as explained
below
If NPER= 0, POTEN and FORCE will run but not execute any hydrodynamic analysis.
This option can be used to test for errors in input files, and to evaluate the hydrostatic
coefficients in the OUT file. If NPER=0 the array PER must be removed from the Potential
Control File.
PER is the array of wave periods T in seconds, or of optional inputs as specified by
the parameter IPERIO. Normally the values of PER must be positive. By using the
optional parameter IPERIO in the configuration file, it is possible to replace the input
array of wave periods by a corresponding array with values of the radian frequencies ω =
2π/T , infinite-depth wavenumbers KL, or finite-depth wavenumbers νL. The wavenumbers
3–8

are nondimensionalized by the length L=ULEN, and defined relative to the gravitational
acceleration g=GRAV. Both ULEN and GRAV are input in the GDF file. The following
table gives the definitions of each input and the corresponding value of IPERIO:
IPERIO Input
1
2
3
4

Definition

Period
Frequency
Infinite-depth wavenumber
Finite-depth wavenumber

T
ω = 2π/T
KL = ω 2 L/g
νL (νL tanh νH = ω 2 L/g)

If the fluid depth is infinite (HBOT ≤ 0), K = ν and there is no distinction between
the inputs for the last two cases. The default case IPERIO=1 is assumed if IPERIO is not
specified in the configuration file. Regardless of the form of these inputs, the output data is
unchanged, with wave periods in seconds and the nondimensional finite-depth wavenumbers
specified in the OUT file.
The limiting values of the added mass coefficients may be evaluated for zero or infinite
period by specifying the values PER= 0.0 and PER< 0.0, respectively. In V6.2-3, the
limiting values of the body pressure and velocity and the field pressure and velocity due
to the radiation solution may be evaluated(see Section 4.13). These special values can be
placed arbitrarily within the array of positive wave periods. These special values are always
associated with the wave period, irrespective of the value of IPERIO and the corresponding
interpretation of the positive elements of the array PER. For example, the effect of the
parameter IPERIO=2 and the array PER with the four inputs 0., 1., 2., -1. is identical to
the default case IPERIO=1 with the array PER equal to 0., 2π, π, -1. These limiting values
can be included in runs where the irregular-frequency removal option is used (IRR≥1) (In
V6.1 and lower, a separate run must be made for these limiting values after setting IRR=0.)
If NP ER < 0 a special convention is followed to assign a total of |NP ER| wave periods,
frequencies, or wavenumbers with uniform increments, starting with the value equal to
PER(1) and using the increment equal to the value shown for PER(2). In this case only
two values PER(1),PER(2) should be included in the POT file. This option is convenient
when a large number of uniformly spaced inputs are required. The two following examples
show equivalent sets of input data in lines 5 and 6 of the POT file:
8
1.0

(NPER)
2.0 3.0

4.0

5.0

6.0

7.0

8.0

(PER array)

and
-8
1.0

(NPER)
1.0

(PER(1), increment)

This convention is applied in the same manner for all IPERIO values, irrespective of
whether PER represents the wave period, frequency, or wavenumber. Special attention is
required when zero-period or zero-frequency inputs are required, following the definitions
as specified above. For the example described in the preceding paragraph with IPERIO=2,
3–9

the inputs NPER=-4 and (PER = -1. 1.) will result in the sequence of wave frequencies
equal to (zero, infinity, 1., 2.).
NBETA is the number of incident wave headings to be analyzed in POTEN. (Additional
heading angles may be specified subsequently in FORCE, as described in Section 3.3.)
NBETA must be an integer. The following alternatives can be used in Version 6.:
NBET A = 0: There are no incident wave heading angles. (IDIFF=-1)
NBET A > 0: Execute the hydrodynamic analysis for NBETA wave angles BETA
NBET A < 0: Execute the hydrodynamic analysis for |NBET A| wave angles as explained below
BETA is the array of wave heading angles in degrees. The wave heading is defined as
the angle between the positive x−axis of the global coordinate system and the direction
in which the waves propagate, as shown in Figure 12.1. The sign of the wave heading is
defined by applying the right-hand rule to the body fixed system. In POTEN the wave
headings specified in the Potential Control File pertain to the solution of the diffraction
problem only. NBETA may be set equal to 0 if IDIFF= −1; in this case wave heading angles
specified in the Potential Control File are ignored in Alternative Form 1. In Alternative
Form 2, described in Section 3.2, if NBETA=0 no data BETA should be included in the
POT file.
If N BET A < 0 a special convention is followed to assign a total of |NBET A| wave
angles, with uniform increments, starting with the value equal to BETA(1) and using
the increment equal to the value shown for BETA(2). In this case only two values
BETA(1),BETA(2) should be included in the POT file. This option is convenient when a
large number of uniformly spaced inputs are required. The two following examples show
equivalent sets of input data in lines 7 and 8 of the POT file:
8
(NBETA)
0.0 45.0 90.0 135.0 180.0 225.0 270.0 315.0

(BETA array)

and
-8
(NBETA)
0.0 45.0

(BETA(1), increment)

3–10

3.2

THE POTENTIAL CONTROL FILE (Alternative Form 2)

If IALTPOT=2 is specified in the configuration file, the data in the Potential Control File
must be in the format listed below:
header
IRR†
HBOT
IRAD IDIFF
NPER
PER(1) PER(2) PER(3)...PER(NPER)
NBETA
BETA(1) BETA(2) BETA(3) ... BETA(NBETA)
NBODY
GDF(1)
XBODY(1,1) XBODY(2,1) XBODY(3,1) XBODY(4,1)
MODE(1,1) MODE(2,1) MODE(3,1) MODE(4,1) MODE(5,1) MODE(6,1)
NEWMDS(1)
GDF(2)
XBODY(1,2) XBODY(2,2) XBODY(3,2) XBODY(4,2)
MODE(1,2) MODE(2,2) MODE(3,2) MODE(4,2) MODE(5,2) MODE(6,2)
NEWMDS(2) ·
·
·
GDF(NBODY)
XBODY(1,NBODY) XBODY(2,NBODY) XBODY(3,NBODY) XBODY(4,NBODY)
MODE(1,NBODY) MODE(2,NBODY) MODE(3,NBODY) ... MODE(6,NBODY)
NEWMDS(NBODY)
The data shown on each line above is read consecutively by corresponding read statements. Thus it is necessary to preserve the line breaks indicated above, but if a large
number of periods (PER) and/or wave heading angles (BETA) are input, these may be
placed on an arbitrary number of consecutive lines.
The definitions of the data on lines 1-8 (header, IRR, HBOT, IRAD, IDIFF, NPER,
PER, NBETA, BETA) are the same as given in Section 3.1 for the Alternative Form 1 POT
File. As in Alternative Form 1, the dagger shown after IRR denotes that this parameter
must be included either here, or in the configuration file, but not in both.
Definitions of the other data above are as follows:
NBODY is the number of bodies. Except for the analysis of multiple bodies (Chapter 7),
NBODY=1.
GDF(K) is the name of the Geometric Data File of the Kth body.
XBODY(1,K), XBODY(2,K), XBODY(3,K) are the dimensional (X, Y, Z) coordinates of the origin of the body-fixed coordinate system of the kth body relative to the
global coordinate system, input in the units of the length ULEN. The incident-wave
3–11

velocity potential is defined relative to the global coordinate system. Consequently, the
phases of the exciting forces, motions, hydrodynamic pressure and field velocity induced
by the incident wave are understood relative to the incident-wave elevation at X = Y = 0.
XBODY(4,K) is the angle in degrees of the x−axis of the body coordinate system of the
Kth body relative to the X−axis of the global system in counterclockwise sense.
NEWMDS(K) is the number of generalized modes for Kth body. Except for the analysis
of generalized modes (Chapter 8), NEWMDS=0.
When the Alternative Form 2 POT file is used, the parameters NBODY and NEWMDS
must be specified in the POT file, even if they have the default values NBODY=1 and/or
NEWMDS=0. The name(s) specified for the GDF file(s) take precedence over the GDF
filename, if any, that is specified in the FNAMES.WAM file.

3–12

3.3

THE FORCE CONTROL FILE (Alternative form 1)

The Force Control File (FRC) is used to input various parameters to the FORCE subprogram. The name of the FRC file can be any legal filename accepted by the operating
system, with a maximum length of 16 ASCII characters, followed by the extension
‘.frc’.
In this Section the first form of the FRC file is described, in which the input of the
body inertia matrix is simplified, and it is assumed that the body is freely floating. (For
the more general alternative form see Section 3.4.)
The data in the Alternative 1 FRC file is listed below:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
VCG
XPRDCT(1,1) XPRDCT(1,2) XPRDCT(1,3)
XPRDCT(2,1) XPRDCT(2,2) XPRDCT(2,3)
XPRDCT(3,1) XPRDCT(3,2) XPRDCT(3,3)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
The index IOPTN(9) was added in Version 5, and must be included in the FRC file. Users
of earlier versions of WAMIT must take special care to add this extra index to old files
before using Version 6 and higher. Since FORTRAN free-format read statements are used
to read this file, if IOPTN(9) is missing its value will be read from the line containing VCG,
VCG will be read as XPRDCT(1,1), etc. If there are no values of BETAH or XFIELD
specified in the FRC file, the final result will be that there are not sufficient lines of data and
an error message will be displayed that there is erroneous data in the FRC file. However if
values of BETAH and/or XFIELD are specified it is possible for all of the data to be read
without detecting an error, with unpredictable results. This possibility can be prevented
by adding a non-numeric comment at the end of the line containing the IOPTN array, as
illustrated in Test Run 01.
The definition of each variable in the Force Control File is as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72. This line is
available for the user to insert a brief description of the file.

3–13

IOPTN is an array of option indices. These indicate which hydrodynamic parameters
are to be evaluated and output from the program. The available options, descriptions and
numeric file names are as follows:
Option
1
2
3
4
5p
5v
6
7
8
9

Description

Filename

Added-mass and damping coefficients
Exciting forces from Haskind relations
Exciting forces from diffraction potential
Motions of body (response amplitude operator)
Hydrodynamic pressure on body surface
Fluid velocity vector on body surface
Pressure/ free-surface elevation at field points
Fluid velocity vector at field points
Mean drift force and moment from momentum
Mean drift force and moment from pressure

frc.1
frc.2
frc.3
frc.4
frc.5p
frc.(5vx,5vy,5vz)
frc.6
frc .(7x,7y,7z)
frc.8
frc.9

The evaluation and output of the above parameters is accordance with the following
choice of the corresponding index:
IOPTN(I) = 0: do not output parameters associated with option I.
IOPTN(I) = 1: do output parameters associated with option I.
Options 4, 5, 6, 7, 8 and 9 may have additional values as listed below:
IOPTN(4)
IOPTN(4) = 0: do not output response amplitude operator, RAO
IOPTN(4) = ±1: do output RAO by Haskind exciting force
IOPTN(4) = ±2: do output RAO by diffraction exciting force
IOPTN(4) = −3: output field data only for specified radiation modes
The use of IOPTN(4)=-1, -2, or -3 is explained in Sections 3.5.
IOPTN(5)
IOPTN(5) = 0: do not output pressure and fluid velocity on the body
IOPTN(5) = 1: do output pressure on the body
IOPTN(5) = 2: do output fluid velocity on the body
IOPTN(5) = 3: do output both pressure and fluid velocity on the body
IOPTN(6)
IOPTN(6) = 0: do not output pressure in the fluid and/or free-surface elevation
IOPTN(6) = 1: do output pressure in the fluid and/or free-surface elevation by the
potential formulation
IOPTN(6) = 2: do output pressure in the fluid and/or free-surface elevation by the
source formulation
3–14

IOPTN(7)
IOPTN(7) = 0: do not output fluid velocity in the fluid
IOPTN(7) = 1: do output fluid velocity in the fluid by the potential formulation
IOPTN(7) = 2: do output fluid velocity in the fluid by the source formulation
IOPTN(8)
IOPTN(8) = 0: do not output mean force and moment from momentum integration
IOPTN(8) = 1: do output mean force and moment only for unidirectional waves
IOPTN(8) = 2: do output mean force and moment for all combinations of wave headings
IOPTN(9)
IOPTN(9) = 0: do not output mean force and moment from pressure integration
IOPTN(9) = 1: do output mean force and moment only for unidirectional waves
IOPTN(9) = 2: do output mean force and moment for all combinations of wave headings
The options IOPTN(5)=2 and 3, IOPTN(6)=2, and IOPTN(7)=2 apply only for the
low-order method (ILOWHI=0), and require the source formulation (ISOR=1). IOPTN(9)=1
and 2 require the source formulation (ISOR=1) if and only if the low-order method
(ILOWHI=0) is used. If the higher-order method (ILOWHI=1) is used, IOPTN(9) can
take any of the values 0,1,2 with ISOR=0.
The settings of the indices IOPTN(I) must be consistent with themselves and with
the indices IRAD, IDIFF, and NBETA set in the Potential Control File. Error messages
are generated if inconsistent indices are input. Otherwise, the indices IRAD, IDIFF and
IOPTN(I), I=1,...,9 can be selected in any way the particular application may suggest.
Three principal applications are as follows:
Forced motions in calm water (the radiation problem). In this case the modes of
possible motion are specified by the MODE(I) indices in the Potential Control File. The
diffraction index IDIFF should be set equal to -1. The corresponding linear force coefficients
are obtained with Option 1. Field pressures, velocities, free-surface elevations and drift
forces follow from the corresponding options 5-9.
Note that in the default case the pressures, velocities, and drift forces are not separated
according to each mode, and the consequence of setting more than one mode to be nonzero
is to superpose all such modes with unit amplitude. For example if IRAD=1 and IDIFF=1, the default outputs from options 5-9 correspond to unit amplitudes of motion in all six
degrees of freedom, with the same phase. Generally this is a nonphysical problem and
care should be taken to avoid it. The simplest procedure to evaluate these outputs for
each mode is to set MODE(I)=1 for only one mode at a time. However this requires a
separate run of WAMIT for each mode. In V6.2-3 it is possible to output separate results
for options 5-7 for each mode of forced motion in a single run, using the configuration
parameters INUMOPT5, INUMOPT6, INUMOPT7, as explained in Section 4.12.)
Diffraction of incident waves by a stationary structure (the diffraction problem).
3–15

In this case the radiation index IRAD should be set equal to -1. To solve the complete
diffraction problem set IDIFF=1, with corresponding outputs from the options 3,5,6,7,8,9
in FORCE. (If IOPTN(4)=0 and IDIFF=1 it is assumed that the body is stationary,
irrespective of IRAD. Thus, after a complete WAMIT run where the P2F file is output
from POTEN and saved, it is possible to execute another run using only FORCE with the
body motions both free and fixed, without re-running POTEN.)
In V6.2-3 it is possible to output separate results for options 5-7 for the diffraction
problem, using the configuration parameters INUMOPT5, INUMOPT6, INUMOPT7, as
explained in Section 4.12.
Body motions in incident waves. In this case the index IRAD and IDIFF are set equal
to 0 (body free only in specified modes) or 1 (body free in all modes). Body motions are
obtained from the solution of the equations of motion using Option 4. The resulting field
data and drift forces are evaluated for this particular combination of the radiation and
diffraction solutions.
VCG Dimensional z-coordinate of the center of gravity of the body relative to the origin
of the body-system, input in the same units as the length ULEN.
XPRDCT is the 3×3 matrix of the body radii of gyration about the body-fixed axes, where
I,J=1,2,3 correspond to (x, y, z) respectively, input in the same units as the length ULEN.
More precisely, the elements of the body inertia matrix mij are evaluated for i, j = 4, 5, 6
according to the algorithm mij = m × XPRDCT(i − 3, j − 3) × |XPRDCT(i − 3, j − 3)|.
Here the body mass m is evaluated from the displaced mass of fluid, and the absolute
value is used in the last factor so that negative products of inertia can be specified. The
remaining elements of mij are evaluated assuming the body is freely floating in equilibrium,
based on the calculated values of the displaced volume and center of buoyancy and on the
specified value of VCG. In practical cases the matrix XPRDCT is symmetric. Zeroes may
be specified if the body motions are not evaluated.
NBETAH is the number of Haskind wave headings, defined below. NBETAH must be an
integer, greater than or equal to zero.
BETAH is an array of length NBETAH defined as the Haskind wave headings in degrees.
The Haskind wave headings may be introduced in the Force Control File as an option, to
enable evaluations to be made of the Haskind exciting forces (Option 2) and body motions
in waves (Option 4) at heading angles not included in the Potential Control File. This
option is feasible since the evaluation of Haskind exciting forces requires only the radiation
potentials already determined by POTEN (see Section 4.3). This is a useful feature since a
relatively small number of wave headings for the diffraction problem may be specified in the
Potential Control File and the time required to solve many diffraction problems in POTEN
greatly exceeds the time required to evaluate the Haskind exciting forces in FORCE. Since
the number of Haskind wave headings will affect the subsequent READ statements for data
in the Force Control File, it is important to ensure that this number corresponds with the
prescribed integer NBETAH. In particular, if NBETAH= 0 no values of BETAH should
be included and NFIELD should appear on the next line of the Force Control File. If
3–16

NBETAH> 0 is specified, the settings of the IOPTN switches are automatically set equal
to 0 for options 3,5,6,7,8.
NFIELD is the number of points in the fluid domain(free surface) where the hydrodynamic
pressure(wave elevation) and/or velocity are to be evaluated. NFIELD must be an integer,
greater than or equal to zero.
XFIELD is a three-dimensional array with dimensions 3 × NFIELD, defining the dimensional global coordinates of field points where the pressure/wave elevation and/or fluid
velocity vector will be evaluated. Here I=1,2,3 correspond to the (X, Y, Z) coordinates. If
Z = 0 the resulting output should be interpreted as the nondimensional wave elevation,
otherwise as the nondimensional pressure. If NFIELD= 0 no input should be made for the
array XFIELD.
Additional data is required to specify the field point array XFIELD when field points are
placed inside internal tanks. This departure from the usual default procedure is described
in Sections 3.7 and 10.7.

3–17

3.4

THE FORCE CONTROL FILE (Alternative form 2)

In this Section the second alternative form of the FRC file is described, where it is possible
to specify separately three independent external force matrices including the mass matrix
of the body, an external damping matrix, and an external stiffness matrix. This permits
the analysis of bodies which are not freely floating in waves, with arbitrary linear external
forces and moments, and also permits the specification of the complete body mass matrix
instead of the simpler radii of gyration (cf. Section 3.3).
The format of the Alternative 2 FRC file is shown below:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
RHO
XCG YCG ZCG
IMASS
EXMASS(1,1) EXMASS(1,2) ... EXMASS(1,6+NEWMDS)
EXMASS(2,1) EXMASS(2,2) ... EXMASS(2,6+NEWMDS)
.
.
EXMASS(6+NEWMDS,1) EXMASS(6+NEWMDS,2) ... EXMASS(6,6+NEWMDS)
IDAMP
EXDAMP(1,1) EXDAMP(1,2) ... EXDAMP(1,6+NEWMDS)
EXDAMP(2,1) EXDAMP(2,2) ... EXDAMP(2,6+NEWMDS)
.
.
EXDAMP(6+NEWMDS,1) EXDAMP(6+NEWMDS,2) ... EXDAMP(6,6+NEWMDS)
ISTIF
EXSTIF(1,1) EXSTIF(1,2) ... EXSTIF(1,6+NEWMDS)
EXSTIF(2,1) EXSTIF(2,2) ... EXSTIF(2,6+NEWMDS)
.
.
EXSTIF(6+NEWMDS,1) EXSTIF(6+NEWMDS,2) ... EXSTIF(6+NEWMDS,6+NEWMDS)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
The recommended procedure to designate Alternative Form 2 is to assign the parameter
IALTFRC=2 in the configuration file, as described in Section 3.7. In earlier versions of
3–18

WAMIT this parameter was assigned differently, by inserting the integer 2 on a separate
line after the header line of the FRC file, as shown below:
header
2
IOPTN(1) IOPTN(2) ...
RHO
.
.

IOPTN(9)

This variant of the FRC file format is still supported in Version 6, to enable old files to be
read correctly. However it is deprecated, and may not be supported for later versions of
WAMIT.
The header, IOPTN array, and all lines beginning with the variable NBETAH, are
identical to the data in the Alternative form 1 FRC file, as defined in Section 3.3. In the
remainder of this Section the data which differ in Form 2 are described.
RHO Dimensional density of the fluid, in the same units as used for the external force
matrices and for GRAV.
XCG YCG ZCG Dimensional coordinates of the body center of gravity in terms of the
body coordinate system and in the same units as ULEN.
IMASS This index is either 0 or 1, to signify if the external mass matrix EXMASS is
read. If the value of the index is zero, the matrix EXMASS is not included in the FRC file
and the program assumes that all values in this matrix are zero. If the value of the index
is one, the matrix EXMASS is included in the FRC file.
EXMASS is the (6+NEWMDS) × (6+NEWMDS) dimensional inertia matrix of the body
about the body-fixed axes. (For a conventional rigid body this is a 6×6 dimensional matrix
as defined in Reference [3], page 149, equation 141.) Each element in this matrix is added
to the corresponding added mass of the body, in setting up the equations of body motions.
IDAMP This index is either 0 or 1, to signify if the external damping matrix EXDAMP
is read. If the value of the index is zero, the matrix EXDAMP is not included in the FRC
file and the program assumes that all values in this matrix are zero. If the value of the
index is one, the matrix EXDAMP is included in the FRC file.
EXDAMP is the (6+NEWMDS) × (6+NEWMDS) dimensional damping matrix of an
arbitrary external force or moment acting on the body, e.g. from a mooring cable subject
to viscous damping. The value of each element in this matrix is added to the corresponding
linear wave damping coefficient of the body, in setting up the equations of body motions.
ISTIF This index is either 0 or 1, to signify if the external mass matrix EXSTIF is read.
If the value of the index is zero, the matrix EXSTIF is not included in the FRC file and
the program assumes that all values in this matrix are zero. If the value of the index is
one, the matrix EXSTIF is included in the FRC file.
3–19

EXSTIF is the (6+NEWMDS) × (6+NEWMDS) dimensional stiffness matrix of an arbitrary external force or moment acting on the body, e.g. from an elastic mooring cable.
In setting up the equations of body motions, the value of each element in this matrix is
added to the corresponding restoring coefficient of the body, including both hydrostatic
pressure and the gravitational moment due to the body’s mass, as defined in Reference
[3], page 293, equation 145. (The vertical inertia force due to heave, EXMASS(3,3), is
assumed equal to the body mass and is used to derive the gravitational restoring moment
of the body. In any situation where this assumption is not satisfied, due to the presence of
an external vertical inertia force, the gravitational restoring moment should be corrected
for this difference via the stiffness matrix EXSTIF.)
The units of EXMASS, EXDAMP, EXSTIF must correspond to those used to specify the
fluid density RHO and the length ULEN, with time measured in seconds. These matrices
must be defined with respect to the body-fixed coordinate system.
It is also possible to use Alternative Form 2 with the external force matrices in separate
files. In this case the index IMASS, IDAMP, and/or ISTIF is set equal to 2, and followed
by the corresponding file name:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
RHO
XCG YCG ZCG
2
MASS (file name containing inertia matrix)
2
DAMP (file name containing damping matrix)
2
STIF (file name containing stiffness matrix)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
MASS is a file name which contains external inertial forces to the bodies and interactive
inertial forces between bodies. For example, the data in MASS for a body are listed below:
header
EXMASS(1,1) EXMASS(1,2) ...
EXMASS(2,1) EXMASS(2,2) ...
.

EXMASS(1,6+NEWMDS)
EXMASS(2,6+NEWMDS)

3–20

.
EXMASS(6+NEWMDS,1) EXMASS(6+NEWMDS,2) ... EXMASS(6+NEWMDS,6+NEWMDS)
For N interacting
bodies, the number of data in MASS are NDFR × NDFR, where
PN
NDFR= n=1 (6 + NEWMDS(n)) is the total number of rigid body modes and generalized modes. Similarly the external damping and stiffness can be specified to the files
DAMP and STIF.
In summary, there are three legal values for IMASS, IDAMP, ISTIFF:
I=0: no matrix or file name is input, matrix is assumed equal to zero
I=1: matrix follows on subsequent lines
I=2: file name follows on next line
The recommended procedure to designate Alternative Form 2 is to assign the parameter
IALTFRC=2 in the configuration file, as described in Section 3.7, and use the formats
above. In earlier versions of WAMIT the parameter IALTFRC was assigned differently, by
inserting the integer 2 on a separate line after the header line of the FRC file, as shown
below:
header
2
IOPTN(1) IOPTN(2) ...
RHO
.
.

IOPTN(9)

This variant of the FRC file format is still supported in Version 6, to enable old files to be
read correctly. However it is deprecated, and may not be supported for later versions of
WAMIT. If a file with this variant is input, Version 6 issues a warning message.

3–21

3.5

DEFINITION OF FIXED OR FREE MODES

It is possible to specify that a sub-set of the modes of body motion analyzed in POTEN
can be fixed in FORCE. As a simple example, consider a single body with six degrees of
rigid-body motions, all of which have been analyzed in POTEN (either by setting IRAD=1
or by setting IRAD=0 and setting all six elements of MODE=1). Normally, in the FORCE
analysis (IOPTN(4)) the body motions in all six degrees of freedom are computed. Now
suppose that the body is restrained in the vertical modes (heave, roll, pitch) as would
be the case for the first-order motions of a tension-leg platform. This condition can be
analyzed in FORCE by modifying the Force Control File in the following manner:
(1) assign a negative value to IOPTN(4) (-1 to use the Haskind exciting force or -2 to
use the diffraction exciting force).
(2) insert two new lines of data after IOPTN (before VCG or RHO):
NDFR
MODE(1),MODE(2),MODE(3), ... MODE(NDFR)
Here NDFR is the total number of possible radiation modes and MODE is an array with
the value of each element 0 if the mode is fixed or 1 if the mode is free. For the example
described above, NDFR=6 and MODE = (1,1,0,0,0,1). Thus surge, sway, and yaw are free
while heave, roll, and pitch are fixed.
When this option is employed the RAO’s output for the free modes are defined in the
conventional manner, as the amplitudes of body motions in the corresponding degrees of
freedom (cf. Section 4.4). For the fixed modes the RAO’s are replaced by the loads acting
on the body in the corresponding directions. In this case the corresponding modal index
in the output file is shown with a negative value, to signify the change. For the example
described above, the output RAO for heave is equal to the vertical load acting on the body
(equal and opposite to the load on the restraining structure), and preceded by the index
-3. The TLP Test Runs (06, 07, 14) described in the Appendix illustrate this application.
For a single body with no generalized modes NDFR=6 in all cases. For the analysis
of multiple bodies (cf. Chapter 7), with no generalized modes, NDFR=6*NBODY. If
generalized modes are analyzed (cf. Chapter 8) NDFR is the total number of modes for all
bodies, including both rigid-body
modes and generalized modes. Thus, in the most general
P
NEWMDS(n).
case, NDFR=6*NBODY+ NBODY
n=1
It is also possible to perform the analysis in FORCE for a problem where no incident
waves exist, after running POTEN with incident waves present and with the diffraction
solution obtained. To suppress the incident waves in FORCE, IOPTN(4)=-3 is assigned
in the FRC file. In that case the body pressure/velocity, field point pressure/velocity, and
mean drift forces are evaluated assuming that the body is oscillating in the specified free
mode(s), without incident waves.

3–22

3.6

FILENAMES LIST ‘FNAMES.WAM’

An optional input file may be used to specify the filenames of the primary input files
CFG, POT, FRC, and GDF. Use of this optional file is recommended, particularly to
facilitate batch processing. The optional file must be named FNAMES.WAM. (The name
‘fnames.wam’ must be used for this file if the system is case sensitive. Source-code users
can redefine this name if it is desired to do so.) The optional file is simply a list of the
other input file names, including their respective extensions ‘.cfg’, ‘.pot’, ‘.frc’, and ‘.gdf’.
If this file does not exist, or if it is incomplete, the user is prompted to supply the missing
filenames interactively. In Version 6 the number and order of these file names is arbitrary,
but they must include the specified extensions. If the alternative (IALTPOT=2) format
is used, as explained in Section 3.2, the GDF filename(s) are specified in the POT file
and it is not necessary to specify any GDF filename in the FNAMES.WAM file. (In this
case a GDF filename that exists in the FNAMES.WAM file is ignored.) In Version 6.2-3
the filename of the configuration file may be input in the FNAMES.WAM file, with the
extension ‘.cfg’, as explained below.

3–23

3.7

CONFIGURATION FILE ‘CONFIG.WAM’ OR ‘∗.CFG’

The configuartion file may be used to specify various parameters and options in WAMIT.
The configuration file can be assigned the generic name ‘config.wam’, as in previous versions
of WAMIT. Alternatively, the filename of this file can be specified by the user, with the
extension ‘.cfg’. The latter option, which is illustrated in the Test Runs in the Appendix,
permits configuration files to be associated with specific names of runs or other input files.
If a configuration file is included in the FNAMES.WAM list with the extension ‘.cfg’ this
file is used. If no configuration file is included in the FNAMES.WAM list, the generic name
CONFIG.WAM is assigned.
The complete list of inputs which may be specified in the configuration file are as follows
(in alphatetic order):
Parameter
IALTFRC
IALTFRCN
IALTPOT
ICTRSURF
IDIAG†
IFIELD ARRAYS
IFORCE
IGENMDS
ILOWGDF
ILOWHI
ILOG†
INUMOPT5
INUMOPT6
INUMOPT7
IPERIO
IPLTDAT
IPNLBPT
IPOTEN
IQUAD†
IQUADI†
IQUADO†
IRR†
ISCATT
ISOLVE
ISOR†
ITANKFPT
KSPLIN
MAXITT
MAXMIT
MAXSCR

Alternate File Default value
1
1,1,1,...
1
0
POT
0
1
0
0
0
POT
0
0
0
1
0
0
1
POT
0
SPL
4
SPL
3
POT
0
0
POT
0
SPL
3
35
8
2

3–24

MODLST
0
MONITR
0
0
NEWMDS(n)†
POT2
NOOUT
1
NPTANK(n)
NUMHDR
0
NUMNAM
0
-1.0
PANEL SIZE
RHOTANK
current directory
SCRATCH PATH
current directory
USERID PATH
XBODY(n)†
POT or GGDF
Parameters marked † must be input once and only once, either in the configuration file,
or in the other files specified following the symbol †. All other parameters in the list above
are optional, with the specified default value assigned when the parameter is not included
explicitly.
The parameters marked † are defined in the Sections pertaining to the respective input
files (Sections 3.1, 6.9, 7.1 respectively for the POT, SPL, GGDF files). POT2 denotes the
Alternative Form 2 POT file format as explained in Section 3.2.
NEWMDS(n), NPTANK(n) and XBODY(n) denote the NEWMDS, NPTANK and XBODY
arrays for body number n. When NBODY=1 these can be written more simply as
NEWMDS, NPTANK and XBODY. If NBODY>1 the number of each body should be
indicated in parenthesis, as illustrated in the example in Section 7.6. If IALTPOT= 1
nonzero values of NEWMDS must be input for each body in the configuration file. If
IALTPOT= 2 the parameters NEWMDS and XBODY must be input for each body in the
POT file as shown in Section 3.2.
Explanations of the other parameters which may be specified in the configuration file
are as follows:
IALTFRC is an integer specifying the alternative forms of the FRC file as explained in
Sections 3.3, 3.4, and 7.3-5.
IALTFRC= 1: use the Alternative Form 1 FRC format shown in Section 3.3,
IALTFRC= 2: use the Alternative Form 2 FRC format shown in Section 3.4,
IALTFRC= 3: use the Alternative Form 3 FRC format shown in Section 7.5.
See Sections 3.4 and 7.5 concerning the specification of Alternative Forms 2 and 3 FRC
files by special modification of the data on line 2, as in earlier versions of WAMIT. This
option is deprecated and may not be supported beyond Version 6.
The default value is IALTFRC= 1.
IALTFRCN is an integer array of length NBODY. The value of each array element, 1 or
2, designates the corresponding Alternative Form of the separate FRC files for each body,
as explained in Section 7.5. This array is only used if IALTFRC=3 and NBODY>1.

3–25

IALTPOT is an integer specifying the alternative forms of the POT file and GGDF file.
IALTPOT= 1: use the Alternative Form 1 POT format shown in Section 3.1, and define
the GDF filename in FNAMES.WAM. For multiple bodies the separate GDF filename for
each body is assigned in the Global GDF as explained in Section 7.1.
IALTPOT= 2: use the POT format shown in Section 3.2, with the GDF filename(s)
assigned in the POT file.
IALTPOT= −1: use the POT format shown in Section 3.1. This is a special alternative
intended to permit using V6 for multiple bodies without changing the Global GDF format
required in earlier versions of WAMIT. Further details are given in Section 7.1. Use of this
option is deprecated, and may not be supported beyond Version 6.
The default value is IALTPOT= 1.
ICTRSURF is an integer specifying the alternative forms to evaluate the drift forces
based on integration of the momentum flux over a user-defined control surface (see Chapter
14).
ICTRSURF= 0: Do not evaluate the control-surface drift forces.
ICTRSURF= 1: Evaluate the control-surface drift forces using Alternative 1 integration
of the momentum flux.
ICTRSURF= 2: evaluate the control-surface drift forces using Alternative 2 integration of
the momentum flux.
The default value is ICTRSURF= 0.
IFIELD ARRAYS is an integer parameter specifying if uniform arrays of field points are
assigned in the FRC control file, using a compressed format as explained in Section 3.10.
IFIELD ARRAYS= 0: Field point data is assigned only as shown in the conventional FRC
files, as explained in Sections 3.3 and 3.4.
IFIELD ARRAYS= 1: Additional field point data is assigned in the conventional FRC
files, as explained in Section 3.10.
The default value is IFIELD ARRAYS= 0.
IFORCE is an integer parameter specifying if the FORCE subprogram is executed during
the WAMIT run.
IFORCE= 0: Do not execute FORCE.
IFORCE= 1: Do execute FORCE.
The default value is IFORCE= 1.
IGENMDS is an integer parameter specifying the option to input the geomeric data
associated with the mode shapes of generalized modes, when NEWMDS > 0. The details
on the use of the generalized mode option are described in Chapter 8.
IGENMDS= 0: use a seperate program DEFMOD to input the geometric data for generalized modes. This option can be used only with the low order method (ILOWHI=0).
3–26

In this case the user prepares a subroutine in DEFINE to describe the generalized mode
shapes.
IGENMDS= 1: use a DLL file containing the subroutine NEWMODES. The user modifies
NEWMODES to specify the mode shape. This option can be used with both the low- and
higher-order options.
The default value is IGENMDS= 0.
ILOWGDF is an integer parameter specifying the option to generate a low-order GDF
output file based on the input geometry (see Section 10.6). If ILOWHI=0, the original
input panels are used. If ILOWHI=1 the low-order output panels are generated from the
panels of the higher-order geometry as defined in Section 6.1. If ILOWGDF> 1, the panels
are subdivided into ILOWGDF×ILOWGDF sub-panels.
ILOWGDF= 0: Do not generate the output file gdf LOW.GDF.
ILOWGDF≥ 1: Generate the output file gdf LOW.GDF.
The default value is ILOWGDF= 0.
ILOWHI is an integer parameter specifying the use of the low-order or higher-order panel
method.
ILOWHI= 0: Low-order panel method (Chapter 5)
ILOWHI= 1: Higher-order panel method (Chapter 6)
INUMOPT5 is an integer parameter specifying the option to evaluate the separate radiation and diffraction components of the body pressure and velocity. (see Section 4.12).
INUMOPT5= 0: output the total body pressure and velocity, as in Version 6.0.
INUMOPT5= 1: output the total body pressure and velocity in the formatted output file
(.out). Output the separate components in the numeric output files (.5p, .5vx, .5vy, .5vz).
INUMOPT6 is an integer parameter specifying the option to evaluate the separate radiation and diffraction components of the pressure at field points.(see Section 4.12).
INUMOPT6= 0: output the total field pressure, as in Version 6.1.
INUMOPT6= 1: output the total field pressure in the formatted output file (.out). Output
the separate components in the numeric output file (.6).
INUMOPT7 is an integer parameter specifying the option to evaluate the separate radiation and diffraction components of the fluid velocity at field points. (see Section 4.12).
INUMOPT7= 0: output the total field velocity, as in Version 6.1.
INUMOPT7= 1: output the total field velocity in the formatted output file (.out). Output
the separate components in the numeric output files (.7x, .7y, .7z).
IPERIO is an integer parameter specifying the input data for PER in the POT file.
IPERIO= 1: Input periods in seconds for PER
IPERIO= 2: Input radian frequencies for PER
3–27

IPERIO= 3: Input infinite-depth wavenumbers for PER
IPERIO= 4: Input finite-depth wavenumbers for PER
The default value is IPERIO= 1.
IPLTDAT is an integer parameter specifying the option to generate the output files
gdf PAN.DAT and gdf PAT.DAT, used for plotting the panel and patch discretizations
(see Section 10.6).
IPLTDAT= 0: Do not generate the output files gdf PAT.DAT and gdf PAN.DAT
IPLTDAT≥ 1: Generate the output files gdf PAT.DAT and gdf PAN.DAT for plotting
the panel and patch discretizations with IPLTDAT×IPLTDAT subdivisions.
The default value is IPLTDAT= 0.
IPNLBPT is an integer parameter specifying the option to evaluate the body pressure at
specified points (x, y, z) listed in the input file gdf.BPI. (see Section 4.11).
IPNLBPT= 0: output the body pressure at the panel centroids, or on a uniform parametric
mesh in the higher-order method, as in Version 6.0.
IPNLBPT≥ 1: output the body pressure at the points listed in the input file gdf.BPI. These
points are defined with reference to the body-fixed dimensional Cartesian coordinates for
each body.
IPNLBPT≤ −1: output the body pressure at the points listed in the input file gdf.BPI.
These points are defined with reference to the global dimensional Cartesian coordinate
system.
The default value is IPNLBPT= 0. When IPNLBPT6=0 and ILOWHI=0, the absolute
value of IPNLBPT specifies the number of panels nearest to each point specified in gdf.BPI.
See Section 4.11 for further details.
IPOTEN is an integer parameter specifying if the POTEN subprogram is executed during
the WAMIT run.
IPOTEN= 0: Do not execute POTEN.
IPOTEN= 1: Do execute POTEN.
The default value is IPOTEN= 1.
ISCATT is an integer parameter specifying whether the diffraction or the scattering problem is solved to obtain the diffraction potential. The diffraction potential may be solved
by the equation (12.12) which we define as the diffraction problem. Alternatively in the
scattering problem, the scattered potential is solved by (12.13) and the diffraction potential
is obtained from equation (12.8): This option is only available in the low-order method
(ILOWHI=0). In the higher-order method (ILOWHI=1) an error message is generated if
ISCATT=1.
ISCATT= 0: Solve the diffraction problem
ISCATT= 1: Solve the scattering problem
The default value is ISCATT= 0.
3–28

ISOLVE is an integer parameter specifying the method of solution for the linear systems
in POTEN.
ISOLVE= 0: Use the iterative solver
ISOLVE= 1: Use the direct solver
ISOLVE= N For N ≥ 2 : Use the block iterative solver with N blocks
The default value is ISOLVE= 0.
The direct solver is based on a partial-pivoting LU decomposition algorithm known as
Gauss elimination. In general, the iterative solver is faster than the direct solver as NEQN,
the size of the linear system, increases. However, there are cases where the direct solver
may have advantage. For applications where NEQN is relatively small the direct solver
may be somewhat faster than the iterative solver. This is particularly the case where the
number of right-hand sides is large, as in cases where the diffraction solution is required
for a large number of wave headings, or where a large number of radiation solutions are
required for the analysis of multiple bodies or generalized modes. The direct solver is also
useful in applications where the iterative solver fails to converge (See Section 12.6). If the
direct solver is used it is necessary for MAXSCR to be at least as large as NEQN.
The block iterative solver is based on the combination of the local LU decomposition
for each diagonal block and the algorithm of the iterative solver. This option may be used
in cases where the the iterative solver does not converge sufficiently fast, but where the
available RAM does not permit assigning MAXSCR=NEQN, or where NEQN is so large
that the direct solver is too time-consuming (the cpu required for the LU decomposition is
proportional to NEQN3). In general, the number of iterations is reduced as the block size
increases.
ISOLVE=0 is recommended when ILOWHI=0. In some cases the iterative method does not
converge after MAXITT iterations (see below). In this case, the user can i) set ISOLVE=1
to invoke the direct solver ii) set ISOLVE=N, (N≥2) to invoke the block iterative method,
iii) keep ISOLVE=0 and increase MAXITT or iv) a combination of ii) and iii).
ISOLVE=1 is recommended when ILOWHI=1 because the iterative method is not effective
in general. For a large number of unknowns, the block-iterative method can be efficient.
But the number of blocks should be chosen carefully so that iteration converges sufficiently
fast.
In the standard test runs described in Appendicies A and B, the iterative or block-iterative
solvers converge within the default number of iterations MAXITT=35, for all cases where
these solvers are used, except for TEST02.
ITANKFPT is an integer parameter specifying the option to input field point (fpt) coordinates in the .frc control file, either in the conventional format (default) or in the format
required when some or all of the field points are located within the internal tanks.
ITANKFPT= 0: All field points are in the exterior domain. Field points are input as
specified in Sections 3.3 and 3.4, with the coordinates of each field point on one line. The
line containing the input data for the n’th field point is
XFIELD(1,n),XFIELD(1,n),XFIELD(1,n)
3–29

ITANKFPT= 1: Field points are listed in the .frc control file with the tank number
preceding the coordinates. Thus, in place of the line shown above, the correponding data
are input as follows:
ITANK(n) XFIELD(1,n) XFIELD(1,n) XFIELD(1,n)
ITANK(n) is an integer which specifies the tank number containing the n’th field point.
If ITANK(n)=0, the field point is in the exterior domain. The numbering of the tanks
corresponds to the order of the NPTANK indices, as explained below.
The default value is ITANKFPT= 0.
MAXITT is the integer parameter used to specify the maximum number of iterations in
the iterative solver of POTEN. (Further information is given above regarding the parameter
ISOLVE.)
The default value MAXITT=35 is recommended for general use.
MAXMIT is the maximum number of iterations in the adaptive integration used to
evaluate the momentum integral for the drift force and moment (Option 8). (The maximum
number of integration ordinates in the azimuthal direction is equal to 2**MAXMIT.)
The default value MAXMIT=8 is recommended for general use. A warning message is
issued if the integration does not coverge to 4 significant digits within the specified number
of iterations.
Often the warning message is issued because of the smaller component of drift force or
moment. This force or moment can be either converged for most practical purposes or too
small to be important in practice. It is recommended to check the practical importance
of this quantity. Further check on the convergence of the result can be made by increasing MAXMIT gradually. Since the computational time increases exponentially, it is not
recommended to use significantly large MAXMIT than the default value.
MAXSCR is the integer parameter used to specify the available RAM for scratch storage
in POTEN. The default value MAXSCR=2 is specified in V6PC to minimize the required
RAM. Users should assign MAXSCR in to have the largest practical value as described in
Section 10.4.
MODLST is the integer parameter used to control the order in which the added-mass and
damping coefficients, exciting forces, and RAO’s for different modes of motion are written
to the output files. (In V5.3 the added-mass and damping coefficients were not included
in this option.)
MODLST= 0: Outputs are in ascending order of the modal indices.
MODLST= 1: Outputs are in the order evaluated for each of the corresponding left-handsides.
These two alternatives differ only if NLHS is greater than one. The default value is
MODLST= 0.
3–30

MONITR is the integer parameter used to control the display of output to the monitor
during the execution of FORCE.
MONITR= 0: Outputs to the monitor are abbreviated, consisting of the header text and
displays of each wave period as it is executed. This option is convenient in long runs of
FORCE with extensive lines of output, to permit monitoring the progress of the execution.
MONITR= 1: Outputs of all data evaluated by FORCE are displayed on the monitor
during execution, in the same format as in the OUT ﬁle.
The default value is MONITR= 0.
NOOUT is an integer array with length 9, used to control the output to the OUT ﬁle.
Each element of the array corresponds to one of the 9 options in FRC. If the element equals
0, the corresponding output is omitted from the OUT ﬁle.
NOOUT(I)= 0 Omit the output corresponding to IOPTN(I) in the OUT ﬁle
NOOUT(I)= 1 Include the output corresponding to IOPTN(I) in the OUT ﬁle
If NOOUT is included in the conﬁguration ﬁle all 9 integers must be speciﬁed. An example
is shown on the next line, which speciﬁes that all outputs are included in the OUT ﬁle
except the pressures and/or ﬂuid velocities on the body panels.
NOOUT= 1 1 1 1 0 1 1 1 1
This option can be useful to avoid very long OUT ﬁles, since the data for option 5 is
generally much more extensive than for the other options. The default value NOOUT(I)= 1
for I = 1, ..., 9 is assigned if NOOUT is not included in the conﬁguration ﬁle. (The data for
each speciﬁed option is always included in the corresponding numeric output ﬁle, regardless
of the array NOOUT.)
NPTANK is an integer array used to specify the panel or patch indices of internal tanks.
The data in this array are in pairs, denoting the ﬁrst and last index for each tank. An
even number of indices must be included on each line. More than one line can be used
for multiple tanks, and/or multiple tanks can be deﬁned on the same line. If NBODY>1,
the body number(s) for each body containing tanks must be appended to the parameter
name. Only integer data are read for the array NPTANK, with spaces separating each
index. Other ASCII characters may be used to delimit the data in pairs. Further details
and examples are given in Section 10.7.
NUMHDR is the integer parameter used to control writing of a one-line header in the
numeric output ﬁles.
NUMHDR= 0: No headers are included.
NUMHDR= 1: A one-line header is included in the numeric output ﬁles specifying the ﬁle
name, date, and time.
The default value is NUMHDR= 0.
NUMNAM is the integer parameter used to control the assignment of ﬁlenames to the
numeric output ﬁles.

3–31

NUMNAM= 0: Numeric ﬁlenames are assigned based on the ﬁlename of the FRC input
control ﬁle. (The same ﬁlename is used for the OUT output ﬁle.)
NUMNAM= 1: Numeric ﬁlenames are assigned as ‘optn’.
The default value is NUMNAM= 0.
PANEL SIZE is a parameter used for automatic subdivision of patches in the higherorder panel method. Further information is given in Chapter 6.
PANEL SIZE ≤ 0.0: subdivide patches into panels as speciﬁed by the parameters NU,NV
in the SPL ﬁle.
PANEL SIZE > 0.0: subdivide patches into panels so that the maximum length of each
panel is approximately equal to the value of this parameter in dimensional units.
RHOTANK is a real array used to specify the density of ﬂuid in internal tanks. The
density speciﬁed is relative to the density ρ of the ﬂuid in the external domain outside
the bodies, as deﬁned in Chapter 4. The data in the array RHOTANK must be input
in the same order as the data in the array NPTANK. Multiple lines of this parameter
may be used, with an arbitrary number of data on each line, but each line must begin
with ‘RHOTANK=’. The total number of tanks NTANKS is derived from the inputs
NPTANK in the POTEN run. If fewer than NTANKS values of RHOTANK are speciﬁed,
the remainder of the array is assigned the last non-negative value. Thus if the density is
the same for all tanks, only the ﬁrst value is required. Zero may be assigned, but negative
values of the density must not be assigned. RHOTANK is only used in the FORCE run,
and may be changed if separate FORCE runs are made using the same POTEN outputs.
Further details and examples are given in Section 10.7.
The following two parameters are applicable to the PC version only.
SCRATCH PATH is the path designating a directory (folder) for storage of some scratch
arrays. If this input is not used all scratch storage is in the default directory where
the program is run. If a diﬀerent directory is speciﬁed, about half of the scratch arrays
will be stored in the default directory, and the remaining arrays will be stored in the
designated alternative directory. This option permits users with two or more disk drives
to distribute the scratch storage, thereby increasing the usable disk storage. The example
below illustrates this option. (The user must make the speciﬁed directory, if it does not
already exist, prior to running the program.) Spaces cannot be used in the pathname, as
explained below.
USERID PATH designates the directory (folder) where the input ﬁle USERID.WAM is
stored. (This input ﬁle is required for users of V6PC.) It is convenient to store USERID.WAM,
the executable WAMIT.EXE, and the dynamic link libraries DFORRT.DLL, GEOMXACT.DLL, NEWMODES.DLL together in one directory (e.g. C:\WAMIT). In this case
USERID PATH should be speciﬁed as in the example below. (Note that the executables
can be run from another directory provided their resident directory is included in the system path speciﬁed in the AUTOEXEC.BAT ﬁle.) Spaces cannot be used in the pathname,
as explained below.
3–32

The syntax of the configuration file is illustrated in the following example. To specify
each of the desired inputs, the corresponding parameter is displayed, followed by an ‘=’
sign, followed by the value of the parameter. These lines may be in any order. Lines
which do not contain an ‘=’ sign are ignored. Comments may be inserted following the
value of a parameter on the same line, separated by at least one blank space. (Since
a blank space is used to designate a comment, the names of directories or folders in
SCRATCH PATH and USER PATH cannot include blank spaces.) All elements of the
arrays IALTFRCN(NBODY), NOOUT(9) and XBODY(4) should be displayed in order on
one line. All other parameters must be input singly on separate lines.

3–33

The following example of a configuration file illustrates all of the possible input parameters for NBODY=1. Section 7.6 shows additional inputs for NBODY>1. For clarity the
parameters are arranged in alphabetic order, but their actual order is arbitrary.
IALTFRC=1 (use FRC format in Section 3.1)
IALTPOT=1 (use POT format in Section 3.3)
ICTRSURF=1 (use Alternative 1 evaluation of control surface drift forces)
IDIAG=0 (omit IDIAG from POT file)
IFIELD ARRAYS=1 (arrays of field point data is assigned, as in Section 3.10)
IFORCE=1 (run the FORCE subprogram)
IGENMDS=1 (run DEFINE subroutine in .dll file for generalized modes)
ILOG=0 (omit ILOG from POT file)
ILOWGDF=0 (do not output low-order GDF from higher-order geometry)
ILOWHI=0 (use low-order method)
INUMOPT5=1 (output the separate components of the body pressure/velocity)
INUMOPT6=1 (output the separate components of the field pressure)
INUMOPT7=1 (output the separate components of the field velocity)
IPERIO=1 (POT file contains wave periods in seconds)
IPLTDAT=0 (do not output data for plotting geometry)
IPNLBPT=1 (evaluate body pressure at points specified in body coordinates)
IPOTEN=1 (run the POTEN subprogram)
QUAD=0 (omit IQUAD from POT file)
IQUADI=4 (inner Gauss quadrature order in higher-order method)
IQUADO=3 (outer Gauss quadrature order in higher-order method)
IRR=0 (omit IRR from POT file)
ISCATT=1 (Solve for scattered potential)
ISOLVE=2 (Use two blocks in iterative solver)
ISOR = 1 (solve for source strength – omit ISOR from POT file)
ITANKFPT=1 (input field points in internal tanks with special format)
KSPLIN = 3 (Use 3rd-order B-splines for potential in higher-order method)
MAXITT = 35 (Maximum number of iterations in solver)
MAXMIT = 8 (Maximum number of iterations in momentum integral)
MAXSCR = 1024 (store ≤ 1024 × 1024 LHS matrix elements in RAM)
MODLST =1 (Outputs in same order as left-hand-sides)
MONITR=1 (display all FORCE output to monitor)
NEWMDS=0 (No generalized modes)
NOOUT=1 1 1 1 0 1 1 1 1 (output all but body pressure/velocity data)
NPTANK= 8-11 12-15 (range of panels/patches on two internal tanks)
NUMHDR=1 (write headers to numeric output files)
NUMNAM=1 (Numeric filenames are assigned as ‘OPTN’)
PANEL SIZE = 0.1 (automatic subdivision of patches in higher-order method)
RHOTANK = 0.6 1.0 (relative fluid densities in the two tanks)
SCRATCH PATH=D:\TEMP
USERID PATH=C:\WAMIT
XBODY = 0.0 0.0 0.0 0.0 (omit XBODY from POT file)

3–34

3.8

FILE NAMES

The names of the input data files should not exceed 20 characters in length including
extensions and should have one period separating the filename and extension. Provisions
are made in WAMIT to guard against unintended loss of old output files. If the names
specified for the P2F and OUT files are identical to existing files, the user is prompted
interactively to choose between (1) changing the new output filename, or (2) overwriting
the old file. If a new filename is specified interactively it must include the desired extension.
For example, if the name CYL.FRC is retained in Session 2 as described in Section 2.2,
instead of a modified name such as CYL2.FRC, the user will be prompted at the start of
the FORCE subprogram with the choice of either overwriting the old file CYL.OUT or
specifying another name for the new OUT file. If the default setting NUMNAM=0 is used,
the same safeguard will apply to the numeric output files, minimizing the possibility that
these are lost during a subsequent run. Otherwise, if NUMNAM=1, the OPTN output
files are assigned the same generic names for all runs, and old OPTN files are overwritten
without warning when a new run is made; this option avoids the proliferation of old output
files, but requires the user to rename or otherwise preserve the contents of OPTN files which
are to be saved.
For batch processing it is important to avoid interactive interrogation from the programs.
Thus the user should delete or rename P2F and/or OUT files from previous runs, if the
same names will be assigned (from the POT and/or FRC control filenames respectively in
a new run.)
In the User Manual capital letters are generally used for file names to provide emphasis and visual correspondence with systems where file names are displayed in capital
letters. A distinction must be made for systems such as UNIX, which are case sensitive. Most filenames and extensions which are assigned by WAMIT are specified in lowercase letters (via assignments of type CHARACTER in the source file modulesc.f, module
MAINC ARRAYS). This applies in particular to the input files ‘fnames.wam’ and ‘config.wam’, and to the extensions ‘.p2f’, ‘.out’, etc. The only exceptions to this convention
are the scratch files opened temporarily by WAMIT with the explicit names ‘SCRATCHA’,
SCRATCHB’, etc.
In previous versions of WAMIT the extensions ‘.gdf’, ‘.pot’, and ‘.frc’ were recommended
but not required for the GDF, POT, and FRC input files. In Version 6 these extensions are
required, unless the corresponding assignments are changed in the source code. (This new
requirement permits a more flexible arrangement of the optional input file fnames.wam.)
In V6.2-3, a separate utility program F2T.EXE is provided. ’INPUTS.F2T’ is reserved
as a generic input filename to this program.

3–35

3.9

FILE FORMAT

The free-format READ statements read only the specified data on a line, or on subsequent
lines if there is insufficient data on the first line. Comments inserted after the specified
data are ignored. Thus it is possible for the user to include comments at the ends of
selected lines in the input files, to identify the data on these lines. Such comments should
be separated from the data by at least one blank space. This format is illustrated in the
input files of Test Run 01. Generally, comments at the ends of appropriate lines, which
contain non-numeric ASCII characters, will ensure that execution is interrupted with an
error message if insufficient data is contained on the line.
When blocks of data are written on multiple lines, and read by a single READ statement,
comments are only permitted after all of the data is read. In the POT file, for example,
comments could be placed after the last elements of the arrays PER and BETA, but not on
intermediate lines which contain these arrays, and similarly for field point coordinates in the
FRC file. A special restriction applies to the format of the GDF file when the higher-order
method is used (ILOWHI=1) and the geometry is defined by MultiSurf (IGDEF=2), or
explicitly (IGDEF <0 or >2). In this case it is possible to input relevant body dimensions
in the GDF file, as illustrated in Section 6.8 and in most of the higher-order test runs. In
this case the data included in the GDF file (starting on line 5) must be restricted to columns
1-80. Note that an extra line of data has been added to the GDF file in Versions 6.2-3, in
all cases where (IGDEF <0 or >1), to specify the number of lines to be read beyond line 5.
Users of Version 6.0 must insert this extra line in GDF files where the higher-order method
is used, except in the cases IGDEF=0 and IGDEF=1 where no changes are required. Also,
IGDEF=2 is reserved in V6.2-3 for MultiSurf inputs and should not be used for explicit
geometry definition as permitted in V6.0.

3–36

3.10

UNIFORM ARRAYS OF FIELD POINTS

In some applications large numbers of field points are required, with a spacing on a rectangular grid. In Version 6.3 a new option has been introduced to facilitate inputting this data
in the FRC file. This option is specified by assigning the parameter IFIELD ARRAYS= 1
in the CFG file, as explained in Section 3.7. When IFIELD ARRAYS= 1, additional data
is appended to the FRC file immediately after the line(s) containing NFIELD and XFIELD.
The following shows the complete Alternative 1 FRC file, replacing the format shown in
Section 3.3:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
VCG
XPRDCT(1,1) XPRDCT(1,2) XPRDCT(1,3)
XPRDCT(2,1) XPRDCT(2,2) XPRDCT(2,3)
XPRDCT(3,1) XPRDCT(3,2) XPRDCT(3,3)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
NFIELD ARRAYS
ITANKFLD(1)
NFX(1) X1(1) DELX(1)
NFY(1) Y1(1) DELY(1)
NFZ(1) Z1(1) DELZ(1)
ITANKFLD(2)
NFX(2) X1(2) DELX(2)
NFY(2) Y1(2) DELY(2)
NFZ(2) Z1(2) DELZ(2)
.
.
ITANKFLD(NFIELD ARRAYS)
NFX(NFIELD ARRAYS) X1(NFIELD ARRAYS) DELX(NFIELD ARRAYS)
NFY(NFIELD ARRAYS) Y1(NFIELD ARRAYS) DELY(NFIELD ARRAYS)
NFZ(NFIELD ARRAYS) Z1(NFIELD ARRAYS) DELZ(NFIELD ARRAYS)

The additional data is defined as follows:
NFIELD ARRAYS is the number of separate arrays. NFIELD ARRAYS must be an
3–37

integer, greater than or equal to zero.
ITANKFLD(n) (n=1,2,...,NFIELD ARRAYS) is an integer which specifies if the array n
is in the exterior fluid domain, or in an internal tank. ITANKFLD=0 denotes the exterior
fluid domain. In cases where the field point array is in an internal tank ITANKFLD is
assigned with the same integer as the number of the tank, as explained in Section 10.7. In
all cases where no internal tanks are included, ITANKFLD=0 must be specified.
NFX, NFY, NFZ are positive integers, specifying the number of elements in the array
parallel to the X,Y,Z axes. (The total number of elements in each array is the product of
these three integers.)
X1, Y1, Z1 are the coordinates of the first point in the array.
DELX, DELY, DELZ are the distances between adjacent points in the array, in each
direction.
If NFX=1, indicating that there is only one point in the direction parallel to the x-axis,
the value of DELX is irrelevant, but must be assigned to prevent a read error. Similarly,
for NFY=1 and/or NFZ=1 the values of DELY and/or DELZ are irrelevant.
The field points assigned using this procedure are augmented to the list of field points (if
any) assigned in the conventional manner as explained in Section 3.3. If the array option
is used for all field points, then NFIELD=0 must be assigned. At runtime NFIELD is
increased to include all field points. The complete list of all field points is output in the
.FPT file.
Test23 illustrates the use of this option.

3–38

Chapter 4

DEFINITION OF QUANTITIES
EVALUATED BY WAMIT
The forces and other quantities evaluated by WAMIT are output in a standard nondimensional form, in terms of the appropriate combinations of the water density ρ, the
acceleration of gravity g, the incident-wave amplitude A, frequency ω,and the length scale
L deﬁned by the input parameter ULEN in the GDF ﬁle. (The volume and coordinates of
the center of buoyancy are exceptions. They are output in a dimensional form.)
Body motions and forces are deﬁned in relation to the origin of the same Cartesian
coordinate system relative to which the panel oﬀsets are deﬁned. Note that this origin
may be located on, above or below the free surface. If planes of symmetry are deﬁned for
the body, the origin must always lie on these planes of symmetry. The x- and y-axes must
be parallel to the mean position of the free surface.
The notation and deﬁnitions of physical quantities here correspond with those in Reference [3], except that in the latter reference the y axis is vertical.
All of the evaluated quantities are included with appropriate identiﬁcation in the OUT
output ﬁle. To facilitate post-processing the same quantities are also saved in the numeric
output ﬁles, following the format deﬁned in Section 4.9.
In the cases of ﬁeld data and mean drift forces (Options 5-9) the deﬁnitions given below
in Sections 4.5-4.8 apply to the complete solution for the combined radiation and diﬀraction
problems. The same quantities can be evaluated separately for either the radiation or
diﬀraction problems, as explained in Section 4.10.
For the sake of simplicity, the deﬁnitions which follow in this Section assume that the
origin of the coordinate system is located on the free surface. Special deﬁnitions apply to
some quantities if vertical walls are deﬁned, as explained in Section 5.3.

4–1

4.1

HYDROSTATIC DATA

All hydrostatic data can be expressed in the form of surface integrals over the mean body
wetted surface Sb , by virtue of Gauss’ divergence theorem.
a) Volume
∀=−


Sb

n1 xdS = −


Sb

n2 ydS = −


Sb

n3 zdS

All three forms of the volume are evaluated in WAMIT, as independent checks of the
panel coordinates, and printed in the summary header of the output ﬁle. The median
volume of the three is used for the internal computations. If it is less than 10−30 , a
warning is displayed and the coordinates of the center of buoyancy are set equal to zero.
For bottom-mounted structures, where panels are not deﬁned on the bottom, the last
integral deﬁned above diﬀers from the correct submerged volume as noted in Section 3.1.
b) Coordinates of center of buoyancy
xb =

−1 
n1 x2dS
2∀ Sb

yb =

−1
2∀



−1
zb =
2∀



Sb

Sb

n2 y 2dS
n3 z 2dS

c) Matrix of hydrostatic and gravitational restoring coeﬃcients


C(3, 3) = ρg Sb n3 dS
C(3, 4) = ρg Sb yn3 dS
C(3, 5) = −ρg
xn3 dS
 Sb
C(4, 4) = ρg Sb y 2 n3dS + ρg∀zb − mgzg
C(4, 5) = −ρg Sb xyn3dS
C(4, 6) = −ρg∀x
+ mgxg
 b 2
C(5, 5) = ρg Sb x n3 dS + ρg∀zb − mgzg
C(5, 6) = −ρg∀yb + mgyg

C̄(3, 3)
C̄(3, 4)
C̄(3, 5)
C̄(4, 4)
C̄(4, 5)
C̄(4, 6)
C̄(5, 5)
C̄(5, 6)

= C(3, 3)/ρgL2
= C(3, 4)/ρgL3
= C(3, 5)/ρgL3
= C(4, 4)/ρgL4
= C(4, 5)/ρgL4
= C(4, 6)/ρgL4
= C(5, 5)/ρgL4
= C(5, 6)/ρgL4

where C(i, j) = C(j, i) for all i, j, except for C(4, 6) and C(5, 6). For all other values of
the indices i, j, C(i, j) = 0. In particular, C(6, 4) = C(6, 5) = 0.
In C(4,4), C(4,6), C(5,5) and C(5,6), m denotes the body mass. When Alternative
form 1 is used for the FRC ﬁle (Section 3.3) the body mass is computed from the relation
m = ρ∀. When Alternative form 2 is used for the FRC ﬁle (Section 3.4) the body mass is
deﬁned by EXMASS(3,3).

4–2

4.2

ADDED-MASS AND DAMPING COEFFICIENTS


i
Aij − Bij = ρ
ω
Āij =

Aij
ρLk

Sb

ni ϕj dS

B̄ij =

Bij
.
ρLk ω

where k = 3 for i, j = 1, 2, 3, k = 4 for i = 1, 2, 3, j = 4, 5, 6 or i = 4, 5, 6, j = 1, 2, 3 and
k = 5 for i, j = 4, 5, 6.

4.3

EXCITING FORCES

a) Exciting forces from the Haskind relations
Xi = −iωρ




Sb



∂ϕ0
ni ϕ0 − ϕi
dS
∂n

b) Exciting forces from direct integration of hydrodynamic pressure
Xi = −iωρ
X̄i =


Sb

ni ϕD dS

Xi
,
ρgALm

where m = 2 for i = 1, 2, 3 and m = 3 for i = 4, 5, 6.

4.4

BODY MOTIONS IN WAVES

Two alternative procedures are followed to evaluate the body motions in waves, corresponding respectively to the Alternative 1 (Section 3.3) and Alternative 2 (Section 3.4)
FRC control ﬁles.
In Alternative 1, which is restricted to a body in free stable ﬂotation without external
constraints, the following relations hold
m = ρ∀
xb = xg ,

yb = yg

where m is the body mass and (xg , yg , zg ) are the coordinates of the center of gravity.
4–3

The inertia matrix is deﬁned as follows.
⎛

M=

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

⎞

m
0
0
0
mzg −myg
0
m
0
−mzg
0
mxg ⎟
⎟
⎟
0
0
m
myg −mxg
0 ⎟
⎟.
0
−mzg myg
I11
I12
I13 ⎟
⎟
mzg
0
−mxg
I21
I22
I23 ⎟
⎠
−myg mxg
0
I31
I32
I33

(4.1)

WAMIT equates the body mass to the mass of the displaced water in free ﬂotation. The
moments of inertia Iij are deﬁned in terms of the corresponding radii of gyration rij , deﬁned
by the relation
Iij = ρ∀rij |rij |.
The array XPRDCT(I,J) input to WAMIT contains the radii of gyration input with the
same units of length as the length scale ULEN deﬁned in the panel data ﬁle.
In the Alternative 2 format of the FRC ﬁle the matrices Mij + MijE , BijE and CijE are
input by the user to include the possibility of external force/moment constraints acting on
the body.
The complex amplitudes of the body’s motions ξj are obtained from the solution of the
6×6 linear system, obtained by applying Newton’s law
6
j=1

E
−ω 2 (Mij + MijE + Aij ) + iω(Bij + Bi,j
) + (Cij + CijE ) ξj = Xi .

where the matrices MijE , BijE and CijE are included only in the Alternative 2 case. Note that
in the Alternative 2 case the user must specify the body inertia matrix Mij and include it
in the total inertia matrix Mij + MijE speciﬁed in the FRC ﬁle.
The non-dimensional deﬁnitions of the body motions are
ξ¯i =

ξi
,
A/Ln

where n = 0 for i = 1, 2, 3 and n = 1 for i = 4, 5, 6.

4–4

4.5

HYDRODYNAMIC PRESSURE

The complex unsteady hydrodynamic pressure on the body boundary or in the ﬂuid domain
is related to the velocity potential by the linearized Bernoulli equation
p = −ρ

∂ϕ
.
∂t

The total velocity potential is deﬁned by
ϕ = ϕD + iω

6
j=1

ξj ϕj ,

where the radiation and diﬀraction velocity potentials are deﬁned in Section 12.1. In order
to render the velocity potential and the hydrodynamic pressure non-dimensional, we deﬁne
p̄ =

6
p
ξ¯j ϕ̄j .
= ϕ̄D + KL
ρgA
j=1

where K = ω 2 /g and
ϕ̄D =

ϕD
,
igA/ω

ϕ̄j =

ϕj
Ln+1

with n = 0 for j = 1, 2, 3 and n = 1 for i = 4, 5, 6.
The body pressure can be evaluated separately for the diﬀraction or radiation problems
by following the procedure described in Section 4.12. When the radiation components are
output separately, the nondimensional pressure due to jth mode is deﬁned by
p̄ =

p
= KLφ̄j
ρgξj Ln

or p̄ =

p
= −φ̄j
ρgaj Ln+1

where aj is the acceleration in the same unit as the gravitional constant speciﬁed in the
GDF ﬁle for j = 1, 2, 3 and the angular acceleration in rad/s2 for j = 4, 5, 6. n = 0 for
j = 1, 2, 3 and n = 1 for j = 4, 5, 6.

4.6

FREE-SURFACE ELEVATION

The free surface elevation is obtained from the dynamic free-surface condition
1
η=−
g



∂ϕ
∂t

and in non-dimensional form
4–5



,
z=0

⎛

⎞

6
η
⎝
η̄ = = ϕ̄D + KL
ξ¯j ϕ̄j ⎠
A
j=1

,

z=0

where ϕ̄ is deﬁned as in Section 4.5. [Note that the non-dimensional ﬁeld hydrodynamic
pressure and wave elevation are equal to the non-dimensional velocity potential at the
respective positions.]
These parameters can be evaluated separately for the diﬀraction or radiation problems
by following the procedure described in Section 4.10. When the radiation components are
output separately, the nondimensional free-surface elevation due to jth mode is deﬁned by
η
= KLφ̄j
ξj Ln
where n = 0 for j = 1, 2, 3 and n = 1 for j = 4, 5, 6.
η̄ =

The evaluation of the pressure or free-surface elevation requires special caution close
to the body surface. Within a distance on the order of the dimensions of the adjacent
panel(s), ﬁeld-point quantities cannot be computed reliably. More speciﬁc limits can be
ascertained by performing a sequence of computations and studying the continuity of the
result. Approaching the body along a line normal to the centroid of a panel will minimize
this problem. See Reference [12] regarding the computation of run-up at the intersection
of the body and free surface.

4.7

VELOCITY VECTOR ON THE BODY AND IN THE FLUID
DOMAIN

The non-dimensional velocities evaluated by WAMIT are deﬁned in vector form by
6
V
¯
¯ ϕ̄j
= ∇ϕ̄D + KL
V̄ =
ξ¯j ∇
igA/(ωL)
j=1

where
¯ = L∇
∇
is the non-dimensional gradient operator.
These parameters can be evaluated separately for the diﬀraction or radiation problems
by following the procedure described in Section 4.10.
The evaluation of the velocity requires special caution close to the body surface, in the
same manner as the pressure and free-surface elevation. (See Section 4.6 above.) When
the radiation components are output separately, the nondimensional velocity due to jth
mode is deﬁned by
4–6

V̄ =

V
¯ ϕ̄j
=∇
vj Ln

where vj denotes the velocity of the body for j = 1, 2, 3 and the angular velocity for
j = 4, 5, 6. n = 0 for j = 1, 2, 3 and n = 1 for j = 4, 5, 6.

4.8

MEAN DRIFT FORCE AND MOMENT

The deﬁnition of the non-dimensional mean drift force and moment in unidirectional waves
is
F̄i =

Fi
ρgA2 Lk

where k = 1 for the forces (i = 1, 2, 3), and k = 2 for the moments (i = 4, 5, 6).
For bi-directional waves of the same period, with complex amplitudes (A1, A2) and
corresponding angles of incidence (β1, β2), the nondimensional outputs F̄i (β1, β2) are the
coeﬃcients such that the total dimensional mean drift force or moment exerted on the
body is given by the equation
Fi (β1, β2) = ρgLk (|A1|2F̄i (β1, β1) + |A2 |2F̄i (β2, β2) + 2Re[A1 A∗2F̄i(β1 , β2)])
Note that F̄i∗(β2, β1 ) = F̄i (β1, β2), where the asterisk (∗) denotes the complex conjugate.
In Option 8, the evaluation of the horizontal drift force and vertical moment is based
on the momentum conservation principle in its general form (see References [4] and [26]),
without the assumption of energy conservation. This permits the analysis of cases where
the body motions are aﬀected by non-conservative eﬀects, such as external damping. The
azimuthal integration required to evaluate the momentum ﬂux is performed by an adaptive
quadrature formula in subroutine MDRFTM. [The integration is performed iteratively,
with convergence speciﬁed by the criterion of absolute or relative errors in each drift force
less than TOL=10−4 . The maximum number of iterations is controlled by the parameter
MAXMIT=8, which is speciﬁed in the main program. A warning message is displayed in
the event that this convergence criterion is not satisﬁed.]
In Option 9, the evaluation of the drift force and moment is based on integration of
the pressure over the body surface, using the relations in [10] and [17], as summarized in
Section 12.7. When ICTRSURF=1 the drift force and moment are also evaluated based
on the momentum ﬂux across the control surface, using equations (12.57-60).
These parameters can be evaluated separately for the diﬀraction or radiation problems
by following the procedure described in Section 4.10.

4–7

4.9

FORMAT OF NUMERIC OUTPUT FILES

For each of the nine options in the FORCE subprogram, separate output ﬁles of numeric
data are generated as listed in Section 3.3. The hydrodynamic parameters in these ﬁles
are output in the same order as in the OUT ﬁle, and listed in the following format:
OPTN.1:
PER I
J
Āij
B̄ij
OPTN.2:
PER BETA I
Mod(X̄i ) Pha(X̄i ) Re(X̄i ) Im(X̄i )
OPTN.3:
PER BETA I
Mod(X̄i ) Pha(X̄i ) Re(X̄i ) Im(X̄i )
OPTN.4:
PER BETA I
Mod(ξ¯i ) Pha(ξ¯i ) Re(ξ¯i ) Im(ξ¯i )
OPTN.5P:
PER BETA M K Mod(p̄)
Pha(p̄)
Re(p̄)
Im(p̄)
OPTN.5VX: PER BETA M K Mod(V̄x ) Pha(V̄x ) Re(V̄x) Im(V̄x )
OPTN.5VY: PER BETA M K Mod(V̄y ) Pha(V̄y ) Re(V̄y ) Im(V̄y )
OPTN.5VZ: PER BETA M K Mod(V̄z ) Pha(V̄z ) Re(V̄z ) Im(V̄z )
OPTN.6:
PER BETA L
Mod(p̄)
Pha(p̄)
Re(p̄)
Im(p̄)
OPTN.7X:
PER BETA L
Mod(V̄x ) Pha(V̄x ) Re(V̄x) Im(V̄x )
OPTN.7Y:
PER BETA L
Mod(V̄y ) Pha(V̄y ) Re(V̄y ) Im(V̄y )
OPTN.7Z:
PER BETA L
Mod(V̄z ) Pha(V̄z ) Re(V̄z ) Im(V̄z )
OPTN.8: PER BETA1
OPTN.9: PER BETA1
[PER BETA1

BETA2 I Mod(F̄i ) Pha(F̄i ) Re(F̄i )
BETA2 I Mod(F̄i ) Pha(F̄i ) Re(F̄i )
BETA2 -I Mod(F̄io ) Pha(F̄io ) Re(F̄io )

Im(F̄i)
Im(F̄i)
Im(F̄io)]

(Depending on the value of NUMNAM, the ﬁlenames OPTN will be replaced by frc.)
If option 5 is speciﬁed and INUMOPT5=1, as explained in Section 4.12, the numeric
output ﬁles .5P, .5VX, .5VY, .5VZ contain the separate components of the radiation and
diﬀraction pressure and velocity in the following modiﬁed format:
OPTN.5P: PER M K Re(p̄1) Im(p̄1 ) Re(p̄2 ) Im(p̄2 ) ...
PER BETA M K Re(p̄D ) Im(p̄D )
Here ... denotes the remaining components for modes 3,4,5,6 if the six rigid-body modes are
speciﬁed for a single body. More generally when diﬀerent sets of modes are evaluated for
one or multiple bodies, these are output in sequence. For each wave period the radiation
pressures are listed for all values of M and K before the diﬀraction pressures. Corresponding formats apply for the ﬂuid velocity components in the ﬁles OPTN.5VX, OPTN.5VY,
OPTN.5VZ.
Starting in Version 6.3, the supplementary output ﬁle out.hst is created in the following
format, to output values of the hydrostatic matrix Cij :
out.hst: I

J

C(I,J)

If Option 5 is speciﬁed and IPNLBPT=0, the supplementary output ﬁle gdf.PNL is
created in the following format:
gdf.PNL: M

K XCT YCT ZCT AREA nx ny nz (r × n)x (r × n)y (r × n)z

If option 6 is speciﬁed and INUMOPT6=1 or option 7 is speciﬁed and INUMOPT7=1,
the numeric output ﬁles .6, or .7X, .7Y and 7Z contain the separate components of the
4–8

radiation and diﬀraction pressure and velocity in the following modiﬁed format:
OPTN.6: PER L
Re(p̄1 ) Im(p̄1) Re(p̄2) Im(p̄2 ) ...
PER BETA L
Re(p̄D ) Im(p̄D )
Here ... denotes the remaining components for modes 3,4,5,6 if the six rigid-body modes are
speciﬁed for a single body. More generally when diﬀerent sets of modes are evaluated for
one or multiple bodies, these are output in sequence. For each wave period the radiation
pressures are listed for all values of L before the diﬀraction pressures. Corresponding formats apply for the ﬂuid velocity components in the ﬁles OPTN.7X, OPTN.7Y, OPTN.7Z.
(See section 4.12.)
If Option 6 or 7 is speciﬁed, the supplementary output ﬁle frc.FPT will be created in
the following format:
OPTN.FPT: L XFIELD(L)

YFIELD(L)

ZFIELD(L)

Except as noted below, the deﬁnitions of parameters in these ﬁles are as follows:
I, J: Mode indices
M: Index for quadrant (2 planes of symmetry) or half (1 plane of symmetry).
(If no planes of symmetry are speciﬁed, or if IPNLBPT>0, then M=1.)
K: Index for panels on the body surface
L: Index for ﬁeld points
PER: Period
BETA: Wave heading
BETA1, BETA2: Two wave headings for the mean drift forces and moments
XCT, YCT, ZCT: Dimensional global coordinates of panel centroid.
AREA: Dimensional value of the area of a panel
nx , ny , nz : Components of the unit vector normal to K-th panel in local coordinate system
(r × n)x, (r × n)y , (r × n)z : Components of the cross product of the position vector to the
centroid of the K-th panel and it’s normal vector, in the local coordinate system. Here r
is given in dimensional units.
XFIELD, YFIELD, ZFIELD: Dimensional global coordinates of the ﬁeld point
All output quantities are nondimensionalized as deﬁned in Sections 4.2-8. Complex
quantities are deﬁned by the magnitude (Mod), phase in degrees (Pha), and also in terms
of the real (Re) and imaginary (Im) components. The phase is relative to the phase of an
incident wave at the origin of the global coordinates system.
In Option 5, when IPNLBPT= 0, the index M refers to the body index and K refers to
the body point in the order listed in the .bpi input ﬁle and .bpo output ﬁle. The ﬁle gdf.pnl
is only output when IOPTN(5)>0 and IPNLBPT=0. If ILOWHI=1 the data output in
this ﬁle diﬀer from those shown above as follows:
4–9

K: Index for points on the body surface (See section 4.10)
XCT, YCT, ZCT: Dimensional global coordinates of points
AREA: Product JδUδV where J is the Jacobian at the point, and δU, δV denote the
diﬀerential increments between points in parametric coordinates.
nx , ny , nz : Components of the unit vector normal to the body surface at each point
(r × n)x , (r × n)y , (r × n)z : Components of the cross product of the position vector at
each point.
In Option 8, the mean force and moment are output only for modes I=1, 2 and 6,
corresponding to the two horizontal forces and yaw moment, respectively.
In Option 9, the six components of the mean forces and moments, F̄i , are output on
the ﬁrst six lines, with positive indices (i=1,2,...,6). These are the components of the force
and moment vectors, deﬁned with respect to the inertial reference frame corresponding to
the mean position of the body coordinate system. When IRAD= −1, three additional
components of the moment F̄io , are output and identiﬁed by negative indices (i=-4,-5,-6).
These are the components of the moment about the moving origin, denoted by ‘o’ in Figure
12-2. In all cases the components of the vector force and moment are deﬁned with respect
to the inertial (mean) coordinate system.
In Option 9, if ICTRSUFT=1, the drift forces evaluated from momentum ﬂux on the
control surface are output in the numeric output ﬁle OPTN.9c, in the same format as
shown above for OPTN.9.
If NBODY>1, the panels of all bodies are merged with a common index K, following
the same order as the body numbers in the global GDF ﬁle (See Chapter 7).

4–10

4.10

BODY PRESSURE OUTPUT FOR THE HIGHER-ORDER
METHOD

If the higher-order method is used (ILOWHI=1), Option 5 is selected in the FRC ﬁle,
and IPNLBPT=0, the pressure and the ﬂuid velocity on the body surface are output at
the points corresponding to equally spaced points in parametric space. These points are
deﬁned in parametric space as the midpoints of the set of (KU + 1) ∗ (KV + 1) panel
subdivisions on each patch (see Chapter 6). The coordinates, the extended normal vector
corresponding to 6 rigid body modes and the Jacobian are output in the .pnl ﬁle. The
value of the Jacobian at the prescribed point replaces the panel area in the format shown
in Section 4.9. The pressure and the ﬂuid velocity vector at these points are output in the
ﬁles 5p, 5vx, 5vy and 5vz in the same format as shown in Section 4.9. ( If IPNLBPT=0
is assigned in the conﬁguration ﬁle an alternative option is utilized with the points on the
body surface speciﬁed by the user, as described in Section 4.11.)
When the above options are speciﬁed a second output ﬁle .5pb is also generated. This
ﬁle contains the B-spline coeﬃcients and other relevant parameters for the evaluation of
the pressure and its derivatives on the body surface. The total pressure coeﬃcient (φ),
the diﬀraction pressure (φD ) and the radiation pressure (φR ) are output separately. The
radiation pressure has as many components as the number of modes speciﬁed in the POT
ﬁle, including generalized modes. Following the deﬁnition of the nondimensional pressure
(Section 4.5) these three components are related by the equation
φ = φD + KL

j

ξj φj

where KL is the nondimensional inﬁnite depth wavenumber, ξj is the nondimensional
motion amplitude and j is the mode index.
The total pressure coeﬃcient is output always. The diﬀraction pressure coeﬃcient is
output when IRAD> −1 and IDIFF>-1. Since the total pressure is the same as the diﬀraction pressure if IRAD=-1, the diﬀraction pressure is not output in this case. The radiation
pressure coeﬃcient is output when IRAD> −1.
The data in the .5pb ﬁle is useful for special post-processing purposes, such as for
interfacing with structural loads analyses. The content of the .5pb numeric output ﬁle is
listed below:
HEADER
ISX,ISY
ULEN
NPATCH
IRAD,IDIFF
NPER,NBETA
NEQN
NLHS
NDFR
NBODY
4–11

((XBODY(L,J),L=1,4),J=1,NBODY)
((XBCS(L,J),L=1,2),J=1,NBODY)
(IBPTH(L),L=1,NPATCH)
(IBMOD(L),L=1,NBODY)
(IGEO(J),J=1,8)
(ILHS(J),J=1,4)
(IFLAT(L),L=1,NPATCH)
(KU(L),KV(L),NU(L),NV(L) ,L=1,NPATCH)
(NMDS(J),J=1,4)
(ICOL(J),J=1,NDFR)
((MDS(L,J),L=1,NDFR),J=1,4)
(BETA(NB),NB=1,NBETA) (omit if IDIFF=-1)
Loop over number of periods (repeat NPER times)
PER(IP),WVNFIN(IP),WVNUM,IFREQ
IF block starts (if IFREQ=0)
IF block starts (if IRAD>-1 and IDIFF>-1)
Loop over wave-headings starts (repeat NBETA times)
(WRAO(IM,NB),IM=1,NDFR)
Loop over wave-headings ends (repeat NBETA times)
Loop over wave-headings starts(repeat NBETA times)
Loop over number of symmetric images (repeat MXNLHS times)
Loop over number of patches (repeat NPATCH times)
(WPRS(I,M,NB),I=NP+1,NQ) (omit if IFLAT(L)=-1)
End of the loop over number of patches
End of the loop over symmetric images
Loop over wave-headings ends (repeat NBETA times)
IF block ends (if IRAD>-1 and IDIFF>-1)
IF block starts (if IDIFF >-1)
Loop over wave-headings starts (NBETA times)
Loop over number of symmetric images (repeat MXNLHS times)
Loop over number of patches (repeat NPATCH times)
(WBD(I,M,NB),I=NP+1,NQ) (omit if IFLAT(L)=-1)
End of the loop over number of patches
End of the loop over symmetric images
Loop over wave-headings ends (NBETA times)
IF block ends (if IDIFF >-1)
IF block ends (if IFREQ=0)
IF block starts ( if IRAD>-1 )
Loop over left hand side starts (repeat NLHS times)
Loop over number of modes for each left-hand-side
MDI (mode index)
Loop over number of symmetric images (repeat MXNLHS times)
Loop over number of patches (repeat NPATCH times)
(WBR(I,ICOL(MDI)),I=NP+1,NQ) (omit if IFLAT(L)=-1)

4–12

End of the loop over number of patches
End of the loop over symmetric images
End of the loop over number of modes for each left-hand-side
Loop over left hand side ends (repeat NLHS times)
IF block ends ( if IRAD>-1 )
End of the loop over number of periods
NP+1 and NQ are the pointers of the ﬁrst and the last B-spline coeﬃcients of the unknown
velocity potential on patch L.
HLINE: header line
ISX,ISY: Symmetry index (1/0 = symmetric/asymmetric).
ULEN: Characteristic length speciﬁed in GDF.
NPATCH: Number of patches.
IRAD, IDIFF: Radiation/diﬀraction problem indices.
NPER, NBETA: Number of periods and wave headings.
NEQN : The total number of unknown B-spline coeﬃcients.
NLHS : Number of components to be solved when the total solution is decomposed into
symmetry and antisymmetry components for the body having geometric symmetry
NDFR : The total number of degrees of freedom. It equal to the sum of the degrees of
freedom of each body.
NBODY : Total number of bodies.
XBODY : Normalized coordinates of the origin of body coordinate system and its orientation relative to the global coordinates system.
XBCS : XBCS(1,I) and XBCS(2,I) are cosine and sine of XBODY(4,I)
IBPTH(L) : Body index for patch index L.
IBMOD(N) : Global modes counter. Number of modes prior to the present body N.
IGEO : Parameter used to determine the sign of the pressure/velocity on the reﬂected
patches (see MODE.F)
ILHS : Pointer of the given LHS among NLHS components
IFLAT : Index for patches on the free surface.(IFLAT=-1, patches on interior free surface.
IFLAT=1, patches for ﬂat physical surface on the free surface. IFLAT=0, patches not on
the free surface.)
KU,KV,NU,NV: Orders and panels
NMDS : For given LHS, total number of modes of radiation problem.
MDS : For given LHS, MDS stores NMDS modes indices.
ICOL : The solution such as motion amplitude is stored in the order which is not ascending
from mode 1 (surge). ICOL stores the pointer in that sequence for all modes.
BETA : Wave headings
PER WVNFIN WVNINF IFREQ: Period, ﬁnite depth wave number, inﬁnite depth
4–13

wave number, period index (IFREQ=0: normal period, IFREQ=1: inﬁnite or zero period).
When IFREQ=1, the total and diﬀraction pressure coeﬃcient are not output in .5pb.
WRAO (I,J): Complex motion amplitude (I: modes, J:wave heading)
WPRS: Total pressure coeﬃcient (I: unknown coeﬃcient, M:reﬂection J:wave heading)
WBD: Diﬀraction pressure coeﬃcient (I: unknown coeﬃcient, M:reﬂection J:wave heading)
MDI: Mode index.
WBR: Radiation pressure coeﬃcient (I: unknown coeﬃcient, ICOL(MDI):pointer of mode
MDI)

4–14

4.11

BODY PRESSURE AND FLUID VELOCITY AT SPECIFIED POINTS

If IOPTN(5)> 0, the hydrodynamic pressure and ﬂuid velocity on the body surface can be
evaluated. The points where the pressure is evaluated depend on the parameter IPNLBPT
in the conﬁguration ﬁle. In the default case IPNLBPT=0, the pressure is evaluated at
the panel centroids in the low-order method (ILOWHI=0) or at a set of uniformly spaced
parametric points on each patch in the higher-order method (ILOWHI=1).
If IPNLBPT= 0 the body pressure is evaluated at points on the body which are speciﬁed
by the user in a special input ﬁle gdf.bpi (Body Point Input). The format of this ﬁle is as
follows:
header
NBPT
X(1) Y(1) Z(1)
X(2) Y(2) Z(2)
.
.
X(NBPT) Y(NBPT) Z(NBPT)
Note that the ﬁlename of this ﬁle must be the same as the ﬁlename of the .GDF ﬁle.
If IP NLBP T > 0, the data in the .bpi ﬁle is read and interpreted to be in dimensional
body-ﬁxed coordinates. If IP NLBP T < 0, the data in the .bpi ﬁle is read and interpreted
to be in dimensional global coordinates. (The relationship between these two coordinate
systems is deﬁned by the array XBODY, as deﬁned in Section 3.1.)
The procedure followed to evaluate the body pressure at these speciﬁed points is different in the low-order (ILOWHI=0) and higher-order (ILOWHI=1) solutions. These are
described separately below.
If ILOWHI=0, the solution is based on piecewise constant values of the potential on
each panel based on colocation at the panel centroids. In order to evaluate the pressure at
other points an interpolation procedure is adopted. This interpolation is based on a userspeciﬁed number NNEAR of nearest panel centroids. The parameter NNEAR is determined
from the absolute value of the input parameter IPNLBPT. IPNLBPT=4 is recommended,
when the input points are in body-ﬁxed coordinates. In this case the program searches and
identiﬁes the four nearest panel centroids to each speciﬁed input point, and assigns weights
to each of these panels based on the (inverse) distance to each centroid. The pressure is
output in the .5p numeric output ﬁle with the following format:
OPTN.5P: PER

BETA

IBODY IPOINT Mod(p̄)

Pha(p̄)

Re(p̄) Im(p̄)

This format and the deﬁnitions of the data are the same as in Section 4.9, except that
the index IBODY is used to specify the body index and IPOINT is used to specify the index
of the input point in the .bpi ﬁle (J=1,2,...,NBPT) for each body. Similar output ﬁles .5vx,
4–15

.5vy, .5vz contain the components of the ﬂuid velocity on the body surface in the same
format, when IOPTN(5)≥2. In addition to these hydrodynamic outputs, a supplementary
output ﬁle gdf.bpo (Body Point Output) is created with the following format:
gdf.BPO: M N1 R1 N2 R2 N3 R3 N4 R4
Here M is the quadrant index, N1 is the panel index of the nearest panel and R1 is the radial
distance from the speciﬁed point (x, y, z) to the centroid of the panel. Successive pairs (Ni,
Ri) are the index and radial distance to the other panel centroids, where (i=1,2,...,NNEAR).
(In the example shown above NNEAR=4.)
In the higher-order method (ILOWHI=1) the solution for the velocity potential and
pressure is represented by continuous B-splines on each patch. For each speciﬁed input
point (x, y, z) the program searches for the patch index and (U, V ) coordinates of the
point on this patch which is closest to the input point. The pressure is evaluated at the
corresponding point (U, V ) and output in the .5p and .5v∗ ﬁles with the same format as
shown above. In this caes the supplementary output ﬁle gdf.bpo contains the following
data for each point:
gdf.BPO: K M

NP U

V R

I XI XN

Here K is the body point index, M is the quadrant index, NP is the patch index, and (U,V)
are the parametric coordinates on the patch. R is the radial distance from the point (U,V)
on the patch to the speciﬁed (x, y, z) point. An iterative procedure is used to ﬁnd (U,V),
with a speciﬁed convergence tolerance of 1.0E-4 for the radial distance in nondimensional
Cartesian coordinates. (When the length scale of the patch is larger than 1.0 the tolerance
is increased by a factor equal to this length scale, estimated from the Jacobian of the
parametric transformation at the center of the patch.) I is the number of iteration. A
maximum of 16 iterations are used in this search, and if I=17 this indicates noncovergence
of the search. A warning message is generated if nonconvergence occurs for one or more
input points, showing the total number of unconverged points. XI is the position vector
of the output point on the body surface and XN is the normal vector at XI, both in body
coordinates system.
In both the low-order and higher-order implementations, the input data in the .bpi ﬁle
should correspond to points which lie as close as possible to the body surface.
If points in the BPI ﬁles are very close to intersections of adjacent patches in the
higher-order method, the index NP in the BPO ﬁle should be checked to verify that the
correct patch is used, especially in cases where there is ambiguity between the pressure
on a conventional patch and the pressure jump on a dipole patch. Similarly, in the loworder method, the panel indices, N1, N2, ... in the BPO ﬁle can be checked to verify the
corresponding output at the points in BPI ﬁles is the pressure obtained from those on the
conventional body panels or the pressure jump on the dipole panels.

4–16

4.12

RADIATION AND DIFFRACTION COMPONENTS OF
THE PRESSURE AND VELOCITY

If IOPTN(5)> 0, the parameter INUMOPT5 in the conﬁguration ﬁle can be used to
control the outputs of the body pressure in the numeric output ﬁles. In the default case
INUMOPT5=0 the body pressure output in the numeric output ﬁles is the same total
pressure as in the formatted .out output ﬁle, as deﬁned in Section 4.5. Alternatively, if
INUMOPT5=1, the separate components KLϕ̄j and ϕ̄D are output in the numeric output
ﬁle .5p and the corresponding components of the ﬂuid velocity on the body surface are
output in the ﬁles .5vx, .5vy, .5vz. Here ϕ̄j is the nondimensional potential in mode j,
and ϕ̄D is the nondimensional potential of the diﬀraction problem with the body ﬁxed, as
deﬁned in Section 4.5. In this case the format of the numeric output ﬁles is modiﬁed, as
shown in Section 4.9.
If IOPTN(6) > 0, the parameter INUMOPT6 in the conﬁguration ﬁle can be used
to control the outputs of the ﬁeld pressure in the numeric output ﬁle. In the default
case INUMOPT6=0, as in previous versions of WAMIT, the ﬁeld pressure output in the
numeric output ﬁles is the same total pressure as in the formatted .out output ﬁle, as
deﬁned in Section 4.5. Alternatively, if INUMOPT6=1, the separate components KLϕ̄j
and ϕ̄D are output in the numeric output ﬁle .6. Similarly, if IOPTN(7) > 0, the parameter
INUMOPT7 in the conﬁguration ﬁle can be used to control the outputs of the ﬁeld velocity
in the numeric output ﬁles, .7x, .7y and .7z. The format of the numeric output ﬁles is shown
in Section 4.9.

4.13

RADIATION PRESSURE AND VELOCITY FOR ZERO
AND INFINITE PERIODS

In the two limiting cases of zero and inﬁnite period (or equivalently, inﬁnite and zero
frequency), it is possible to evaluate the pressure and velocity on the body (Option 5) and
in the ﬂuid (Options 6 and 7). This extension is particularly important in the context
of evaluating the corresponding time-domain impulse response functions, as explained in
Chapter 13. These extended outputs are only included in the numeric output ﬁles if the
corresponding parameters INUMOPT5, INUMOPT6, INUMOPT7 are assigned with value
1 in the .cfg ﬁle. The formats of the corresponding numeric output ﬁles are explained in
Section 4.9.
Special deﬁnitions are applied to the radiation pressure and velocity in the case of zero
frequency, which is identiﬁed in the output ﬁles by a negative value of the wave period.
In general, for nonzero ﬁnite values of the frequency, the nondimensional outputs for the
radiation pressure and velocity are as deﬁned in Sections 4.5 and 4.7. Thus the output
¯ ϕ̄j .
pressure for each radiation mode is KLϕ̄j and the output velocity for each mode is KL∇
However for the two limiting cases, where KL = 0 or KL = ∞, the factor KL is omitted
from the outputs for options 5,6,7. The following table summarizes these deﬁnitions:

4–17

frequency

period

pressure

ω=0

PER< 0

ϕ̄j

velocity
¯ ϕ̄j
∇

KLϕ̄j

¯ ϕ̄j
KL∇

ϕ̄j

¯ ϕ̄j
∇

0 < ω < ∞ PER=(2π/ω)
ω=∞

PER=0

4–18

Chapter 5

THE LOW-ORDER METHOD
(ILOWHI=0)
This Chapter includes speciﬁc topics which are applicable when the low-order method is
used, as in earlier versions of WAMIT. The essential features of this method are (a) the
geometry of the body is represented by an ensemble of ﬂat quadrilateral panels, or facets,
and (b) the solutions for the velocity potential, and optionally for the source strength, are
approximated by piecewise constant values on each panel.
The geometry of the body is speciﬁed in this case by a Geometric Data File (GDF)
which includes the Cartesian coordinates of each vertex of each panel, listed sequentially.
In addition the GDF ﬁle speciﬁes the characteristic length ULEN used for nondimensionalization of outputs, the value of the gravitational acceleration constant GRAV in the same
units of measurement, the number of panels NPAN, and two symmetry indices ISX, ISY,
as described in Section 5.1. The syntax for data in this ﬁle follows the same requirements
outlined for the generic input ﬁles in Chapter 3.
When the low-order method is used there are three options which can be used for
special purposes. These include the source formulation, analysis of a body near one or two
orthogonal vertical walls, and the analysis of bodies with thin elements such as damper
plates or strakes.
When the source formulation is used, as described in Section 5.2, the solution is computed both for the velocity potential and for the equivalent source distribution which
generates this potential. The source formulation must be used, in the low-order method, if
the mean drift force and moment are evaluated by pressure integration, and more generally
if the ﬂuid velocity is required on the body surface.
The option to include one or two vertical walls, described in Section 5.3, is based on the
method of images. The body geometry is speciﬁed in the usual manner by a GDF ﬁle, and
the program automatically represents the image body (or bodies, if two walls are present).
If the body has thin elements, there are two possible approaches. The ﬁrst is to panel
both sides of these elements, with a ﬁnite thickness to separate the two sides. The disadvantage of this approach is that, as a general rule, the size of the panels must be comparable
5–1

to the thickness, and thus a very large number of small panels may be required to achieve
accurate results. The second approach is to reduce the thickness to zero, and represent the
corresponding elements of the body by special ‘dipole panels’. This approach is analogous
to the thin-wing approximation in lifting-surface theory [21]. Version 6 permits the user
to specify a set of dipole panels, as described in Section 5.4. This option facilitates the
analysis of bodies with damper plates, strakes, and similar thin elements, without the need
to use very large numbers panels or to artiﬁcially increase the thickness.
Starting in V6.2, the same extension is made for ILOWHI=1 and the users may specify
dipole patches to represent thin structures as described in Section 6.10.

5–2

5.1

THE GEOMETRIC DATA FILE

In the low-order method the wetted surface of a body is represented by an ensemble of
connected four-sided facets, or panels. The Geometric Data File contains a description
of this discretized surface, including the body length scale, gravity, symmetry indices, the
total number of panels speciﬁed, and for each panel the Cartesian coordinates x, y, z of
its four vertices. A panel degenerates to a triangle when the coordinates of two vertices
coincide. The order in which the panels are deﬁned in the ﬁle is unimportant, but each
panel must be described completely by a set of 12 real numbers (three Cartesian coordinates
for each vertex) which are listed consecutively, with a line break between the last vertex of
each panel and the ﬁrst vertex of the next. The value of gravity serves to deﬁne the units
of length, which apply to the body length scale, panel oﬀsets, and to all related parameters
in the other input ﬁles. The coordinate system x, y, z in which the panels are deﬁned is
referred to as the body coordinate system. The only restrictions on the body coordinate
system are that it is a right-handed Cartesian system and that the z−axis is vertical and
positive upward.
The name of the GDF ﬁle can be any legal ﬁlename accepted by the operating system,
with a maximum length of 16 ASCII characters, followed by the extension ‘.gdf’.

Figure 5.1: Discretization of a circular cylinder showing the convention for panel vertex numbering. The
perspective view is from above the free surface, showing portions of the exterior and interior of the cylinder
(lower and upper portions of the figure, respectively). The view of panel i is from the ‘wet side’, inside
the fluid domain, so the vertex ordering appears anti-clockwise. The view of panel j is from the ‘dry side’
outside the fluid domain, so the vertex ordering appears clockwise.

The data in the GDF ﬁle can be input in the following form:
5–3

header
ULEN GRAV
ISX ISY
NPAN
X1(1) Y1(1) Z1(1) X2(1) Y2(1) Z2(1) X3(1) Y3(1) Z3(1) X4(1) Y4(1) Z4(1)
X1(2) Y1(2) Z1(2) X2(2) Y2(2) Z2(2) X3(2) Y3(2) Z3(2) X4(2) Y4(2) Z4(2)
.
.
.
.
.
. . . . .
.
. . . . X4(NPAN) Y4(NPAN) Z4(NPAN)
Each line of data indicated above is input by a separate FORTRAN READ statement,
hence line breaks between data must exist as shown. Additional line breaks between data
shown above have no eﬀect on the READ statement, so that for example the user may elect
to place the twelve successive coordinates for each panel on four separate lines. (However
the format used above is more eﬃcient regarding storage and access time.)
Input data must be in the order shown above, with at least one blank space separating
data on the same line.
The deﬁnitions of each entry in this ﬁle are as follows:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72. This line is
available for the user to insert a brief description of the ﬁle, with maximum length 72
characters.
ULEN is the dimensional length characterizing the body dimension. This parameter
corresponds to the quantity L used in Chapter 4 to nondimensionalize the quantities output
from WAMIT. ULEN can be input in any units of length, meters or feet for example, as
long as the length scale of all other inputs is in the same units. ULEN must be a positive
number, greater than 10−5 . An error return and warning statement are generated if the
last restriction is not satisﬁed.
GRAV is the acceleration of gravity, using the same units of length as in ULEN. The units
of time are always seconds. If lengths are input in meters or feet, input 9.80665 or 32.174,
respectively, for GRAV.
ISX, ISY are the geometry symmetry indices which have integer values 0, +1. (A negative
value is assigned when a vertical wall is present. This case is discussed in Section 5.3.) If
ISX and/or ISY =1, x = 0 and/or y = 0 is a geometric plane of symmetry, and the input
data (panel vertex coordinates X,Y,Z and their total number NPAN) are restricted to one
quadrant or one half of the body, namely the portion x > 0 and/or y > 0. Conversely, if
ISX=0 and ISY=0, the complete submerged surface of the body must be represented by
panels.
ISX = 1: The x = 0 plane is a geometric plane of symmetry.

5–4

ISX = 0: The x = 0 plane is not a geometric plane of symmetry.
ISY = 1: The y = 0 plane is a geometric plane of symmetry.
ISY = 0: The y = 0 plane is not a geometric plane of symmetry.
For all values of ISX and ISY, the (x, y) axes are understood to belong to the body system.
The panel data are always referenced with respect to this system, even if walls or other
bodies are present.
NPAN is equal to the number of panels with coordinates deﬁned in this ﬁle, i.e. the
number required to discretize a quarter, half or the whole of the body surface if there exist
two, one or no planes of symmetry respectively.
X1(1), Y1(1), Z1(1) are the (x, y, z) coordinates of vertex 1 of the ﬁrst panel, X2(1),
Y2(1), Z2(1) the (x, y, z) coordinates of the vertex 2 of the ﬁrst panel, and so on. These
are expressed in the same units as the length ULEN. The vertices must be numbered in
the counter-clockwise direction when the panel is viewed from the ﬂuid domain, as shown
in Figure 3.1. The precise format of each coordinate is unimportant, as long as there
is at least one blank space between coordinates, and the coordinates of the four vertices
representing a panel are listed sequentially.
There are two situations when panels lie on the free surface, and thus all four vertices
are on the free surface: (1) the discretization of a structure which has zero draft over
part or all of its submerged surface, and (2) the discretization of the interior free surface
for the irregular frequency removal as described in Chapter 9. For the ﬁrst case, where
the panels are part of the physical surface, the panel vertices must be numbered in the
counter-clockwise direction when the panel is viewed from the ﬂuid domain as in the case
of submerged panels. For the second case, where the panel is interior to the body and
non physical, the vertices must be numbered in the clockwise direction when the panel is
viewed from inside the structure (or in the counter-clockwise direction when the panel is
viewed from above the free surface). Details of the discretization of the interior free surface
are provided in Chapeter 9.
Although the panels on the free surface are legitimate in these two special cases, a
warning message is displayed by WAMIT when it detects panels which have four vertices
on the free surface. This is to provide a warning to users for a possible error in the
discretization other than the above two exceptional cases. The run continues in this case,
without interruption. An error message is displayed with an interruption of the run when
the panels have only three vertices on the free surface, unless two adjacent vertices are
coincident. (The latter provision permits the analysis of a triangular panel with one side
in the free surface.)
The three Cartesian coordinates of four vertices must always be input for each panel, in
a sequence of twelve real numbers. Triangles are represented by allowing the coordinates
of two adjacent vertices to coincide, as in the center bottom panels shown in Figure 3.1.
5–5

Two adjacent vertices are deﬁned to be coincident if their included side has a length less
than ULEN × 10−6 . An error return results if the computed area of any panel is less than
ULEN2 × 10−10 .
The input vertices of a panel do not need to be co-planar. WAMIT internally deﬁnes
planar panels that are a best ﬁt to four vertices not lying on a plane. However it is advisable
to discretize the body so that the input vertices deﬁning each panel lie close to a plane,
in order to achieve good accuracy in the computed velocity potentials. An error message
is printed if a panel has two intersecting sides. A warning message is printed if a panel is
‘convex’ (the included angle between two adjacent sides exceeds 180 degrees).
The origin of the body coordinate system may be on, above or below the free surface.
The vertical distance of the origin from the free surface is speciﬁed in the Potential Control
File. The same body-system is also used to deﬁne the forces, moments, and body motions.
(See Chapter 5 regarding the change in reference of phase relations when walls are present.)
Only the wetted surface of the body should be paneled, and then only half or a quarter
of it if there exist one or two planes of symmetry respectively. This also applies to bodies
mounted on the sea bottom or on one or two vertical walls. The number of panels NPAN
refers to the number used to discretize a quarter, half or the whole body wetted surface if
two, one or no planes of symmetry are present respectively.
The displaced volume of the structure deserves particular discussion. Three separate
algorithms are used to evaluate this quantity, as explained in Section 4.1. Except for
the special case where the structure is bottom-mounted, the three evaluations (VOLX,
VOLY, VOLZ) should be identical, but they will generally diﬀer by small amounts due
to inaccuracies in machine computation and, more signiﬁcantly, to approximations in the
discretization of the body surface.
For a bottom-mounted structure VOLZ is less than the true volume due to the missing
panels in the bottom. In this case a substantial reduction should be observed for VOLZ,
but VOLX and VOLY should be nearly equal. For the same reason, substantial diﬀerences
may occur if a body is mounted in a vertical wall unless the origin of the body coordinates
is in the plane of the wall. With these exceptions, substantial diﬀerences between the three
volumes may indicate errors in the GDF input data.
A unique value of the displaced volume is required in computing the hydrostatic parameters, and in evaluating the body inertia for the motions of the freely-ﬂoating body.
In these cases the displaced volume of ﬂuid is based on the median (middle value of the
three when ranked according to value) of VOLX, VOLY, and VOLZ. A warning message
is displayed by WAMIT if the median volume is less than 10−30 .
A general-purpose pre-processor has been developed for preparation of GDF ﬁles, using
the MultiSurf geometric modelling program.1

1 AeroHydro,

Inc., 54 Herrick Rd., Southwest Harbor, Maine 04679 USA 207-244-4100 (www.aerohydro.com)

5–6

5.2

USE OF THE SOURCE FORMULATION (ISOR=1)

This section describes the evaluation and use of the source strength, in the context of
calculating the ﬂuid velocity components on the body and the mean drift force and moment
based on pressure integration in uni- and bi-directional waves.
In order to evaluate eﬀectively the tangential components of the ﬂuid velocity on the
body (and hence the second-order mean pressure), the solution for the velocity potential
based on Green’s theorem is augmented if ISOR=1 by the corresponding solution for the
source distribution on the body surface.
Setting the parameter ISOR=1 in the POT ﬁle or in the CONFIG.WAM ﬁle speciﬁes that the source-distribution integral equation is solved in addition to the velocitypotential integral equation. Extended values are assigned to the option switches IOPTN(5),
IOPTN(6), IOPTN(7) and IOPTN(9) in the FRC ﬁle, to control the evaluation of the ﬂuid
velocity on the body and the pressure-integrated drift force/moment, respectively. Explanations of the new output quantities and changes in input are explained in the following
section. A brief description of the theory is provided in Section 12.5. (Further details are
given in [10] and [26].)
Values of the horizontal drift force and vertical drift moment can be compared with
the corresponding outputs evaluated using momentum conservation. In general the results
obtained from integration of the second-order pressure will require a ﬁner discretization on
the body surface, particularly in the vicinity of sharp corners.
Body symmetries can be exploited to minimize computing time. Special attention must
be given to the evaluation of the drift forces, since these are dependent on quadratic
products of the ﬁrst-order solution. For example, if the body has two planes of symmetry
the vertical ﬁrst-order exciting force and heave response can be evaluated simply by setting
IRAD=-1, IDIFF=0, MODE(3)=1, and the remaining MODE indices equal to zero. This
will not give the correct vertical drift force on the body, however, since the components of
the diﬀraction potential and body motions which are odd functions of x and y have not
been evaluated. In general the drift forces should be evaluated only after evaluating all
components of the ﬁrst-order potential, i.e. by setting IDIFF=1 for the stationary body
and IRAD=1 and IDIFF=1 for the freely ﬂoating body in the POT ﬁle. [An example of
a valid short cut exists if both the body geometry and the hydrodynamic ﬂow ﬁeld are
symmetrical about a plane of symmetry; then it is not necessary to evaluate ﬁrst-order
potentials which are odd about that plane since these would vanish. For example, if the
body is symmetrical about y = 0 and the incident-wave heading angle is either zero or
180◦ , the drift force and moment can be obtained by setting MODE(n)=0 for n = 2, 4, 6.]
To calculate the mean drift forces it is necessary to evaluate the runup or, equivalently,
the velocity potential at the waterline. Since the program utilizes the velocity potentials
at the centroids of the panels adjacent to the waterline for the runup, it is advisable to use
panels with small vertical dimensions near the waterline.
In the Force Control File (FRC) extended options are assigned to IOPTN(5), IOPTN(6),
IOPTN(7) and IOPTN(9), as follows (See also Section 3.3):

5–7

IOPTN(5) is the index used to specify whether the pressure and/or velocity are evaluated
on the body boundary:
IOPTN(5) = 0: Do not output either the pressure or velocity.
IOPTN(5) = 1: Output the pressure only.
IOPTN(5) = 2: Output the velocity only (requires ISOR= 1).
IOPTN(5) = 3: Output both the pressure and the velocity (requires ISOR= 1).
IOPTN(6) is the index used to specify whether the pressure in the ﬂuid and/or free
surface elevation are evaluated.
IOPTN(6) = 0: do not output pressure and/or free-surface elevation
IOPTN(6) = 1: do output pressure and/or free-surface elevation by the potential formulation
IOPTN(6) = 2: do output pressure and/or free-surface elevation by the source formulation
IOPTN(7) is the index used to specify whether the velocity in the ﬂuid is evaluated.
IOPTN(7) = 0: do not output ﬂuid velocity in the ﬂuid
IOPTN(7) = 1: do output ﬂuid velocity in the ﬂuid by the potential formulation
IOPTN(7) = 2: do output ﬂuid velocity in the ﬂuid by the source formulation
IOPTN(9) is the index used to specify whether the mean drift forces are evaluated from
pressure integration:
IOPTN(9) = 0: do not evaluate the mean force and moment
IOPTN(9) = 1: do evaluate the mean force and moment for each wave heading (requires
ISOR= 1).
IOPTN(9) = 2: do evaluate the mean force and moment for all combinations of the wave
headings (requires ISOR= 1).
The non-dimensional ﬂuid velocity is evaluated in vector form with respect to the global
coordinate system at the centroid of each panel. The non-dimensional deﬁnition of the ﬂuid
velocity is given in Section 4.7.
Three coordinate systems are used in the analysis of the second order mean forces and
moments. The ﬁrst is the global coordinate system (an inertial coordinate system) whose
origin should be on the free-surface with the vertical axis positive upward. The phases of
time-harmonic quantities in WAMIT are deﬁned with respect to the phase of the incident
wave at the origin of the global coordinate system. The second coordinate system is the
body-ﬁxed coordinate system. The position and orientation of this coordinate system
at rest are deﬁned relative to the global coordinates by the input array XBODY (see
Chapter 3). The third coordinate system, introduced to provide additional ﬂexibility in
the mean-force analysis, is the inertial coordinate system which coincides with the bodyﬁxed coordinate system at rest. This third system is illustrated in Section 12.5, Figure
12.2.
5–8

The non-dimensional deﬁnitions of mean drift forces and moments are deﬁned as in
Section 4.8. Typical outputs are provided in Test03 described in the Appendix (Section
A.3). The following features should be noted in the output:
i) On the line showing the number of iterations required for the solution, the second number
inside the parenthesis indicates the maximum number of iterations required for the solution
of the linear system for the evaluation of the source strength.
ii) The components of the ﬂuid velocity on the body surface are deﬁned in reference to the
global coordinate system.
iii) The three drift force and moment components evaluated by momentum conservation
are deﬁned with reference to the axes of the global coordinate system.
iv) The six components of the drift force and moment evaluated by pressure integration
are deﬁned with reference to the axes of the body coordinate system at rest (O of
Figure 12.2). Also listed in the OUT ﬁle, in the two right columns, are the components
of the moment vector deﬁned with respect to the body-fixed coordinate system (o of
Figure 12.2).
The format of the numerical output ﬁles is given in Section 4.9.

5–9

5.3

BODY NEAR VERTICAL WALLS

WAMIT includes the option to account for images of the body in the presence of one
vertical wall, or two vertical walls which intersect at a right angle.
Figure 5-2 deﬁnes two coordinate systems, one ﬁxed on the body and the other on the
wall. The axes of the former are denoted by (x, y, z) and those of the latter by (X, Y, Z). In
the presence of wall(s), the global coordinate system deﬁned in Section 3.1 must coincide
with the wall coordinate system as deﬁned in Figure 5-2. Only one body may be analyzed
in the presence of walls (NBODY=1). The drift force and moment cannot be evaluated by
momentum integration.

Y

Y

XBODY(4)

(XBODY(1),XBODY(2))
X

X

O

O
Figure 5.2: Definition sketch of coordinates

In the GDF ﬁle the panels are deﬁned as in Section 5.1, in terms of the body coordinate
system. Negative values (-1, -2) are assigned to the symmetry indices ISX, ISY in place
of the conventional corresponding values (0, 1) to indicate the presence of either one or
two walls, as follows:
ISX = −1 or −2: The X = 0 plane is a plane wall.
ISY = −1 or −2: The Y = 0 plane is a plane wall.
If ISX or ISY is assigned the value −1, no local plane of symmetry is assumed and the
body must be represented completely (or half of it must be represented in the case where
the complementary plane of symmetry is speciﬁed). Conversely, if the body has a local
plane of symmetry x = 0 and/or y = 0, ISX and/or ISY can be assigned the value −2,
with panels deﬁned in the GDF ﬁle only for the portion of the body where x ≥ 0 and/or
5–10

y ≥ 0. For example, ISX = −2, ISY = 0 signify that a wall is present on the X = 0 plane
and that the body is symmetric about the local x = 0 plane. In this case only the half of
the body needs to be discretized by the input panels. When ISX = 1, ISY = −2, a wall is
present on the Y = 0 plane and the body has two local planes of symmetry.
In the Potential Control File the vector XBODY(1),XBODY(2),XBODY(3) speciﬁes the dimensional (X, Y, Z) coordinates of the origin of the body-ﬁxed coordinate system
relative to wall system, in the units of the length ULEN. XBODY(4) is the angle in degrees formed by the body x-axis and the X-axis of the wall system, as deﬁned in Figure
5-2. The values of the incident wave heading angles BETA are deﬁned with respect to the
positive X-axis.
An important detail to note is the deﬁnition of the incident-wave amplitude and its
physical interpretation. The ‘incident-wave’ is deﬁned as the incoming wave component
prior to reﬂection from the wall(s), and A is the corresponding amplitude. After reﬂection
the resulting wave ﬁeld, in the absence of the body, is an oblique standing wave with
maximum free-surface elevation 2A. In the special case β = 0 the incident wave propagates
parallel to the wall, without a distinct reﬂected component, but the physical amplitude of
this wave is 2A. Some consequences of this deﬁnition are noted in the comparison of Test
Runs 04 and 19 (see Sections A.4 and A.19).
In the Force Control File the array IOPTN is unchanged from the deﬁnitions in Section
3.3. Since momentum integration cannot be used to determine the mean drift force and
moment, IOPTN(8)=1 is ignored. IOPTN(9) is used in the normal manner to evaluate
the drift force and moment from pressure integration. The Haskind wave heading angles
BETAH are deﬁned with respect to the walls in the same manner as the array BETA
above. The coordinates of ﬁeld points XFIELD where the pressure, wave elevation, and
velocity are evaluated, are deﬁned as in Section 3.3 relative to the wall-mounted system.
The incident-wave velocity potential is deﬁned relative to the wall-mounted coordinate
system. Consequently, the phases of the exciting forces, motions, hydrodynamic pressure
and ﬁeld velocity induced by the incident wave are understood relative to the incident-wave
elevation at X = Y = 0. In addition the ﬂuid velocity vector components are given with
respect to the wall-mounted coordinate system.
The other deﬁnitions of output quantities in Chapter 4 are unchanged.

5–11

5.4

BODIES WITH THIN SUBMERGED ELEMENTS

In Version 6 an extension has been developed to analyze bodies which consist partially (or
completely) of elements with zero thickness. These elements are represented by panels in
the same manner as conventional body panels (Section 5.1). Figure 5-3 shows a typical
example of a ﬂoating spar with thin helical strakes. This structure is analyzed in Test Run
09, described in the Appendix (Section A.9).

Figure 5.3: Perspective view of a floating spar with three helical strakes, showing the conventional panels
on the spar and dipole panels on the strakes.

The velocity potential on the zero-thickness elements is represented by dipoles alone,
with no corresponding sources. The panels are referred to as ‘dipole panels’. Since both
sides of the dipole panels adjoin the ﬂuid, the order of the vertex coordinates shown in
Figure 5-1 (clockwise or counter-clockwise) is irrelevant.

5–12

In the GDF ﬁle the dipole panels must be included separately, after the conventional
panels which represent the remainder of the body surface (and also the interior free surface,
if IRR=1). The format of the GDF ﬁle is as follows:
header
ULEN GRAV
ISX ISY
NPANC
X1(1) Y1(1)
X1(2) Y1(2)
.
.
.
.
. .
NPAND
X1(1) Y1(1)
X1(2) Y1(2)
.
.
.
.
. .

Z1(1) X2(1) Y2(1) Z2(1) X3(1) Y3(1) Z3(1) X4(1) Y4(1) Z4(1)
Z1(2) X2(2) Y2(2) Z2(2) X3(2) Y3(2) Z3(2) X4(2) Y4(2) Z4(2)
. . .

.

. . . . X4(NPANC) Y4(NPANC) Z4(NPANC)

Z1(1) X2(1) Y2(1) Z2(1) X3(1) Y3(1) Z3(1) X4(1) Y4(1) Z4(1)
Z1(2) X2(2) Y2(2) Z2(2) X3(2) Y3(2) Z3(2) X4(2) Y4(2) Z4(2)
. . .

.

. . . . X4(NPAND) Y4(NPAND) Z4(NPAND)

Here NPANC is the number of conventional panels, NPAND is the number of dipole panels,
and NPAN=NPANC+NPAND is the total number of panels in the GDF ﬁle. The ﬁrst
part of the GDF ﬁle, ending with the line which includes X4(NPANC), is identical to a
conventional GDF ﬁle for the same body without the dipole panels. The remainder of the
ﬁle, starting with NPAND, deﬁnes the dipole panels which represent the zero-thickness
elements.
The same format can be extended to multiple bodies, with dipole panels speciﬁed for
some or all of the bodies, following the procedure described in Chapter 7.
The source formulation cannot be used if dipole panels are included. Thus the ﬂuid
velocity on the body cannot be evaluated, and the mean drift force/moment can only be
evaluated by the momentum method. An error message is output and the run is terminated
if ISOR=1 with dipole panels speciﬁed in the GDF ﬁle.
It is possible to analyze bodies which consist entirely of zero-thickness elements, by
setting NPANC=0.
A symmetry plane can be used when there are ﬂat thin elements represented by dipole
panels on the plane of symmetry. As an example, when a keel on the centerplane y = 0 is
represented by dipole panels, either the port or starboard side of the vessel can be deﬁned
in the GDF ﬁle with ISY=1

5–13

Chapter 6

THE HIGHER-ORDER METHOD
(ILOWHI=1)
The higher-order method which is introduced as an option in Version 6 is fundamentally
diﬀerent from the low-order panel method described in Chapter 5. The body geometry
can be represented by diﬀerent techniques including ﬂat panels, B-spline approximations,
geometry models developed in MultiSurf, and explicit analytical formulae. The velocity
potential on the body is represented by B-splines in a continuous manner, and the ﬂuid
velocity on the body can be evaluated by analytical diﬀerentiation. In most applications
this provides a more accurate solution, with a smaller number of unknowns, compared to
the low-order method.
This higher-order method is developed from the earlier research code HIPAN, which has
been described in References [18-20].
A brief outline of the method is provided in Sections 6.1-6.3, to give the necessary background for several input parameters which must be speciﬁed. This includes the subdivision
of the body surface into patches, the further subdivision of the patches into panels, and
the use of B-splines to develop approximations on these surfaces. It is important to note
in this context that a panel is not restricted to be a ﬂat quadrilateral in physical space,
but can be a general surface in space with continuous curvature to ﬁt the corresponding
portion of the body as precisely as is appropriate.
The number of patches NPATCH is speciﬁed in the GDF ﬁle. Various options exist to
specify the other input parameters which determine the number or size of the panels, order
of the B-splines, and order of the Gauss quadratures used to integrate over each panel.
Section 6.4 describes the data in the Geometric Data File (GDF) which is common to all
applications of the higher-order method. Sections 6.5-6.8 describe the four diﬀerent options
for describing the body geometry, and the corresponding inputs. Section 6.9 describes the
procedure for modifying the GEOMXACT subroutine to represent the geometry of userspeciﬁed bodies.
If the body has thin elements, there are two possible approaches as in the low-order
method (see Chapter 5). The ﬁrst is to represent both sides of these elements with patches.
6–1

As a general rule, this approach requires a large number of panel subdivisions of the patches
as the thickness of the elements decreases and thus becomes ineﬃcient. The second is
to reduce the thickness to zero and represent the elements by special ’dipole patches’,
analogous to the thin-wing approximation in lifting-surface theory [21]. Version 6.2 permits
the user to specify a set of dipole patches, as described in Section 6.10.
Section 6.11 describes the optional Spline Control File (SPL) which can be used to deﬁne
the orders of the B-Splines, Gauss quadratures, and the numbers of panel subdivisions on
each patch.
The maximum size of the panels (measured in dimensional units) can be speciﬁed in
the conﬁguration ﬁle, instead of specifying the number of panels on each patch in the SPL
ﬁle. This is particularly convenient to achieve a panel size that is commensurate with the
body dimensions and wavelength. Default values of the remaining parameters in the SPL
ﬁle (B-spline and Gauss quadrature orders) are assigned automatically, if not input by the
user. Section 6.12 describes this procedure, which permits users to exploit the ﬂexibility
and eﬃciency of the higher-order method with a minimum of inputs.

6–2

6.1

SUBDIVISION OF THE BODY SURFACE IN PATCHES
AND PANELS

The body surface is ﬁrst deﬁned by one or more ‘patches’, each of which is a smooth
continuous surface in space. Contiguous patches meet at a common edge, where the coordinates are continuous but the slope may be discontinuous. A simple illustrative example
is provided by the circular cylinder of ﬁnite draft shown in Figure 6.1. (The same cylinder
is shown in Figure 5.1 as it would be represented by low-order panels.) Since there are two
planes of geometric symmetry we consider only one quadrant, represented by the shaded
portion of Figure 6.1. Two patches are used, one for the ﬂat horizontal bottom and the
other for the curved cylindrical side. The important properties of the patches are that (a)
the surface is smooth, with continuous coordinates and slope, on each patch, and (b) the
ensemble of all patches represents the complete body surface (or one half or quarter of that
surface, if one or two planes of symmetry exist).

Figure 6.1: Representation of the circular cylinder by two patches on one quadrant, shown by the shaded
portion, with reﬂections about the two planes of symmetry.

On each patch a pair of parametric coordinates (u, v) are used to deﬁne the position.
6–3

The parametric coordinates are normalized so that they vary between ±1 on the patch.
Continuing with the example in Figure 6.1, denoting the cylinder radius R and the draft
D and deﬁning conventional circular cylindrical coordinates (r, θ, z), appropriate choices
for the parametric coordinates are
u=

4θ
− 1,
π

v=1−2

r
R

(6.1)

on the bottom, where 0 ≤ θ ≤ π/2 and 0 ≤ r ≤ R, and
u=

4θ
− 1,
π

v = −2

z
−1
D

(6.2)

on the side, where 0 ≤ θ ≤ π/2 and −D ≤ z ≤ 0. In order to give a consistent deﬁnition
for the normal vector we impose the right-hand convention: if the ﬁngers of the right hand
are directed from +u toward +v the thumb should point out of the ﬂuid domain and into
the interior domain of the body. With these deﬁnitions the Cartesian coordinates (x, y, z)
of any point on either patch can be expressed in terms of the parametric coordinates (u, v).
More generally, any physically relevant body surface can be represented by an ensemble
of appropriate patches, where the Cartesian coordinates of the points on each patch are
deﬁned by the mapping functions
x = X(u, v),

y = Y (u, v),

z = Z(u, v)

(6.3)

This is the fundamental manner in which the body surface is represented for the higherorder option of WAMIT. Alternative methods for prescribing these mapping functions are
described separately in Sections 6.4-6.7.
In order to provide a systematic procedure for reﬁning the accuracy of approximations
on each patch, a set of smaller surface elements are deﬁned, as described in Section 6.2. For
this purpose each patch is sub-divided in a rectangular mesh, in parametric space. These
elements are referred to as panels. Note that while these panels are ﬂat and rectangular in
parametric space, they are unrestricted in physical space except for the requirement that
they represent a subdivided element of the corresponding patch. Thus, in general, these
panels are curved surfaces in physical space. (In some references, such as [22], panels are
called ‘sub-patches’, or simply ‘patches’). Figure 6.2 shows the example where the side and
bottom of the shaded quadrant in Figure 6.1 are each subdivided into four panels.

6–4

Figure 6.2: Subdivision of one quadrant of the cylinder shown in Figure 6,1 into panels. In this case
Nu = Nv = 2 on both patches.

6.2

B-SPLINE REPRESENTATION OF THE SOLUTION

The other important subject to consider is the manner in which the velocity potential is
represented on each patch. Desirable properties of this representation are that it should
be smooth and continuous, corresponding to the physical solution for the ﬂuid ﬂow over
the surface, with control over the accuracy. B-splines are used for this purpose. More
speciﬁcally, the velocity potential is represented by a tensor product of B-spline basis
functions
φ(u, v) =

Mv 
Mu

j=1 i=1

φij Ui (u)Vj (v)

(6.4)

Here Ui(u) and Vi (v) are the B-spline basis functions of u and v, and Mu and Mv are
the number of basis functions in u and v, respectively. The unknown coeﬃcients φij are
determined ultimately by substituting this representation in the integral equation for the
6–5

potential, as described in Chapter 12. The total number of unknowns on a patch is Mu ×Mv .
In the low order panel method, the accuracy of the numerical solution depends on the
number of panels. (To a lesser extent, the panel arrangement, such as cosine spacing, may
also aﬀect the accuracy of the solution.) In the higher-order method the accuracy depends
on two parameters: the order of the basis functions and their number Mu and Mv . Order
is deﬁned as the degree of the polynomial plus one. For example, a quadratic polynomial
u2 + au + b is of order three. We denote the order of U (u) and V (v) by Ku and Kv ,
respectively. Further information regarding the B-spline basis functions can be found in
Reference [22].
While Ku and Kv are input parameters speciﬁed by users, Mu and Mv are not direct
input parameters to WAMIT. Instead, users may specify the number of panel subdivisions
on each patch, Nu and Nv . (In standard B-spline terminology, these correspond to knots.)
Alternatively, users can specify the desired size of each panel in physical space, and the
program will automatically assign the corresponding inputs Nu and Nv on each patch to
achieve this objective.
The relations between the number of basis functions and the number of panels are as
follows:
(6.5)
M u = N u + Ku − 1 M v = N v + Kv − 1
Since Ku = Kv = 1 in the low-order panel method, the number of unknowns is the same
as the number of panels. Chapter 6 of [18] contains examples showing how the accuracy
of the solution depends on K and N for various geometries.

6–6

6.3

ORDER OF GAUSS QUADRATURES

Another topic which must be considered is the integration over patch surfaces. Since
the Galerkin method is used to solve the boundary integral equation, as described in
Chapter 12, this integration is carried out ﬁrst with respect to the source point, and then
with respect to the ﬁeld point. These are referred to respectively as the inner and outer
integrations which are carried out in parametric space. For this purpose, each patch is
sub-divided into Nu × Nv panels, and Gauss-Legendre quadrature is applied on each panel.
The orders of the Gauss quadratures are speciﬁed by input parameters. Experience with a
variety of applications has shown that it is suﬃcient to set the order of the outer integrals
with respect to (u, v) equal to (Ku , Kv ) and the order of the inner integrals equal to
(Ku + 1, Kv + 1).

6–7

6.4

THE GEOMETRIC DATA FILE

In the higher-order method the ﬁrst part of the GDF ﬁle is as follows:
header
ULEN GRAV
ISX ISY
NPATCH IGDEF

Subsequent data may be included in the GDF ﬁle after these four lines, depending on
the manner in which the geometry of the body is represented. (See Sections 6.5-6.8.)
The data on the ﬁrst three lines are identical to the low-order method as described in
Section 5.1. Thus:
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72.
ULEN is the dimensional length characterizing the body dimensions, used to nondimensionalize the quantities output from WAMIT.
GRAV is the acceleration of gravity, using the same units of length as in ULEN.
ISX, ISY are the geometry symmetry indices which have integer values 0, +1 to denote
no symmetry, or symmetry about the plane x = 0 or y = 0 respectively.
The data on line 4 of the GDF ﬁle are deﬁned as follows:
NPATCH is equal to the number of patches used to describe the body surface, as explained
in Section 6.1. If one or two planes of symmetry are speciﬁed, NPATCH is the number
of patches required to discretize a half or one quadrant of the whole of the body surface,
respectively.
IGDEF is an integer parameter which is used to specify the manner in which the geometry
of the body is deﬁned. Four speciﬁc cases are relevant, corresponding respectively to the
representations explained in Sections 6.5, 6.6, 6.7 and 6.8:
IGDEF = 0: The geometry of each patch is a ﬂat quadrilateral, with vertices listed in
the GDF ﬁle.
IGDEF = 1: The geometry of each patch is represented by B-splines, with the corresponding data in the GDF ﬁle.
IGDEF = 2: The geometry is deﬁned by inputs from a MultiSurf .ms2 ﬁle.

6–8

IGDEF < 0 or > 2: The geometry of each patch is represented explicitly by a special
subroutine, with optional data in the GDF ﬁle

6–9

6.5

GEOMETRY REPRESENTED BY LOW-ORDER PANELS
(IGDEF=0)

The simplest option to deﬁne the body geometry is appropriate if each patch of the body
surface is a ﬂat quadrilateral in physical space. In this case the vertices of each patch are
input via the GDF ﬁle in the same format as described in Section 5.1 for the low-order
method:
header
ULEN GRAV
ISX ISY
NPATCH 0
X1(1) Y1(1) Z1(1) X2(1) Y2(1) Z2(1) X3(1) Y3(1) Z3(1) X4(1) Y4(1) Z4(1)
X1(2) Y1(2) Z1(2) X2(2) Y2(2) Z2(2) X3(2) Y3(2) Z3(2) X4(2) Y4(2) Z4(2)
.
.
.
.
.
. . . . .
.
. . . . X4(NPATCH) Y4(NPATCH) Z4(NPATCH)
The data in the ﬁrst four lines are deﬁned above, in Section 6.4. Note that IGDEF=0 is
assigned on line 4. The patch vertices (X1, Y 1, Z1, ... , X4, Y 4, Z4) are deﬁned in precisely
the same manner as the panel vertices in Section 5.1. The convention deﬁned in Figure
5-1 must also be applied here, with the vertices numbered in the anti-clockwise direction
when the patch is viewed from the ﬂuid domain.
This option is particularly useful in the case of structures which consist of a small number
of ﬂat surfaces. Examples include rectangular barges, similar vessels with rectangular
moonpools, the Hibernia platform (a star-shaped bottom-mounted cylinder), etc. In such
cases it is not necessary or desirable to use a large number of small patches on each ﬂat
surface, as would be necessary to achieve accurate results with the low-order method.
The most eﬃcient procedure is to use the smallest number of patches which permits a
complete representation of the structure. For a simple rectangular barge, one quadrant
can be represented with three patches (bottom, side, end). If a rectangular moonpool is
centered amidships, 6 patches are required with two on the bottom and two on the walls
of the moonpool.
This option also might be useful to check the accuracy of a low-order application, using
the same GDF ﬁle for both (except that IGDEF=0 must be assigned for the higher-order
input). This provides a more general scheme for improving the accuracy compared to the
low-order option IQUAD=1. Two caveats should be noted in this context. First, since
each low-order panel is replaced by a patch, the number of patches may be quite large; this
will result in substantially longer run times and memory requirements as compared with
the low-order method. Secondly, if the ﬂat low-order panels do not correspond exactly to
the body surface, this part of the low-order approximation is not reﬁned by such a check.
6–10

6.6

GEOMETRY REPRESENTED BY B-SPLINES (IGDEF=1)

The most general approach to represent the geometry in the higher-order method is the
same as that which was ﬁrst developed in [18,19]. In this approach each patch of the
body is represented by B-splines, in an analogous manner to the representation of the
velocity potential (Section 6.2). The panel subdivision (knot vector) and the order of
the B-splines can be assigned independently between the geometry and the potential. If
the subdivisions and orders are the same, this is analogous to the isoparametric approach
in ﬁnite-element analysis. In V6.2, the domain of the parameters of the B-SPLINES
representing the geometry is no longer limited to (−1, 1). Arbitrary limits can be used and
they are normalized to (−1, 1) in the program.
More speciﬁcally, the mapping function X = (X, Y, Z) deﬁned by Equation (6.3) is
represented on each patch in the tensor-product form
(g)

X(u, v) =

(g)

M
M
v
u


j=1 i=1

Xij Ui (u)Vj (v)

(6.6)

Here Ui(u) and Vi (v) are the B-spline basis functions of u and v, and Mu(g) and Mv(g) are
the number of basis functions in u and v, respectively. (The superscripts are used to distinguish these geometric parameters from the corresponding parameters used to represent
the potential in Section 6.2.) As in (6.5),
Mu(g) = Nu(g) + Ku(g) − 1 Mv(g) = Nv(g) + Kv(g) − 1

(6.7)

where Ku(g) and Kv(g) are the orders of the respective B-splines. These parameters, and the
values of the unknown coeﬃcients Xij , are assigned for each patch in the GDF ﬁle.
The format of the GDF ﬁle is as follows:
header
ULEN GRAV
ISX ISY
NPATCH 1
NUG(1) NVG(1)
KUG(1) KVG(1)
VKNTUG(1,1) ... VKNTUG(NUA(1),1)
VKNTVG(1,1) ... VKNTVG(NVA(1),1)
XCOEF(1,1) XCOEF(2,1) XCOEF(3,1)
XCOEF(1,2) XCOEF(2,2) XCOEF(3,2)
·
·
·
XCOEF(1,NB(1)) XCOEF(2,NB(1)) XCOEF(3,NB(1))
·
6–11

·
·
NUG(NPATCH) NVG(NPATCH)
KUG(NPATCH) KVG(NPATCH)
VKNTUG(1,NPATCH) ... VKNTUG(NUA(NPATCH),NPATCH)
VKNTVG(1,NPATCH) ... VKNTVG(NVA(NPATCH),NPATCH)
XCOEF(1,1) XCOEF(2,1) XCOEF(3,1)
XCOEF(1,2) XCOEF(2,2) XCOEF(3,2)
·
·
·
XCOEF(1,NB(NPATCH)) XCOEF(2,NB(NPATCH)) XCOEF(3,NB(NPATCH))

Here IGDEF=1 is assigned on line 4 to specify the B-spline representation of the geometry.
NUG(I) and NVG(I) are the numbers of panel subdivisions of the u and v coordinates
on I-th patch.
KUG(I) and KVG(I) are the orders of B-splines
VKNTUG(J,I) is the B-spline knot vector in u on patch I. J=1,2,...NUA(I)
NUA(I)=NUG(I)+2*KUG(I)-1.
VKNTVG(J,I) is the B-spline knot vector in v on patch I. J=1,2,...NVA(I)
NVA(I)=NVG(I)+2*KVG(I)-1.
XCOEF(1,K)) XCOEF(2,K) XCOEF(3,K) are the components of the vector coeﬃcient Xij in (6.6). These are deﬁned in terms of the single array index K, where
K=1,2,...,NB(I). Here NB(I) is the total number of coeﬃcients on patch I, given by the
relation NB(I)=(NUG(I)+KUG(I)-1)×(NVG(I)+KVG(I)-1).
TEST11 (Appendix, Section A.11) is an example of this type of GDF input ﬁle.

6–12

6.7

GEOMETRY REPRESENTED BY MULTISURF (IGDEF=2)

Version 6.2 includes the option to import .ms2 geometry database ﬁles from the CAD program MultiSurf directly into WAMIT, and to represent the geometry during execution of
WAMIT by linking to the MultiSurf kernel. A detailed description of this option is contained in Reference 24. The principal advantages of this option are (a) the representation
of the geometry can be developed using the CAD environment of MultiSurf, and (b) this
representation can be transferred to WAMIT without signiﬁcant eﬀort or approximations.
Two special .dll ﬁles are required: RGKERNEL.DLL and RG2WAMIT.DLL. The ‘real’
versions of these ﬁles are not included in the standard WAMIT license. Users who intend
to use this option may license RGKERNEL and RG2WAMIT as part of an extended
version of WAMIT, or separately. The standard distribution of WAMIT Version 6.2PC
includes a ‘dummy’ ﬁle with the name ”rg2wamit.dll”. This enables WAMIT to be executed
without the ‘real’ ﬁles. As explained in Section 2.1, the PC-executable version of WAMIT
(wamit.exe) must be accompanied by four .dll ﬁles (geomxact.dll, newmodes.dll, dforrt.dll,
rg2wamit.dll). The dummy version of rg2wamit.dll can be distinguished from the real
version in two ways: (a) the dummy ﬁlename uses lower-case letters (rg2wamit.dll), and
(b) the size of this ﬁle is smaller, as indicated in the following table:
version

name

dummy
real

rg2wamit.dll
RG2WAMIT.DLL

size (approximate)
24Kb
78Kb

Note that the size of these ﬁles is approximate, and may change with updates and
subsequent versions, but the disparity in size will serve to distinguish the dummy and real
ﬁles.
To proceed with this option a user should ﬁrst prepare the MultiSurf model for the body
following the procedure in the MultiSurf documentation. A special appendix ‘Using the
WAMIT-RGKernel Interface’ is included in this User Manual (Appendix C). The output
ﬁle from MultiSurf will include a ﬁlename speciﬁed by the user and the extension ‘.ms2’.
This ﬁle will be referred to below as ‘body.ms2’. If the .ms2 ﬁle is missing or cannot
be found, a WAMIT runtime error message ‘Error return from subroutine RGKINIT’ is
generated, and the log ﬁle ‘RGKLOG.TXT’ will contain a statement that the designated
.ms2 ﬁle could not be opened.
In its simplest form, the GDF input ﬁle required to run WAMIT should be in the
following format:
header
ULEN GRAV
ISX ISY
NPATCH
2
3
6–13

(path)body.ms2
*
0 0 0

The ﬁrst four lines are explained in Section 6.4. IGDEF=2 is assigned by the second
integer on line 4. Line 5 contains an integer specifying the number of subsequent lines to
be read from the .gdf ﬁle. Line 6 contains the name of the .ms2 ﬁle, and may include the
optional path if this ﬁle is in a diﬀerent directory (folder). The asterisk (∗) on line 7 is
a default speciﬁer to indicate that all visible surfaces in the .ms2 ﬁle are to be included;
alternatively if only a subset of these surfaces are submerged these may be designated by
following the instructions in Appendix C. Line 8 includes three integer parameters with
default values zero, which may be used to control the accuracy of the geometry evaluation
in RGKernel, and also to modify the convention regarding the direction of the unit normal.
Further information is contained in Appendix C. TEST11C and TEST20 in Appendix A are
examples showing typical WAMIT runs for a circular cylinder and for a barge. Additional
examples are included in Reference 24.
A pre-processor utility GDF2MS2.EXE has been developed by AeroHydro Inc.1 to
convert low-order WAMIT .gdf input ﬁles to .ms2 geometry database ﬁles for MultiSurf.
Its results depend on the organization and content of the .gdf ﬁle. In general this utility
will create correctly dimensioned points for building a surface model in MultiSurf; and if
the .gdf ﬁle is suitably structured it is possible to create appropriate surface patches for
higher-order analysis with the IGDEF=2 option.

1 AeroHydro,

Inc., 54 Herrick Rd., Southwest Harbor, Maine 04679 USA 207-244-4100 (www.aerohydro.com)

6–14

6.8

ANALYTIC REPRESENTATION OF THE GEOMETRY

This option can be used in cases where the geometry of the body can be deﬁned explicitly,
with the fundamental advantage that the deﬁnition of the body geometry is exact and that
the only numerical approximation which remains is in the representation of the velocity
potential. Further details and examples based on this method are contained in Reference
25. The domain of parameters must be (-1.,1) in analytic representation.
The formulae required to deﬁne the geometry must be coded in FORTRAN, in the ﬁle
GEOMXACT.F. This ﬁle can be compiled separately as a .dll ﬁle and linked with WAMIT
at runtime. This special arrangement makes it possible for users of the PC executable
code to modify GEOMXACT for their own particular applications. Another feature of
this option is the possibility to input relevant body dimensions in the GDF ﬁle. Thus the
body dimensions can be changed without modiﬁcation of the code.
In the version of GEOMXACT.F and GEOMXACT.DLL as supplied with the WAMIT
software, there are several subroutines to produce various generic body shapes as listed
in the table below. Most of these subroutines are illustrated in the higher-order test
runs described in Appendix A. The dimensions of these generic bodies can be modiﬁed
by introducing appropriate data in the GDF ﬁle. Thus there is a variety of possibilities
for exploiting this option with or without special programming eﬀorts. Several diﬀerent
subroutines can be collected in a library, and identiﬁed with speciﬁc reserved values of
the index IGDEF which is input in the GDF ﬁle. The WAMIT software includes the
FORTRAN library ﬁle GEOMXACT.F, where several examples of these subroutines are
included. Note that IGDEF=0 or 1 are reserved for the options described in Sections 6.5
and 6.6, and thus IGDEF≥ 2 or IGDEF≤ −1 are appropriate values to select for the
analytic representation option. In the WAMIT software package as distributed, several
negative values IGDEF≤ −1 have been used for the test runs, and for other pertinent
examples which may be useful. Thus it is recommended that any new additions to this
library developed by users should be identiﬁed with positive values IGDEF≥ 2.
Continuing with the example of the circular cylinder shown in Figures 6.1 and 6.2, the
subroutine CIRCYL can be used without modiﬁcation. CIRCYL is included in the source
ﬁle GEOMXACT.F and selected by specifying IGDEF=-1. The relevant dimensions are
the radius and draft (and also ULEN and GRAV), which are speciﬁed in the GDF ﬁle in
the following format:
header
ULEN GRAV
1 1
2
-1
2
RADIUS DRAFT
INONUNIF

6–15

Here the symmetry indices ISX=1 and ISY=1 have been assigned, as well as the parameters
NPATCH=2 and IGDEF=-1. The number 2 on line 5 indicates that two lines follow in the
ﬁle to be read as input data. In addition to the dimensions of the cylinder, the parameter
INONUNIF is used in subroutine CIRCYL to specify either uniform (INONUNIF=0) or
nonuniform (INONUNIF=1) mapping between the parametric coordinates (U,V) and the
Cartesian coordinates (X,Y,Z).
Uniform mapping uses linear functions to transform V to the vertical coordinate on
the side, and to the radial coordinate on the bottom (and interior free surface). When
the nonuniform mapping option is selected the vertical coordinate on the side is a cubic
polynomial in V, and the radial coordinate on the other patches is a quadratic polynomial
in V, such that the ﬁrst derivatives vanish at the corner and at the intersection of the
side and free surface. This nonuniform mapping is analogous to the use of ‘cosine spacing’
in the low-order panel method, to achieve a ﬁner discretization of the solution near these
boundaries. The motivation for using the nonuniform mapping option is discussed in
Appendix A.11, where both options are compared, and in more detail in Reference 25.
The code in the subroutine CIRCCYL may be used as a guide for other geometries where
nonuniform mapping is desirable.
Before using this GDF ﬁle the user should assign appropriate values for the parameters
ULEN, GRAV, RADIUS, DRAFT, INONUNIF, and an appropriate header. As noted in
Section 3.7, this data must be contained within columns 1-80 of the GDF ﬁle.
In the normal case described above, NPATCH=2, corresponding to the side and bottom
of the cylinder. Two other situations exist where the same subroutine can be used: (1)
for a bottom-mounted cylinder NPATCH=1 and DRAFT is assigned with the same value
as the ﬂuid depth HBOT, and (2) if NPATCH=3 the interior free surface is included to
permit the removal of irregular-frequency eﬀects (IRR=1) as described in Chapter 9. The
restriction DRAFT1. Figure 6.3 illustrates the
patch numbering to achieve this ﬂexibility.

6–16

Figure 6.3: One quadrant of the cylinder shown in Figure 6.1 showing the patch numbering system which
permits using the subroutine CIRCYL with NPATCH=1 (bottom-mounted caisson), NPATCH=2 (ﬂoating
cylinder of ﬁnite draft), or NPATCH=3 (ﬂoating cylinder with a patch on the interior free surface to remove
irregular-frequency eﬀects). The view is from above the free surface, looking toward the interior of the
cylinder.

Other subroutines are also included in GEOMXACT.F to deﬁne a variety of bodies, in all
cases with IGDEF<0 so that positive values of IGDEF>2 will be reserved for users. Twelve
of the subroutines are listed in the table below and explained in detail here. Several of
these are used for the higher-order Test Runs described in Appendix A. Several additional
subroutines are also included in GEOMXACT for possible use by users. The details of
these other subroutines are explained in comments which are inserted in the source code
of each subroutine in the ﬁle GEOMXACT.F.
The following table lists the twelve subroutines which are described in more detail below:

6–17

IGDEF SUBROUTINE

NPATCH GDF INPUTS

-1

CIRCCYL

1,2,3

-2
-3
-4
-5
-6

ELLIPCYL
SPHERE
ELLIPSOID
BARGE
BARGEMP

1,2,3
1,2
1,2
2,3,4
6,7

-7
-8
-9

CYLMP
TORUS
TLP

3,4
1,2
11,12

-10

SEMISUB

-11

FPSO

-12

SPAR

4,6

RADIUS,DRAFT
INONUMAP
A,B,DRAFT
RADIUS
A,B,C
HALFLEN,HALFBEAM,DRAFT
HALFLEN,HALFBEAM,DRAFT
XMP,YMP
RADIUS,DRAFT,RADMP
RCIRC,RAXIS,ZAXIS
RADIUS,DRAFT,HSPACE
WIDTH,HEIGHT
XL,Y1,Y2,Z1,Z2
DCOL,RCOL,NCOL
XBOW,XMID,XAFT
HBEAM,HTRANSOM
DRAFT,DTRANSOM
RADIUS DRAFT
WIDTH THICKNESS TWIST NSTRAKE
IRRFRQ
IMOONPOOL, RADIUSMP
IMPGEN

The last column indicates the dimensions to be included in the GDF ﬁle. Where a
second line of dimensions is shown in the table (barge with moonpool, TLP, and semisub)
the GDF ﬁle should follow the same format as illustrated in the test runs TEST14 and
TEST15.
Brief descriptions of each subroutine are given below. More speciﬁc information is
included in the comments of each subroutine. In all cases the subroutine deﬁnes one
quadrant of a body with two planes of symmetry (ISX=ISY=1). These bodies can be
combined for multiple-body analysis, as described in Chapter 7, without modiﬁcations of
the subroutines.
CIRCCYL deﬁnes a circular cylinder as explained above.
ELLIPCYL deﬁnes an elliptical cylinder with semi-axes A,B. If A=B=RADIUS the results are identical to using CIRCCYL. The options NPATCH=1 (bottom mounted) and
NPATCH=3 (IRR=1) are the same as for CIRCCYL. The semi-axes A and B coincide
with the x− and y−axis of the body coordinate system, respectively.
SPHERE deﬁnes a ﬂoating hemisphere, with one patch on the body surface. If NPATCH=2
the interior free surface is included for use with the irregular-frequency option (IRR=1).
ELLIPSOID deﬁnes an ellipsoid with semi-axes A,B,C, ﬂoating with its center in the
plane of the free surface. (C is equal to the draft.) If A=B=C=RADIUS the results are
6–18

identical to using SPHERE. The semi-axes A, B and C coincide with the x−, y− and
z−axis of the body coordinate system, respectively.
BARGE deﬁnes a rectangular barge with length equal to 2×HALFLEN and beam equal
to 2×HALFBEAM. If NPATCH=2 and DRAFT=HBOT the barge is a bottom-mounted
rectangular caisson. If NPATCH=4 the interior free surface is included for use with the
irregular-frequency option (IRR=1). The longitudinal and transverse directions coincide
with the x− and y−axis of the body coordinate system, respectively.
BARGEMP deﬁnes a rectangular barge with a rectangular moonpool at its center. The
moonpool is bounded by vertical walls x = ±XMP and y = ±YMP. Other dimensions are
the same as for BARGE. In the normal case, NPATCH=6, separate patches are on the end
and side, two patches on the bottom, and two patches for the moonpool walls. Optionally,
if NPATCH=7, the moonpool free surface is represented by an additional patch; this is an
alternate scheme for the analysis of moonpools, using generalized modes to describe the
free surface so that resonant modes can be damped. TEST17B illustrates this scheme.
The longitudinal and transverse directions coincide with the x− and y−axis of the body
coordinate system, respectively.
CYLMP deﬁnes a spar-type structure consisting of a circular cylinder with a concentric
moonpool of constant radius RADMP. In the normal case, NPATCH=3, separate patches
are on the outer side of the cylinder, on the bottom, and on the interior wall of the moonpool. Optionally, if NPATCH=4, the moonpool free surface is represented by an additional
patch; this is an alternate scheme for the analysis of moonpools, using generalized modes
to describe the free surface so that resonant modes can be damped. TEST17B gives a
description of this scheme.
TORUS deﬁnes a ﬂoating or submerged torus, as illustrated in Figure 6.4. The sections
of the torus are circles of radius RCIRC, with their axes on a circle of radius RAXIS in
the horizontal plane z =ZAXIS. If -RCIRC0 in the top ﬁgure corresponds to the axis above the free surface. In the middle
ﬁgure the axis is in the plane of the free surface, and the sections are semi-circles. In these two ﬁgures
the torus is ﬂoating, with the upper edges of the body in the plane of the free surface. The bottom ﬁgure
shows a complete submerged torus. The dark lines indicate the boundaries between adjacent quadrants,
with one patch on each quadrant.

6–20

SEMISUB deﬁnes a generic semi-submersible with two rectangular pontoons and NCOL
equally-spaced circular columns on each pontoon. The pontoon dimensions include the
total length XL, transverse coordinates of the inner/outer pontoon sides Y1, Y2, and
vertical coordinates of the bottom and top horizontal surfaces Z1, Z2. Note that 00
is assigned in the CONFIG.WAM ﬁle (See Section 3.9). In that case the program
automatically assigns appropriate values to NU and NV on each patch with the objective
that the maximum physical length of each panel is equal to PANEL SIZE. This parameter
is speciﬁed in the same dimensional units of length as the data in the GDF ﬁle. This option
is especially convenient for convergence tests, where the size of all panels can be reduced
simultaneously.
Similarly, KU/KV, IQUO/IQVO and IQUI/IQVI (marked by †) should not be speciﬁed
in the SPL ﬁle when nonzero values are assigned to KSPLIN, IQUADO and IQUADI,
6–25

respectively, in the CONFIG.WAM ﬁle. (See Section 3.9.) In this case, the program sets
KU and KV equal to KSPLIN, IQUO and IQVO to IQUADO, and IQUI and IQVI to
IQUADI.
If these parameters are assigned in the SPL ﬁle, separate assignments must be made for
each patch as indicated in the above format. Conversely, parameters which are assigned
in CONFIG.WAM are global, with the same value assigned to all patches and all bodies.
Similarly, if KU/KV, IQUO/IQVO or IQUI/IQVI are included in the SPL ﬁle, separate
values are assigned to the u and v coordinates whereas if these parameters are assigned
via global parameters KSPLIN, IQUADO, IQUADI the same values are used for both
coordinates.
Experience using the higher-order method indicates that quadratic (KSPLIN=3) or
cubic (KSPLIN=4) B-splines are generally appropriate to represent both the geometry and
velocity potential, with the former (KSPLIN=3) preferred when the body shape is relatively
complex and the latter (KSPLIN=4) when the body is smooth and continuous (e.g. a
sphere). Most of the test runs described in the Appendix use KSPLIN=3. Experience
also suggests that eﬃcient choices for the inner and outer Gauss integrations are equal to
KSPLIN+1 and KSPLIN, respectively. Tests for accuracy and convergence can be achieved
most easily and eﬀectively by increasing the numbers of panels, either by increasing NU
and NV or by decreasing the parameter PANEL SIZE. This procedure permits systematic
convergence tests to be made easily and eﬃciently, without simultaneously changing the
other parameters or inputs.

6–26

6.12

THE USE OF DEFAULT VALUES TO SIMPLIFY INPUTS

Experience with the higher-order method indicates that for typical applications the global
parameters deﬁned above may be assigned the values KSPLIN=3, IQUADO=3, IQUADI=4.
These default values are assigned by the program automatically, if they are not assigned in
the CONFIG.WAM ﬁle and if there is no SPL input ﬁle available to open and read with the
same ﬁlename as the GDF ﬁle. In the latter case, however, the parameter PANEL SIZE
must be speciﬁed with a nonzero positive value in CONFIG.WAM. This is the simplest way
to use the higher-order method since it does not require the user to input the B-spline and
Gauss quadrature orders either locally in the SPL ﬁle or globally in the CONFIG.WAM
ﬁle.
The following table summarizes the options for inputting these parameters:
gdf .spl

conﬁg.wam

NU,NV
KU,KV
IQUO,IQVO
IQUI,IQVI

PANEL SIZE
KSPLIN
IQUADO
IQUADI

NONE
error
3
3
4

Here the ﬁrst column indicates inputs in the optional SPL ﬁle and the second column
indicates the corresponding inputs in the CONFIG.WAM ﬁle. The third column indicates
the default values which are set if there is no SPL ﬁle and if the parameters are not included
in the CONFIG.WAM ﬁle. It is important not to specify the same parameters in
both the SPL and CONFIG.WAM files, since this will cause errors reading the
data in the SPL file.
In summary, the simplest way to use the higher-order method is to specify PANEL SIZE
only, in the CONFIG.WAM ﬁle, and ignore all of the other parameters shown in this table.
The values of these parameters are displayed for each patch in the header of the .out ﬁle.
When the parameter PANEL SIZE is used, its value is also displayed on the line indicating
that the higher-order method is used.

6–27

6.13

ADVANTAGES AND DISADVANTAGES
OF THE HIGHER-ORDER METHOD

Some advantages and disadvantages of the higher-order method in comparison of the loworder method are listed below.
Advantages:
1. The higher-order method is more eﬃcient and accurate in most cases. More precisely,
the higher-order method converges faster than the low-order method, when the number
of panels is increased in both. (Comparisons for various geometries can be found in
[18,19]). Thus accurate solutions can be obtained more eﬃciently with the higherorder method.
2. Various forms of geometric input are possible, including the explicit representation.
When it is possible to use this approach it is relatively simple to input the geometry
and modify its dimensions for each run.
3. The pressure and velocity on the body surface are continuous. Continuity of the
hydrodynamic pressure distribution is particularly useful for the analysis of structural
loads.
Disadvantages:
1. The linear system which must be solved for the velocity potentials is not as well
conditioned in the low-order method. Thus the iterative method for the solution of
the linear system fails to converge in many cases. The direct or block-iterative solution
options are recommended in these cases. Since the size of the linear system (number
of unknowns) is signiﬁcantly smaller than for the low-order method, this generally
does not impose a substantial computational burden.
2. The second-order pressure due to the square of the ﬂuid velocity is unbounded at
sharp corners. The approximation of this pressure by higher-order basis functions is
more diﬃcult than in the low-order method. The result may be less accurate unless
the mapping accounts for the ﬂow singularity near the corner.

6–28

Chapter 7

ANALYSIS OF MULTIPLE BODIES
(NBODY>1)
WAMIT includes the capability to analyze multiple bodies which interact hydrodynamically and mechanically. Each of the separate bodies may oscillate independently with up
to six degrees of freedom. (Additional generalized modes can be deﬁned for each body,
as described in Chapter 8.) The bodies may be freely ﬂoating, ﬁxed, or constrained by
external forces. The basic theory for multibody interactions with waves is similar to that
of a single body as described in Chapter 12. The principal extension is to increase the
maximum number of degrees of freedom from 6 for a single body, to 6N for N bodies (N
is hereafter used to denote the number of bodies and the index K = 1, 2, ..., N is used to
denote each of the N bodies. ). For example, when two bodies are present the maximum
possible number of degrees of freedom becomes 12, 6 for each body. In this example, modes
7, 8, 9 represent translation of body 2 in the direction of the x,y and z-axes of the coordinate system ﬁxed on that body. These modes correspond to surge, sway and heave for the
second body, respectively. Modes 10, 11, 12 represent rotation about the same axes. The
extension of this convention to more bodies is evident. Thus, some output quantities are
given in vector or matrix form with dimensions 6N or (6N ) × (6N ), respectively.
Separate GDF ﬁles must be input for each body. The speciﬁcation of the GDF ﬁlenames and coordinates of each body is diﬀerent in the two alternative POT ﬁle formats
(Sections 3.1 and 3.2). Further details are included in the corresponding sections below.
The individual GDF ﬁles for each body are unchanged from the case where N = 1, thus
GDF ﬁles can be combined without modiﬁcation to analyze multiple-body conﬁgurations.
However the units of measurement, identiﬁed by the parameter GRAV, must be the same
for all bodies. The program assigns GRAV based on the value in the GDF ﬁle for the ﬁrst
body (K=1). If the value of GRAV for another body diﬀers from this by more than a small
tolerance (GTOL=0.1) the run is terminated with an appropriate error message.
WAMIT assumes that there are no planes of hydrodynamic symmetry when N > 1.
If geometric symmetry is speciﬁed for individual bodies, via their respective GDF input
ﬁles, the program reﬂects about the corresponding planes and increases the number of
panels accordingly. The total number of unknowns is the sum of the number required to
7–1

describe each body, including reﬂections. Thus the run times and memory requirements
are substantially increased.
Walls are not allowed when N > 1.
The multiple-body extensions are essentially the same for analyses based on the loworder and higher-order methods. However it is impossible to use both methods simultaneously for diﬀerent bodies. If NBODY>1 and ILOWHI=1 the option IALTPOT=2 must
be used.
The GEOMXACT subroutines described in Section 6.8 can be used for multiple bodies,
but special attention is required unless all of the bodies using the same subroutine have
identical dimensions. Most of the existing GEOMXACT subroutines read in the appropriate dimensions from the GDF ﬁle in scalar form, to initialize parameters within the
subroutine for subsequent calls. If the same subroutine is initialized again for another
body, the dimensions are overwritten. As a result the dimensions in the last GDF ﬁle
are applied to all of the bodies using the same subroutine. In cases where bodies with
diﬀerent dimensions are represented by the same subroutine, the dimensions used in subsequent calls should be saved within the subroutine as arrays with dimension NBODY. The
subroutine FPSO12 in GEOMXACT illustrates this procedure for two FPSO hulls with
diﬀerent dimensions. To call attention to this problem, a warning message is issued if two
or more bodies use the same value of IGDEF in GEOMXACT.
There are three alternative ways to input parameters to the FORCE subprogram. Alternative Form 1 can be used for freely ﬂoating bodies, and the more general Alternative
2 Form can be used for bodies subject to external forces. Alternative 3 includes a Global
Force Control ﬁle (GFRC) and a separate FRC ﬁle for each body, each being in either
Form 1 or 2. These three alternative are described respectively in Sections 7.3, 7.4, and
7.5.

7–2

7.1

INPUT TO POTEN (IALTPOT=±1)

If IALTPOT=1 is speciﬁed in the conﬁguration ﬁle, or if this parameter is not speciﬁed
and is set by default, a Global Geometric Data File (GGDF) must be input. The name of
this GGDF ﬁle is speciﬁed in the FNAMES.WAM ﬁle, or interactively if no GDF ﬁlename
is included in FNAMES.WAM. The ﬁlenames of each individual body are input in the
GGDF ﬁle. This option can only be used for the low-order method (ILOWHI=0).
If IALTPOT=1 the data in the Global Geometric Data File (GGDF) are listed below:
header
-1.0 GRAV
NBODY
GDF(1)
XBODY(1,1) XBODY(2,1) XBODY(3,1) XBODY(4,1)
MODE(1,1) MODE(2,1) MODE(3,1) MODE(4,1) MODE(5,1) MODE(6,1)
.
.
.
GDF(N)
XBODY(1,N) XBODY(2,N) XBODY(3,N) XBODY(4,N)
MODE(1,N) MODE(2,N) MODE(3,N) MODE(4,N) MODE(5,N) MODE(6,N)
If IALTPOT=-1, corresponding to earlier versions of WAMIT, the data in the Global
Geometric Data File (GGDF) are listed below:
header
-1.0 GRAV
NBODY
GDF(1)
XBODY(1,1) XBODY(2,1) XBODY(3,1) XBODY(4,1)
IRAD(1) IDIFF(1)
MODE(1,1) MODE(2,1) MODE(3,1) MODE(4,1) MODE(5,1) MODE(6,1)
.
.
.
GDF(N)
XBODY(1,N) XBODY(2,N) XBODY(3,N) XBODY(4,N)
IRAD(N) IDIFF(N)
MODE(1,N) MODE(2,N) MODE(3,N) MODE(4,N) MODE(5,N) MODE(6,N)
The only diﬀerence between these two formats is the inclusion of separate values IRAD,IDIFF
for each body when IALTPOT=-1. The latter is generally not necessary, and is considered redundant with the assignment of IRAD and IDIFF in the POT ﬁle. However users
of earlier versions who wish to do so may continue to use old GGDF ﬁles by assigning
7–3

IALTPOT=-1. This option is deprecated and may not be supported in later versions of
WAMIT.
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72.
-1.0 is the ﬂag which indicates that the ﬁle is a Global GDF ﬁle. (Any real number less
than or equal to zero is acceptable.) (Note that in an ordinary GDF ﬁle this parameter is
ULEN, which must be positive.)
GRAV is the acceleration of gravity, using the same units of length as in ULEN(K) of
the GDF ﬁles. If lengths are input in meters or feet, input 9.80665 or 32.174, respectively
for GRAV. (The same units of length must be used in each GDF ﬁle if N > 1.)
NBODY is the total number of bodies.
GDF(K) is the name of the K-th Geometric Data File.
XBODY(1,K),XBODY(2,K),XBODY(3,K) are the (X, Y, Z) coordinates of the origin of the body-ﬁxed coordinate system of the K-th body, relative to the global coordinate
system, input in the dimensional units of the length ULEN(K). The global coordinate
system is used in place of the body coordinate system to deﬁne ﬁeld-point data (ﬂuid pressures, velocities, and free-surface elevation). The origin of the global coordinate system
must be located on the free surface.
XBODY(4,K) is the angle in degrees between the x−axis of the body coordinate system
of the K-th body and the X−axis of the global system (see Figure 5.2). The direction
of the global Z-axis, and of the z-coordinate in each body coordinate system, must be
positive upward and perpendicular to the undisturbed free surface.
IRAD(K), IDIFF(K) are indices used to specify the components of the radiation and
diﬀraction problems to be solved for the K-th body. The following options are available,
depending on the values of IRAD(K) and IDIFF(K):
IRAD(K)= 1: Solve for the radiation velocity potentials due to all six rigid-body modes
of motion of the K-th body.
IRAD(K)= 0: Solve the radiation problem only for those modes of motion speciﬁed by
setting the elements of the array MODE(I,K)=1 (see below).
IRAD(K)= −1: Do not solve any component of the radiation problem.
IDIFF(K)= 1: Solve for all diﬀraction components, i.e. the complete diﬀraction problem.
IDIFF(K)= 0: Output only the exciting forces in the modes speciﬁed by MODE(I,K)=1.
IDIFF(K)= −1: Do not solve the diﬀraction problem.
MODE(I,K) is a six-element array of indices for the K-th body, where I=1,2,3 correspond
to the surge, sway and heave translational modes along the body-ﬁxed (x, y, z) axes, and
I=4,5,6 to the roll, pitch and yaw rotational modes about the same axes, respectively. Each
of these six indices should be set equal to 0 or 1, depending on whether the corresponding
radiation mode(s) and diﬀraction component(s) are required. (See the options IRAD(K)=0
and IDIFF(K)=0 above.)
The data in the GDF ﬁle for each individual body is as described in Section 3.1. Panel
coordinates and symmetry indices are deﬁned in terms of the local body coordinates of
7–4

the respective body, as for a single-body application. For each body there is a respective
dimensional length ULEN(K), and a corresponding value of gravity GRAV(K). ULEN(K)
must be input in the same dimensional units for every body. ULEN(K) must be a positive
number greater than 10−5 . An error return and warning statements are generated if the
last restriction is not satisﬁed. ULEN(1), the characteristic length of the ﬁrst body, is used
to nondimensionalize the outputs as described in Chapter 4.
The values GRAV(K) in all GDF ﬁles are read, and used to check for consistency of
units among the diﬀerent bodies. An error return occurs if any GRAV(K) diﬀers from
GRAV(1) by more than 0.1 units in absolute value (0.1 m/s2 or 0.1 ft/s2 if these units
are used.) (The value of GRAV(1) is used for all calculations and outputs.) The Potential
Control File (POT) is described in Section 3.1, and is unchanged when N > 1. However
the values of the array XBODY in the POT ﬁle (or in the optional conﬁguration ﬁle), and
array MODE which are speciﬁed in the POT ﬁle are all ignored, since these are superseded
by the individual values for each body in the GGDF ﬁle. (If IALTPOT=-1, the same
statement applies to IRAD, IDIFF.)

7.2

INPUT TO POTEN (IALTPOT=2)

If IALTPOT=2 is assigned in the conﬁguration ﬁle, the POT ﬁle is as described in Section
3.2. This Alternative Form of the POT ﬁle includes all of the relevant data for each body
as is described above for the GGDF ﬁle when IALTPOT=1. (This is considered to be a
more eﬃcient scheme to adopt for multiple-body analyses.)
To summarize, when NBODY>1 and IALTPOT=2, the POT ﬁle should include the
applicable value of NBODY, followed by appropriate data for each body, as shown in
Section 3.2. A separate GDF ﬁle must be input for each body. No other changes are
required in the input ﬁles.

7–5

7.3

INPUT TO FORCE (IALTFRC=1)

If IALTFRC=1 the format of the FRC ﬁle is as shown below:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
VCG(1)
XPRDCT(1,1,1) XPRDCT(1,2,1) XPRDCT(1,3,1)
XPRDCT(2,1,1) XPRDCT(2,2,1) XPRDCT(2,3,1)
XPRDCT(3,1,1) XPRDCT(3,2,1) XPRDCT(3,3,1)
VCG(2)
XPRDCT(1,1,2) XPRDCT(1,2,2) XPRDCT(1,3,2)
XPRDCT(2,1,2) XPRDCT(2,2,2) XPRDCT(2,3,2)
XPRDCT(3,1,2) XPRDCT(3,2,2) XPRDCT(3,3,2)
.
.
VCG(N )
XPRDCT(1,1,N) XPRDCT(1,2,N) XPRDCT(1,3,N)
XPRDCT(2,1,N) XPRDCT(2,2,N) XPRDCT(2,3,N)
XPRDCT(3,1,N) XPRDCT(3,2,N) XPRDCT(3,3,N)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
The only diﬀerence relative to the case of a single body (Section 3.3), is that the VCG and
3 × 3 matrix of each body’s radii of gyration are entered in succession.

7–6

7.4

INPUT TO FORCE (IALTFRC=2)

If IALTFRC=2 the format of the FRC ﬁle is the same as described in Section 3.4 for
a single body, except that the array specifying (XCG,YCG,ZCG) is extended to include
all bodies, and the external force matrices have dimensions NDFR × NDFR. NDFR=
N
n=1 (6 + NEWMDS(n)) is the total number of degrees of freedom including all rigid body
modes and generalized modes. The normal format is as follows:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
RHO
XCG(1) YCG(1) ZCG(1) XCG(2) YCG(2) ZCG(2) ...
... XCG(N) YCG(N) ZCG(N)
IMASS
EXMASS(1,1) EXMASS(1,2) ... EXMASS(1,NDFR)
EXMASS(2,1) EXMASS(2,2) ... EXMASS(2,NDFR)
.
.
EXMASS(NDFR,1) EXMASS(NDFR,2) ... EXMASS(NDFR,NDFR)
IDAMP
EXDAMP(1,1) EXDAMP(1,2) ... EXDAMP(1,NDFR)
EXDAMP(2,1) EXDAMP(2,2) ... EXDAMP(2,NDFR)
.
.
EXDAMP(NDFR,1) EXDAMP(NDFR,2) ... EXDAMP(NDFR,NDFR)
ISTIF
EXSTIF(1,1) EXSTIF(1,2) ... EXSTIF(1,NDFR)
EXSTIF(2,1) EXSTIF(2,2) ... EXSTIF(2,NDFR)
.
.
EXSTIF(NDFR,1) EXSTIF(NDFR,2) ... EXSTIF(NDFR,NDFR)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
As in Section 3.4, the integers (IMASS,IDAMP,ISTIF) are set equal to one if the matrix follows, and equal to zero if no corresponding external matrix is included in the ﬁle.
Omitting the matrix is equivalent to including the matrix with zero values for all elements.
7–7

The same format can be used with the external force matrices in separate ﬁles and with
the corresponding ﬁlenames replacing the matrices in the FRC ﬁle. This option is speciﬁed
by the values (IMASS,IDAMP,ISTIF)=2:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
RHO
XCG(1) YCG(1) ZCG(1) XCG(2) YCG(2) ZCG(2) ...
... XCG(N) YCG(N) ZCG(N)
2
MASS (filename containing inertia matrix)
2
DAMP (filename containing damping matrix)
2
STIF (filename containing stiffness matrix)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
The separate external force data ﬁles MASS, DAMP, STIF contain a one-line header plus
the three corresponding matrices shown in the ﬁrst format.
As in Section 3.4, it is also possible in Version 6 to specify the Alternative Form 2 by
inserting the integer 2 in the second line, but that option is deprecated.
The ﬁrst line of this ﬁle, and all lines beginning with the variable NBETAH, are identical
to the data in Alternative 1 FRC ﬁle. The data which diﬀer in Alternative 2 are described
in Section 3.4.

7–8

7.5

INPUT TO FORCE (IALTFRC=3)

Alternative Form 3 includes one Global FRC ﬁle (GFRC) and N FRC ﬁles. The FRC
ﬁle for each body can take either the form of Alternative 1 or Alternative 2. With this
option existing FRC ﬁles for single bodies can be used without modiﬁcation. Note however
that this precludes the consideration of external mass, damping and stiﬀness forces which
produce coupling interactions between the bodies.
If IALTFRC=3 the input parameters in the GFRC ﬁle are listed below:
header
IOPTN(1) IOPTN(2) IOPTN(3) IOPTN(4) IOPTN(5) IOPTN(6) IOPTN(7) IOPTN(8) IOPTN(9)
RHO
FRC(1)
FRC(2)
.
.
FRC(N)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
XFIELD(1,2) XFIELD(2,2) XFIELD(3,2)
XFIELD(1,3) XFIELD(2,3) XFIELD(3,3)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)
The ﬁrst three lines of this ﬁle, and all lines beginning with the variable NBETAH, are
identical to the data in the Alternative Form 2 FRC ﬁle.
FRC(K) is the name of the FRC ﬁle for body K. The Form of each separate ﬁle must be
1 or 2, and this is speciﬁed by the optional array IALTFRCN in the conﬁguration ﬁle, as
described in the following section.
Some of the data given in N FRC ﬁles are read but neglected, if the same data is given in
the GFRC ﬁle. For example, the data IOPTN and RHO in the FRC ﬁles for each body
are neglected and the corresponding parameters provided by the GFRC ﬁle are used.
As with IALTFRC=2, it is possible to specify IALTFRC=3 by inserting the integer 3 on
line 2 of this ﬁle, but this practice is deprecated in Version 6 and may not be supported in
later versions.

7–9

7.6

PARAMETERS IN THE CONFIGURATION FILE

The conﬁguration ﬁle described in Section 3.9 includes a few inputs that are relevant to the
analysis of multiple bodies. These include NBODY, XBODY, NEWMDS, and IALTFRCN.
If IALTFRC=3 the array IALTFRCN must be speciﬁed in conﬁguration, unless the default
values 1 are applicable for all bodies. The parameters NBODY, XBODY, and NEWMDS
can be speciﬁed either in conﬁguration or in other input ﬁles, as indicated in Section 3.9.
A typical conﬁguration ﬁle used for all of these parameters is as follows, with explanatory
comments on each line:
IALTFRC=3 (use FRC format in Section 7.5)
IALTFRCN=2 1 2 (Form of FRC ﬁle for each body)
IALTPOT=1 (use POT format in Section 3.1)
IDEFINE=1 (run DEFINE subroutine in .dll ﬁle for generalized modes)
ILOWHI=0 (use low-order method)
IPERIO=1 (POT ﬁle contains wave periods in seconds)
NEWMDS(1)=0 (Body 1: No generalized modes)
NEWMDS(2)=4 (Body 2: 4 generalized modes)
NEWMDS(3)=0 (Body 3: No generalized modes)
XBODY(1) = -5.0 0.0 0.0 0.0 (Body 1 coordinates)
XBODY(2) = 0.0 0.0 0.0 0.0 (Body 2 coordinates)
XBODY(3) = 5.0 0.0 0.0 90.0 (Body 3 coordinates)

7–10

7.7

OUTPUT

The nondimensionalizations given in Chapter 4 hold for all output quantities. L=ULEN(1),
the dimensional length for Body 1, is the characteristic length and is used for the nondimensionalization of the output quantities.
The added mass (Aij ), damping coeﬃcients (Bij ) and hydrostatic coeﬃcients(C(i, j))
are matrices of dimension up to 6N × 6N . These quantities are deﬁned in the direction
of the axes of the corresponding body coordinate systems. For example A1,9 is the added
mass in the direction of the x-axis of the coordinate system of Body 1 (surge added mass)
due to the motion of the Body 2 in the direction of the z-axis of Body 2 (heave motion).
The forces (Xi ) and the motion amplitudes (ξi ) are vectors of dimension up to 6N .
These quantities are also deﬁned in the direction of the axes of the coordinates system of
the corresponding body. For example, X17 is the pitch exciting moment about the y-axis
of Body 3. The phases of the forces and motion amplitudes and of the ﬁeld quantities
such as the ﬁeld pressure and ﬁeld velocity, are deﬁned relative to the phase of the incident
wave at the origin of the global coordinate system.
The pressure drift force and moment (Option 9) returns values for each body in its
respective body coordinate system.
When Option 8 is speciﬁed (momentum drift force and moment) the quantities calculated are the global horizontal drift forces and mean yaw moment acting on the entire
ensemble of bodies. It is possible to compare these outputs with the total drift force and
moment from pressure integration, by summing the latter outputs for the forces and moments on each body. This provides a useful check on consistency. Special attention is
required if the body coordinates are not parallel to the global coordinates system.

7–11

Chapter 8

GENERALIZED BODY MODES
(NEWMDS>0)
WAMIT includes the capability to analyze generalized modes of body motion, which extend
beyond the normal six degrees of rigid-body translation and rotation. These generalized
modes can be deﬁned by the user to describe structural deformations, motions of hinged
bodies, and a variety of other modes of motion which can be represented by speciﬁed
distributions of normal velocity on the body surface. To simplify the discussion it will be
assumed that only one body is analyzed, i.e. NBODY=1. The analysis of multiple bodies
with generalized modes is discussed below in Section 8.5.
Each generalized mode is deﬁned by specifying the normal velocity in the form
ϕjn = nj = uj nx + vj ny + wj nz

(8.1)

where j > 6 is the index of the mode. (The ﬁrst six indices j = 1, 2, ..., 6 are reserved
for the conventional rigid-body modes.) The displacement vector (uj , vj , wj ) is deﬁned by
the user in a special subroutine which can be accessed and modiﬁed by the user. The
displacement vector can be any physically relevant real function of the body coordinates
(x, y, z) which can be deﬁned by FORTRAN code.
Corresponding to these modes are the generalized hydrodynamic force components,
which are deﬁned as in Sections 4.2 and 4.3 with the extended normal-vector ni and the
corresponding radiation solutions ϕi . Further discussion of the pertinent theory may be
found in References [13] and [26].
The following four examples, which are intended to illustrate applications, are simpliﬁed
from the computational examples in [13].
1. A ship with simpliﬁed transverse and vertical bending modes described by the Legendre polynomial of order 2:
u7 = 0,

v7 = P2 (q),

u8 = 0,

v8 = 0,
8–1

w7 = 0

(8.2)

w8 = P2(q)

(8.3)

Here NEWMDS=2, P2 (q) = 32 q 2 − 12 is the Legendre polynomial and q = 2x/L is the
normalized horizontal coordinate varying from -1 to +1 over the length L of the ship.
2. A vertical column, bottom mounted, with three orthogonal cantilever modes described
by shifted Jacobi polynomials:
u7 = q 2 ,

v7 = 0,

u8 = 6q 3 − 5q 2 ,
4

3

w7 = 0

v8 = 0,
2

u9 = 28q − 42q + 15q ,

(8.4)

w8 = 0

v9 = 0,

w9 = 0

(8.5)
(8.6)

Here NEWMDS=3, and (q = z/HBOT + 1) is the normalized vertical coordinate
varying from 0 at the bottom to 1 at the free surface.
3. Two bodies connected by a simple hinge joint at the origin, permitting each body
to pitch independently about the y-axis. In this case the six rigid-body modes are
deﬁned as if the hinge is rigid, and the new mode (j = 7) is speciﬁed by the vector
u7 = znx sgn(x),

v7 = 0,

w7 = |x|nz

(8.7)

with NEWMDS=1. Here sgn(x) is equal to ±1 according as x > 0 or x < 0.
4. An array of ﬁve identical bodies, all described by panels in the same GDF ﬁle as if for a
single body with ﬁve separate elements; the bodies are centered at transverse positions
y = −4w, −2w, 0, 2w, 4w to simulate images of the central body in the presence of
tank walls at y = ±w. The surge, heave, and pitch exciting forces on the central body
are then speciﬁed by the exciting force coeﬃcients X7 , X8 , X9 . The corresponding
new modes are deﬁned which have the same normal velocities on the central body
and zero on the images. Here NEWMDS=3, and the vectors (uj , vj , wj ) are all zero,
except in the range −w < y < w where
u7 = 1

(8.8)

w8 = 1

(8.9)

u9 = z

w9 = −x

(8.10)

(With these deﬁnitions the exciting force coeﬃcients can be evaluated either from
the Haskind relations or direct pressure integrals as deﬁned in Section 4.3, with the
surface integrals extended over the ensemble of the central body and its images.)
In the generalized mode analysis one or two planes of geometric symmetry can be exploited to reduce the computational burden, when the body geometry permits. In such
cases it is necessary to deﬁne the generalized modes to be either symmetrical or antisymmetrical. These symmetries must be speciﬁed in the subroutine by assigning one of the
integers (1-6) to the array ISYM for each of the generalized modes. The value of this
integer signiﬁes that the symmetries of the generalized mode are the same as the corresponding rigid-body mode. Since the symmetries of surge (j=1) and pitch (j=5) are always
the same, and likewise for sway (j=2) and roll (j=4), specifying ISYM=1 or 5 is equivalent,
8–2

and similarly ISYM=2 or 4 is equivalent. If there are no planes of symmetry, any integer
in the range (1-6) may be assigned to ISYM.
Any problem which can be analyzed with the NBODY option can also be analyzed with
the generalized mode option. If geometric symmetry planes exist for the ensemble of all
bodies, the use of the generalized mode method is more eﬃcient computationally. On the
other hand, the preparation of input ﬁles generally is simpler in the NBODY approach.
Complex generalized modes can be analyzed by superposition of the separate real and
imaginary parts, each of which is treated as a separate real mode. For example, the
speciﬁcations w7 = cos(kx), w8 = sin(kx) deﬁne two vertical deﬂections which can be
superposed with a phase diﬀerence of 90◦ to represent a snake-like traveling wave along
the body.
Two alternative program units have been provided in the WAMIT software package to
facilitate the use of generalized modes. The ﬁrst method, also used with previous versions
of WAMIT, uses a separate program DEFMOD to evaluate the geometric data associated
with generalized modes. DEFMOD contains a subroutine DEFINE, which can be modiﬁed
by the user to compute the displacement vector for diﬀerent generalized modes. This
method is retained in Version 6 and can be used with the low-order method (ILOWHI=0).
The second method, which is applicable for both the low-order and higher-order methods,
uses a DLL ﬁle containing a special subroutine NEWMODES with a library of lower-level
routines where diﬀerent types of generalized modes can be deﬁned. Since NEWMODES
is contained in a DLL ﬁle it can be modiﬁed by the user, in a similar manner to the
GEOMXACT ﬁle for deﬁning bodies analytically (cf. Section 6.9).
The ﬁrst method, using the program DEFMOD, can be used with any suitable FORTRAN compiler. Three separate runs must be made, ﬁrst with WAMIT to set up the
input ﬁle for DEFMOD, then with DEFMOD, and ﬁnally with WAMIT to solve for the
potentials in the usual manner. In the second method only one run of WAMIT is required, however users of the PC executable code must compile the DLL ﬁle following the
instructions below.
In Section 8.1 the input ﬁles are described for performing the generalized mode analysis
for a single body. Section 8.2 describes the use of DEFMOD, and Section 8.3 describes
the alternative use of the DLL ﬁle NEWMODES. The deﬁnitions of hydrostatic restoring
coeﬃcients are described in Section 8.4 and the analysis of multiple bodies (NBODY>1)
is described in Section 8.5.
Several test runs are used to illustrate the use of generalized modes, including the use of
both DEFMOD and NEWMODES and the appropriate input ﬁles. TEST08 (ILOWHI=0)
and TEST18 (ILOWHI=1) analyze a bottom-mounted column with bending modes. TEST16
analyzes a rectangular barge with bending modes. TEST17(a,b) illustrates the use of generalized modes to analyze damped motions of a moonpool. TEST23 uses generalized modes
to analyse a bank of ‘paddle’ wavemakers. TEST 24 analyses the motions of a vessel with
ﬁve separate segments connected by hinges. Further information is contained in the Appendix.

8–3

8.1

INPUT FILES

Two input parameters NEWMDS and IGENMDS control the implementation of the generalized mode option. NEWMDS speciﬁes the number of generalized modes, with the
default value zero. Instructions for inputting NEWMDS in the POT ﬁle (if IALTPOT=2)
or in the conﬁguration ﬁle (if IALTPOT=1) are given in Sections 3.2 and 3.7 respectively.
IGENMDS is an integer used to specify the deﬁnition of the generalized modes, using either
the DEFMOD program or the NEWMODES subroutine library. IGENMDS is input in
the conﬁguration ﬁle. The default value IGENMDS=0 is used if the program DEFMOD
is used, as explained in Section 8.2. If IGENMDS is nonzero, the DLL ﬁle NEWMODES
is used. In the latter case, the value of IGENMDS can be used to identify appropriate
subroutines within the NEWMODES library, as explained in Section 8.3.
The deﬁnition of nondimensional outputs corresponding to each mode of motion cannot
be speciﬁed in general, without prescribing the dimensions of each mode. To avoid this
complication, the parameter ULEN should be set equal to 1.0 in the GDF ﬁle whenever
generalized modes are analyzed. Except for this restriction, the GDF and POT input ﬁles
are not changed.
In the POT ﬁle the six rigid-body modes can be speciﬁed as free or ﬁxed in the usual
manner, by appropriate choices of the index IRAD and array MODE (see Section 3.2). The
program assumes that all generalized modes (j > 6) are free, and sets the array elements
MODE(j) equal to one for these modes during the computations.
The options in FORCE have the same eﬀect for generalized modes as for the rigidbody modes, except for restrictions on the mean drift forces evaluated by direct pressure
integration (Option 9) and control surfaces (Option 9c). The two horizontal drift forces and
the vertical drift moment can be evaluated, including all pertinent motions of the body in
the rigid-body and generalized modes, using the momentum analysis (Option 8). Options
9 and 9c cannot be used for bodies with generalized modes. In the analysis of multiple
bodies (NBODY>1), where some but not all of the bodies have generalized modes, Options
9 and 9c can be used for the bodies with no generalized modes.
The Alternative Form 2 of the FRC ﬁle (IALTFRC=2) should be used to specify the
appropriate mass, damping, and stiﬀness matrices for the body including its extended
modes. For example, in case 1 above, to account for the mass and stiﬀness of the ship
hull it is necessary to include corresponding 8 × 8 matrices which correctly specify these
coeﬃcients for the distribution of internal mass within the ship and for its bending motion.
The FRC ﬁles used for Test Runs 08, 16, 17, and 18 illustrate appropriate forms.

8.2

USING DEFMOD WITH THE LOW-ORDER METHOD

To facilitate the deﬁnition of the vectors (uj , vj , wj ) by users, a pre-processor program
DEFMOD is provided in Fortran source code to input the values of the normal velocity
for each generalized mode, at the centroid of each body panel. DEFMOD includes a short
8–4

subroutine DEFINE, which should be modiﬁed by the user for each application. In the
DEFMOD subroutine as provided, DEFINE evaluates the bending modes of the vertical
column used in the Test Run TEST08. (The same subroutine is included in NEWMODES
and used for TEST18.) The four examples itemized in the introduction of this Chapter are
included in separate ﬁles DEFINE.1, DEFINE.2, DEFINE.3, and DEFINE.4 to illustrate
the preparation of appropriate subroutines. (Additional modes are included in these ﬁles.)
The evaluation of the normal velocity (8.1) for generalized modes requires a speciﬁcation
of the vectors u, v, w, and normal components nx, ny , nz at the centroid of each panel. The
ﬁrst WAMIT run is aborted after writing these data to a ﬁle, and also the panel areas
which are required to evaluate the hydrostatic coeﬃcients for the generalized modes. The
user is then requested to run DEFMOD. After DEFMOD is run, WAMIT may then be rerun to complete the analysis in the normal manner. This procedure is described in greater
detail below, and illustrated by the ﬂow chart in Figure 8.1. There are two input/output
ﬁles associated with DEFMOD, both denoted by the ﬁlename (gdf) of the GDF ﬁle. The
ﬁle gdf.PRE (for PRE-processing) contains for each panel the centroid coordinates x, the
area, and the six components of n and x × n. The ﬁle gdf.MOD (for MODes) contains
the generalized normal velocity on each panel, and the hydrostatic coeﬃcients. These two
ﬁles are prepared automatically by WAMIT and DEFMOD, and do not require special
attention by the user.
The ﬁrst run of WAMIT is made with NEWMDS > 0 speciﬁed in the conﬁguration
ﬁle. The pre-processor ﬁle gdf.PRE is output to the hard disk, and execution of WAMIT
is interrupted with the following message:
‘PREMOD run completed – now run DEFMOD’
This ﬁrst run of WAMIT is interrupted if NEWMDS > 0 and if there is not already in
the default directory an input ﬁle with the name gdf.MOD. For this reason the user must
rename or delete old MOD ﬁles prepared with the same gdf ﬁlename.
The output ﬁle gdf.PRE includes a one-line header for identiﬁcation, followed by one
line containing the symmetry indices (ISX, ISY), number of input panels (NEQN), and
the maximum number of degrees of freedom which can be assigned in the WAMIT run
(MAXDFR). Each of the remaining lines of this ﬁle corresponds to a panel, in the same
order as the GDF ﬁle, and includes the coordinates x=(x, y, z) of the panel centroid, its
area, and the six components of the normal vector n and cross-product x×n.
Either before or after the ﬁrst run of WAMIT the user should modify the subroutine
DEFINE in the program DEFMOD, specifying the number of generalized modes, their
symmetries with respect to the geometric planes of symmetry of the body, and appropriate
code for computing the vectors (uj , vj , wj ). DEFMOD should then be compiled and linked
with an appropriate FORTRAN compiler. (Since DEFMOD is a self-contained FORTRAN
ﬁle, linking is a trivial operation.)
DEFMOD is executed after the ﬁrst run of WAMIT, using the gdf.PRE ﬁle as input
to generate the output ﬁle gdf.MOD. This MOD ﬁle generated by DEFMOD includes the
normal velocities of each generalized mode at each panel centroid. Also included at the
end of this ﬁle are the generalized hydrostatic coeﬃcients computed from equation (8.12).

8–5

After creating the gdf.MOD input ﬁle, execute WAMIT again to continue the run. From
this on to the completion of the run the procedure is identical to that where NEWMDS=0.
The ﬁrst and second runs are distinguished by the absence or presence, respectively,
of the ﬁle gdf.MOD. Thus the existence of an old ﬁle with the same name is important.
If changes are made only in the POT ﬁle (e.g. changing the wave periods), an existing
MOD ﬁle can be reused without repeating the ﬁrst run of WAMIT. On the other hand, if
changes are made in the GDF ﬁle (e.g. changing the number of panels), and if the same
gdf ﬁlename is used for the new GDF ﬁle, the old MOD ﬁle must be renamed or deleted
before running WAMIT. If changes are made in DEFMOD (e.g. changing the deﬁnitions
and/or number of new modes), the old PRE ﬁle can be used as input to create the new
MOD ﬁle. A warning message is issued by DEFMOD before overwriting an old MOD ﬁle
with the same name. No warning message is issued by WAMIT before overwriting an old
PRE ﬁle with the same name.

8.3

USING THE DLL SUBROUTINE NEWMODES

The source code for the DLL ﬁle NEWMODES.F is provided with the WAMIT software
to facilitate the speciﬁcation of generalized modes by users, including users of the WAMIT
PC executable version. Source-code users may prefer to compile and link NEWMODES
together with WAMIT, or alternatively to keep it as a separate DLL ﬁle. PC executable
users must compile their modiﬁed versions of NEWMODES, following the instructions
below. This procedure is analogous to the modiﬁcation of the DLL ﬁle GEOMXACT, as
described in Section 6.9.
The ﬁle NEWMODES.F includes the main subroutine NEWMODES, and a library of
speciﬁc subroutines used for diﬀerent applications. The library can be modiﬁed or extended
by users to describe generalized modes for other applications. In all cases the calls to these
speciﬁc subroutines are made from NEWMODES. Thus the user has the capability to make
appropriate modiﬁcations or extensions and to implement these with the executable version
of WAMIT, without the necessity to access to the remaining source code for WAMIT.
The principal inputs to NEWMODES are the Cartesian coordinates (X,Y,Z) of a point
on the body surface, speciﬁed in the vector form X(1),X(2),X(3), and the corresponding
components of the unit normal vector XN at the same point. These inputs are provided
by the calling unit of WAMIT, and the user does not need to be concerned with providing
these inputs. The principal outputs, which the user must specify in an appropriate subroutine for each generalized mode, are (1) the symmetry index of the mode, (2) the normal
component of the displacement, and (3) the vertical component of the displacement. The
symmetry indices, which are deﬁned above in the Introduction, identify the symmetry of
each mode (assuming the body has one or two planes of symmetry) by assigning the values (1,2,3,...,6) to indicate the same symmetries as the corresponding rigid-body modes.
The normal component of the displacement, denoted VELH in NEWMODES, is computed
from the product of the displacement vector (U,V,W) and the normal vector XN. The
8–6

vertical component of the displacement, denoted ZDISP in NEWMODES, is identical to
the component W of the displacement vector (U,V,W).
Several other inputs are included in the argument list of NEWMODES to simplify
its use and increase its computational eﬃciency. These include the body index IBI, the
vector IBMOD which speciﬁes the starting index for each body in the global array of mode
indices, the patch/panel index IPP, the vector NEWMDS which speciﬁes the number of
generalized modes for each body, the integer IGENMDS speciﬁed in the conﬁguration ﬁle,
and an integer IFLAG which speciﬁes the required outputs from each call. These inputs
are explained in more detail below. Unit numbers for three ﬁles are also included in the
argument list to facilitate input of user-deﬁned data, and the output of error messages.
The body index IBI is useful for multiple-body computations, where diﬀerent generalized
modes may be speciﬁed on each body. If NBODY=1, the index IBI=1.
For the analysis of a single body, the mode index j is assigned consecutively as explained
after equation (8.1), with the ﬁrst generalized mode j = 7 and the last generalized mode j =
6+N EW M DS(1). For multiple bodies the arrays for each body are concatenated in succession. The vector IBMOD(1:NBODY) points to the last mode index j used for the preceding
body. Thus, in general, for body IBI, the ﬁrst generalized mode is j = IBMOD(IBI) + 7
and the last generalized mode is j = IBM OD(IBI) + 6 + N EW M DS(IBI). Thus the
pointer IBMOD is useful to assign generalized modes correctly when multiple bodies are
analyzed.
The patch/panel index IPP can be used to identify speciﬁc portions of the body. In the
low-order method (ILOWHI=0), IPP identiﬁes the local panel number where the input
point X,Y,Z is located (usually at the centroid of this panel). In this case, if NBODY>1,
IPP is the local panel number of the body IBI, as listed in the corresponding GDF ﬁle. In
the higher-order method (ILOWHI=1) IPP identiﬁes the global patch number where the
input point is located. (If NBODY=1 the local and global patch numbers are identical.
If NBODY>1 the global patch numbers are assigned consecutively for all bodies.) When
NBODY > 1 and a part of the bodies are input, WAMIT automatically reﬂects the surface.
The following convention should be followed to ﬁnd the appropriate patch numbers after
the reﬂections. For each body where one reﬂection occurs, the original patches IPP=1 to
NP are reﬂected to NP+1 to 2NP. When two reﬂections occurs, the ﬁrst reﬂection is about
x = 0. Then the second reﬂection occurs for the patches from IPP=1 to IPP=2NP+1,
4NP about y = 0. For example, if the ﬁrst quadrant of a body is deﬁned by one patch, the
next three patches will be in the order of the second, fourth and the third quadrant.
The subroutine MOONPOOL FS used for the test runs TEST17a/b is an example where
IPP is useful. In this case the generalized modes are deﬁned as nonzero only on the free
surface inside the moonpool, identiﬁed as patch 4 for this geometry (see Section 6.8). Thus
the use of the patch index is simpler and more eﬃcient than performing a test based on
the values of the input coordinates (X,Y,Z).
The input IGENMDS should be speciﬁed in the conﬁguration ﬁle with a nonzero value,
when NEWMODES is used to deﬁne generalized modes. Diﬀerent values of IGENMDS can
be useful in NEWMODES to identify diﬀerent subroutines of the library. This is illustrated
in the version supplied with WAMIT, where the integers 16, 17, 18 are used to identify the
8–7

corresponding test runs and associated subroutines.
The input IFLAG is generated internally by the calling routine, with three possible
values. Initially NEWMODES is called once with IFLAG=-1, to assign the array IMODE
specifying the symmetry index for each mode. In subsequent calls where only the normal
component of the displacement vector VELH is required, IFLAG=1. If both VELH and
the vertical component ZDISP are required (for computations of hydrostatic coeﬃcients)
IFLAG=2. In the higher-order method a large number of calls are made to NEWMODES
with IFLAG=1. If computational eﬃciency is important this should be considered in
modifying or extending the subroutines in NEWMODES.
The comments inserted in the NEWMODES ﬁle should be consulted for further details.
In order to use NEWMODES for any of the purposes described in this Chapter, the ﬁle
NEWMODES.DLL must be in the same directory as WAMIT.EXE.
Instructions for making new DLL ﬁles are included in Section 10.5.
In some cases it is useful to input data from a special input ﬁle, so that the subroutines
in NEWMODES can be used with diﬀerent values of relevant parameters. For example, the
subroutine WAVEMAKER inputs the depth of the pivot axis from a special ﬁle WAVEMAKER DEPTH.DAT, as illustrated in TEST23, and the subroutine HINGE MODES
inputs data from XHINGE.DAT, as illustrated in TEST24. The NEWMODES argument
IFILEDLL speciﬁes a ﬁle number (also known in FORTRAN as a unit number) which is
reserved for this purpose. Users who modify the DLL subroutines or make new DLL subroutines should follow the code in these two subroutines for guidance in inputting data from
an external ﬁle. In order to make it possible to copy this data ﬁle to the log ﬁle WAMITLOG.TXT, as is done for other input ﬁles, the ﬁle number of the log ﬁle IWAMLOG is
included in the argument list. Error messages which are generated in DLL subroutines can
be added to the error ﬁles ERRORP.LOG and ERRORF.LOG by using the ﬁle number
IERROR.

8.4

HYDROSTATICS

To evaluate the motions of a body including generalized modes, it is necessary to evaluate
the corresponding hydrostatic coeﬃcients. In general these are deﬁned by the matrix
(Reference 13, equation 2.17)
cij = ρg



Sb

nj (wi + zDi )dS

(8.11)

Here Di denotes the divergence of the vector (ui ,vi ,wi ), assumed to be continuous in the
vicinity of the body surface.
In cases where Di = 0 the hydrostatic matrix can be evaluated uniquely from the vertical
component wi . For these cases the simpliﬁed hydrostatic matrix
cij = ρg



Sb

8–8

nj wi dS

(8.12)

is computed and no further steps are required by the user. This computation is performed
in DEFMOD if that program is used, or internally in WAMIT if the DLL subroutine
NEWMODES is used.
In special applications where Di = 0, the hydrostatic coeﬃcients can be programmed
specially by modifying the code in the main program of DEFMOD. Alternatively, the
extra contribution from the last term in (8.11) can be included as an ‘external’ force, in
the stiﬀness matrix of the FRC ﬁle.
The hydrostatic coeﬃcients cij are output, as part of the complete hydrostatic matrix,
in the ﬁle out.hst, with the format indicated in Section 4.9. (Here out denotes the ﬁlename
of the .out ﬁle for the run.)

8.5

NBODY ANALYSIS

When it is desirable to do so, the NBODY and generalized mode analyzes can be combined.
An example where this might be eﬀective is if two separate bodies are in close proximity,
one or both of them are undergoing structural deﬂections, and there are no planes of
geometric symmetry for the ensemble of two bodies.
If NBODY>1, separate values of NEWMDS must be speciﬁed for each body. If IALTPOT=1 this is done in the conﬁguration ﬁle using a separate line for each body with the
syntax ‘NEWMDS(m) = n’. Here m is the body index and n is the number of new modes
for that body. For example if there are three bodies, and body 1 has 3 generalized modes,
body 2 has none, and body 3 has 2, two lines should be added to the conﬁguration ﬁle as
follows:
NEWMDS(1)=3
NEWMDS(3)=2
(The additional line NEWMDS(2)=0 may be added for clarity but is not required.)
If IALTPOT=2 the value of NEWMDS for each body is speciﬁed in the POT ﬁle, as
shown in Section 3.2.
In the approach described in Section 8.2, prior to the NBODY run of WAMIT, the
subroutine DEFMOD must be executed for each body, to prepare the corresponding MOD
ﬁle. This procedure is carried out separately for each body for which generalized modes
are speciﬁed, with an appropriate subroutine DEFINE corresponding to the generalized
modes of that body. The procedure for doing this is identical to that described in Section
8.2 for a single body.
In the approach described in Section 8.3, the subroutines in NEWMODES should be
organized in a logical manner so that the generalized modes for each body are deﬁned.
Usually this can be done most eﬀectively by using separate subroutines for each body, but
that is not necessary. The body index IBI is used to identify the body for each call.
The numbering sequence for these modes in the output ﬁles is with the new modes of
each body following the six rigid-body modes of the same body. Thus, in the example
above, the nine exciting-force coeﬃcients of body 1 are denoted by Xj (j = 1, 2, ..., 9, the
8–9

six conventional coeﬃcients of body 2 by (j = 10, 11, ..., 15, and the eight coeﬃcients of
body 3 by (j = 16, 17, ..., 23).

8–10

gdf.GDF

POTEN

pot.POT
gdf.PRE

FNAMES.WAM
gdf.GDF
pot.POT
frc.FRC

DEFMOD

CONFIG.WAM
(NEWMDS>0)

gdf.MOD

gdf.GDF
POTEN

pot.POT
pot.P2F

Figure 8.1: Flow chart showing the use of WAMIT and DEFMOD to deﬁne and analyze generalized modes
in the low-order method (ILOWHI=0). In this example IALTPOT=1, and the number of generalized modes
NEWMDS is speciﬁed in the conﬁguration ﬁle. On the ﬁrst run of WAMIT the centroid coordinates, area,
normal vector n, and cross-product x×n are output for each panel to the ﬁle gdf.PRE, for pre-processing
by the program DEFMOD. After modiﬁcation by the user to specify the desired modes, DEFMOD is run
with the input ﬁle gdf.PRE, to produce the output ﬁle gdf.MOD containing the normal velocity of each new
mode at the panel centroids, and also the hydrostatic coeﬃcients. After this pre-processing is completed,
WAMIT is run again in the normal manner, as in the ﬂow chart of Figure 1-1. Note that WAMIT reads
the GDF and POT input ﬁles on both runs, although the data in the POT ﬁle is only used on the second
run.

8–11

Chapter 9

USE OF IRREGULAR
FREQUENCY OPTION
WAMIT includes a method for removing the eﬀects of irregular frequencies on both the
velocity potential and the source strength. An outline of the method is given in Section
12.8, with more detailed information in References [8], [16] and [26]. In this method, the
computational domain includes the interior free surface of the body and it is necessary to
discretize it as well as the body surface. The integer switch IRR controls the removal of
the eﬀect of irregular frequencies. This parameter is required either in the POT ﬁle or in
the conﬁguration ﬁle, as explained in Sections 3.1, 3.2 and 3.9. Depending on the value
of IRR, the discretization of the interior free surface may be provided by the user or it
may be done automatically by the program. Explanation of the parameter IRR and the
discretization of the interior free surface are given in the following Sections.

9.1

INPUT PARAMETERS

The parameter IRR may be speciﬁed in the POT ﬁle as shown in Section 3.1 and 3.2, or
it may be assigned in the conﬁguration ﬁle as shown in Section 3.9. The deﬁnition of IRR
is as follows.
IRR is the integer used to specify whether the eﬀect of the irregular frequency is removed
or not. The values IRR=(0,1,2,3) are legal in the low-order method (ILOWHI=0). Only
the values IRR=(0,1) are legal in the higher-order method (ILOWHI=1).
IRR= 0: Do not remove the eﬀect of the irregular frequencies.
IRR= 1: Do remove the eﬀect of the irregular frequencies.
In the low-order method (ILOWHI=0) the user must discretize the free surface. The
coordinates of these panel vertices must be included in GDF ﬁle. (These must precede
dipole panels, as explained in Section 5.4. Otherwise, the locations of the free-surface
9–1

panels are arbitrary, within the GDF ﬁle.) The vertices of the free surface panels must
be numbered in the clockwise direction when the panel is viewed from inside the body or
in the counter-clockwise direction when the panel is viewed from above the free surface.
The whole, half or a quadrant of the interior free surface must be discretized in accordance
with the discretization of the body surface when there is geometric symmetry. The number
of equations, the parameter NEQN in the GDF ﬁle, should be the sum of the number of
panels on the body and the free surface.
In the higher-order method (ILOWHI=1) the user must represent the interior free surface
with one or more patches, in the same manner as for the body surface. This must be done
with the normal pointing downward, in the negative z−direction. In the context of the
right-hand-rule stated in Section 6.1, this is equivalent to requiring that the parametric
coordinates (u, v) for the free-surface patch must be deﬁned such that the positive normal
vector in the (u, v) plane points downward from the free surface into the body. The
subroutine CIRCCYL in GEOMXACT.F is an example where the interior free surface is
represented analytically as one extra patch. The locations of the free-surface patches are
arbitrary, within the GDF ﬁle.
IRR= 2: Do remove the eﬀect of the irregular frequencies.
The program projects the body panels onto the free surface to generate panels on the free
surface. IRR=2 should not be speciﬁed for a certain body shape such as a Tension Leg
Platform with pontoons. For these bodies, a vertical line intersects the body surface more
than once and thus the projection of two body panels overlap on the free surface. The
GDF ﬁle contains only the body panels (it is the same as the GDF ﬁle used with IRR= 0).
This option cannot be used with the higher-order method.
IRR= 3: Do remove the eﬀect of the irregular frequencies.
With IRR= 3, the program automatically generates a discretized interior free surface, and
stores the panel vertices in a special output ﬁle gdf.IDF. In this ﬁle, the vertices of the free
surface panels are appended to the data in the GDF ﬁle. An outline of this procedure is
described below in Section 9.2. Since this automatic discretization cannot accommodate
abnormal waterline shapes, the user should visualize the paneling on the interior free
surface using the IDF ﬁle, to check the quality of the interior free surface discretization.
This option cannot be used with the higher-order method.
When IRR≥ 1, the convergence rate of the iterative solver (ISOLVE=0) is reduced. It is
recommended to use either the block iterative solver or the direct solver.
The parameter ILOG in the POT file or the configuration file should be set
equal to 1 when IRR≥ 1. The special limits of zero or negative (infinite) wave
period should not be included in the POT file when IRR≥1 .
The generation of free-surface panels in the low-order method, or free-surface patches in the
higher-order method, can generally be facilitated by using MultiSurf. Various techniques
may be considered for this purpose, including the higher-order integration option described
in Section 6.7 and Appendix C.

9–2

Figure 9.1: Automatic discretizations on the interior free surface of a circular cylinder. The figure on the
left shows the result of the discretization algorithm used when ISOR=0. The figure on the right shows
the result of the special algorithm used when ISOR=1 to ensure continuity of the panels adjacent to the
waterline.

9.2

AUTOMATIC FREE-SURFACE DISCRETIZATION
(IRR=3)

The program discretizes the interior free surface based on i) the algorithm described in [8]
when ISOR=0 and ii) in a regular pattern when ISOR=1 as described below. Figure 9.1
shows these two discretizations on the interior free surface of a truncated cylinder. In i)
(ISOR=0) the interior free surface is discretized with triangular panels. In general, the
side of the triangular panels is similar to the average length of the waterline segments. due
to the O(1) aspect ratio of the panels, the number of panels on the interior free surface is
not so large as in i).
In ii) (ISOR=1) the connectivity between the free surface panels and the waterline segments is enforced. The necessity of this connectivity in the source formulation is explained
in [16]. The program ﬁrst ﬁnds the centroid of the waterplane area enclosed by a waterline
contour formed by a set of waterline segments. Then the two nodes of each waterline
segment are connected to the centroid forming triangles with vertices on the centroid and
on the two nodes. The triangles are discretized by quadrilateral panels (two sides of each
panel are on the lines from the centroid to the nodes) from the waterline segments to the
centroid with denser discretization toward the waterline. The last panel is a small triangle
which is used in the subsequent part of the program as a ﬂag indicating the end of the
connectivity. In this way, the connectivity is guaranteed between certain groups of panels
starting from quadrilateral panels contiguous to waterline segment. This option should
not be used for any body where a ray from the centroid of the waterplane area
intersects the waterline more than once. In that case the IRR=1 option should
be used.

9–3

When IRR=1 and ISOR=1, the interior free surface panels should be discretized in
a following manner. First, deﬁne a set of interior panels which are contiguous with the
waterline segments, one panel for each segment, and having aspect ratios similar to the
adjoining body panels. Continue this process recursively toward the centroid of the waterplane area. As the process continues moving inward toward the centroid, the panel size
can be increased and the continuity between adjacent panels is less important.
When IRR=1 and ISOR=0 it is recommended to discretize the interior free surface with
the panels having the following properties: i) Panels have O(1) aspect ratio. ii) The length
of the sides of the panels is similar to the waterline segments.
When the option IRR=3 is used, special care is required in preparing the GDF ﬁle
to ensure that there are no signiﬁcant gaps between the vertices of adjacent panels at the
waterline. In addition it is desirable to avoid panels at the waterline which are of extremely
small dimensions. These requirements are best understood in the context of the procedure
used by the program to deﬁne the waterline contour from the data in the GDF ﬁle.
As the ﬁrst step for the automatic discretization of the free surface, the program searches
for and identiﬁes all of the waterline segments between the panel vertices which are in
the plane of the free surface or within a speciﬁed small distance of this plane. If the
nondimensional absolute value of the vertical coordinate is less than TOL=1.E-5 the vertex
is deﬁned to be in the plane of the free surface. (This parameter TOL can be modiﬁed in
the source code, in the subroutine WTLPN of the ﬁle ‘getip.f’.)
To avoid including very small segments, which would produce a nonuniform discretization, waterline segments are neglected if their lengths are smaller than a speciﬁed minimum
length. The minimum length for this purpose is the product of another prescribed tolerance
(TOL=10E-3, deﬁned in subroutine WLSORT of the same ﬁle) times the average length of
the waterline segments. (In computing this average length, segments which are very small
are neglected.) The program then rearranges the waterline segments in consecutive order,
based on testing the distance between the end points of the segments. If this distance is less
than the product of TOL times the average length, the program assumes the two segments
are contiguous. When there is a gap larger than this product, the program outputs the
error message ‘Waterline panels do not form a closed contour’.
Since waterline segments with a nondimensional length less than TOL are neglected,
it is possible for this error message to be output even when the actual gap is less than
TOL, due to the simultaneous occurrence of one or more gaps and/or small panels. More
speciﬁcally, the error message is output when the cumulative length1 of successive small
waterline segments is larger than the prescribed tolerance for the gap.
When this error message is encountered, the user is advised to re-discretize the body
surface, to avoid panels with very small waterline segments and/or large gaps between
adjacent panels, or alternatively to use the IRR=1 option which requires the user to discretize the free surface. It is also possible to overcome this problem on an ad hoc basis by
modifying the parameter TOL in the subroutine WLSORT of the ﬁle ‘genﬁ.f’. Modiﬁcation
of TOL is potentially dangerous, if there are relatively short waterline segments.
1 The

cumulative length is defined here as the length of the vector sum of the waterline segments

9–4

The parameter SCALEH is deﬁned in WAMIT with the value 1.4. This parameter is
used when IRR=3 and ISOR=0, to set the typical length ratio between the sides of the
triangular panels on the free surface and the side of body panels.

9–5

Chapter 10

SPECIAL TOPICS
10.1

ERROR MESSAGES AND LOG FILES

Numerous checks are made in WAMIT for consistency of the data in the three input ﬁles.
Appropriate error messages are displayed on the monitor to assist in correcting erroneous
inputs. Output ﬁles containing warning and error messages are created after each execution
of the subprograms POTEN and FORCE. errorp.log contains messages from POTEN
and errorf.log from FORCE. The existing .LOG ﬁle, in the directory where the program
runs, is overwritten with every new run. When the program runs successfully without any
warning or error, the .LOG ﬁle contains two lines: a header line including the date and
time when the program starts to run and a line indicating the completion of the run.
Error messages are associated with problems where the program execution is halted.
Warning messages indicate that a possible error may occur, but under certain circumstances
the results may be correct. (Examples include failure of the convergence tests for various
numerical integrations, which sometimes result from inappropriate choices of characteristic
length scales or of overly conservative convergence tolerances. Another example is in the
case of diﬀraction by a body with one or two planes of symmetry, where it is possible to
compute the exciting forces at certain heading angles without solving for all components of
the diﬀraction potential; in this case the warning message states that the solution is nonphysical, whereas at some heading angles the exciting forces will be correctly evaluated.)
Error and warning messages generated during execution of the POTEN sub-program
are output to the ﬁle errorp.log, and messages generated during execution of the FORCE
sub-program are output to the ﬁle errorf.log. The same outputs are displayed on the
monitor. Since some of these messages may be lost on the monitor due to scrolling of other
outputs, a special warning message is generated at the end of the run to alert users when
signiﬁcant messages are contained in these two ﬁles.
Starting in V6.2, a new ﬁle wamitlog.txt is output during the run. This ﬁle is intended
to provide an archival record of the run. The ﬁle includes the starting and ending time and
date for each sub-program, copies of the principal input ﬁles, and copies of the outputs in
the ﬁles errorp.log and errorf.log. (Since the GDF input ﬁles are relatively long in the
10–1

low-order method, and also in the higher-order method when the geometry is deﬁned by
low-order panels or B-splines (ILOWHI=0 or 1, respectively), in these cases only the ﬁrst
10 lines of the GDF ﬁle are copied to the ﬁle wamitlog.txt. Another point to note is that
the maximum width of lines of data is truncated to 80 characters in wamitlog.txt. The
existing wamitlog.txt ﬁle, in the directory where the program runs, is overwritten with
every new run.
Additional data is included in the ﬁle wamitlog.txt as explained below.
When a warning message occurs indicating that the ‘Number of subdivisions exceeds
MAXSQR’ for the Rankine integration over a higher-order panel, the Cartesian coordinates
of the ﬁeld point and source point are output to wamitlog.txt so that the user can more
easily check if there is a singularity or inconsistency in the geometry deﬁnition in the
vicinity of these points.
When the input data in the Force Control Files for one or more bodies are in the
Alternative form 1, as deﬁned in Section 3.3, the nondimensional inertia matrix for each
body is included in the ﬁle wamitlog.txt. This is particularly useful when the analysis
of a body is ﬁrst performed using Alternative form 1, and then changed for subsequent
extensions to Alternative form 2, for example when external damping is imposed on an
otherwise freely ﬂoating body. The normalizing factors for the nondimensional inertia
matrix are the products of the ﬂuid density, and appropriate powers of the characteristic
length parameter ULEN. In preparing a force control ﬁle for Alternative form 2, as deﬁned
in Section 3.4, these normalizing factors must be included in the inputs EXMASS when
these are derived from the nondimensional inertia matrix.

10.2

RESERVED FILE NAMES

To avoid conﬂicting ﬁlenames, users are advised to reserve the extensions GDF, POT, FRC,
SPL, P2F, OUT, PNL, FPT, PRE, MOD, HST, CSF, CSP, 1, 2, 3, 4, 5p, 5vx, 5vy, 5vz,
6, 7x, 7y, 7z, 8, 9, and 9c for WAMIT input and output. Other reserved ﬁlenames include
CONFIG.WAM, FNAMES.WAM, ERRORP.LOG, ERRORF.LOG, WAMITLOG.TXT,
SCRATCH* (here *=A,B,C,...,O), as well as WAMIT.EXE, DEFMOD.FOR, DEFMOD.EXE,
and for V6PC, the DLL ﬁles GEOMXACT.F and NEWMODES.F. Source-code users can
modify the extensions by editing the appropriate assignments (in the source ﬁle modulesc.f,
module MAINC ARRAYS).

10–2

10.3

SCRATCH FILES

Two types of temporary scratch ﬁles are opened during execution of the subprogram
POTEN. One group are opened formally as scratch ﬁles using the FORTRAN convention, with ﬁlenames which are assigned by the compiler. The second group are opened
with the temporary ﬁlenames SCRATCHA, SCRATCHB, ..., SCRATCHO. All of these
ﬁles are deleted prior to the end of the run, but if execution is interrupted by the user (or
by power interruption to the system) some or all of the above scratch ﬁles may remain on
the hard disk. In the latter case the user is advised to delete these ﬁles manually.

10.4

MEMORY REQUIREMENTS AND NUMBER OF UNKNOWNS

The system memory requirements of WAMIT depend on the number of number of separate
solutions for the velocity potential (and source strength, if ISOR=1), and the number of
unknowns in each solution. This section is intended to provide the user with an understanding of these relationships, and of the manner in which WAMIT optimizes the solution
of a given problem.
The number of simultaneous equations NEQN, equal to the number of unknowns, is
used here to denote the dimension of the linear system solved for the determination of the
radiation and diﬀraction velocity potentials on the body surface, and the corresponding
source strengths when the option ISOR=1 is selected. The value of NEQN for each run is
listed in the header of the OUT output ﬁle.
In the low-order method (ILOWHI=0) the number of equations NEQN is equal to the
number of panels NPAN speciﬁed in the GDF input ﬁle(s), except when reﬂections about
body planes of symmetry are required as explained below. If dipole panels are input, as
explained in Section 5.4, NPAN is the sum of the conventional panels and dipole panels.
If the irregular-frequency option is used, additional panels are required on the interior
free surface inside of the body waterline, as described in Chapter 9, and the number of
equations is equal to the total number of panels including the interior free surface, regardless
of whether this is included in the GDF ﬁle and parameter NPAN explicitly (IRR=1), or
the interior free surface panels are added by the program (IRR=2 or IRR=3).
In the higher-order method (ILOWHI=1) NEQN depends on the number of patches,
panels, and the order of the B-splines:
NEQN =

NP


(NU(i) + KU(i) − 1) × (NV(i) + KV(i) − 1)

(10.1)

i=1

Here NP =NPATCH is the total number of patches, NU and NV are the numbers of panels
on each patch, and KU and KV are the orders of the B-splines used to represent the
solution. NU and NV are speciﬁed by the user in the CONFIG.WAM or SPL ﬁles, as
explained in Sections 6.9-10. KU and KV are speciﬁed in the CONFIG.WAM or SPL ﬁles,
or assigned the default value KU=KV=3. When reﬂections about planes of symmetry are
10–3

required, NPATCH is increased in the same manner as for NPAN.
If the parameter PANEL SIZE in the CONFIG.WAM ﬁle is positive, NU and NV are
determined internally in the program, as explained in Sections 3.7 and 6.9. In this case
it is not generally possible to determine the exact values assigned to NU and NV on each
patch. NU and NV are determined such that the estimated maximum physical dimension
of each panel is less than PANEL SIZE. The value of NEQN printed in the header of the
OUT ﬁle can be used to verify estimates of NU and NV in this case.
When planes of geometric symmetry (x = 0 and/or y = 0 of the body coordinate
system) do not coincide with the X = 0 and/or Y = 0 planes of the global coordinate
system, WAMIT assumes no hydrodynamic symmetry with respect to those planes. In
these cases the program reﬂects about the corresponding planes and increases the number
of panels accordingly. The other case where planes of hydrodynamic symmetry are not
utilized is the analysis of multiple interacting bodies (Chapter 7). In this case the number
of unknowns NEQN is the total number of panels required to describe the entire bodies,
and NLHS=1. The following list summaries the cases where the number of unknowns is
increased due to reﬂection:
1. When planes of geometric symmetry (x = 0 and/or y = 0) of the body coordinate
system do not coincide with the X = 0 and/or Y = 0 planes of the global coordinate system, due to nonzero values of the input parameters XBODY(1), XBODY(2),
XBODY(4) in the POT ﬁle or conﬁg.wam ﬁle. In this case the program assumes that
there are no planes of hydrodynamic symmetry, and the body geometry is reﬂected
about its speciﬁed planes of geometric symmetry.
2. In the multiple-body analysis described in Chapter 7 (NBODY Option), the same
procedure applies as in (1) above. (In some special cases the conﬁguration of multiple
bodies may be symmetrical about one or two global planes, but the program does
not check for that possibility. In such cases more economical computations can be
performed using generalized modes.)
3. In the analysis of a body near vertical walls described in Chapter 5.3, the same
procedure applies as in (1) above.
‘Total number of panels’ refers to the number of panels used to represent the entire
body surface. WAMIT takes into account ﬂow symmetries in setting up the linear systems, therefore the number of unknowns and total number of panels are diﬀerent if body
geometry symmetry planes are present. If 0, 1, or 2 planes of symmetry are speciﬁed, the
total number of panels is equal to NPAN, 2×NPAN, or 4×NPAN, respectively. Since the
computational burden of solving the linear system of equations is at least proportional to
NEQN2, a substantial reduction in computational eﬀort is achieved by imposing the planes
of symmetry when this is physically appropriate.
Provision is made in WAMIT to specify a subset of modes to be analyzed separately,
thus reducing the memory required and the run time. If the user anticipates the analysis
of more than one mode it is more eﬃcient to only once, for all modes of interest.

10–4

Since the issue of hydrodynamic symmetry is so important, it should be emphasized
that the separate analysis of symmetric and antisymmetric modes of motion applies not
only to the obvious cases of radiation modes, such as surge, sway, and heave, but also
to the more complex solution of the diﬀraction problem, even in oblique waves. This is
achieved in WAMIT by decomposing the complete diﬀraction (or scattering) solution as
the sum of four separate components that are respectively even or odd functions of the
horizontal coordinates. (Physically these can be interpreted as the solutions of problems
where standing waves are incident upon the body.)
In considering memory requirements a distinction must be made between storage in
RAM and on the hard disk. RAM requirements are aﬀected by NEQN, and are diﬀerent
in the low-order and higher-order methods. Usually NEQN can be signiﬁcantly smaller
when the higher-order method is used, compared to the analysis of the same problem with
the low-order method. But when ILOWHI=0, RAM can be used only for arrays which
are linear in NEQN and thus it is possible to analyze structures with a large number of
panels and unknowns, and to analyze simultaneously the diﬀerent radiation and diﬀraction
solutions of interest. When ILOWHI=1, one complex matrix of dimension NEQN, requiring
8*NEQN2 bytes, is stored in RAM. All other arrays stored in RAM are linear in NEQN.
The most important parameter which aﬀects the space required on the hard disk is
the number of equations, and unknowns, NEQN. For large values of NEQN the required
amount of scratch storage on the hard disk is proportional to NEQN2. If the low-order
method is used (ILOWHI=0), the number of bytes required for these arrays on the hard
disk can be estimated from the equation
[(4 × NLHS) × (1 + 4 × ISOR) × (3 + ILOG) + 2]NEQN2 − 8[min(MAXSCR, NEQN)]2
where min(MAXSCR,NEQN) denotes the minimum of these two parameters. If the higherorder method is used (ILOWHI=1), the number of bytes can be estimated from the equation
[(4 × NLHS) × (3 + ILOG)]NEQN2
In these equations the parameter NLHS is the number of left-hand-sides appropriate to
the analysis. If all modes of motion are studied simultaneously, for a body with 0, 1, or 2
planes of symmetry speciﬁed, NLHS is equal to 1, 2, or 4, respectively. As an example, the
truncated vertical cylinder described in the Chapter 5, which has 2 planes of symmetry and
a total of 1024 panels, requires about 3.4 megabytes of scratch storage on the hard disk to
analyze all modes of motion simultaneously, i.e. for the run described in the Chapter 5.
If the storage requirements of a run exceed the available disk space a system error will
be encountered; in this event the user should either increase the available disk space or
reduce the number of panels or solutions.
Subroutine ITRCC (the iterative solver) reads matrix elements in each iteration step
from the hard-disk. The run time in this subroutine can be reduced substantially by storing
some or all of elements in available RAM. The parameter MAXSCR deﬁnes the dimension of
a square sub-array which can be stored temporarily in available RAM. Since the coeﬃcients
10–5

of this array are complex, the corresponding storage requirement in RAM is 8*MAXSCR2
bytes, for systems which use 8 bytes for a single-precision complex number. Thus the
parameter MAXSCR should be determined initially by estimating the size of excess RAM,
after the program is loaded, and setting the largest integer value of MAXSCR such that
8*MAXSCR2 does not exceed the excess RAM which is available. If a value of MAXSCR
is speciﬁed which is larger than NEQN, MAXSCR is reduced by the program at run time,
and set equal to NEQN.
If the option ISOLVE= 1 is selected, to utilize the direct solver for the linear system
of equations, the entire left-hand-side matrix must be stored in RAM. This can only be
achieved if MAXSCR≥NEQN. Otherwise, if ISOLVE= 1, an appropriate error message is
generated at run time.
If the option ISOLVE> 1 is selected, to utilize the block iterative solver, the diagonal
block matrices must be stored in RAM, each block at a time for the local LU decomposition. This can only be achieved if MAXSCR is equal to or greater than the dimension
of the diagonal blocks. Otherwise the size of the diagonal blocks is reduced to MAXSCR
internally.

10.5

MODIFYING DLL FILES

The ﬁles GEOMXACT.F and NEWMODES.F can be modiﬁed by users following the
instructions in Sections 6.8 and 8.3. This makes it possible for users of the PC-executable
code to develop special subroutines for the deﬁnitions of the body geometry and generalized
modes, respectively, and to link these subroutines with WAMIT at runtime. Users of the
source code have two alternative options: (1) maintain these two ﬁles separately from the
remainder of the code, and use the DLL option if it is available with their compiler, or (2)
compile and link the two source ﬁles directly with the remainder of the code.
The following procedure should be followed by PC-executable users to make a modiﬁed
version of GEOMXACT.DLL, using Compaq Visual Fortran. For NEWMODES.DLL the
procedure is identical.
• Open a new project ‘geomxact’ as a Fortran Dynamic Link Library
• Add geomxact.f to the project
• Build a release version of geomxact.dll
• Copy the new version of geomxact.dll to the working directory for WAMIT
It may be possible to use other FORTRAN compilers to make the DLL ﬁle, but certain
conventions in calling subroutines must be consistent with those of Compaq Visual Fortran.
Further information is provided in [23], Chapters 8 and 18.

10–6

10.6

AUXILIARY OUTPUT FILES FOR THE GEOMETRY

Version 6.2 includes options to generate three auxiliary geometry output ﬁles, with the
names gdf PAN.DAT, gdf PAT.DAT, and gdf LOW.GDF. (Here gdf is the ﬁlename of
the GDF input ﬁle. For a run with NBODY>1 the GDF ﬁlename for the ﬁrst body (N=1)
is used.) These ﬁles contain the Cartesian coordinates of panels or patches in formats
suitable for perspective plotting, and the coordinates of new low-order panels which are
derived from the input geometry. In all cases the coordinates are dimensional, and deﬁned
in the same units as speciﬁed in the input GDF ﬁle(s).
The data in gdf LOW.GDF has the same deﬁnitions and format as a conventional loworder GDF ﬁle (Section 5.1). The coordinates of the panel vertices are deﬁned with respect
to the body coordinate system, corresponding to the original GDF inputs. If NBODY>1
the ﬁle gdf LOW.GDF represents the body identiﬁed as N = 1. If the original body panels
are reﬂected by the program, the ﬁle gdf LOW.GDF will include subdivided panels for the
reﬂected body. (This will occur if NBODY>1, if walls are present, or if the body is not
symmetric with respect to the global coordinate system.) If ILOWGDF=1 and the body
is reﬂected by the program, gdf LOW.GDF will contain the original body panels (without
subdivision) plus their images about the reﬂected planes of symmetry.
The data in gdf PAN.DAT and gdf PAT.DAT are deﬁned with respect to the global
coordinate system. In a WAMIT run with NBODY>1 the data for all of the bodies are
included. The ﬁgures in Sections A5 and A13 illustrate this feature.
These auxiliary ﬁles are described separately below for the low-order and higher-order
methods.
In the low-order method (ILOWHI=0) the vertex coordinates of the body panels are
stored in the output ﬁle gdf PAN.DAT in a format suitable for input to plotting programs
such as Tecplot. This facilitates the use of perspective plots to illustrate and check the
GDF inputs. If (ILOWHI=0) the ﬁle gdf PAN.DAT is output in the ﬁnite-element format
FEPOINT, as speciﬁed for use with the Tecplot program. Examples of these plots are
included in the Appendix for each low-order test run (Sections A1-A9). The integer parameter IPLTDAT in the ﬁle CONFIG.WAM is used to specify whether or not to generate
this output ﬁle. In the default case (IPLTDAT=0) no ﬁle is generated. If (IPLTDAT>0)
the ﬁle is generated.
In the low-order method (ILOWHI=0) the optional output ﬁle gdf LOW.GDF is controlled by the integer parameter ILOWGDF in the ﬁle CONFIG.WAM. If ILOWGDF> 0
the output ﬁle gdf LOW.GDF is generated, with all of the original panels subdivided into
ILOWGDF×ILOWGDF sub-divisions. The ﬁrst three lines are copied from the GDF input ﬁle. The total number of sub-divided panels is included on line 4. This option can be
used to increase the number of panels, and hence to increase the accuracy of the solution
for the potential or source strength. However this subdivision scheme does not increase
the accuracy of the geometric representation of the body, since the subdivided panels are
coplanar with the original panels. Only one body can be subdivided in this manner.
In the higher-order method (ILOWHI=1) the optional output ﬁles gdf PAT.DAT and
gdf PAN.DAT specify the vertex coordinates of both the patches and panels, as deﬁned in
10–7

Chapter 6. If (ILOWHI=1) these ﬁles are in the ordered-list format POINT, as speciﬁed
for use with the Tecplot program. Examples of these plots are included in the Appendix
for each higher-order test run (Sections A11-A19). The integer parameter IPLTDAT in
the ﬁle CONFIG.WAM is used to specify whether or not to generate these output ﬁles.
In the default case (IPLTDAT=0) no ﬁles are generated. If (IPLTDAT>0) the ﬁles are
generated. The number of panel subdivisions on each patch is determined by the parameters NU and NV in the SPL ﬁle, as explained in Section 6.11. If IPLTDAT=1 the data
ﬁle gdf PAT.DAT contains only the four vertices of each patch, and the ﬁle gdf PAN.DAT
contains only the four vertices of each panel. If IPLTDAT>1, each element is subdivided
into IPLTDAT×IPLTDAT sub-elements. Subdivision of the elements is useful when perspective plots are constructed for bodies with curved boundaries of the patches and panels.
When the curvature is large, IPLTDAT should be increased to give a more accurate plot.
(IPLTDAT=5 is used for the plots shown in Sections A11-A19.)
In the higher-order method (ILOWHI=1) the optional output ﬁle gdf LOW.GDF is controlled by the integer parameter ILOWGDF in the ﬁle CONFIG.WAM. If ILOWGDF> 0 a
low-order GDF ﬁle is generated, using the panel vertices of the higher-order geometry with
ILOWGDF×ILOWGDF sub-divisions. The ﬁrst three lines are copied from the higherorder GDF input ﬁle. The total number of sub-divided panels is included on line 4. This
option can be used to generate low-order GDF ﬁles for any of the geometries which can be
input to the higher-order method, including geometries represented by a small number of
ﬂat patches (Section 6.5), B-splines (Section 6.6), and geometries which are deﬁned in the
subroutine GEOMXACT (Section 6.8). In each case the number of low-order panels can
be increased by increasing the value of ILOWGDF. The coordinates of the panels are in
the same body-ﬁxed dimensional system as the original input data.
These optional ﬁles are generated in the POTEN subprogram, after reading the geometry input ﬁles and before looping over the wave periods. If NPER=0 these ﬁles can be
generated quickly, without the extra time required to solve for the potential and hydrodynamic parameters.

10–8

10.7

INTERNAL TANKS

Version 6.2 includes the options to analyze the linear hydrodynamic parameters for a ﬂuid
inside an oscillatory tank, or to analyze the coupled problem where one or more tanks are
placed within the interior of one or more bodies, including their dynamic coupling. Usually
the ﬂuid in each tank will be bounded above by a free surface, but in special cases a rigid
boundary surface can be placed above the ﬂuid to represent a tank entirely ﬁlled by ﬂuid.
The following discussion pertains to the situation where a free surface is present in each
tank. The free surface boundary condition in each tank is linearized in the same manner as
for the exterior free surface. Special attention is required near the eigenfrequencies of the
tanks, where nonlinear eﬀects are signiﬁcant. A two-dimensional study of this problem,
including nonlinear motions in the tank, is presented in Reference [26].
The tank geometry is deﬁned in the same manner as for the exterior surface of each body,
using either the low-order method with conventional panels, or the higher-order method.
In both cases it is essential that the normal vector points away from the ‘wet’ side of the
tank surface, as explained in Section 5.1 (ILOWHI=0) and Section 6.1 (ILOWHI=1). In
the context of a tank, this means that the normal vector points out of the tank and into the
interior space of the body. In the low-order method the tank panels are included with the
conventional body panels in the .GDF ﬁle. In the higher-order method the tank patches are
deﬁned in the same manner as the body surface, using one of the options listed in Sections
6.5 to 6.9. In both cases the tank panels or patches are identiﬁed by their starting and
ending indices, which must be listed in the .CFG ﬁle using the parameter name NPTANK,
as explained in Section 3.7.
Test 22 is an example, where the body is an FPSO containing two rectangular internal
tanks. Patches 1-7 represent the exterior surface of the FPSO, patches 8-11 represent tank
1, and patches 12-15 represent tank 2. (When IRR=1, this convention is modiﬁed, with
patches 8-10 used for the interior free surface of the FPSO and patches 11-18 used for the
tanks, as explained in the header of the GEOMXACT.DLL subroutine FPSOINT.) One
side of each tank is represented by four rectangular patches, with their vertex coordinates
included in the GDF ﬁle TEST22.GDF.
The panels or patches which represent each tank must be contiguous. Separate tanks
can be grouped together, or interspersed arbitrarily within the description of the exterior
surface. It is recommended to group all of the tanks together, either at the beginning
or at the end of the panels/patches which deﬁne the exterior surface. When IRR=1 and
ILOWHI=0, the interior free surface panels must be placed at the end of the GDF ﬁle, as
explained in Section 9.1, and thus if tanks are included they must be located before the
panels which deﬁne the interior free surface.
The vertical positions of tanks can be speciﬁed arbitrarily, in other words the free surface
of each tank can be deﬁned independently of the other tanks and the exterior free surface.
For each tank the free surface is deﬁned to coincide with the highest point of the panel
vertices or patch corners deﬁning that tank. In TEST 22, the free surface of one tank
coincides with the exterior free surface, i.e. the plane z = 0, and the free surface of the
other tank is elevated.
10–9

The solutions for the velocity potential (or source strength) are performed independently
for the exterior ﬂuid domain, and for each interior tank. Thus the mutual locations of these
surfaces are irrelevant. Small gaps between them do not cause problems, and they may even
coincide without introducing numerical diﬃculties. When irregular-frequency removal is
used (IRR>0), the entire interior free surface inside the body should be described, ignoring
any possible intersections with tanks. Thus the same interior free surface should be used
with or without the presence of tanks.
Two types of tank parameters must be included in the CFG ﬁle, as explained in Section
3.7:
NPTANK is an integer array used to specify the panel or patch indices of internal tanks.
The data in this array are in pairs, denoting the ﬁrst and last index for each tank. An
even number of indices must be included on each line. More than one line can be used
for multiple tanks, and/or multiple tanks can be deﬁned on the same line. If NBODY>1,
the body number(s) for each body containing tanks must be appended to the parameter
name. Only integer data are read for the array NPTANK, with spaces separating each
index. Other ASCII characters may be used to delimit the data in pairs.
RHOTANK is a real array used to specify the density of ﬂuid in internal tanks. The
density speciﬁed is relative to the density ρ of the ﬂuid in the external domain outside
the bodies, as deﬁned in Chapter 4. The data in the array RHOTANK must be input
in the same order as the data in the array NPTANK. Multiple lines of this parameter
may be used, with an arbitrary number of data on each line, but each line must begin
with ‘RHOTANK=’. The total number of tanks NTANKS is derived from the inputs
NPTANK in the POTEN run. If fewer than NTANKS values of RHOTANK are speciﬁed,
the remainder of the array is assigned the last non-negative value. Thus if the density is
the same for all tanks, only the ﬁrst value is required. Zero may be assigned, but negative
values of the density must not be assigned. RHOTANK is only used in the FORCE run,
and may be changed if separate FORCE runs are made using the same POTEN outputs.
Further details and examples are given in Section 10.8.
The following are equivalent formats for the required lines in the ﬁle TEST22.CFG:
NPTANK= (8-11) (12-15) (patch indices for two tanks)
RHOTANK= 0.6 0.6 (fluid densities for tanks one and two)
NPTANK= 8 11 (patch indices for tank one)
NPTANK= 12 15 (patch indices for tank two)
RHOTANK= 0.6
These examples illustrate the following rules:
1. Only integer data are recognized in NPTANK. Arbitrary ASCII characters other than
these can be used both as comments and to delimit the pairs of indices for each
tank, according to the user’s preferences. At least one blank space must be used to
10–10

separate pairs of indices. Comments appended to these lines must not include integer
characters.
2. The total number NTANKS of tanks to be included is determined by the number
of pairs of indices in NPTANK inputs, in this case NTANKS=2. The same number
of densities RHOTANK is required for the analysis, but it is not necessary to input
repeated values if all of the densities are the same (or if all of the densities after a
certain point are the same). The order of the tank densities must correspond to the
order of the index pairs in NPTANK.
3. The data in RHOTANK are real numbers. After specifying all NTANKS values,
arbitrary comments can be appended as in the ﬁrst example above, but if fewer than
RHOTANK real numbers are assigned the remainder of the line should be left blank,
as in the second example above.
When ﬁeld pressures and/or velocities are required for ﬁeld points inside the tanks
(Options 6-7) special inputs are required. In this case, when the ﬁeld point coordinates
XFIELD are input as explained in Section 3.3, the parameter ITANKFPT=1 must be
speciﬁed in the .cfg ﬁle, as explained in Section 3.7, and the format of the XFIELD inputs
in the .frc ﬁle must include the corresponding number of each tank, or zero for the exterior
domain. When all of the ﬁeld points are deﬁned by arrays, using the option described in
Section 3.10, the parameter ITANKFPT is not used and may be deleted from the .cfg ﬁle.
Inputs which relate to the body’s mass including VCG and the radii of gyration XPRDCT
(IALTFRC=1), or XCG, YCG, ZCG and the inertia matrix EXMASS (IALTFRC=2) refer
to the mass of the body alone, without the tanks (or with the tanks empty). The same
deﬁnitions apply to the outputs of these quantities in the header of the .out ﬁle and in
the log ﬁle wamitlog.txt. When IALTFRC=1 is used, the body mass is derived from the
displaced ﬂuid mass corresponding to the body volume, minus the ﬂuid mass in the tanks.1
The body volumes, center of buoyancy, and hydrostatic restoring coeﬃcients displayed
in the header of the .out ﬁle are calculated from the exterior wetted surface of the body, and
are not aﬀected by the tanks. The volumes, densities, values of ZTANK, and hydrostatic
restoring coeﬃcients for the tanks are listed separately after the corresponding data for
the hull. When the higher-order option (ILOWHI=1) is used, the header of the .out ﬁle
includes a list of all the patches, and the corresponding tank numbers ITANK for patches
which are deﬁned as interior tanks.
Hydrodynamic parameters which are physically relevant for the tanks alone (options
1,5,6,7,9) can be computed by inputting only the patches or panels for the tanks. In
this case IDIFF=-1 should be used, and the outputs for options 5,6,7,9 correspond to the
combination of radiation modes speciﬁed by the IMODE array, with unit amplitude of each
mode. The damping coeﬃcients for the tanks should be practically zero. From momentum
1 In WAMIT Versions prior to 6.312 the inertia matrix output in the log file wamitlog.txt when IALTRFRC=1 is not
reduced by the fluid mass in the tanks. The correct inertia matrix is used in the equations of motion and in all hydrodynamic
outputs which depend on the RAO’s. Users of earlier versions can correct the output in the log file by multipyling each
element of the inertia matrix by the factor 1 − (RHOTANK × VOLT )/VOLB ). Here VOLT is the sum of the volumes of all
tanks in the body and VOLB is the volume of the body. These quantities are all specified in the header of the .out file.

10–11

conservation the outputs from option 8 are zero, and the horizontal drift forces and vertical
drift moment from option 9 should be practically zero.
Internal tanks aﬀect the hydrostatic restoring coeﬃcients Cij , as described in Section
12.10. The complete hydrostatic matrix, which is output in the out.hst ﬁle, includes the
eﬀects of the tanks. Note that these outputs are not the same as the hydrostatic matrix
for the hull alone as described above, and displayed in the .out ﬁle. For example the
hydrostatic coeﬃcient C33 in the .out ﬁle is equal to the total area of the waterplane inside
the waterline of the body, whereas the coeﬃcient C33 in the .hst ﬁle is reduced by the
product of the free-surface area and density for each tank.
When both tanks and the external hull surfaces are included, hydrodynamic and hydrostatic parameters which are relevant for the hull alone, with no internal tanks, can be
computed by setting RHOTANK=0.0

10.8

RADIATED WAVES FROM WAVEMAKERS IN TANK
WALLS

Version 6.3 includes the option to analyze the waves generated by one or more wavemakers
in the plane(s) of symmetry x = 0 and/or y = 0. This conﬁguration corresponds to a wave
tank with rectangular walls in the planes of symmetry. The opposite walls of the wave
tank are assumed to have absorbing beaches, represented here by open domains extending
to inﬁnity. If the symmetry indices ISX=1, ISY=1 are input, with wavemakers in one
or both planes of symmetry, the walls extend to inﬁnity in both the x and y directions.
Alternatively, with ISX=1 and ISY=0 and wavemakers in the plane x = 0, or conversely,
one wall extends to both ± inﬁnity corresponding to a wave tank of inﬁnite width. In
all cases it is assumed that the wall(s) are planes of symmetry, and the ﬂuid motion is
symmetrical about these planes. Thus the solution for the velocity potential in the ﬂuid
domain can be represented by a distribution of sources of known strength, proportional to
the normal velocity of the wavemaker, and it is not necessary to solve the integral equation
for the velocity potential on the wavemakers. This saves considerable computational time,
and also avoids the singular solution that would otherwise occur for bodies of zero thickness
in the plane of symmetry.
The geometry of each wavemaker is deﬁned in the .gdf ﬁle. In the low-order method
(ILOWHI=0) a suﬃcient number of panels must be included on each wavemaker to ensure
a converged solution. In the higher-order method (ILOWHI=1) only one patch is required
on each wavemaker. If the wavemakers are rectangular (or quadrilateral), the higher-order
analysis can be carried out most easily using the option IGDEF=0, as explained in Section
6.5.
The parameter ISOLVE=-1 is used to indicate that the wavemakers are in planes of
symmetry, and that the solution of the integral equation should be skipped. Suitable generalized modes must be deﬁned, as explained in Chapter 8, to represent the normal velocity
of each wavemaker. IRAD=0 is recommended, with IMODE(1:6)=0, to avoid computing
the 6 rigid-body modes. IDIFF=-1 is required. The separate wavemaker elements are
10–12

considered to be part of one ‘body’, with appropriate generalized modes used to represent
the independent motion of each element, and NBODY=1. No other bodies can be present
within the ﬂuid domain.
The principal outputs are WAMIT options 6 and 7 (potential and ﬂuid velocity at
speciﬁed ﬁeld points). If the ﬁeld point is on the free surface the potential is equivalent
to the wave elevation. No other options are supported. If multiple wavemakers are run
together with separate modes for each wavemaker, the parameter INUMOPT6=1 should
be speciﬁed in the .cfg ﬁle to provide separate outputs for each mode. In that case only the
complex amplitude is output, with a separate pair of columns for each mode, as indicated
in Section 4.9. A post-processor should be used to combine these outputs, for arbitrary
combinations of simultaneous motion of all wavemakers.
In the NEWMODES DLL ﬁle the subroutine WAVEMAKER is included, to analyze one
or more paddle-type wavemaker segments, with pitching motions about a horizontal axis
in the lower edge of each wavemaker. The depth of this axis must be speciﬁed in the input
ﬁle WAVEMAKER DEPTH.DAT. In the subroutine the depth of the axis is the same
for all wavemaker elements. The number of wavemakers is arbitrary, but each separate
mode of motion corresponds to one patch (or panel) of the geometry, in the same order
as these are deﬁned in the .gdf ﬁle. In the low-order method this eﬀectively restricts the
use of the subroutine to only one panel per wavemaker. Thus it is strongly recommended
to use the higher-order method (ILOWHI=1) when using the subroutine WAVEMAKER.
NEWMDS=NPATCH must be speciﬁed in the pot or cfg ﬁle, with the same value as the
number of patches in the gdf ﬁle. To specify the use of WAVEMAKER the parameter
IGENMDS=21 is deﬁned in the CFG ﬁle.
TEST23, described in Appendix A23, illustrates the use of this option for a bank of
eight wavemakers along the wall x = 0, with symmetry about both x = 0 and y = 0.
The generated wave elevations for each wavemaker are evaluated over a square array of
8 × 8 = 64 ﬁeld points. The depth of the horizontal axis is speciﬁed by the parameter
ZHINGE=-2m in the input ﬁle WAVEMAKER DEPTH.DAT.

10–13

Chapter 11

VERSION 6.3S (Second-order
module)
This chapter is not completed and will be updated with the release of V6.3S.
WAMIT Version 6.3S designates a program which extends the capabilities of WAMIT
Version 6.3 to include the complete analysis of second-order hydrodynamic quantities, as in
the previous version of the second-order module, V5.3S. V6.3S has all of the capabilities of
V6.3, as described in Chapters 1-10. The extended capabilities of V6.3S include the sumand diﬀerence-frequency components of the second-order forces and moments (Quadratic
Transfer Functions, or QTF), the second-order hydrodynamic pressure on the body and in
the ﬂuid domain, the second-order wave elevation and the second-order Response Amplitude Operator (RAO), all in the presence of bichromatic and bidirectional waves and one
or more structures. The structures can be either freely ﬂoating, constrained or ﬁxed.
The list of major updates from V5.3S and V6.1S is as follows.
1. The higher-order method is implemented and the second-order solution can now be
evaluated either by the low-order method or higher-order method. All of the geometric
input formats described in Chapters 5 and 6 can be used for the second-order solution.
When the higher-order method is speciﬁed (ILOWHI=1), the second-order solution is
represented by B-splines in the same manner as for the linear solution. But the forcing
on the body and the free surface are integrated in a piecewise manner as in the low-order
method. Using this hybrid approach, the continuous second-order solution is obtained with
eﬃcient evaluation of the integral of the forcing.
2. An option for automatic free-surface discretization is implemented. The discretization is
made in an optimum manner based on the geometric data in GDF ﬁles. This simpliﬁes the
use of the second-order extension, particularly for the analysis of multi-body interactions.
3. V6.3S is written in Fortran 90/95. The parameter statements in the earlier versions are
removed and arrays are allocated dynamically at runtime.
4. The second-order hydrodynamic pressure can be output at user-speciﬁed points in the
same manner as the linear pressure.

11–1

5. The second-order quadratic pressure force acting along the waterline is output along
with the body pressure. This output may be useful when the pressure force is integrated
locally to evaluate the cross-sectional force/moment of structures. The geometric data of
the waterline segments are output in the PNL ﬁle.
The second-order output may not be available in some situations. They include i)
the generalized mode (NEWMDS) option is not allowed, ii) ﬁxed/free modes option is not
allowed, iii) walls are not allowed and iv) the zero-thickness structure is not allowed. When
the second-order RAO of constrained structures are evaluated, the external constraints may
not be accounted fully. More speciﬁcally, the quadratic interaction between the external
constraints and the linear body motion can not be included as an input to the program.
This Chapter describes the special features in input and output of V6.3S which diﬀer
from those of V6.3. It is recommended to review the previous chapters, if users are not
familiar with V6.3. In the following, the preparation of input ﬁles is explained followed by
the description of the output from V6.3S. The description of the theoretical background
of the second-order analysis can be found in References [9], [17], [26] and other documents
cited therein.
All input ﬁles prepared for V6.3 are required in order to run V6.3S. These ﬁles are CFG,
GDF, POT, FRC, optional FNAMES.WAM and, when ILOWHI=1, optional SPL. The
input ﬁles to V6.3 can be used without modiﬁcations, when only the linear output and
mean drift force are required.
To evaluate the second-order output, new parameters must be added to the CFG (or
CONFIG.WAM) and FRC ﬁles. In addition, one or two new input ﬁles must be prepared
as instructed below. The extensions of new ﬁles are .PT2 (Potential Control ﬁle 2) and
.FDF (Free surface Data File). The ﬁlenames of PT2 and FDF can be speciﬁed in the
optional FNAMES.WAM in order to avoid interactive input during batch runs.

11.1

THE CONFIGURATION FILE (CFG)

The parameter I2ND must be speciﬁed to evaluate the second-order output.
I2ND=0: Do not compute the second-order solution (0 is the default value).
I2ND=1: Compute the second-order solution for the sum- and diﬀerence-frequencies listed
in the PT2 ﬁle.
When I2ND=1, the dimension of the NOOUT array parameter in the CFG ﬁle must be
increased from 9 to 16. The last 7 options correspond to second-order output as described
in the next section. For example, by setting
NOOUT=1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1
all linear and second-order outputs would be printed in the OUT ﬁle except the linear and
the second-order pressure on the body surface (IOPTN(5) and IOPTN(13)).
When ILOWHI=0 and I2ND=1, the source formulation must be activated by setting
ISOR=1. Execution of the program will be aborted with an appropriate error message, if
11–2

ISOR=0.

11.2

THE FORCE CONTROL FILE (FRC)

If I2ND=1 in the CFG ﬁle, the dimension of the IOPTN array in FRC must be increased
from 9 to 16. The last seven elements control the second-order output. (If I2ND=0, only
the ﬁrst 9 elements would be read in the program. The trailing IOPTNs must be on the
same line with IOPTN(9). Otherwise the ﬁrst IOPTN on the next line would be read in
by the program as VCG.) As an example, the Alternative Form 1 of Force Control File is
shown below.
header
IOPTN(1)...IOPTN(9)IOPTN(10)IOPTN(11)IOPTN(12)...IOPTN(16)
VCG
XPRDCT(1,1) XPRDCT(1,2) XPRDCT(1,3)
XPRDCT(2,1) XPRDCT(2,2) XPRDCT(2,3)
XPRDCT(3,1) XPRDCT(3,2) XPRDCT(3,3)
NBETAH
BETAH(1) BETAH(2) ... BETAH(NBETAH)
NFIELD
XFIELD(1,1) XFIELD(2,1) XFIELD(3,1)
.
.
XFIELD(1,NFIELD) XFIELD(2,NFIELD) XFIELD(3,NFIELD)

The deﬁnitions of IOPTN(I), for I=1,...,9 are the same as those of V6.3 as listed in Section 3.3. The deﬁnitions of IOPTN(I), for I=10,...,16, and the associated numeric output
ﬁlenames are shown below.
IOPTN(I) = 0 : do not output parameters associated with option I=10,...,15.
IOPTN(I) = 1 : do output parameters associated with option I=10,...,15.
IOPTN(16) can take one of three value 0, 1, or 2:
IOPTN(16) = 0 : do not output the second-order RAO.
IOPTN(16) = 1 : do output the second-order RAO computed using the ‘indirect’ exciting
force.
IOPTN(16) = 2 : do output the second-order RAO computed using the ‘direct’ exciting
force.

11–3

Option Description

File name

10
Quadratic second-order forces
frc.10s, frc.10d
11
Total second-order forces by indirect method
frc.11s, frc.11d
12
Total second-order forces by direct method
frc.12s, frc.12d
13
Second-order hydrodynamic pressure on the body frc.13s, frc.13d
14
Second-order hydrodynamic pressure in the ﬂuid frc.14s, frc.14d
15
Second-order wave elevation
frc.15s, frc.15d
16
Second-order response amplitude operator
frc.16s, frc.16d
The last letters, ‘s’ and ‘d’, of the numeric ﬁlenames signify that the ﬁles contain the
sum-frequency output and diﬀerence-frequency output, respectively.
When IOPTN(13)=1, the second-order hydrodynamic pressure on the body surface is
output. If the body is surface piercing, the normalized hydrodynamic pressure force per
unit length at the midpoint of the waterline segments is output as well.
When IOPTN(14)=1 or IOPTN(15)=1, the number of ﬁeld points NFIELD and their
Cartesian coordinates XFIELD should be speciﬁed in the FRC ﬁle. The second-order wave
elevations are computed only for the points on the free surface. (A point is deﬁned to be
on the free surface if the vertical distance from the free surface, nondimensionalized by
ULEN, is less than 10−6 in absolute value.)
The total second-order forces of the output Options 11 and 12 are the sum of the two
force components. One component is the quadratic force of Option 10 and the other is
the second-order potential force. The latter obtained by the ‘indirect’ method is added to
the quadratic force in the output Option 11. In the output Option 12, the second-order
potential force is obtained by the ‘direct‘ method. The output options Option 11 and 12
are the same physical forces computed by diﬀerent approaches and they must converge to
each other with ﬁner discretization. The description of these two methods can be found in
References [9] and [17].
When two linear wave periods are the same, the diﬀerence frequency quadratic force of
Option 10 is the same as the mean drift force of Option 9 and Option 8. Since the secondorder potential force is trivial in this case, the output Options 11 and 12 are also the
same as the mean drift force. The second-order RAO of Option 16 is set to zero in this
case. Although the output options 11d, 12d and 16d are trivial, they are included in the
corresponding numerical output ﬁles. (Note that, in earlier versions of the second-order
module, the output was omitted from Options 11d, 12d and 16d, when two periods are
identical.)

11.3

THE POTENTIAL CONTROL FILE 2 (PT2)

It is not necessary to prepare the PT2 ﬁle, when all of the 2nd-order output options, from
10 to 16, are suppressed.
The PT2 ﬁle contains two set of parameters. One speciﬁes mode indices for which the
second-order output is calculated. The other speciﬁes period/wave heading pairs for which

11–4

the second-order output is calculated.
The input data in the Potential Control File 2 (PT2) are listed below:
header
IRAD2(1) IDIF2(1)
MODE2(1,1) MODE2(2,1) MODE2(3,1) MODE2(4,1) MODE2(5,1) MODE2(6,1)
IRAD2(2) IDIF2(2)
MODE2(1,2) MODE2(2,2) MODE2(3,2) MODE2(4,2) MODE2(5,2) MODE2(6,2)
.
.
IRAD2(NBODY) IDIF2(NBODY)
MODE2(1,NBODY) MODE2(2,NBODY) MODE2(3,NBODY) ... MODE2(6,NBODY)
IXSUM IXDIF
NSUMP
IPER(1) JPER(1) NBETA2(1)
IBETA(1,1) JBETA(1,1)
.
IBETA(1,NBETA2(1)) JBETA(1,NBETA2(1))
.
.
.
IPER(NSUMP) JPER(NSUMP) NBETA2(NSUMP)
IBETA(NSUMP,1) JBETA(NSUMP,1)
.
IBETA(NSUMP,NBETA2(NSUMP)) JBETA(NSUMP,NBETA2(NSUMP))
NDIFP
IPER(NSUMP+1) JPER(NSUMP+1) NBETA2(NSUMP+1)
IBETA(NSUMP+1,1) JBETA(NSUMP+1,1)
.
IBETA(NSUMP+1,NBETA2(NSUMP+1)) JBETA(NSUMP+1,NBETA2(NSUMP+1))
.
.
.
IPER(NPER2) JPER(NPER2) NBETA2(NPER2)
IBETA(NPER2,1) JBETA(NPER2,1)
.
IBETA(NPER2,NBETA2(NPER2)) JBETA(NPER2,NBETA2(NPER2))
IRAD2(N) is an index for the computation of the radiation solution of the N-th body at
the sum- or diﬀerence frequency.
IRAD2(N)=-1 Do not compute the second-order radiation solution.
IRAD2(N)= 0 Compute the second-order radiation solution for the modes speciﬁed by
MODE2.
IRAD2(N)=1: Compute the second-order radiation solution for all modes.
11–5

IDIF2(N) is an index for the computation of the second-order diﬀraction solution at the
sum- or diﬀerence- frequency.
IDIF2(N)=-1: Do not compute the second-order diﬀraction solution.
IDIF2(N)=0: Compute the second-order diﬀraction solution for the modes speciﬁed by
MODE2.
IDIF2(N)=1: Compute the second-order diﬀraction solution for all modes.
MODE2(1∼6) represent the modes for which the second-order radiation and diﬀraction
solutions are to be computed.
IXSUM, IXDIF are indices controlling the selection of periods and wave headings to
form pairs.
IXSUM=0: Do not compute the sum-frequency second-order solution.
IXSUM=1: Compute the sum-frequency second-order solution only for selected combinations of the wave periods and headings listed below.
IXSUM=2: Compute the sum-frequency second-order solution for all combinations of the
wave periods and headings (PER and BETA) listed in the POT ﬁle.
IXDIF=0: Do not compute the diﬀerence-frequency second-order solution.
IXDIF=1: Compute the diﬀerence-frequency second-order solution only for selected combinations of the wave periods and headings listed in the PT2 ﬁle.
IXDIF=2: Compute the diﬀerence-frequency second-order solution for all combinations of
the wave periods and headings (PER and BETA) listed in the POT ﬁle.
When IXSUM=1 or IXDIF=1, the number of combinations and indices of selected periods
and headings should be speciﬁed in the PT2 ﬁle through the following parameters:
NSUMP is the total number of combinations of wave periods for the sum-frequency
solution.
NDIFP is the total number of combinations of wave periods for the diﬀerence-frequency
solution.
The dimension NPER2, used in the last arrays in PT2, is the total number of period
combinations; NPER2 = NSUMP + NDIFP.
IPER, JPER are indices which identify the 1st-order periods selected from an array PER
(1∼ NPER).
NBETA2(I) is the total number of wave-heading pairs for the I-th period combination.
IBETA, JBETA are indices which identify the wave headings selected from an array
BETA (1∼NBETA).

11–6

11.4

FREE SURFACE DATA FILE (FDF)

It is not necessary to prepare an FDF ﬁle if IOPTN(10)=1 but all other second-order
output are suppressed.
The FDF ﬁle contains all requisite data to perform the integration of the quadratic forcing over the entire free surface exterior to the bodies. The integration is carried out by
numerical quadratures over the free surface close to the body. Away from the body, the
integration can be performed eﬃciently using asymptotic approximation of the forcing. For
this reason, the free surface is divided into two regions by a ’Partition Circle’. The radius
of the partition circle, which will be denoted by Rpart , should be suﬃciently large so that
the asymptotic expansion of the forcing is valid outside of this circle.
For better computational eﬃciency, the region inside Rpart may be divided into two subregions by an inner circle which is smaller than the partition circle. One subregion is the
free surface inside the inner circle and exterior to the bodies. The other is an annular
region between the inner and partition circles. The inner region is discretized into quadrilateral panels and the integration is performed piecewise for each panel. The intermediate
region may be subdivided into one or more annuli and the integration performed based
on the Gauss Chebyshev quadrature in the azimuthal direction and the Gauss Legendre
quadrature in the radial direction over each annulus. In the outer region, semi-analytic
integration is performed based on the asymptotic approximation of the integrand.
There are two alternative forms for the FDF ﬁle. In a simpler form, the free surface panels
in the inner region is generated automatically inside the program. In the other form, the
user must specify the horizontal coordinates of the panel vertices.
The data in the FDF ﬁle should be given relative to the global coordinate system.
11.4.1

ALTERNATIVE Form 1 FDF

When this form is used, the discretization of the inner region is carried out automatically
inside the program. The data in this form of FDF are
header
PARTR
NPF SCALE
NAL DELR NCIRE NGSP
PARTR is the dimensional radius of the inner circle with the center at the origin of the
global coordinate system. PARTR is measured in the same units as the length ULEN.
If PARTR is smaller than 1.25 times the maximum distance from the body surface to the
origin of the global coordinate system, the program issues a warning and increases PARTR
automatically.
NPF is an integer ﬂag specifying Alternative forms of the FDF ﬁle. In Alternative 1
format, NPF must be a negative integer.
11–7

SCALE is a real number used as a scaling factor of the size of the free surface panels relative
to the average length of the waterline panels on the body. For example, if SCALE=1.5,
the length scale of free surface panels will be 1.5 times the waterline panels.
The following four input parameters are relevant to the integration on the intermediate
region:
NAL is the total number of annuli subdividing the intermediate annular region. When
NAL=0, the parameters DELR, NCIRE and NGSP need not be speciﬁed.
DELR is the dimensional radial increment of each annulus (in the same units as ULEN).
NCIRE is the integer exponent of 2 such that the total number of nodes is equal to
2NCIRE + 1 for the azimuthal integration based on Gauss-Chebyshev quadrature.
NGSP is the number of nodes used for the Gauss-Legendre radial quadrature on each
annulus.
The Gauss quadrature over the intermediate region may be more eﬃcient than the integration using quadrilateral panels, especially when the partition circle is large compared to
the bodies. However, choosing appropriate values for the parameters NCIRE and NGSP
may not be simple. Unlike the piecewise integration using panels, inadequate values of
these parameters may produce the result with gross error. It is recommended to check the
dependence of the results to these parameters.
11.4.2

ALTERNATIVE Form 2 FDF

The data in this form of FDF should be input in the following form:
header
PARTR
NPF NTCL
NAL DELR NCIRE NGSP
VERX(1,1) VERX(2,1) VERX(3,1) VERX(4,1)
VERY(1,1) VERY(2,1) VERY(3,1) VERY(4,1)
.
.
.
VERX(1,NPF) VERX(2,NPF) VERX(3,NPF) VERX(4,NPF)
VERY(1,NPF) VERY(2,NPF) VERY(3,NPF) VERY(4,NPF)
PARTR: same as Alternative 1 FDF
NPF is the total number of free surface panels deﬁned by the user.
NTCL is the total number of segments on the inner circle
NAL : same as Alternative 1 FDF
DELR: same as Alternative 1 FDF
11–8

NCIRE: same as Alternative 1 FDF
NGSP: same as Alternative 1 FDF
VERX(K,I), K=1−4 is the dimensional x coordinate of the K-th vertex of the I-th panel.
VERY(K,I), K=1−4 is the dimensional y coordinate of the K-th vertex of the I-th panel.

For a single body, one quadrant of, half of, or the entire inner region must be discretized
according to the symmetry of the problem. Likewise, the number of segments NTCL
should be speciﬁed over the range of one quadrant, half, or the entire circle depending on
the symmetry.
For multiple bodies, the entire inner region must be discretized with respect to the global
coordinate system, regardless of the symmetry of the individual bodies. The inner circle
must enclose all of the bodies.
When there are multiple waterlines, the distance between each pair of waterlines must
be greater than twice the average length of waterline segments. This may require ﬁne
discretization of the body near the free surface when the gap between the waterlines are
small. In this case, larger value of SCALE must be used in order to avoid ﬁne discretization
inside inner circle.
11.4.3

PARTITION CIRCLE AND INNER CIRCLE

As noted above, the radius of the partition circle, Rpart =PARTR+NAL×DELR, should be
suﬃciently large so that the asymptotic expansion of the potentials is valid outside of the
circle. An appropriate estimation of the radius of the partition circle, Rpart , is Rpart ∼ O(h)
(h=water depth) for shallow water and Rpart ∼ O(λ) (λ= longest wavelength involved) for
deep water (h >> λ). The actual constant of proportionality Rpart /λ may have to be
substantially larger than one to achieve accuracy in deep water (see Reference [9] and
[12]).
If NAL>0, the radius of the inner circle PARTR should be determined with care. If PARTR
is too close to the body, the integration over the intermediate region may not be eﬃcient
as intended. The Gauss quadrature converges slowly for the integration of the inﬂuence of
Rankine sources on the free surface, at nearby points on the body.
If we denote the maximum distance to the body surface from the center of the circle as
Rbody , the following table shows the required number of nodes in the azimuthal integration
as PARTR varies. In the table, NCIR=2NCIRE + 1 is the number of nodes of the GaussChebyshev quadrature between [0, 2π]. The recommended procedure to determine PARTR
is i) ﬁrst to select the parameter NCIRE which would be dependent on the wavelengths of
the linear and second-order waves and ii) select Rpart from the table.

11–9

Rbody/PARTR

NCIRE NCIR

0.1
3
9
0.4
4
27
0.5
5
33
0.8
6
65
0.9
7
129
0.95
9
257
When ﬁeld quantities are evaluated on the free surface, such as the second-order wave
elevation, similar consideration applys. Thus PARTR should be suﬃciently large so that
the maximum radial distance to the ﬁeld points from the origin is substantially less than
PARTR. The above table can be used to estimate PARTR after replacing Rbody with the
maximum radial distance to the ﬁeld points.
11.4.4

RESTRICTION ON VERY LONG WAVES

When ki R, kj R or k2±R is less than BOUND in the DATA statement of subroutine FARFS.F
and RHSFFS.F, the program will continue to run but neglect the integration over the outer
region in the total second-order exciting forces. Here R is the partition radius and ki, kj
and k2± are wavenumbers associated with the frequencies ωi , ωj and ωi ± ωj . This case
may be encountered when a very long wavelength is involved in the diﬀerence-frequency
solution. If this occurs a warning message will appear in the .OUT ﬁle. To avoid this
condition the user may either increase the partition radius or increase the spread between
ﬁrst-order wave periods.
In V6.3S, BOUND is set to 10−3 . The parameter BOUND should be as small as possible,
depending on the range of the double-precision ﬂoating-point decimal exponent of the
computing system. The source code users can modify this value following the guideline
that when the exponent is in the range ± 64 × n, the recommended value of BOUND is
10−n .
11.4.5

APPROXIMATION WITHOUT FREE SURFACE FORCING

Users can compute the second-order solution without the evaluation of the free surface
integral. In light of the large computational eﬀort for the evaluation of this integral, this
option provides an eﬃcient way to compute an approximation to the complete second-order
solution. However, since the quality of the approximation depends on the problem, this
option must be used with discretion. It is not recommended to use this option for the
sum-frequency problem where the free-surface forcing is relative important. (Reference
[15] provides an example of computational results showing the relative importance of the
free-surface integral for a particular structure.)
To use this option, the parameters NPF, NTCL and NAL in the FDF ﬁle should be set
equal to zero as follows. (PARTR can be any number.)
header
11–10

PARTR
00
0

11.4.6

HIGHER-ORDER METHOD (ILOWHI=1)

With the low-order option speciﬁed (ILOWHI=0), the body and the free surface are represented by quadrilateral panels. The unknown potential on the body surface is assumed
constant over each panel. Similarly the linear or quadratic forcing on the body and/or free
surface is assumed constant on each panel and the integration of this forcing is performed
in a piecewise manner.
With the higher-order option speciﬁed (ILOWHI=1), the potential on the body surface
is represented by B-splines in both the linear and second-order analyses. In the linear
analysis, the forcing on the body surface is assumed continuous and the integration is
carried out by Gauss quadrature. In the second-order analysis, however, the integration of
the quadratic forcing on the body and the free surface is performed piecewise in the same
manner as for ILOWHI=0.
The body surface is virtually discretized into quadrilateral panels by dividing the space
between the knot vectors into smaller line segments. The number of line segments, L, is
L = K + 1, where K is the order of B-spline. If K is the same for all patches, the virtual
panels are the same as the low order geometric data in gdf low.GDF which is output by the
program, with the parameter ILOWGDF= K + 1. These panels can be visualized using a
program such as TECPLOT with the input data gdf pan.DAT, by setting IPLTDAT=K+1
in the CFG ﬁle.
11.4.7

VISUALIZING FDF

If Alternative Form 1 of FDF is used, the program identiﬁes the panels (virtual panels when
ILOWHI=1) on the waterline. Then the free surface panels are generated in a continuous
manner starting from the sides of body panels on the waterline by sharing the same vertices.
It is advised to arrange the free surface panels in a similar manner near the waterline, when
Alternative 2 format is used.
When the Alternative Form 1 of FDF is used, the program outputs the ﬁle f df new.FDF.
This is an Alternative Form 2 format of FDF containing the coordinates of the vertices
of the free surface panels generated by the program. For both Alternatives of the FDF
ﬁle, the program output f df fdf.dat contains the free surface panels in TECPLOT data
format. (f df denotes the original ﬁlename of Alternative 1 format FDF).

11–11

11.5

FNAMES.WAM

When FNAMES.WAM is used to list the input ﬁles, the latter should contain the PT2 and
FDF ﬁle names as well as CFG, GDF, POT, FRC as in the following example:
FNAMES.WAM:
test.cfg
test.gdf
test.pot
test.frc
test.pt2
test.fdf
If all second-order outputs are suppressed, pt2 and fdf can be omitted. If all second-order
outputs are suppressed except Option 10, fdf can be omitted.

11.6

DEFINITIONS OF OUTPUT

The non-dimensional deﬁnitions of the second-order output from V6.3S are listed in this
Section. Note that the second-order outputs satisfy corresponding symmetry relations.
For example, the wave excitation Quadratic Transfer Functions (QTF) at the sum- and
∗
diﬀerence-frequencies satisfy the relations Fij+ = Fji+ , and Fij− = Fji− , respectively. Here ∗
denotes the complex conjugate. (In the diﬀerence-frequency problem, ωi ≥ ωj is assumed.
Otherwise, the indices are interchanged automatically within the program.)
1. The second-order wave forces and moments at sum- and diﬀerence-frequencies are
deﬁned as
F+
ρgLAiAj

and

F̄ − =

F−
ρgLAi A∗j

M+
ρgL2Ai Aj

and

M̄ − =

M−
ρgL2Ai A∗j

F̄ + =
M̄ + =

where L=ULEN is the characteristic body length, and A, g and ρ represent the complex
ﬁrst-order incident-wave amplitude, gravitational acceleration and ﬂuid density, respectively.
2.1 The second-order hydrodynamic pressure at sum- and diﬀerence-frequencies is deﬁned
as
p̄+ =

p+
ρgAiAj /L

and

p̄− =

p−
ρgAi A∗j /L

2.2 The second-order hydrodynamic force per unit length acting on the waterline at sumand diﬀerence-frequencies is deﬁned as
11–12

f¯w+ =

fw+
ρgAiAj

f¯w− =

and

fw−
ρgAi A∗j

3. The second-order wave elevation at sum- and diﬀerence-frequencies is deﬁned as
η̄ + =

η+
Ai Aj /L

η̄ − =

and

η−
Ai A∗j /L

4. The second-order response amplitude operator at sum- and diﬀerence-frequencies is
deﬁned as
ξ¯k+ =

ξk+
Ai Aj /Ln

ξ̄k−

and

ξk−
=
Ai A∗j /Ln

where n = 1 for the translational modes k = 1, 2, 3 and n = 2 for the rotational modes
k = 4, 5, 6

11.7

NUMERIC OUTPUT FILES

All requested outputs are listed collectively in the .OUT ﬁle and in the numeric output
ﬁles. The data in the numeric output ﬁles are listed as follows:
OPTN.10s, OPTN.11s and OPTN.12s:
PER(i) PER(j) BETA(i) BETA(j)

I

MOD(F̄ij+ )

PHS(F̄ij+ ) Re(F̄ij+ )

Im(F̄ij+ )

OPTN.10d, OPTN.11d and OPTN.12d:
PER(i) PER(j) BETA(i) BETA(j)

I

MOD(F̄ij− )

PHS(F̄ij− ) Re(F̄ij− )

Im(F̄ij− )

OPTN.13S:
PER(i) PER(j)
(PER(i) PER(j)

BETA(i) BETA(j)
BETA(i) BETA(j)

M
M

K
K

MOD(p̄+ )
MOD(f¯w+ )

OPTN.13D:
PER(i) PER(j)
(PER(i) PER(j)

BETA(i) BETA(j)
BETA(i) BETA(j)

M
M

K
K

MOD(p̄− ) PHS(p̄− ) Re(p̄− ) Im(p̄− )
(MOD(f¯w− ) PHS(f¯w− ) Re(f¯w− ) Im(f¯w− ))

PHS(p̄+ ) Re(p̄+ )
PHS(f¯w+) Re(f¯w+ )

Im(p̄+ )
Im(f¯w+ ))

OPTN.14S:
PER(i) PER(j) BETA(i) BETA(j)

L

MOD(p̄+ )

PHS(p̄+ ) Re(p̄+ )

Im(p̄+ )

OPTN.14D:
PER(i) PER(j) BETA(i) BETA(j)

L

MOD(p̄− )

PHS(p̄− ) Re(p̄− )

Im(p̄− )

11–13

OPTN.15S:
PER(i) PER(j) BETA(i) BETA(j)

N

MOD(η̄ + ) PHS(η̄ + ) Re(η̄ + ) Im(η̄ + )

OPTN.15D:
PER(i) PER(j) BETA(i) BETA(j)

M

MOD(η̄ − ) PHS(η̄ − ) Re(η̄ − ) Im(η̄ − )

OPTN.16S:
PER(i) PER(j) BETA(i) BETA(j)

I

MOD(ξ¯+ )

PHS(ξ¯+) Re(ξ¯+ )

Im(ξ¯+ )

OPTN.16D:
PER(i) PER(j) BETA(i) BETA(j)

I

MOD(ξ¯− )

PHS(ξ¯−) Re(ξ¯− )

Im(ξ¯− )

Here I is mode index, M indicates the quadrant or half when the body has two or one
planes of symmetry, K is the index of the panels on the body, L is the index of the ﬁeld
points.
The numeric output ﬁles optn.13s and optn.13d contain the second-order hydrodynamic
pressure on the body surface. The points where the pressure is calculated are speciﬁed in
gdf.PNL (see Section 4.9 for the contents in PNL ﬁle). If Option 5 is speciﬁed, the linear
pressure is output for the same set of points.
For the surface-piercing bodies, the second-order pressure force acting on the waterline
is of interest. This will be referred to as the waterline pressure force and denoted by fw±.
When IPNLBPT=0, fw± is output to optn.13s and optn.13d for the surface-piercing bodies.
fw± is appended after the output of the pressure(p± ) on the body surface.
The PNL ﬁle is also extended accordingly and contains the Cartesian coordinates, the
length of waterline segments and normal vector corresponding to the points where fw± is
calculated. The new additional parameters in the PNL ﬁle are shown below.
gdf.PNL: M K XCT YCT ZCT LENGTH nx ny nz (r × n)x (r × n)y (r × n)z
where LENGTH is the length of the waterline segments. The deﬁnitions of other parameters are the same as those in Section 4.9.
Only the hydrodynamic pressure is output for Option 13 (also Option 5). It would
be straightforward to calculate the static pressure, however, using available output from
±
normalized in a same
the program. The second-order quadratic hydrostatic pressure, P̄sq
manner as the hydrodynamic pressure, can be calculated from the following expression.
1
±
= [(ξ4i ξ6±j + ξ4±j ξ6i )x + (ξ5i ξ6±j + ξ5±j ξ6i )y − (ξ4i ξ4±j + ξ6i ξ6±j )z]
P̄sq
4
where ξk is the linear RAO output in optn.4. The subscripts 4,5,6 correspond to the
roll, pitch and yaw modes, respectively. ξ − denotes the complex conjugate of ξ = ξ + . The
subscripts i and j denote the indices of the periods of the linear incident-wave pair. The
normalized position vector of the points on the body surface (x, y and z) can be obtained
from the data in the PNL ﬁle. Note that the position vector in PNL must be normalized
11–14

by ULEN(1) and converted into each body coordinate system before using in the above
expression.
The hydrostatic pressure due to the second-order RAO can be calculated from the
following expression where ξ ± is the second-order RAO output optn.16s and optn.16d.
±
P̄sp
= ξ3± + ξ4± × y − ξ5± × x

The linear static pressure can be calculated from the above expression after replacing
the second-order RAO with the linear RAO.
The body motion is taken into account in the evaluation of fw± and it includes the eﬀects
of both hydrodynamic and static pressure.

11.8

INSTALLATION OF V6.3S-PC Executable

V6.3S-PC Executable can be installed following the same procedure for the installation of
V6.3PC Executable as described in Chapter 2. An additional set of standard Test Runs
for the second-order analysis is provided in the directory TESTRUNS 2ND.

11–15

Chapter 12

THEORY
In this Chapter the theoretical basis for WAMIT is described. Further information can be
found in Reference [26], and in the references cited below.

12.1

THE BOUNDARY-VALUE PROBLEM

Figure 12.1 illustrates a three-dimensional body interacting with plane progressive waves
in water of ﬁnite water depth H. The objective of WAMIT is to evaluate the unsteady
hydrodynamic pressure, loads and motions of the body, as well as the induced pressure and
velocity in the ﬂuid domain. The free-surface and body-boundary conditions are linearized,
the ﬂow is assumed to be potential, free of separation or lifting eﬀects. A harmonic time
dependence is adopted.
z
y

O

β

x

H

Figure 12.1: Sketch defining the coordinates, fluid depth, and wave-heading angle.

12–1

The Cartesian coordinate system (x, y, z) deﬁned in Figure 12.1 is stationary relative
to the undisturbed position of the free surface and body. Its origin may lie on, above or
below the free surface as the particular application requires. The body geometry input to
WAMIT is deﬁned relative to that system. Here, the origin is located on the free surface
for the sake of simplicity.
The assumption of a potential ﬂow permits the deﬁnition of the ﬂow velocity as the
gradient of the velocity potential Φ satisfying the Laplace equation
∇2 Φ = 0

(12.1)

in the ﬂuid domain. The harmonic time dependence allows the deﬁnition of a complex
velocity potential ϕ, related to Φ by
Φ = Re(ϕeiωt ),

(12.2)

where Re denotes the real part, ω is the frequency of the incident wave and t is time.
The ensuing boundary-value problem will be expressed in terms of the complex velocity
potential ϕ, with the understanding that the product of all complex quantities with the
factor eiωt applies. The linearized form of the free-surface condition is
ϕz − Kϕ = 0,

z=0

(12.3)

where K = ω 2 /g, and g is the acceleration of gravity. The velocity potential of the incident
wave is deﬁned by
ϕ0 =

igA cosh[ν(z + H)] −iνx cos β−iνy sin β
,
e
ω
cosh νH

(12.4)

where the wavenumber ν is the real root of the dispersion relation
ω2
= ν tanh νH,
g

(12.5)

and β is the angle between the direction of propagation of the incident wave and the
positive x−axis as deﬁned in Figure 12.1.
The linearization of the problem permits the decomposition of the velocity potential ϕ
into the radiation and diﬀraction components
ϕ = ϕR + ϕD ,
ϕR = iω

6

j=1

ξ j ϕj ,

ϕD = ϕ0 + ϕ7 .

12–2

(12.6)
(12.7)
(12.8)

The constants ξj denote the complex amplitudes of the body oscillatory motion in its six
rigid-body degrees of freedom, and ϕj the corresponding unit-amplitude radiation potentials. The velocity potential ϕ7 represents the scattered disturbance of the incident wave by
the body ﬁxed at its undisturbed position. We will refer to the sum (12.8) as the diﬀraction
potential ϕD .
On the undisturbed position of the body boundary, the radiation and diﬀraction potentials are subject to the conditions,
ϕjn = nj ,

(12.9)

ϕDn = 0,

(12.10)

where (n1 , n2, n3 ) = n and (n4 , n5, n6) = x × n, x = (x, y, z). The unit vector n is normal
to the body boundary and points out of the ﬂuid domain.
The boundary value problem must be supplemented by a condition of outgoing waves
applied to the velocity potentials ϕj , j = 1, . . . , 7.

12.2

INTEGRAL EQUATIONS FOR THE VELOCITY POTENTIAL

In WAMIT the boundary value problems (12.1-10) are solved by using Green’s theorem to
derive integral equations for the radiation and diﬀraction velocity potentials on the body
boundary. The integral equation satisﬁed by the radiation velocity potentials ϕj on the
body boundary takes the form


2πϕj (x) +

Sb

ϕj (ξξ )


∂G(ξξ ; x)
dξξ =
nj G(ξξ ; x)dξξ .
∂nξ
Sb

(12.11)

where Sb denotes body wetted surface at calm water.
The corresponding equation for the total diﬀraction velocity potential ϕD is


2πϕD (x) +

Sb

ϕD (ξξ )

∂G(ξξ ; x)
dξξ = 4πϕ0 (x).
∂nξ

(12.12)

The diﬀraction potential may also be obtained from equation (12.8) after solving for
the scattered potential ϕ7 . The equation for the scattered velocity potential is


2πϕ7 (x) +


∂G(ξξ ; x)
∂ϕo (ξ)
G(ξξ ; x)dξξ .
ϕ7 (ξξ )
dξξ = −
Sb
∂nξ
Sb
∂n

(12.13)

From the computational point of view, equation (12.12) has some advantages over equation
(12.13) in terms of cpu time and the requirement of storage space, because of the relative
simplicity of the right-hand side.
12–3

The Green function G(x; ξ ) is referred to as the wave source potential. It is the velocity
potential at the point x due to a point source of strength −4π located at the point ξ . It
satisﬁes the free-surface and radiation conditions, and in inﬁnite water depth is deﬁned by
G(x; ξ ) =

1
1
2K  ∞
ek(z+ζ)
+ +
J0 (kR)
dk
r r
π 0
k −K

(12.14)

r 2 = (x − ξ)2 + (y − η)2 + (z − ζ)2

(12.15)

r 2 = (x − ξ)2 + (y − η)2 + (z + ζ)2 ,

(12.16)

where J0 (x) is the Bessel function of zero order. In ﬁnite depth, the Green function is
deﬁned by

G(x; ξ ) =

1
1
+  + 2
r r

 ∞
0

dk

(k + K) cosh k(z + H) cosh k(ζ + H) −kH
J0 (kR)
e
k sinhkH − K cosh kH

(r  )2 = (x − ξ)2 + (y − η)2 + (z + ζ + 2H)2 .

(12.17)

(12.18)

In both expressions (12.14) and (12.17) the Fourier k−integration is indented above the
pole on the real axis in order to enforce the radiation condition. Eﬃcient algorithms for the
evaluation of the inﬁnite and ﬁnite-depth wave-source potentials and their spatial derivatives, have been developed in [1] and [11]. These algorithms are coded in the subroutines
VGRN00 and HGRN00 which are included in WAMIT. (These subroutines supersede the
earlier versions of the subroutines FGRN89, HGRN89 and IGRN89 or VGRN94, HGRN94,
VGRN98 and HGRN98.)
Special attention must be given to the singular components of the Green function for
small values of r, r  and r . The source-like singularities 1/r, 1/r  and 1/r  and their normal
derivatives can be integrated analytically over a quadrilateral panel, as described in [2]. In
addition, the ascending series expansion of the wave source potential for small values of r 
(Ref. [1], eq.(5)), reveals the presence of the logarithmic singularity,
G(x; ξ ) =

1 1
+ − 2K log K(r  + |z + ζ|) + O(1),
r r

(12.19)

(The derivation of this result in [1] is for the inﬁnite-depth case, but it can be shown
from the analysis of the ﬁnite-depth case in the same reference that the same singularity
applies.) Provision has been made in WAMIT to permit the logarithmic singularity and
its derivatives to be integrated analytically in the solution of the integral equations when
the source and ﬁeld points are close to each other and to the free surface. Further details
are given in Section 12.4.

12–4

12.3

INTEGRAL EQUATIONS FOR THE SOURCE FORMULATION

In this Section a brief description is given of the source distribution method, which is used
to derive the ﬂuid velocity components on the body surface in the low-order method. The
velocity are required to evaluate the mean second-order pressure, from which the mean
drift forces and moments can be evaluated in the manner described in Section 12.7.
In source distribution method, the velocity potential is expressed by a distribution of
sources only
ϕ(x) =


Sb

σ(ξξ )G(x; ξ )dξξ

(12.20)

After discretizing the body boundary with plane panels with constant source strength on
each panel, the potential can be expressed by
ϕ(xi ) =

N

j=1

σ(xj )


Sj

G(xi ; ξ)dξξ

(12.21)

Denote the normal vector as n and the two tangential unit vectors as s and t on each
panel. The three components of the velocity are then given in the (n,s,t) coordinate system
as follows:
ϕn (xi ) = 2πσ(xi ) +
ϕs (xi ) =
ϕt(xi ) =

12.4

N

j=1
N

j=1

N

j=1

σ(xj )

σ(xj )
σ(xj )


Sj


Sj


Sj

Gn (xi ; ξ)dξξ

(12.22)

Gs (xi ; ξ )dξξ

(12.23)

Gt (xi ; ξ )dξξ

(12.24)

DISCRETIZATION OF THE INTEGRAL EQUATIONS
IN THE LOW-ORDER METHOD (ILOWHI=0)

The mean position of the body wetted surface is approximated by a collection of quadrilaterals. Each quadrilateral is deﬁned by four vertices, lying on the body surface. Their
Cartesian coordinates are input to WAMIT. They are numbered in the counter-clockwise
direction when the panel is viewed from the ﬂuid domain. Instructions on how to input
the vertex coordinates are given in Chapter 5.
In general the quadrilaterals deﬁned above are not plane, but if a suﬃciently ﬁne discretization is used for a boundary surface with continuous curvature, each element will
12–5

approach a plane surface. In this circumstance a plane approximation of the general
quadrilateral is deﬁned by the midpoints of each side, which always lie in the same plane.
Each panel is deﬁned by projecting the four vertices onto this plane. If the coordinates of
two adjacent vertices coincide, the quadrilateral panel reduces to a triangular panel.
For bodies of general shape, gaps may exist between panels. Experience suggests that
they do not signiﬁcantly aﬀect the accuracy of the velocity potential and the hydrodynamic
forces.
The radiation and diﬀraction velocity potentials are taken to be constant over each
panel. The discretization errors associated with the selection of plane panels and a piecewise
constant variation of the velocity potential are of the same order, if the integration of the
singular components of the wave source potential over the panels are carried out with
suﬃcient accuracy.
Based on this discretization, the continuous integral equations (12.11) and (12.12) can
be reduced to a set of linear simultaneous equations for the values of the velocity potentials
over the panels. For the radiation potentials we obtain
2πϕ(xi ) +

N


Dik ϕk =

k=1

N




Sik

k=1

∂ϕ
∂n



,

(12.25)

k

where i = 1, . . . , N , N being the number of panels. For the total diﬀraction potential
2πϕ(xi ) +

N

k=1

Dik ϕk = 4πϕ0 (xi ).

(12.26)

The matrices Dik and Sik are deﬁned by
Dik =
Sik =


sk


sk

∂G(ξξ , xi)
dξξ ,
∂nξ

(12.27)

G(ξξ , xi)dξξ ,

(12.28)

where sk denotes the surface of the k−th panel. The ‘collocation’ points xi , where the
integral equations are enforced, are located at the panel centroids.
The rest of this section describes the computation of the inﬂuence coeﬃcients Dik and
Sik .
If the index IDIAG=0 in the potential control ﬁle (see Section 3.1), the characteristic
length scale DIAG(2) for a given panel is equal to the square root of its area; if IDIAG=1,
DIAG(2) is equal to the length of the largest diagonal of the panel. The setting of the
index IDIAG applies to all the panels. The selection IDIAG=1 is more conservative, and
results in an increase in the number of pairs of panels for which the Rankine singularities
are integrated analytically. The Rankine and logarithmic singularities of the wave source
potential are subtracted and integrated analytically over the panels if the distance between
the centroids of the i−th and j−th panels is less than 10×DIAG(2). This comparison is
12–6

based on the maximum DIAG(2) of the pair of panels. Assuming that the collocation
point is located on the i−th panel, it is reﬂected about the free surface and the bottom
(if present). The distance of its reﬂections from the centroid of the j-th panel are denoted
by r  (above the free surface) and r  (below the bottom). The integrals of the 1/r  and
1/r  singularities are evaluated as for the 1/r singularity, using analytic integration over the
j−th panel, when these distances are less than 10×DIAG(2). When the index ILOG=1 the
logarithmic singularity is subtracted and integrated analytically along with 1/r  . When any
of the Rankine and logarithmic singularities is not subtracted and integrated analytically,
it is incorporated in the evaluation of the wave source potential by VGRN00.
The index ILOG, also set in the potential control ﬁle, controls the analytical integration
of the logarithmic singularity. When ILOG=1, the logarithmic singularity is subtracted
and integrated analytically for the same pairs of panels for which the Rankine singularity
1/r  is integrated analytically. When ILOG=0 the singularity is included in the evaluation
of the wave-source potential and integrated by quadrature.
The integrals of the 1/r, 1/r  , log[r  − (z + ζ)] and 1/r  singularities over the panels are
frequency independent and are carried out once for all frequencies for the relevant pairs of
panels. These integrals are stored in a ﬁle for use at each frequency. The evaluation of the
Rankine integrals is accompanied by the set-up of an integer array of indices to indicate
which singularities have been subtracted for each pair of panels. These indices are input to
VGRN00, which in turn evaluates the appropriate component of the wave source potential.
The analytic integration of the Rankine source potentials and their derivatives follows
the theory outlined in [2]. Depending on the radial distance to the ﬁeld point, relative
to the panel length-scale DIAG(2), the evaluations are performed either from the exact
formulation or from multipole expansions including either second- or fourth-moments of
the panel area. The radial distances for transitions between these three algorithms are
based on the requirement of six-decimal relative accuracy. The formulae used for the
analytic integration of the logarithmic singularity are derived in [6].
The integration over each panel of the regular part of the wave source potential (evaluated in VGRN00) is carried out by quadrature. Two alternative quadrature formulae are
used in WAMIT, depending on the input parameter IQUAD. If IQUAD= 0 the centroid
integration formula is used, where the integral over each panel is approximated by multiplying the value of the integrand at the centroid by the panel area. If IQUAD= 1 the Gauss
integration formula is used with four nodes and corresponding weights over each panel, by
mapping their surface to a unit square using a bilinear transformation. The wave source
potential can be integrated over the panels by either method. When the centroid integration is selected, WAMIT exploits the symmetry of the wave source potential with respect
to the position of the source and ﬁeld points to evaluate the wave source potential once
for each pair of panels. Thus, if there are N panels and no planes of symmetry, N 2/2 calls
are made to VGRN00. The selection of the four-node Gauss quadrature formula requires
4N 2 calls to VGRN00 since the wave source potential symmetries cannot be utilized. For
a suﬃciently large number of panels, the use of the centroid integration has been found to
be suﬃciently accurate, and is recommended.

12–7

12.5

DISCRETIZATION OF THE INTEGRAL EQUATIONS
IN THE HIGHER-ORDER METHOD (ILOWHI=1)

The mean body surface is deﬁned by ‘patches’, as explained in Chapter 6. Each patch
must be a smooth continuous surface. The Cartesian coordinates x = (x, y, z) of a point
on the patch are expressed in term of two parametric coordinates u = (u, v). The latter is
normalized so that they vary in ±1. Details of geometric description of the body surface
are provided in Section 6.1.
The velocity potential on each patch is represented by a product of B-spline basis
functions U (u) and V (v) as shown in equation (6.4). The total number of B-spline basis
functions on each patch is Mu times Mv .
Upon substituting equations (6.3) and (6.4), (12.11) can be expressed in a form
2π

Mu
Mv 

m=1 l=1

(φj )k Uk (uf ) +

Np  


du

n=1


Np

=

du

n=1

where, Uk = Ul(u)Vm (v),



du =

number of patches.

 1

 1

−1

dv

−1

Mu
Mv 

m=1 l=1

(φj )k Uk (u)

∂G(u; uf )
J(u)
∂n(u)

∂φj
(u)G(u; uf )J (u)
∂n

(12.29)

du, J(u) is Jacobian and Np denotes the

Following the Galerkin method, (12.29) is multiplied by Ui and integrated over each
patch. This results in the linear system
2πdH
ik (φj )k +

N

k=1

H
Dik
(φj )k = SiH

(12.30)

for the radiation potential. Similarly, from (12.12), we obtain
2πdH
ik (φD )k +

N

k=1

H
Dik
(φD )k = IiH

(12.31)

for the diﬀraction potential.
In the above equations, (φj )k and (φD )k are unknown coeﬃcients of basis function for
H
the radiation and diﬀraction potentials, respectively. The matrices dH
ik and Dik and vectors
SiH and IiH are deﬁned by

dH
ik =

H
Dik



=

SiH =



duf Ui (uf )Uk (uf )

duf Ui (uf )



duf Ui(uf )



duUk (u)
du

12–8

∂G(u; uf )
J(u)
∂n(u)

∂φj
(u)G(u; uf )J(u)
∂n

(12.32)
(12.33)
(12.34)

IiH =



duf Ui (uf )ϕ0 (uf )

(12.35)

As explained in Chapter 6, a set of B-spline basis functions is deﬁned by the order of
the polynomial (Ku and Kv ) and the number of panels (Nu and Nv ). In general, the basis
function has nonzero value over a part of the patch. For example, Ul(u)Vm (v) is nonzero
on the panels from l − Ku + 1- (or 1, if l − Ku + 1 < 1) to l-th panels (or Nu , if l > Nu) in
u and m − Kv + 1- (or 1, if m − Kv + 1 < 1) to m-th (or Nv , if m > Nv ) panels in v.
The integration in uf over each panel is referred to as the ‘outer’ integration. This is
carried out by Gauss-Legendre quadrature. The order of the quadrature is speciﬁed by
the input parameters IQUO and IQVO in the SPL ﬁle, for u and v respectively, or by the
parameter IQUADO in CONFIG.WAM. The order of the basis functions are speciﬁed by
the parameters KU and KV in the SPL ﬁle, or KSPLIN in CONFIG.WAM.
The inner integrations in u in equations (12.33) and (12.34) are carried out as described
below, for each abscissa of the Gaussian quadrature for the outer integral, uf .
The integration of the regular part of the wave source potential (evaluated in VGRN00)
is carried out by Gauss-Legendre quadrature. The order of the inner quadrature is speciﬁed
by the input parameters IQUI and IQVI in the SPL ﬁle or by the parameter IQUADI in
CONFIG.WAM. Numerical tests suggest that the order of the quadrature should be one
order higher than the order of the basis function.
The integrals involving the Rankine source and normal dipole are evaluated in the
manner explained next. If the ﬁeld point uf is on the panel, the integrand is singular at
this point. Otherwise the integrand is regular throughout the domain of the panel. In the
singular case, the panel is subdivided into a small square subdomain centered at the ﬁeld
point uf , and one or more rectangular subdivisions adjoining the square as required to
cover the remainder of the panel. The integrals over the latter subdivisions are treated in
the same manner as for the other panels where the integrand is regular.
The integrals where the integrand is regular are evaluated by Gauss-Legendre quadrature. If the ﬁeld point is near the panel, the panel is subdivided into four smaller panels.
This subdivision is repeated until the size of the subdivided panel is less than a prescribed
multiple of the physical distance from uf to the centroid of the panel. For this purpose
the size of the panel is deﬁned as the maximum physical length from the center of the
panel
√ to its four vertices. The maximum size permitted without further subdivision is
1/ 2.25 = 2/3. In some cases a large number of subdivisions are required, particularly
when uf is close to the sides or vertices of the panel. In this case, the program terminates
subdivision after the domain is subdivided into 2048 subdomains. The program issues
a warning message to the monitor and error log ﬁle but continues without interruption.
Experience indicates that this warning message is most likely to occur when the mapping
of a physical surface onto a patch is singular, as at the poles of a spherical or spheroidal
surface, and that the accuracy of most relevant hydrodynamic outputs are not aﬀected
signiﬁcantly by this problem.
The singular integral over the square subdomain centered at uf is explained below. The
integration of the dipole is deﬁned in the ‘principal-value’ sense excluding a vanishingly
small area around the origin. With this deﬁnition for the dipole integral, both the source
12–9

and dipole distributions can be evaluated in a same manner.
After an appropriate normalization of the length and use of a local coordinates with the
origin at (uf , vf ), the integral takes a form
 1  1

I=

A(u, v)
dudv
−1 |x(u, v)|

−1

(12.36)

where A(u, v) is regular function. |x(u, v)| is the distance between the source and ﬁeld
points in physical space.
∂x
∂x ∂x
∂x
·
= 0
For simplicity, we consider only a case where | | = 1, | | = 1 and
∂u
∂v
∂u√ ∂v
but the analysis below can be applied directly to the general case. Since |x|/ u2 + v 2 is
regular and thus (12.36) can be expressed in the form
I=

 1  1

f (u, v)
√
dudv
−1
u2 + v 2

−1

(12.37)

where f (u, v) is regular.
The singularity at the origin is removed by subdividing the square domain into 4 isosceles
triangles with a common vertex at the origin and evaluating the integral separately over
each triangle. For example, the integral over a triangle with a side on u = 1 is
 1

 u

f (u, v)
dv √ 2
0
−u
u + v2
After the change of variables u = p, v = pq, we have
I

(1)

=

du

 1

I (1) =

0

dp

 1

f (p, pq)
dq √
−1
1 + q2

(12.38)

(12.39)

After adding the contributions from other three triangles, we have
I=

 1
−1

dp

 1
−1

dq

f (p, pq) + f (pq, p)
√
1 + q2

(12.40)

Next we remove the square root function from (12.40). By change of the variables
q = sinh(as), we have
I=a

 1

−1

dp

 1

−1

dsf (p, psinhas) + f (psinhas, p)

(12.41)

where a = sinh−11. The integral (12.41) is evaluated by applying the Gauss-Legendre
quadrature in p and s coordinates.
The integral of the log singularity in the free-surface component of the source potential
can be evaluated either in a similar manner as for the Rankine source, or together with the
regular part of the wave source potential. The index ILOG, set in the potential control ﬁle
or in CONFIG.WAM controls these options. When ILOG=1, the logarithmic singularity
is subtracted from the wave-source potential and integrated separately. When ILOG=0
it is included in the evaluation of the wave-source potential and integrated by the same
quadrature.
12–10

12.6

SOLUTION OF THE LINEAR SYSTEMS

The procedure used to solve the linear systems (12.25-6) or (12.30-1) is essentially the
same. The dimension of these linear systems is denoted by NEQN (number of equations).
In the low-order method NEQN is the same as the number of panels. In the higher-order
method NEQN depends on the number of patches, panels, and on the order of the basis
functions, as explained in Section 12.5.
If the body geometry has one or two planes of symmetry, half or a quarter of its wetted
surface is represented with panels, respectively. Each of the radiation potentials is symmetric or anti-symmetric with respect to the symmetry planes, depending on the rigid-body
mode, and the diﬀraction potential is decomposed into similar components. Thus for one
or two planes of symmetry either two or four separate components of the diﬀraction problem are deﬁned on the basis of their symmetry or anti-symmetry with respect to the xand y-planes. With this decomposition the dimensions of all linear systems are the same
as NEQN.
Flow symmetries and anti-symmetries are enforced in the solution of the integral equations by the method of images. The collocation point xi in the argument of the wave source
potential, is reﬂected about the geometry symmetry planes accompanied by a multiplication by +1 or −1 for symmetric and antisymmetric ﬂow, respectively.
To avoid unnecessary computations, the architecture of WAMIT permits the analysis
of any desired sub-set of the rigid-body modes and of the corresponding diﬀraction components, based on the settings of the MODE(I) indices in the potential control ﬁle (see
Section 3.1). For example, if only the heave mode is speciﬁed in conjunction with the
solution of the diﬀraction problem, and if there are planes of symmetry, only the symmetric component of the diﬀraction potential is evaluated. For this reason it is necessary to
specify the complete diﬀraction solution (IDIFF= 1) to evaluate ﬁeld data (free surface
elevation, pressure, and ﬂuid velocity) or to evaluate the drift forces.
With the default value of the input parameter ISOLVE=0, WAMIT solves the linear
systems in the radiation and diﬀraction problems by means of a special out-of-core iterative
solver. The maximum number of iterations is controlled by the parameter MAXITT in
the conﬁg.wam ﬁle (Section 3.7), with the default value equal to 35. When convergence
is not archived within the limit, a warning message is printed to the on monitor, and the
computation continues without interruption. If the number of iterations displayed in the
output is equal to MAXITT, this also indicates that convergence does not occur.
A supplementary direct solver for the linear systems may be used for the case for which
the iterative solver is nonconverging or slowly converging. It is based on standard Gauss
reduction, with partial pivotting. The LUD algorithm is employed for eﬃciency in solving
several linear systems simultaneously, with diﬀerent right-hand sides. The direct solver
also can be used to conﬁrm the iterative solution, and in cases where the number of
equations (NEQN) is relatively small the direct solver can result in reduced computing
time, particularly if the number of right-hand sides is large.
WAMIT also includes a block-iterative solver to provide increased options in the solution
of the linear system. This solver is based on the same algorithm as the iterative solver,
12–11

but the local LU decompositions are performed for speciﬁed blocks adjacent to the main
diagonal. At each iteration the back substitution is performed for each block, at each stage
of iterations. This accelerates the rate of convergence, and as the dimension of the blocks
increases the limiting case is the same as the direct solver. The opposite limit is the case
when the dimension of the blocks is one, which is the result of setting ISOLVE=NEQN; in
this case the result is identical to using the iterative solver without blocks (ISOLVE=0).
The convergence of the iterative method is quite diﬀerent for the low- and higher-order
methods. The primary reason for the slow convergence of the higher-order method is that
the linear system loses diagonal dominance as the order of basis function increases as shown
in the expression for dH
ik in (12.32).
Usually in the low-order method the number of iterations required to obtain convergence
is in the range 10-20, and the iterative solver is most eﬀective. In the higher-order method
experience indicates that the convergence rate is reduced, and it is generally advisable
to use the direct solver (ISOLVE= 1) or, in cases where NEQN is very large, the blockiterative solver (ISOLVE> 1). For special problems where hydrodynamic resonances occur,
the convergence rate is reduced and it is necessary to use the direct or block-iterative solver
also with the low-order method.
Experience using the low-order method has shown that slow convergence is infrequent,
and limited generally to special applications where there either is a hydrodynamic resonance
in the ﬂuid domain, as in the gap between two adjacent barges, or in the non-physical
domain exterior to the ﬂuid volume. An example of the latter is a barge of very shallow
draft, where the irregular frequencies are associated with non-physical modes of resonant
wave motion inside the barge. These types of problems can often be overcome by modifying
the arrangement of the panels or increasing the number of panels.
For the low-order method various convergence tests have been published in References
5, 6, 9, 10 and 12, based on applications of WAMIT. The accuracy of the evaluated
quantities has been found to increase with increasing numbers of panels, thus ensuring the
convergence of the discretization scheme. The condition number of the linear systems is
relatively insensitive to the order of the linear systems, and suﬃciently small to permit
the use of single-precision arithmetic. Convergence tests for the higher-order method are
reported in References 18, 19, 20, 24 and 25.

12.7

MEAN DRIFT FORCE AND MOMENT BASED ON PRESSURE INTEGRATION

The instantaneous position vector (X ) in an inertial coordinate system of the point ﬁxed
on the body ﬁxed coordinate system (x) is given by
X = x + ξ + α × x + Hx

(12.42)

For the present purpose we consider only the ﬁrst order quantities of the translational
modes ξ, the rotational modes α and the velocity potential φ. The roll-pitch-yaw sequence
of rotations is adopted, and the transformation matrix is given by
12–12

z
Z

y

x
Y

o

n

X

O

Figure 12.2: Coordinate systems fixed with respect to the body (dashed lines) and its mean position (solid
lines)
⎛

⎞

2
2
0
0
1 ⎜ −(α2 + α3 )
⎟
2
2
H= ⎝
−(α1 + α3 )
0
2α1 α2
⎠
2
2α1 α3
2α2 α3
−(α1 2 + α22 )

(12.43)

The normal vector in the inertial coordinate system can be expressed by
N = n + α × n + Hn

(12.44)

where n is the normal vector on the body boundary in the body ﬁxed coordinate system.
The pressure at the instantaneous position X given in the equation (12.42) can be
expressed by
P =

−ρ[g(z + Zo) + (φt + g(ξ3 + α1 y − α2x))
+( 12 ∇φ · ∇φ + (ξ + α × x) · ∇φt + gH x · ∇z)]

(12.45)

where Zo is the vertical coordinate of O relative to the free-surface.
The mean forces and moments are obtained by time averaging the following expressions
for the forces and moments:


F (2) =
M

(2)



=

NP ds

(12.46)

(X × N )P ds

(12.47)

S̃b

S̃b

In equations (12.46) and (12.47), the integrations are over the exact body wetted surface
S̃b .
12–13

After substituting (12.42), (12.44), and (12.45) and integrating the hydrostatic components,

F

(2)

1 
=
n [ζ 2 − 2ζ(ξ3 + α1y − α2x)]dl
ρg
2
WL

nz
1
ρg k
[ζ − (ξ3 + α1y − α2x)]2 dl
+
2
2
WL
1 − nz
− ρ



1
n( ∇φ · ∇φ + (ξ + α × x) · ∇φt )ds
Sb
2

+ α × (−ρ
(2)

Sb

nφt ds)

+ FS

(12.48)

where FS(2) = −ρgAwp [α1α3 xf + α2α3 yf + 12 (α21 + α22)Zo ]k
1 
ρg
(x × n)[ζ 2 − 2ζ(ξ3 + α1y − α2x)]dl
2
WL
nz
1 
ρg
(x × k)[ζ − (ξ3 + α1y − α2 x)]2 dl
+
2
2
WL
1 − nz

M (2) =

− ρ



1
(x × n)( ∇φ · ∇φ + (ξ + α × x) · ∇φt )ds
2
Sb


− ρα ×

Sb

+ ξ × (−ρ

(x × n)φt ds



Sb

nφt ds)

+ MS(2)

(2)

MS = ρg{

(12.49)

[
+
−
+
+
[
+
+
+

(ξ2 α2 − ξ3 α3)Awp xf − ξ2α1 Awp yf
3
1
α1α2Awp xf Zo − ( α21 + α22 )Awp yf Zo
2
2
ξ2ξ3 Awp − ξ3 α1Awp Zo
α2α3(L11 − L22) − 2α1α3 L12
1
α1α2∀xb − (α21 + α23 )∀yb ]i
2
−ξ1 α2Awp xf + (ξ1 α1 − ξ3 α3)Awp yf
1
3
( α21 + α22 )Awp xf Zo − α1 α2Awp yf Zo
2
2
ξ1ξ3 Awp − ξ3 α2Awp Zo
α1α3(L11 − L22) + 2α2 α3L12
12–14

+

1 2
(α + α23)∀xb ]j}
2 2

(12.50)

ζ is the ﬁrst order runup, Awp is the waterplane area and ∀ is the volume of the body. In
addition (xf , yf ) are the coordinates of the center of ﬂotation, (xb , yb, zb ) are the coordinates
of the center of buoyancy, (i, j, k) are positive unit vectors relative to the x, y, z coordinates,
and Lij = wp xi xj ds denotes the moments of the waterplane area.

12.8

REMOVAL OF IRREGULAR FREQUENCIES

The irregular frequency eﬀects are removed from the velocity potential and the source
strength using the extended boundary integral equations. The details of discussion on the
method are provided in Reference [8] and [16].
The extended boundary integral equation for the potential formulation (12.11) takes a
form


2πϕ(x) +

−4πϕ (x) +

Sb

ϕ(ξξ )


Sb


∂G(x; ξ)
∂G(x; ξ )
dξξ +
ϕ(ξξ )
dξξ
∂nξ
∂nξ
Si

∂ϕ(ξξ )
G(x; ξ )dξξ
x ∈ Sb
=
Sb ∂nξ

ϕ(ξξ )

(12.51)


∂G(x; ξ )
∂G(x; ξ )
dξξ +
ϕ (ξξ )
dξξ
∂nξ
∂nξ
Si

∂ϕ(ξξ )
G(x; ξ )dξξ
x ∈ Si
=
Sb ∂nξ

(12.52)

Here ϕ(x) is the unknown velocity potential on the body surface Sb and ϕ (x) is the
nonphysical velocity potential on the interior free surface Si .
The corresponding equation for the source formulation (12.22) takes a form


2πσ(x) +

−4πσ(x) +

∂G(x; ξ )
σ(ξξ )
dξξ +
Sb
∂nx





∂G(x; ξ )
σ(ξξ )
dξξ +
Sb
∂nx

Si

σ(ξξ )


Si

∂G(x; ξ )
∂ϕ(ξξ )
dξξ =
∂nx
∂nx

σ(ξξ )

12–15

∂G(x; ξ )
∂ϕ(ξξ )
dξξ =
∂nx
∂nx

x ∈ Sb

x ∈ Si

(12.53)

(12.54)

12.9

INTEGRAL EQUATION FOR BODIES WITH ZERO THICKNESS

Green’s integral equations in Section 12.2 tend to be singular as the thickness of the body
decreases. To avoid this singularity in the discretized problem, the panel size should be
of the same order as the thickness, or smaller, in order to render the linear system wellconditioned. Thus the size of the linear system becomes large as the thickness decreases.
An alternative form, which is nonsingular, can be obtained from Green’s integral equation for the limit when the thickness vanishes. In this modiﬁed form of the integral equation,
the velocity potential is represented by a distribution of dipoles only, without sources. The
dipole strength is equal to the diﬀerence of the velocity potential on two opposite sides of
the zero-thickness surface, denoted by ∆φ below.
When bodies consist partly of thin parts, a combination of Green’s integral equation
and the dipole distribution can be applied. It takes the following form:


2πφ(x) +

Sb

φGnξ dSξ +


Sd

∆φGnξ dSξ =

when x is on the normal body surface Sb and


Sb

φGnξ nx dSξ +



Sd

∆φGnξ nx dSξ = −4πφnx +



φnξ GdSξ

(12.55)

φnξ Gnx dSξ

(12.56)

Sb


Sb

when x is on the surface of zero thickness Sd .
In the low-order method this integral equation can be solved by WAMIT, to solve for
the velocity potential on Sb and for ∆φ on Sd . This extension is restricted to the low-order
method (ILOWHI=0) and to the potential solution (ISOR=0). Instructions for using this
option are in Section 5.4, and TEST09 is an example of its use (Section A.9).

12.10

MEAN DRIFT FORCE AND MOMENT USING CONTROL SURFACE

The mean drift force and moment are evaluated by one the following two alternatives
depending on the value of ICTRSURF in the CFG ﬁle.
Alternative 1:

1 
ρg
nζ 2 dl − ρg k
ζ[(ξ + α × x) · n ]dl
2
CL
WL

nz
1
ρg k
[ζ − (ξ3 + α1 y − α2x)]2 dl
+
2
2
WL
1 − nz

F (2) = −

− ρ



Sc

[∇φ

∂φ 1
− n(∇φ · ∇φ)]ds
∂n 2
12–16

+ ρk


Sf

(ζ

∂φt 1
(2)
+ ∇φ · ∇φ)ds + FS
∂z
2



(12.57)



1
ρg
(x × n)ζ 2 dl − ρg
ζ[(ξ + α × x) · n ](x × k)dl
2 CL
WL
nz
1
(x × k)[ζ − (ξ3 + α1y − α2x)]2 dl
+
ρg
2
2
WL
1 − nz

M (2) = −



∂φ 1
− (x × n)(∇φ · ∇φ)]ds
∂n 2
Sc

∂φt 1
(x × k)(ζ
+ ∇φ · ∇φ)ds + MS(2)
+ ρ
∂z
2
Sf
− ρ

[(x × ∇φ)

(12.58)

Alternative 2:

F

(2)


1 
 2
=
n ζ dl − ρg
ζ∇ζds
ρg
2
WL
Sf

− ρg k



WL

ζ[(ξ + α × x) · n]dl


1
ρg k
2
WL

+

nz
n2z

[ζ − (ξ3 + α1y − α2x)]2 dl

1−
∂φ 1
− n(∇φ · ∇φ)]ds
− ρ
[∇φ
∂n 2
Sc

∂φt 1
(2)
+ ∇φ · ∇φ)ds + FS
(ζ
+ ρk
∂z
2
Sf


M (2) =

(12.59)


1 
ρg
(x × n)ζ 2 dl − ρg
ζ(x × ∇ζ)ds
2
WL
Sf

− ρg



WL

ζ[(ξ + α × x) · n ](x × k)dl

1 
ρg
+
2
WL


nz
1−

n2z

(x × k)[ζ − (ξ3 + α1y − α2 x)]2 dl

∂φ 1
− (x × n)(∇φ · ∇φ)]ds
∂n 2
Sc

∂φt 1
(2)
+ ∇φ · ∇φ)ds + MS
(x × k)(ζ
+ ρ
∂z
2
Sf

− ρ

[(x × ∇φ)

(12.60)

In these equations Sc is the submerged part of the control surface, Sf is the part of
the control surface on the free surface, W L is the body waterline and CL is the common
boundary of Sc and Sf . n denotes the two-dimensional normal vector in the horizontal
plane on the body waterline, pointing into the body, ∇ is the gradient in the horizontal
12–17

(2)

(2)

plane, and k is the unit vector pointing vertically upward. FS and MS are the same as
those in the equation (12.48) and (12.49). The derivations of the equations (12.57-12.60)
from (12.48-12.49) are given in Reference [28].

12.11

INTERNAL TANK EFFECTS

The solution for the velocity potential in each tank, and the resulting forces and moments,
are computed in a similar manner as for the exterior domain outside the body or bodies.
Since there is no diﬀraction potential to consider, the velocity potential φ in each tank is
φ = iω



ξj φj

(12.61)

and the ﬁrst-order pressure at a ﬁxed point on the tank surface is given by
P = −ρg(z − ZT + ξ3 + α1y − α2x) − ρφt

(12.62)

Here ZT is the vertical coordinate of the tank free surface above the origin.
The solution for the velocity potential φ in each tank is computed simultaneously with
the potential in the exterior ﬂuid domain outside the hull, using one extended linear system
which includes all of the ﬂuid domains (exterior plus interior of all tanks). The principal
modiﬁcation is to impose the condition that there is no inﬂuence between the separate
ﬂuid domains. Thus the elements in the extended inﬂuence coeﬃcient matrix are set equal
to zero if the row and column correspond to diﬀerent domains. Further details are given
in Reference [27].
The force and moment exerted by the tank ﬂuid on the vessel are given by
F=
M=

 
ST

 
ST

P NdS =

P (X × N) dS =

 
ST

 

P (n + α × n) dS

(12.63)

P (x + ξ + α × x) × (n + α × n) dS

(12.64)

ST

where n is the normal pointing out of the tank (away from the ﬂuid domain of the tank)
and double integrals are over the submerged surface of the tank.
After some vector analysis these equations give relations similar to (12.46-12.47) for the
contributions from the hydrostatic pressure:
 

CT (3, 3) = ρg
CT (3, 4) = ρg

 

ST

CT (3, 5) = −ρg
CT (4, 4) = ρg

 
ST

n3dS

ST

yn3dS

 

ST

xn3dS

y 2 n3dS − ρg∀T zc

12–18

CT (4, 5) = −ρg

 
ST

xyn3dS

CT (4, 6) = ρg∀T xc
CT (5, 5) = ρg

 

x2 n3 dS − ρg∀T zc

ST

CT (5, 6) = ρg∀T yc
All other elements of the matrix CT are equal to zero. Here
 

∀T = VOLTANK(1) =

ST

 

∀T = VOLTANK(2) =

ST

 

∀T = VOLTANK(3) =

ST

 

xn1dS
yn2 dS

(z − ZT )n3 dS

1
x2 n1 dS
xc =
2∀T
ST
 
1
y 2 n2dS
yc =
2∀T
ST
1 
(z2 − ZT2 )n3 dS
zc =
2∀T
ST
Subroutine GEOMSTAT evaluates the hydrostatic parameters of the hull and tanks
separately. Thus VOL, C(i,j) are evaluated for the hull ignoring tank patches/panels
and their values are the same with or without tanks, as deﬁned in Section 4.1. The
corresponding tank parameters VOLTNK(1:3,1:NTANK) and CTANK(1:9,1:NTANK) are
evaluated separately for each tank (the second index is omitted for simplicity of notation):
 

CTANK(1) =
CTANK(2) = −

 

 

CTANK(3) =
CTANK(4) =

ST

ST

 

ST

 

CTANK(5) =

ST

 

CTANK(6) =

n3dS

ST

ST

12–19

yn3dS
x2n1 dS
y 2 n2dS

(z2 − ZT2 )n3 dS

 

CTANK(7) =

xn3 dS

ST

x2n3 dS

CTANK(8) = −

 

 

CTANK(9) =

ST

ST

xyn3dS

y 2 n3dS

It is necessary to consider the implications of planes of symmetry with respect to the
tanks. If all of the tanks are symmetric about a plane of symmetry, then it is appropriate
to use that option (assuming the hull is also symmetric about the same plane). Thus, for
example, for a hull with symmetry about y=0, and with all tanks symmetric about the
same plane, it is appropriate to set IS(2)=1. But if there are two or more tanks along
the x-axis, IS(1)=0 is required. When planes of symmetry are appropriate, the following
integrals (which are originally evaluated over one half or one quarter of the tank, with
nonzero values), should be set equal to zero:
If IS(1)=1, CTANK(2) = 0,

CTANK(4) = 0,

CTANK(8) = 0

If IS(2)=1, CTANK(3) = 0,

CTANK(5) = 0,

CTANK(8) = 0

All other elements are multiplied by IMUL=2 or 4 to account for the planes of symmetry,
in the same manner as C(I,J).
In GETIF, after CALL OPHEAD to output the hydrostatic matrix for the hull, the
subroutine TANKFS is called when tanks are present. In TANKFS the following assignments are made for the hydrostatic restoring coeﬃcients, where the extra terms added for
the tanks are summed over all tanks associated with each body:
C(3, 3) = C(3, 3) + (ρT /ρ)CTANK(1)
C(3, 4) = C(3, 4) + (ρT /ρ)CTANK(3)
C(3, 5) = C(3, 5) + (ρT /ρ)CTANK(2)



C(4, 4) = C(4, 4) + (ρT /ρ) CTANK(9) − 12 CTANK(6)
C(4, 5) = C(4, 5) + (ρT /ρ)CTANK(8)



C(5, 5) = C(5, 5) + (ρT /ρ) CTANK(7) − 12 CTANK(6)
Also, if IALTFRC=2,
C(4, 6) = C(4, 6) + 12 (ρT /ρ)CTANK(4)
C(5, 6) = C(5, 6) + 12 (ρT /ρ)CTANK(5)

The extra terms in C(4, 6) and C(5, 6) are omitted for a freely-ﬂoating body since these
are balanced by the vessel’s corresponding buoyancy moments.
Next consider the inertia forces and moments due to the body mass. If IALTFRC=1
the body mass m is calculated from VOL and all of the inertia coeﬃcients are propotional
to m. Since VOL is the total displaced volume of the hull, the static mass of the ﬂuid
in the tanks is included in m. However the same inertia eﬀects are represented (more
12–20

correctly for dynamic conditions) by the added mass of the tank, to be discussed below.
For this reason, if IALTFRC=1, the mass matrix BFRCND is reduced by the sum

(ρT /ρ)×VOLTANK/VOLM, where the sum is over all tanks. Likewise, the terms −mgzg
in the restoring coeﬃcients C(4, 4) and C(5, 5) are reduced. If IALTFRC=2 these corrections are not made, and the external mass matrix should exclude the ﬂuid in the tanks. If
IALTFRC=1 the radii of gyration should be estimated ignoring the ﬂuid in the tanks.
The added mass and damping coeﬃcients are evaluated globally, by integrating the
corresponding components of the pressure over both the external hull surface and the
internal tank surfaces. The only modiﬁcation for tank panels/patches is to multiply their
contributions by RHOTANK, the relative density of the tank ﬂuid compared to the external
ﬂuid. Since there is no radiation from the tanks, the damping coeﬃcients should be zero.
In test calculations they are generally very small, except near tank resonances. (A useful
check is to verify that the damping coeﬃcients of the hull with tanks are equal to the
damping coeﬃcients without tanks.)
Special attention is required for the vertical modes (heave, roll, pitch), where there is
a ﬁctitious hydrostatic contribution to the added mass. First consider heave, where the
relevant boundary conditions are φn = n3 on Sb and Kφ − φz = 0 on the free surface. Here
K = ω 2 /g. Thus the heave potential is given by φ3 = z + 1/K, where z is a local vertical
coordinate with its origin in the tank free surface. The heave added-mass coeﬃcient is
A33 = ρ

 
ST

n3φ3 dS = ρ∀T + CT (3, 3)/(gK)

(12.65)

Also derivable from the same potential are
A34 = ρ
A35 = ρ

 

 

(yn3 − zn2 )φ3 dS = ρ∀T yc + CT (3, 4)/(gK)

(12.66)

(zn1 − xn3)φ3 dS = −ρ∀T xc + CT (3, 5)/(gK)

(12.67)

ST

ST

Since ω 2 = gK, the last terms are cancelled by the hydrostatic restoring coeﬃcients.
Thus, in the limit ω → 0, there are no contributions to the equations of motion for the LHS
elements associated with the vertical force or vertical translation, as expected on physical
grounds.
In the classical hydrostatic analysis of ships, the tank free-surface eﬀect is evaluated by
considering the second moments of the tank free surface about a local origin at the centroid
of the free surface, whereas in the expressions for CT these moments are about the global
origin. This diﬀerence can also be explained in terms of the corresponding added mass
coeﬃcients, in an analogous manner to the analysis above.

12–21

Chapter 13

THE F2T UTILITY
The Fortran utility F2T (Frequency-to-Time-domain) is a post-processor to transform
frequency-domain WAMIT output to time-domain impulse-response functions (IRFs). This
program is intended to provide a utility which can be used for general purposes, based on
standard WAMIT outputs. This program accepts as input all of the ﬁrst-order (linear)
outputs from WAMIT, including any combinations of Options 1-7 (added mass/damping,
Haskind exciting forces, Diﬀraction exciting forces, RAO’s, body pressures/velocities, ﬁeldpoint pressures/velocities). In principle there are no restrictions regarding the numbers of
rigid-body modes, generalized modes, or bodies. The computed IRFs are saved in output
ﬁles which are analogous to the input ﬁles for each option and use the same ﬁlename
extensions.
The Fourier transforms from the frequency-domain to the time-domain are evaluated
in F2T by Filon numerical integration. This method provides relatively accurate results
for large values of the time variable t. A fundamental requirement is that the frequencydomain data must be evaluated by WAMIT for a large number N of uniformly-spaced
frequencies ωn where n=1,2,3,...,N. Special attention is necessary to ensure that the increment ∆ω = ωn+1 − ωn is suﬃciently small (to preserve the accuracy of the numerical integration) and that the highest ﬁnite frequency ωN is suﬃciently large to span the
physically-signiﬁcant range of frequencies for the application (or from the mathematical
standpoint, to ensure that the truncated Fourier integrals are reasonable approximations
of the inﬁnite integrals). In view of the need to include high frequencies in the WAMIT
analysis it is usually advisable to use the irregular-frequency option (IRR=1), unless the
body is submerged or its waterplane area is very small. The requirement of accuracy over
a broad range of frequencies means that either a large number of low-order panels should
be used, or alternatively that the higher-order method is used with appropriate control of
the panel subdivision indices NU,NV or the global parameter PANEL SIZE.
Reference [26] contains illustrative results from the use of F2T, including comparisons
with the results from the time-domain panel program TiMIT.

13–1

13.1

DEFINITIONS OF RADIATION AND DIFFRACTION
OUTPUTS

The outputs from WAMIT and F2T are considered to be of either the radiation or diffraction type, depending on whether they are caused by forced motions in calm water or
by incident waves, respectively. The simplest physical distinction between these two types
is in terms of the incident wave amplitude: if the response is proportional to the wave
amplitude it is of the diﬀraction type, and vice versa.
The added-mass and damping coeﬃcients (Option 1) are of the radiation type, whereas
the exciting forces and RAO’s (Options 2,3,4) are of the diﬀraction type. Except as noted
in Section 13.5, the pressures and ﬂuid velocities on the body (Option 5) and in the ﬂuid
(Options 6,7) are of the diﬀraction type, since these are deﬁned in the WAMIT convention
as total responses with the body free to respond (or ﬁxed) in incident waves.
A fundamental distinction between the two types of outputs is in terms of their limits at
inﬁnite frequency or zero period. In this limit the radiation outputs are generally real and
nonzero, corresponding to the added mass, pressure, and ﬂuid velocity induced by forced
motions of the body without wave eﬀects on the free surface. Conversely, in the same limit
there are no diﬀraction eﬀects since the ‘incident waves’ have vanishingly small wavelength
and cause no disturbance of either the body or the ﬂuid.
When radiation IRFs are evaluated it is necessary to evaluate the corresponding frequencydomain coeﬃcients for ω0 = 0 and ωN +1 = ∞, using the special instructions in the WAMIT
User Manual (Section 3.1, page 3-9 of the V6.1 User Manual). Thus, in V6.1, two WAMIT
runs are required. V6.1 also restricts the ω0 = 0 and ωN +1 = ∞ evaluations to Option 1,
so that V6.1 cannot be used with F2T to analyse radiation type outputs in Options 5,6,7.
WAMIT V6.2 removes this restriction, and will also permit the analysis of all frequencydomain outputs to be made in a single run. To simplify the instructions below, Version 6.2
or higher is assumed, and instructions are given for making only one WAMIT run. (Users
of V6.1 should make two separate runs, one for ﬁnite wave periods and one for zero and
inﬁnity, and for the latter only Option 1 should be speciﬁed.)

13.2

ACQUIRING INPUT DATA FOR F2T WITH WAMIT

The frequency-domain input data for F2T is evaluated by WAMIT. The algorithms used
to evaluate the Fourier transforms in F2T require that the input data is restricted to a
uniformly-spaced set of frequencies ωn = n∆ω, where (n=1,2,3,...,NPER), augmented by
the limiting frequencies 0 and ∞. In the WAMIT run this is done most easily by setting
the parameter IPERIO=2 in the conﬁguration ﬁle (inputs are radian frequencies), and by
using the option to write the data -NPER and ω1 , ∆ω on the lines normally used to specify
NPER and the array PER. The simplest procedure is to make one run, with ω1 = −∆ω,
so that both inﬁnite and zero wave periods are included in the run. (Note in this case that
NPER must include the total number of wave periods including zero and inﬁnity.)
The input ﬁles used for the tests of the ISSC TLP can be regarded as an example. These
13–2

ﬁles are listed in Appendix A14a. They are essentially the same as for TEST14, with the
exception of IPERIO=2 in TEST14.CFG, and the speciﬁcation of the input frequencies.
When the input ﬁles to F2T are read, the data is sorted so that the frequencies are listed
in ascending order, regardless of their order in the WAMIT output ﬁles. Thus the order of
the periods (-1.0, 0.0) is irrelevant, and it is possible to patch together two or more separate
sets of output ﬁles from WAMIT, e.g. one with a coarse set of wave frequencies and the
other with intermediate frequencies, to provide a ﬁner set, without concern regarding their
order.

13.3

HOW TO USE F2T

The program F2T can be executed after the appropriate WAMIT output ﬁles are available.
The user must specify the ﬁlenames of these ﬁles and a small number of input parameters,
either interactively in response to appropriate runtime prompts or by preparing the special
input ﬁle inputs.f2t. The following example of this special input ﬁle corresponds to the
TEST14a test run described above:
header line for inputs.f2t control file, TLP example
TEST14a
0 0 1 (IRAD IDIFF NUMHDR)
0 0 0 (INUMOPT5 INUMOPT6 INUMOPT7)
0.2 100 (DT NT time step and number of time steps)
0 (IOUTFCFS, output both cosine and sine transforms)
43.125 9.80665 ULEN GRAV
These inputs are described for each line as follows:
Line 1 is an ASCII header dimensioned CHARACTER*72 as in most WAMIT input
ﬁles. This line should be used to insert a brief description of the ﬁle.
Line 2 is a list of the ﬁlenames (not the extensions) of the primary and secondary
WAMIT output ﬁles. F2T attempts to open all numeric output ﬁles with the same ﬁlenames, and includes all of these ﬁles in the analysis. Thus the determination of which
options to be included depends on the available WAMIT output ﬁles. In this example
where the TEST14.FRC control ﬁle was used as in the standard WAMIT test runs, Options 1,2,3,4 will be included in the F2T analysis. If all of the input data is included in
the primary ﬁle it is not necessary to list other ﬁlenames. Additional secondary ﬁles can
also be included, up to a maximum limit of 256 ASCII characters for the complete line.
At least one blank space must be used to separate each ﬁlename.
Lines 3 and 4 contain the six WAMIT control parameters identiﬁed by the comments
in parenthesis. These parameters must have the same values as in the WAMIT runs. (No
distinction is made between IRAD,IDIFF=0 or 1, and the only important value to specify
correctly is −1. For any input values of IRAD,IDIFF other than -1 the results are the same
as for 0 or 1.) NUMHDR, which is optional in WAMIT with the default value 0, must
be speciﬁed heree with the value 0 (no headers) or 1 (one line of headers) to indicate the
13–3

presence or absence of a header line in the WAMIT numeric output ﬁles. INUMOPT5, the
optional conﬁguration parameter introduced in WAMIT Version 6.1 to permit outputting
separate components of the body pressure and velocity, and also INUMOPT6, INUMOPT7
introduced in Version 6.2 must be speciﬁed here with the value 0 (default in WAMIT) or
the separate-components values 1.
Line 5 contains the time step and number of time steps for the computation and tabulation of the time-domain response functions. The radiation IRF’S are computed and
tabulated for t=0 and for NT positive times DT, 2DT, 3DT, ..., NT∗DT. The diﬀraction IRFs are evaluated for both positive and negative times, starting with −NT∗DT and
ending with +NT∗DT.
Line 6 contains the optional parameter IOUTFCFS, with the following options for its
value:
IOUTFCFS=1: output only the cosine transform of radiation irf’s
IOUTFCFS=2: output only the sine transform of radiation irf’s
IOUTFCFS=0 (or any other integer except 1 or 2): output both cosine and sine transforms
(These transforms are redundant, as explained below.) The default value IOUTFCS=0
is used if line 6 is missing from the ﬁle. Thus the use of this parameter is optional.
Line 7 contains the optional parameters ULEN and GRAV, which are the same characteristic length scale and gravitational acceleration parameters as input in the GDF ﬁle.
These parameters are only required when Options 5, 6, or 7 are included, and when the radiation outputs are speciﬁed, as explained in Section 13.5 below. In all other circumstances
the parameters ULEN and GRAV can be omitted from inputs.f2t. When ULEN and
GRAV are included in inputs.f2t it is essential to also include IOUTFCS on
line 6.
The use of the special ﬁle inputs.f2t is optional. If this ﬁle does not exist, or if the
ﬁrst ﬁve lines cannot be read with the appropriate data, the user is prompted to specify all
of the above inputs interactively. The special ﬁle can also be used in a partial form with
some but not all of the above lines, but the lines included must be in the same order as
above. This permits the user to interactively input diﬀerent values of the time step and
number, simply by omitting Line 5 from the special ﬁle.
The numeric data in the special ﬁle is read with free format READ statements, separately for each line. Any additional text on the same lines is ignored, so that comments
may be inserted as in the example above. The ﬁlenames on Line 2 are read as ASCII
text of unknown length (maximum of 256 characters, all on one line) and no additional
comments may be included on this line.

13.4

OUTPUT FILES

The output ﬁles from F2T are in two complementary formats with duplication of the output
data in the two formats. The ﬁlename assigned to all of the output ﬁles is primary, with
13–4

diﬀerent extensions. The ﬁrst set of output ﬁles have appended ﬁlenames including IR
followed by the same extensions as the WAMIT output ﬁles. The second set have the
appended ﬁlenames including JR. The ﬁrst set follow the same format as the WAMIT
numeric output ﬁles of the same number, except that the period is replaced by the time step
and the WAMIT force coeﬃcients are replaced by their Fourier cosine and sine transforms.
Diﬀerent modes and mode combinations are listed on separate lines with the identifying
mode indices, just as in the numeric output ﬁles of WAMIT.
To facilitate plotting and separation of the diﬀerent modes and wave angles (BETA),
all of the Fourier cosine/sine transforms are listed on one line in the output ﬁles denoted
by JR, in the same order of mode combinations but without explicit mode indices. The
cosine/sine transforms are listed as pairs, unless one or the other is ommitted by setting
IOUTFCFS equal to 1 or 2 as explained in the following paragraph. Column one of the
JR ﬁle contains the value of time t.
Either the cosine transforms of the added mass or the sine transforms of the damping
can be used to evaluate the radiation IRFs (cf. equations 3 and 4 below). These two sets
of data can be checked to verify their accuracy and consistency, in much the same way
that the Haskind and Diﬀraction exciting forces or cross-coupling coeﬃcients are compared.
Alternatively, to achieve more compact output ﬁles, one of these transforms can be omitted
using the parameter IOUTFCFS.
One more output ﬁle is produced with the extension .KR1, containing the impulseresponse functions Kij which are evaluated in TiMIT. These alternative IRF’s are evaluated
in F2T by numerical diﬀerentiation of the IRF’s Lij which are deﬁned below.
The diﬀraction ﬁles JR are diﬀerent from the radiation ﬁles in two respects, to facilitate
their use. First, the time steps begin with −NT∗DT, and end with +NT∗DT. Secondly,
the cosine and sine transforms are combined (adding for t < 0 and subtracting for t > 0)
to give the actual IRFs for the corresponding exciting forces and RAOs (cf. equation 8
below).
For practical purposes the .JRn ﬁles will be most useful, and the .IRn ﬁles may be useful
only to clarify the identity of the diﬀerent columns in the .JRn ﬁles.
Some experience and/or trial computations will be needed to determine appropriate values of the input frequencies and time steps. The dimensions of these parameters correspond
to GRAV in the WAMIT run.

13.5

OPTIONS 5,6,7

The F2T utility has been developed primarily for use with Options 1 to 4 (global forces
and RAO’s). Local pressures, velocities, and wave elevations have not been tested, and
these may be diﬃcult to transform accurately, due to limited or non-convergence of the
Fourier transforms at high frequencies.
If IDIFF=−1 is speciﬁed in the WAMIT run, signifying that there are no incident waves,
the outputs from Options 5,6,7 are the total responses from superposition of all speciﬁed

13–5

radiation modes; in this case all of the available outputs are of the radiation type. In
this case, if more than one mode is considered, the output is for nonzero ﬁnite frequencies
only and is not suitable for transform to the time domain. If the conﬁguration parameters
INUMOPT5, INUMOPT6, INUMOPT7 = 1 in the WAMIT run, the corresponding Option
5,6,7 outputs are separated into radiation components for each mode of forced motion plus
the diﬀraction component, as explained in Sections 4.9 and 4.12; in this case F2T analyses
the radiation components and diﬀraction components separately, according to their types.
The procedure using INUMOPT5, INUMOPT6, INUMOPT7 = 1, as described in Section 4.12, should be followed if F2T is used to transform the radiation components of the
pressures, velocities, and wave elevations.

13.6

THEORY

The fundamental relations between the time- and frequency-domain express the addedmass coeﬃcient Aij and damping coeﬃcient Bij in terms of Fourier transforms of the
impulse-response function Lij (t):
 ∞

Aij (ω) − Aij (∞) =
Bij (ω) = ω

0

 ∞

Lij (t) cos ωt dt

Lij (t) sin ωt dt

0

(13.1)
(13.2)

The inverse-transforms of (13.1-2) give complementary relations for the impulse-response
function:
2
Lij (t) =
π

 ∞
0

Lij (t) =

[Aij (ω) − Aij (∞)] cos ωt dω
2
π

 ∞
0

Bij (ω)
sin ωt dω
ω

(13.3)

(13.4)

Similar relations exist for the exciting forces and RAOs. Deﬁne one of these quantities by
the complex function Xi (ω) The corresponding impulse-response function is real, denoted
by Ki (t). The appropriate physical ranges are (0 ≤ ω < ∞) and (−∞ < t < ∞). Then
the complex Fourier transform pairs are as follows:
Xi (ω) =

 ∞

and
2πKi (t) =

−∞

Ki (t)e−iωt dt

 ∞
−∞

Xi (ω)eiωt dω

Formally, since Ki is real, Xi (−ω) = Xi∗ (ω), and thus

13–6

(13.5)
(13.6)

2πKi (t) =
or

 ∞
0



Xi (ω)eiωt + Xi∗ (ω)e−iωt dω

1∞
Ki(t) =
[Re(Xi ) cos ωt − Im(Xi ) sin ωt] dω
π 0

(13.7)
(13.8)

The principal task is to evaluate (13.3), (13.4), and (13.8). This is done by truncating
the inﬁnite integrations at the largest value of the evaluated frequency, and using Filon
quadratures to evaluate the resulting ﬁnite integrals. A truncation correction is derived
below, and applied to (13.3).
Usually the most signiﬁcant truncation error is associated with the transform of the
added-mass (13.3). From partial integration of (13.1) it follows that


1 ∞ 
Aij (ω) − Aij (∞) = −
Lij (t) sin ωt dt  −L(0)ω −2
(13.9)
ω 0
where the neglected integral is of order ω −3 . If (13.3) is truncated at a ﬁnite frequency
ωN = Ω, the truncation correction is


2 ∞
[Aij (ω) − Aij (∞)] cos ωt dω
π Ω
This can be approximated, using (13.9), if Ω is suﬃciently large:
Λij (t) =

(13.10)



∞
2
2 
L (0) [cos Ωt + Ωt si(Ωt)]
Λij (t)  − Lij (0)
ω −2 cos ωt dω = −
(13.11)
Ω
π
πΩ ij
Here we follow the notation of Abramowitz & Stegun (equation 5.2.26) for the sine integral

 ∞
z

sin t
dt = −si(z)
t

The constant Lij (0) can be evaluated from the fact that Lij (0) = 0, and thus


2 Ω
2 
[Aij (ω) − Aij (∞)] dω  −
(13.12)
L (0)
Λij (0) = −
π 0
πΩ ij
Combining (13.11) and (13.12) gives the truncation correction in (13.10) in the form


Ω
2
[Aij (ω) − Aij (∞)] dω
(13.13)
Λij (t)  − [cos Ωt + Ωt si(Ωt)]
π
0
However this procedure suﬀers from the slow algebraic convergence of the last integral. An
alternative procedure, which is adopted in F2T, is based instead on diﬀentiating (13.4) to
give the relations
2∞
2Ω

Bij (ω) dω 
Bij (ω) dω
(13.14)
Lij (0) =
π 0
π 0
 Ω
4
Λij (t)  − 2 [cos Ωt + Ωt si(Ωt)]
Bij (ω) dω
(13.15)
π Ω
0
Equations (13.12) and (13.14) are complementary, but (13.14) is more robust since the integrand of (13.14) is positive-deﬁnite and converges to zero more rapidly than the integrand
of (13.12).

13–7

Chapter 14

MEAN DRIFT FORCES USING
CONTROL SURFACE
Starting with V6.3, option .9c is included in WAMIT to evaluate the mean drift forces
(and moments). In this option, the mean drift forces are evaluated from the momentum
ﬂux through a control surface surrounding the body. If IOPTN(9)>0 in the Force Control
File, and ICTRSURF>0 in the Conﬁguration File, the drift forces are evaluated both from
the pressure-integration method (with output in the .9 numeric output ﬁle) and from the
control-surface momentum-ﬂux method (with output in the .9c ﬁle). When NBODY> 1, a
control surface surrounding each body is required, and the drift force acting on each body
is evaluated separately as in Option 9.
The advantages of using the control surface are i) all six components of the mean drift
forces and moments, on a single body or on each body in the multiple body interaction, are
evaluated as in the pressure integration, and ii) the computational results are more accurate
than the pressure integration method when the body surface is not smooth, especially for
bodies with sharp corners. The disadvantages are i) the user must specify the control
surface as an additional input, and ii) the evaluation of the momentum ﬂux at a suﬃciently
large number of ﬁeld points on the control surfaces increases the run time of the FORCE
module. This option is recommended when the accuracy of the mean forces and moments
evaluated by pressure integration is uncertain, due to slow or lack of convergence with
respect to the discretization of the body.
The drift forces and moments using a control surface are evaluated by one of two
alternatives, depending on the integer parameter ICTRSURF in the conﬁguration ﬁle.
When ICTRSURF=1, Alternative 1 is used, based on equations (12.57) and (12.58) When
ICTRSURF=2, Alternative 2 is used, based on equations (12.59) and (12.60). These two
alternatives are analytically equivalent. Alternative 1 is derived by transfer of the integral on the waterline (WL) in Alternative 2 to the line integral along the intersection of
the free surface and the control surface (CL). Alternative 1 is generally more accurate for
the horizontal forces and yaw moment, because it does not include a waterline integral.
However, Alternative 2 must be used when CL is very close to the body. This is because
the evaluation of the momemtum ﬂux (pressure and/or velocity) very close to the body is
14–1

either inaccurate or not possible. An example for which the Alternative 2 may be required
is when the gap between two adjacent bodies is very small.
To evaluate the mean drift forces and moments using this option, i) the parameter
ICTRSURF must be speciﬁed in the CFG ﬁle, as explained in Section 3.7, and ii) the
CSF(Control Surface File) deﬁning the geometry of the control surface must be prepared.
The CSF ﬁle must have the same ﬁlename as the corresponding geometric data ﬁle for the
body, with the extension .csf, i.e. gdf.csf.
The control surface must be a closed surface surrounding the body in the ﬂuid. In
general, for a ﬂoating body which intersects the free surface, the control surface must
start from the body’s waterline, either extending outward on the free surface or downward
away from the waterline into the ﬂuid. Simple examples include a hemisphere or circular
cylinder with suﬃciently large dimensions so that the body is entirely within the interior
of this surface, together with the intermediate portion of the free surface between the outer
control surface and the body waterline. For multiple bodies, the control surface for each
body should not include or intersect with other bodies, but it can intersect with other
control surfaces.
In principle, the position and shape of the control surface are arbitrary. From a practical
standpoint, the control surface should be suﬃciently far from the body to ensure robust
evaluation of the ﬁeld velocity and pressure, but not so far as to require a very large number
of ﬁeld point evaluations. Simple geometrical description of the control surface is usually
desirable.
If the Alternative 1 method is used, there is no contribution to the horizontal drift force
and vertical drift moment from any part of the control surface that is in the plane of the
free surface (z = 0). Thus, if these are the only required components of the drift forces,
the control surface can be completely separated from the body surface without the need
to include the intermediate portion of the free surface. This simpliﬁes the deﬁnition of
the control surface, especially for bodies with complicated geometry of the waterline. In
addition, the numerical errors are generally smaller for ﬁeld points that are not too close
to the body surface.

14.1

CONTROL SURFACE FILE (CSF)

The geometry of the control surface can be described in the same manner as the body
geometry. Similar options exist to deﬁne the control surface, and diﬀerent options can
be used for the control surface and for the body. In the low-order method, speciﬁed by
inputting the parameter ILOWHICSF=0 on line 2 of the CSF ﬁle, the control surface is
described with quadrilateral panels in the same manner as is described for the body in
Chapter 5. In the higher-order method, the parameter ILOWHICSF=1 on line 2 of the
CSF ﬁle and the control surface is described in the same manner as is described for the
body in Chapter 6, using any of the options available for higher-order representation of the
body surface, including ﬂat panels, B-splines, MS2 ﬁles and analytically using subroutines
in GEOMXACT. The format of the CSF ﬁle is almost identical with the GDF ﬁle. The
14–2

principal diﬀerence is on line 2, where the parameters ULEN and GRAV in the GDF ﬁle are
replaced by ILOWHICSF. Also, when ILOWHICSF=1, the parameter PSZCSF is speciﬁed
in the CSF ﬁle to control the accuracy of the numerical integration.
The control surface is deﬁned in terms of the body coordinates. The normal vector is
deﬁned to point into the interior of the control surface (toward the ﬂuid in the domain
between the control surface and the body). If part of the control surface coincides with
the plane of the free surface, the normal on this surface is positive downwards.
The accuracy of the numerical integration of the momentum ﬂux depends not only on
the accuracy of the ﬁeld quantities on the control surface, but also on the discretization of
the control surface. If a low-order control surface is used, the integral of the momentum
ﬂux is calculated as the sum of the product of the ﬂux at the centroid of each panel and the
area of the panel. If a higher-order control surface is used, the control surface is subdivided
into ‘higher-order’ panels. On each panel, the momentum ﬂux is calculated based on Gauss
quadrature assuming quadratic variation of the ﬂux.
The input ﬁles for the test runs 5, 13, and 22, shown in Appendix A, illustrate the
diﬀerent methods for deﬁning the control surfaces, as noted below. As noted above, it is
possible to use diﬀerent values of ILOWHI and ILOWHICSF. In TEST05, which uses the
low-order analysis for two bodies, the control surface around the cylinder is deﬁned by loworder panels while the control surface around the spheroid is deﬁned by the higher-order
analytical method using a special subroutine in the GEOMXACT DLL ﬁle.
Since the description of the control surface in FORCE is completely separate from the
solution in POTEN, arbitrary combinations of ILOWHI in POTEN and ILOWHICSF in
FORCE can be used together. One or two planes of symmetry can be used to simplify
the deﬁnition of the control surface, even when the body is not symmetric with respect
to the same plane(s). However the opposite is not true: if the body is deﬁned with one
or two planes of symmetry, the control surface must use the same plane(s) of symmetry.
(The case NBODY>1 is an exception, since symmetric bodies are reﬂected and analyzed
without any planes of symmetry; thus it is possible in this case to use a non-symmetric
control surface with a symmetric body, but this is not likely to be useful.)

14.2

LOW-ORDER CONTROL SURFACE FILE

In the low-order method the control surface is represented by an ensemble of panels. The
CSF ﬁle contains a description of this discretized surface in the same format as in the GDF
ﬁle except that ULEN and GRAV are not speciﬁed in CSF.
The data in the CSF ﬁle can be input in the following form:
header
0 (ILOWHICSF)
ISXCSF ISYCSF
NPANCSF
X1(1) Y1(1) Z1(1) X2(1) Y2(1) Z2(1) X3(1) Y3(1) Z3(1) X4(1) Y4(1) Z4(1)
14–3

X1(2) Y1(2) Z1(2) X2(2) Y2(2) Z2(2) X3(2) Y3(2) Z3(2) X4(2) Y4(2) Z4(2)
.
.
.
.
.
. . . . .
.
. . . . X4(NPANCSF) Y4(NPANCSF) Z4(NPANCSF)
‘header’ denotes a one-line ASCII header dimensioned CHARACTER∗72.
ISXCSF, ISYCSF are the geometry symmetry indices which have integer values 0 or 1.
If ISXCSF and/or ISYCSF =1, x = 0 and/or y = 0 is a geometric plane of symmetry, and
the input data are restricted to one quadrant or one half of the control surface. Conversely,
if ISXCSF=0 and ISYCSF=0, the complete control surface must be represented by panels.
ISXCSF = 1: The x = 0 plane is a geometric plane of symmetry.
ISXCSF = 0: The x = 0 plane is not a geometric plane of symmetry.
ISYCSF = 1: The y = 0 plane is a geometric plane of symmetry.
ISYCSF = 0: The y = 0 plane is not a geometric plane of symmetry.
For all values of ISXCSF and ISYCSF, the (x, y) axes are understood to belong to the
body system of the corresponding GDF ﬁle and the panel data are always referenced with
respect to this system.
NPANCSF is equal to the number of panels with coordinates deﬁned in this ﬁle.
XI(J), YI(J), ZI(J) are the Cartesian coordinates (x, y, z) of I-th vertex of the J-th panel.
The four vertices of a panel are speciﬁed in the anti-clockwise direction when the panel is
viewed from outside of the control surface, as in the case of the body surface illustrated in
Figure 5.1.
Further description of the data in the CSF ﬁle can be found in Section 5.1, which
describes the low-order GDF ﬁle.

14.3

HIGHER-ORDER CONTROL SURFACE FILE

In the higher-order method the ﬁrst part of the CSF ﬁle is as follows:
header
1 (ILOWHICSF)
ISXCSF ISYCSF
NPATCSF ICDEF

PSZCSF

14–4

Subsequent data may be included in the CSF ﬁle after these four lines, depending on the
manner in which the geometry of the control surface is represented, in the same manner
as for the GDF ﬁle (See Sections 6.5-6.8.)
header, ISXCSF and ISYCSF are the same as those in the CSF ﬁle for ILOWHI=0
above.
NPATCSF is equal to the number of patches used to describe the control surface. If one
or two planes of symmetry are speciﬁed, NPATCSF is the number of patches required to
discretize a half or one quadrant of the whole of the control surface, respectively.
ICDEF is an integer parameter which is used to specify the manner in which the geometry
of the control surface is deﬁned. Four speciﬁc cases are relevant, corresponding respectively
to the representations explained in Sections 6.5, 6.6, 6.7 and 6.8:
ICDEF = 0: The geometry of each patch is a ﬂat quadrilateral, with vertices listed in the
CSF ﬁle (cf. Section 6.5).
ICDEF = 1: The geometry of each patch is represented by B-splines, with the corresponding data in the CSF ﬁle (cf. Section 6.6).
ICDEF = 2: The geometry is deﬁned by inputs from a MultiSurf .ms2 ﬁle (cf. Section 6.7).
ICDEF < 0 or > 2: The geometry of each patch is represented explicitly by a subroutine
in the library GEOMXACT, with optional data in the CSF ﬁle (cf. Section 6.8).
In the last case (ICDEF < 0 or > 2), the parameter ICDEF is used in the same manner
as IGDEF, to select the appropriate subroutine. Any of the existing subroutines which
are normally used to deﬁne body geometry can be used to deﬁne the control surface, for
example specifying ICDEF=-1 deﬁnes the control surface as a circular cylinder with speciﬁed radius and draft. It is important to use diﬀerent subroutines for the body geometry
and control surface, with ICDEF=IGDEF. (The reason for this restriction is that, when
parameters are input by the same subroutine from the GDF ﬁle and CSF ﬁle, these parameters may be overwritten. If it is desired to use the same subroutine for both geometries,
a duplicate copy of the subroutine with a diﬀerent name and assigned value of IGDEF
should be added to GEOMXACT. )
PSZCSF is a parameter which controls the accuracy of the numerical integration over the
control surface, in the same manner that PANEL SIZE is used on the body (cf. Chapter
6). Thus the control surface is subdivided into elements with the approximate length scale
of each element equal to PSZCSF. If the parameter PSZCSF is negative, the subdivision
of the control surface is determined by the parameters NU,NV in the ﬁle gdf .CSP, in an
analogous manner to the use of the same parameters in the spline control ﬁle gdf .SPL
(Section 6.11). TEST22 is an example where the latter procedure is used.

14–5

14.4

OUTPUT

The mean forces and moments using a control surface are output in the OUT ﬁle and in the
numeric output ﬁle optn.9c (or frc.9c) in the same format as optn.9. When IPLTDAT> 0
in the CFG ﬁle, the auxiliary ﬁle gdf csf.dat is output which can be used for visualization
of the control surface using Tecplot. When ILOWHICSF= 1 and ILOWGDF> 0 in the
CFG ﬁle, gdf low.csf is created which contains the data of the control surface in the format
for the low order method.
TEST05, TEST13 and TEST22 show examples of using the control surface in the evaluation of the mean forces and moments.

14–6

REFERENCES
1. J.N. Newman. “Algorithms for the Free-Surface Green Function”. Journal of Engineering Mathematics, Vol. 19, pp. 57-67, 1985.
2. J. N. Newman. “Distribution of Sources and Dipoles over a Quadrilateral”. Journal
of Engineering Mathematics, Vol. 20, pp. 113-126, 1986.
3. J. N. Newman. Marine Hydrodynamics. MIT Press, 1977.
4. J. N. Newman. “The Drift Force and Moment on Ships in Waves”. Journal of Ship
Research, Vol. 11, pp. 51-60, 1967.
5. F. T. Korsmeyer, C.-H. Lee, J. N. Newman and P. D. Sclavounos. “The Analysis
of Wave Eﬀects on Tension-Leg Platforms”. Invited paper of OMAE ‘88 Conference,
Houston, TX, 1988.
6. J. N. Newman, and P. D. Sclavounos. “The Computation of Wave Loads on Large
Oﬀshore Structures”. BOSS ‘88 Conference, Trondheim, Norway, 1988.
7. R. Eatock Taylor and E. R. Jeﬀerys. “Variability of Hydrodynamic Load Predictions
for a Tension Leg Platform”. Ocean Engineering, Vol. 13, No. 5, pp. 449-490, 1986.
8. X. Zhu, “Irregular Frequency Removal from the Boundary Integral Equation for the
Wave-body Problem”, Master Thesis, Dept. of Ocean Eng., MIT, 1994.
9. C.-H. Lee, J.N. Newman, M.-H. Kim & D.K.P. Yue. “The computation of secondorder wave loads”. OMAE ‘91 Conference, Stavanger, Norway, 1991.
10. C.-H. Lee, and J.N. Newman. “First- and second-order wave eﬀects on a submerged
spheroid,” Journal of Ship Research, 1991.
11. J.N. Newman. “The approximation of free-surface Green functions,” in Wave Asymptotics, P. A. Martin & G. R. Wickham, editors, Cambridge University Press, 1992.
12. J.N. Newman and C.-H. Lee “Sensitivity of wave loads to the discretization of bodies”
BOSS ’92, London, England, 1992
13. J.N. Newman “Wave eﬀects on deformable bodies”, Applied Ocean Research, 16, 1,
47-59, 1994

14–1

14. C.-H. Lee and X. Zhu “Second-order diﬀraction and radiation solutions on ﬂoating
bodies” 8th Int’l Workshop on Water Waves and Floating Bodies, St. John’s, Newfoundland, Canada 1993
15. C.-H. Lee, and J.N. Newman. “Second-order Wave Eﬀects on Oﬀshore Structures”
BOSS’94, MIT, 1994
16. C.-H. Lee, J.N. Newman and X. Zhu “An extended boundary-integral-equation method
for the removal of irregular-frequency eﬀects,” International Journal for Numerical
Methods in Fluids, 23, 637-660, 1996.
17. C.-H. Lee, “WAMIT Theory Manual”, Report 95-2, Dept. of Ocean Engineering,
MIT, 1995
18. H. Maniar, “A three dimensional higher-order panel method based on B-splines.
Ph.D. Thesis, Department of Ocean Engineering, MIT, Cambridge, Massachusetts,
1995.
19. C.-H. Lee, H. Maniar, J.N. Newman and X. Zhu, “Computation of wave loads using
a B-spline panel method” Proceedings, 21st Symposium on Naval Hydrodynamics,
Trondheim, Norway, 1996.
20. C.-H. Lee and J. N. Newman, “HIPAN V2.1 User Manual”, MIT, 1999
21. J.E. Kerwin and C.S. Lee, “Prediction of Steady and Unsteady Marine Propeller
Performance by Numerical Lifting-Surface Theory.” Transactions, Society of Naval
Architects and Marine Engineers, 86, 1978.
22. M. E. Mortenson Geometric Modeling, Second Edition, Wiley, 1997. Digital Press,
1999.
23. M. Etzel and K. Dickinson, Digital Visual Fortran Programmer’s Guide, Digital Press,
1999.
24. C.-H. Lee, J. S. Letcher, Jr., R. G. Mack II, J. N. Newman, D. M. Shook and E.
Stanley. “Integation of Geometry Deﬁnition and Wave Analyis Software”. OMAE
2002 Conference, Oslo, June 2002.
25. C.-H. Lee and J. N. Newman, “Boundary-Element Methods in Oﬀshore Structure
Analysis”. OMAE 2001 Conference, Rio de Janeiro, 2001. Also published in Journal
of Oﬀshore Mechanics and Arctic Engineering, Vol. 124, pp 81-89 (2002).
26. C.-H. Lee and J. N. Newman, “Computation of wave eﬀects using the panel method”.
In Numerical Modeling in Fluid-Structure Interaction, Edited by S. Chakrabarti. WIT
Press, 2004.
27. Newman, J.N., “Wave Eﬀects on Vessels with Internal Tanks”. 20th International
Workshop on Water Waves and Floating Bodies, Spitsbergen, Norway, 2005.
28. Lee, C.-H., “Evaluation of quadratic forces using control surfaces”. 2005 WAMIT
Consortium Report, www.wamit.com\report05.pdf, 2005.
14–2

Appendix A

DESCRIPTION OF TEST RUNS
WAMIT V6 includes 23 standard test runs, including 9 low-order and 14 higher-order
applications. These are designed to illustrate various diﬀerent options and features of
WAMIT, and to help users to develop appropriate input ﬁles for their own purposes.
The following table gives relevant features of each test run. In this table the ﬁrst column tst denotes the name of the test run. All of the corresponding input/output ﬁles
(except FNAMES.WAM) are assigned the ﬁlenames TESTtst. (For example, the input
POT ﬁle for the ﬁrst test run listed below is TEST01.POT.) The ﬁrst character of tst is
0 for low-order test runs (ILOWHI=0), and ≥1 for higher-order test runs (ILOWHI=1).
Higher-order test runs which are identical except for diﬀerent input options are assigned
the same number with a letter suﬃx. Thus TEST11 and TEST11a-c describe the same
physical problem using diﬀerent options to represent the geometry (B-splines, exact analytic formulae, MultiSurf, uniform and nonuniform mapping). In TEST14, the ISSC TLP
is analysed and the use of the ﬁxed mode option is illustrated. In TEST14A, the same
geometry is analysed for a large number of input frequencies including zero and inﬁnite
frequencies and the outputs are postprocessed by the F2T utility. In TEST16 the barge is
deﬁned by the subroutine BARGE (IGDEF=-5), and in TEST16a the patches are deﬁned
by ﬂat panels (IGDEF=0). Tests 17, 17a, and 17b illustrate alternative methods for analyzing a body with moonpools, as explained in detail in the corresponding section below.
TEST23 illustrates the use of WAMIT to compute the radiated waveﬁeld from a bank of
wavemakers.

A–1

tst
01
02
03
04
05
06
07
08
09
11
11a
11b
11c
12
13
14
15
16
16a
17
17a
17b
18
19
20
21
22
23
24

geometry
ILOWHI
Circular cylinder
0
Circular cylinder
0
Circular cylinder
0
Barge near wall
0
Cylinder & spheroid
0
ISSC TLP (coarse)
0
ISSC TLP (ﬁne)
0
Elastic column
0
Spar with strakes
0
Circular cylinder
1
Circular cylinder
1
Circular cylinder
1
Circular cylinder
1
Circular cylinder
1
Cylinder & spheroid
1
ISSC TLP
1
Semi-sub
1
Elastic barge
1
Elastic barge
1
Cylinder & moonpool
1
Cylinder & moonpool
1
Cylinder & moonpool
1
Elastic column
1
Catamaran barge
1
MultiSurf barge
1
Spar with strakes
1
FPSO with 2 tanks
1
Bank of wavemakers
1
Motions of a hinged vessel
1

other parameters
IRR=3
ISOR=1
ISY=-1
NBODY=2
NPAN=128
NPAN=1012
NEWMDS=4
NPAND=288
IGDEF=1
IGDEF=-1, INONUMAP=0
IGDEF=-1, INONUMAP=1
IGDEF=2
IGDEF=-1, INONUMAP=0, IRR=1
NBODY=2
IGDEF=-9
IGDEF=-10
IGDEF=-5
IGDEF=0
IGDEF=-7,
IGDEF=-7, NEWMDS=2, IDAMP=0
IGDEF=-7, NEWMDS=2, IDAMP=1
IGDEF=-1, NEWMDS=4
IGDEF=0
IGDEF=2
IGDEF=-12
IGDEF=-21
IGDEF=0
IGDEF=-32

All of the required input ﬁles for each test run, and the labeled output ﬁle (*.out)
are included with the WAMIT software on the CD-ROM disk provided to licensed users.
The same ﬁles can be downloaded with the demonstration programs from the web site
http:://www.wamit.com. The input ﬁles for Test Run tst are named with the ﬁlename
‘testtst’ followed by the extensions .gdf , .pot, and .f rc. The corresponding ﬁles fnames
and config are given the same ﬁlenames with the extension .tst. Thus, before running
TESTtst, the user should copy two ﬁles as follows:
copy fnames.tst fnames.wam
as explained in Chapter 2. Alternatively, the batch ﬁle runtests.bat can be used to run
all tests in succession.
In all of the test runs metric units are used, and the gravitational acceleration is set
equal to 9.80665 meters-per-second2.
A–2

Each test run is described brieﬂy in the following sections. Also included in these sections
are perspective illustrations of the complete underwater geometry, including reﬂections
about the indicated planes of symmetry, and abbreviated listings of the input ﬁles. For the
low-order tests (01-09) the perspective ﬁgures show the subdivisions into panels. For the
higher-order tests (11-22) two perspective ﬁgures are included, to show the subdivisions
into patches (upper or left) and into panels (lower or right).

A–3

A.1 TRUNCATED VERTICAL CYLINDER – TEST01
The added-mass and damping coeﬃcients, exciting forces, motions, wave elevations, ﬁeld
pressures, ﬁeld velocities and drift forces are evaluated for a freely ﬂoating truncated vertical circular cylinder of radius 1 meter and draft 0.5 meters, in inﬁnite water depth for
three wave periods and one wave heading.
The origin of the coordinate system is located at the intersection of the vertical axis of
the cylinder and the undisturbed position of the free surface. Using two planes of symmetry,
only the ﬁrst quadrant of the surface of the cylinder is discretized with 256 panels. 16,
8, and 8 panels are distributed in the azimuthal, radial, and vertical directions with equal
spacing. The characteristic length is set equal to the radius of the cylinder. The cylinder
center of gravity is located at the origin of the coordinate system, and the radii of gyration
relative to its axes are taken equal to 1 meter.

A–4

TEST01.GDF: (lines 1-8 only):
TEST01.GDF -- circular cylinder, R=1, T=0.5, ILOWHI=0
1.000000
9.806650
ULEN, GRAV
1
1
ISX, ISY
256
NEQN
0.0000000E+00 0.0000000E+00 -0.5000000
0.0000000E+00 0.0000000E+00 -0.5000000
0.1243981
1.2252143E-02 -0.5000000
0.1250000
0.0000000E+00 -0.5000000
TEST01.POT:
TEST01.POT -- cylinder R=1, T=0.5, ILOWHI=0, IRR=0
-1. 0. 0. 0. 0.
HBOT, XBODY(1-4)
1
1
IRAD, IDIFF
1 1 1 1 1 1
IMODE(1-6)
3
NPER (array PER follows)
8.971402 2.006403 1.003033
1
NBETA (array BETA follows)
0.
TEST01.FRC:
TEST01.FRC Circular cylinder, ILOWHI=0, IRR=0
1
1
1
1
0
1
1
2
0
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test01.cfg:
maxscr=1024 (assign a maximum block of 1024*1024 RAM for scratch LHS)
ISOR=1
(omit ISOR in POT file, include source formulation)
ISOLVE=0
(use iterative solver)
ISCATT=0
(solve for total diffraction potential, not scattering)
IQUAD=0
(omit IQUAD in POT file, use single-node quadrature)
ILOG=1
(omit ILOG in POT file, integrate log singularity)
IDIAG=0
(omit IDIAG in POT file, panel length based on area)
IRR=0
(omit IRR in POT file, no irregular-frequency removal)
MONITR=0
(do not write FORCE output data to monitor)
NUMHDR=1
(write headers to numeric output files)
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–5

A.2 IRREGULAR-FREQUENCY REMOVAL – TEST02
This test run illustrates the use of the irregular-frequency option described in Chapter 9.
The geometry and most other inputs are the same as in TEST01, as described above. The
parameter IRR= 3 is set to use automatic panelization of the interior free surface. Wave
periods are chosen so that the wave frequencies are near the ﬁrst and second irregular
frequencies of the cylinder.
The maximum number of iteration MAXITT is increased to 100 in the conﬁguration
ﬁle for this test run.
Input GDF is the same as TEST01.GDF. The following ﬁgure shows additional panels
on the free surface which are generated in WAMIT.

A–6

TEST02.GDF: (lines 1-8 only):
TEST02.GDF circular cylinder, R=1, T=0.5, ILOWHI=0, IRR=3,
1.000000
9.806650
1
1
256
0.0000000E+00 0.0000000E+00 -0.5000000
0.0000000E+00 0.0000000E+00 -0.5000000
0.1243981
1.2252143E-02 -0.5000000
0.1250000
0.0000000E+00 -0.5000000
TEST02.POT:
TEST02.POT -- Circular cylinder, ILOWHI=0, IRR=3
-1. 0.0 0.0 0.0 0.0
1
1
1 1 1 1 1 1
2
1.182288
1.003025
1
0.0
TEST02.FRC:
TEST02.FRC -- Circular cylinder, ILOWHI=0, IRR=3
1
1
1
1
0
1
1
1
1
0.000000
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
0
2
1.5 0.0 0.0
1.5 0.0 -0.5
test02.cfg:
IRR=3
ISOR=1
ISOLVE=4
IQUAD=0
ILOG=1
IDIAG=0
NEWMDS=0
MONITR=0
NUMHDR=1
maxscr=1024
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)
MAXITT=100

A–7

A.3 LOCAL PRESSURE DRIFT FORCE – TEST03
This test run is used to illustrate the use of the source formulation (Section 5.2) to determine the mean drift force and moment from local pressure integration. The motions
and the drift forces are evaluated for a freely ﬂoating truncated vertical circular cylinder
of radius 1 meter and draft 1 meter, in a water depth of 7.14 meter for four wave periods
and one wave heading.
The origin of the global coordinate system is located at the intersection of the vertical
axis of the cylinder and the undisturbed position of the free surface. The origin of the body
ﬁxed coordinate system is shifted -0.515 meters under the free-surface. Using two planes of
symmetry, the ﬁrst quadrant of the surface of the cylinder is discretized with 288 panels.
12, 8, and 16 panels are distributed in the azimuthal, radial, and vertical directions with
cosine spacing at the free surface and corner. The characteristic length is set equal to the
radius of the cylinder. The cylinder center of gravity is located at the origin of the body
coordinate system, and the radii of gyration relative to its axes are shown in the FRC ﬁle.
Generally speaking, the evaluation of mean drift forces is more accurate when the momentum conservation method is used, since this does not depend on local velocities on the
body surface. However the momentum method cannot be used to evaluate the vertical
component of the drift force. Tests with larger number of panels are reported in Reference
[12] to show the sensitivity of the results to the discretization of the body. The vertical
drift force predicted by the mean pressure integration shows slow convergence near the
heave resonance frequency (KL=0.66), due to cancelation between two large contributions
of opposite signs (the second integration in equations (12.47) and (12.48)), when the heave
motion amplitude is large.

A–8

TEST03.GDF: (lines 1-8 only):
TEST03.GDF Cylinder, R=T=1, ILOWHI=0, ISOR=1
1.000000
9.806650
1
1
288
0.0000000E+00 0.0000000E+00 -0.4850000
0.0000000E+00 0.0000000E+00 -0.4850000
0.1934213
2.5464399E-02 -0.4850000
0.1950903
0.0000000E+00 -0.4850000
TEST03.POT:
TEST03.POT Cylinder, R=T=1, ILOWHI=0, ISOR=1
7.14 0.0 0.0 -0.515 0.0
1
1
1 1 1 1 1 1
4
2.837491 2.398118 2.006409 1.638226
1
0.0
TEST03.FRC:
TEST03.FRC Cylinder, R=T=1, ILOWHI=0, ISOR=1
0
0
0
1
0
0
0
1
1
0.000000
0.742000
0.000000
0.000000
0.000000
0.742000
0.000000
0.000000
0.000000
1.000000
0
0
test03.cfg:
maxscr=1024
ISOR=1
ISOLVE=0
ISCATT=0
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
MONITR=0
NUMHDR=1
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–9

A.4 BODY NEAR A WALL – TEST04
The option to analyze a body near one or two vertical walls is described in Section 5.3. In
this test run a rectangular barge of length 80m, beam 20m, draft 10m is positioned with
its longitudinal axis parallel to one wall, separated by a gap of 2m. Incident head waves
are considered, and computations are made of the surge, heave, and pitch coeﬃcients,
RAO’s, and drift force and moment in incident waves which propagate parallel to the wall
(BETA=0).
In the GDF ﬁle one half of the barge is discretized, forward of the midship section x = 0.
Both the port and starboard sides of the barge are included in the GDF ﬁle, hence the
appropriate symmetry indices for this case are ISX=1, ISY=-1.
Since the incident waves propagate parallel to the wall this problem is identical to the
‘barge catamaran’ studied in [6], and in TEST19; the only modiﬁcations required in the
latter case are (1) a lateral oﬀset equal to the sum of the half-beam and gap must be added
to the y-coordinates of the panels in the GDF ﬁle; (2) ISY=1; and (3) the forces and
moments calculated for the catamaran are the total acting on both hulls. The deﬁnition of
the incident-wave amplitude diﬀers between these diﬀerent problems, however, due to the
convention for the wave amplitude in the presence of a wall (Section 5.3). In the present
case, where the incident-wave angle is zero and the waves propagate parallel to the wall, the
wave system in the absence of the body is a progressive wave with total physical amplitude
2A.
It also is possible to replicate the present results with the NBODY option, specifying
two independent hulls in place of the rigid constraint implied by the catamaran; this is a
less eﬃcient computational approach since the planes of symmetry are not exploited.
The ﬁgure below shows the catamaran conﬁguration or, equivalently, the original hull
plus its image with respect to the wall.

A–10

TEST04.GDF: (lines 1-8 only):
TEST04.GDF -- Barge near wall, ILOWHI=0
40.00000
9.806650
1
-1
640
3.920686
10.00000
-0.3806022
0.0000000E+00 10.00000
-0.3806022
0.0000000E+00 10.00000
0.0000000E+00
3.920686
10.00000
0.0000000E+00
TEST04.POT:
TEST04.POT -- Barge near wall, ILOWHI=0
-1. 0. 12. 0. 0.
0
0
1 0 1 0 1 0
3
6. 7. 8.
1
0.0
TEST04.FRC:
TEST04.FRC -- Barge near wall, ILOWHI=0
1
1
1
1
0
0
0
0
1
3.0
20.00000
0.000000
0.000000
0.000000
5.000000
0.000000
0.000000
0.000000
20.00000
0
0
test04.cfg:
maxscr=1024
ISOR=1
ISOLVE=0
ISCATT=0
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
MONITR=0
NUMHDR=1
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–11

A.5 MULTIPLE BODIES – TEST05
The NBODY option described in Chapter 7 is illustrated in this test run. Body one is a
circular cylinder of radius 1 meter and draft 2 meters. Body 2 is a spheroid of length 4
meters and maximum radius 0.25 meters. The gap between these two bodies is set equal
to the beam of the spheroid (0.5 meters) and the origin of the global coordinate system
is located at the mid-point of this gap. The relative locations of the two bodies and the
orientation of the spheroid are speciﬁed in the GGDF ﬁle. One quadrant of the cylinder is
discretized with 112 panels. 8,6 and 8 panels are distributed in the azimuthal, radial, and
vertical directions using cosine spacing in radial and vertical directions. One quadrant of
the spheroid is discretized with 64 panels. 8 and 8 panels are distributed in the longitudinal,
and transverse directions using cosine spacing in the longitudinal direction.
The Alternative 1 input format is used for the POTEN subprogram and the Alternative
3 input format is used for FORCE. The separate FRC ﬁles TEST05C, TEST05S are used
with IALTFRC=3. The vector IALTFRCN is included in TEST05.CFG to indicate that
IALTFRC=1 in the separate FRC ﬁles for each body.
The added-mass and damping coeﬃcients, exciting forces, motions, wave elevations,
ﬁeld pressures and ﬁeld velocities, and drift forces are evaluated in inﬁnite water depth for
two wave periods and one wave heading.
The option is used to evaluate the mean drift force and moment using a control surface,
following the instructions in Chapter 14. The parameter ICTRSURF=1 is assigned in
the CFG ﬁle. The control surfaces surrounding the cylinder and spheroid are deﬁned by
the input ﬁles TEST05c.csf and TEST05s.csf. In order to illustrate the alternatives, the
control surface for the cylinder uses low-order panels (ILOWHICSF=0) and the control
surface for the spheroid is generated with the higher-order (ILOWHICSF=1) subroutine
ELLIPSOID CS in the GEOMXACT DLL library. The corresponding output for the mean
drift force and moment is contained in the ﬁle TEST05.9c. It should be noted that the
higher-order control surface for the spheroid does not include the intermediate free surface
patch, and thus the horizontal drift force is correct whereas the vertical drift force is
not complete. The reason for omitting the free surface patch here is that the low-order
solution for the body does not give a suﬃciently robust evaluation of ﬁeld velocities and
wave elevations at points on the free surface that are very close to the body. The low-order
control surface is more suitable for use with low-order body representations, in this respect,
provided the panels on the free surface have dimensions similar to the dimensions of the
adjacent panels on the body.

A–12

TEST05.GDF:
TEST05.GDF -- GGDF file for use with IGDFPOT_OPTIONS=0
-1. 9.80665
2
test05c.gdf
1.25 0.0 0.0 0.0
1
1
1
1
1
1
test05s.gdf
-0.5 0.0 0.0 90.0
1
1
1
1
1
1
TEST05c.GDF: (lines 1-8 only):
Cylinder R=1 T=2 8*(6+8)
1.000000
9.806650
1
1
112
0.0000000E+00 0.0000000E+00
0.0000000E+00 0.0000000E+00
0.2538459
5.0493091E-02
0.2588190
0.0000000E+00

-2.000000
-2.000000
-2.000000
-2.000000

TEST05s.GDF: (lines 1-8 only):

A–13

Spheroid, Slendernes =0.125 Halflength=2m 8*8
2.000000
9.806650
1
1
64
2.000000
-0.0000000E+00 -0.0000000E+00
1.961571
-7.9460625E-09 -4.8772585E-02
1.961571
9.5150545E-03 -4.7835436E-02
2.000000
0.0000000E+00 -0.0000000E+00
TEST05.POT:
TEST05.POT -- Cylinder + spheroid, ILOWHI=0
-1.0 0.0 0.0 0.0 0.0
0
0
0 0 0 0 0 0
2
1.5 2.0
1
0.0
TEST05.FRC:
TEST05.FRC -- Cylinder + spheroid, ILOWHI=0
1 1 1 1 0 1 1 1 1
1.0
test05c.frc
test05s.frc
0
1
0. 0. 0.
TEST05c.FRC:
CYL.FRC
0
0
0
0.000000
1.000000
.0000000
.0000000
0
0
TEST05s.FRC:
SPD.FRC
0
0
0
0.000000
1.000000

0

0

0

.0000000
1.000000
.0000000

0

0

.0000000

0

0

0

.0000000
.0000000
1.000000

0

0

0

.0000000

A–14

0

.0000000
.0000000
0
0

1.000000
.0000000

.0000000
1.000000

TEST05c.csf:
ccylinder R=1.2 T=2.2 -- analytic CONTROL SURFACE (npatch=3)
0
ILOWHICSF
1
1 ISX ISY
160
NPAN
0.12000E+01 0.00000E+00 0.00000E+00
0.12000E+01 0.00000E+00 -0.27500E+00
0.11769E+01 0.23411E+00 -0.27500E+00
0.11769E+01 0.23411E+00 0.00000E+00

TEST05s.csf:
ELLIPSOID CONTROL SURFACE for ellisoid GDF-- igdef=-1003 without free surface portion
1
ILOWHICSF
1 1
ISX ISY
1 -1003 0.5 NPATCH IGDEF PSZCSF
2
NLINES
2.2 0.3 0.3
A, B, C
2.0 0.25 (x and y maximum of ellipsoid GDF)
test05.cfg:
IPLTDAT=1
maxscr=1024
ISOR=0
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
NUMHDR=1
NOOUT=0 1 1 1 0 1 1 0 0
IALTPOT = 1
! GDF names in GGDF file
IALTFRC = 3
! Alternative Form 3 FRC
IALTFRCN= 1 1
ICTRSURF=1
! Evaluate control surface drift forces
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–15

A.6 THE ISSC TENSION-LEG PLATFORM – TEST06
The added-mass, damping coeﬃcients, exciting forces, motions and wave loads are evaluated for the ISSC Tension-Leg-Platform, in a ﬁnite water depth of 450 meters, for three
wave periods and one wave heading. The TLP consists of four circular cylindrical columns
and four rectangular pontoons as shown in the plots of the panel discretization. The radius
of each column is 8.435 meters. The width and height of each pontoon are 7.5 meters and
10.5 meters, respectively. The distance between the centers of adjacent columns is 86.25
meters. Further information is given by Eatock Taylor and Jeﬀerys [7].
Two planes of symmetry are used with 128 panels in one quadrant. Thus there are a
total of 512 panels on the complete surface. The origin of the coordinate system is located
at the intersection of the undisturbed free surface and the two planes of symmetry. The
characteristic length is set equal to 43.125 meters, which corresponds to half of the distance
between the centers of adjacent columns.
Only head seas are considered, with β = 0 speciﬁed in the .pot ﬁle. For this reason,
only the modes (surge, heave, pitch) are analyzed with IRAD=IDIFF=0, and these modes
are speciﬁed on line 3.
In the .frc ﬁle the horizontal modes (1,2,6) are free and the vertical modes (3,4,5) are
ﬁxed, to represent a TLP moored by vertical tendons. The Alternative 1 form is used,
with the result that the body mass is evaluated as if the TLP is freely ﬂoating (see Section
3.5).
The output shows the conventional response amplitude operator for surge, and the wave
loads for heave and pitch.

A–16

TEST06.GDF: (lines 1-8 only):
TEST06.GDF -- ISSC TLP, coarse
43.125
9.80665
1
1
128
49.09267
49.09267
51.56456
51.56456

discretization

37.15733
37.15733
43.12500
43.12500

0.00000
-5.12567
-5.12567
0.00000

TEST06.POT:
TEST06.POT -- ISSC TLP, coarse discretization
450. 0. 0. 0. 0.
0 0
1 0 1 0 1 0
3
5. 10. 15.
1
0.
TEST06.FRC:
TEST06.FRC -- ISSC TLP, coarse discretization
1 1 1 -1 1 0 0 0 0 IOPTN (IOPTN(4)<0 signifies fixed modes)
6
NDFR
1 1 0 0 0 1
IMODE
3.0
VCG
38.876
0.
0.
0.
38.876
0.
0.
0.
42.420
0
NBETAH
0
NFIELD
test06.cfg:
maxscr=1024
ISOR=0
ISOLVE=0
ISCATT=0
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)
A–17

A.7 THE ISSC TENSION-LEG PLATFORM – TEST07
This test run is intended to reﬁne the analysis of the ISSC TLP described in TEST06. 1012
panels are used on each quadrant, resulting in 4048 panels for the complete structure. The
block iterative solver is used (ISOLVE=4) to provide a relatively fast but robust solution.
For the sake of variety in the analysis of the diﬀraction problem, the solution for the
scattered potential is computed (ISCATT=1).
Alternative form 2 of .FRC is used, but the mass is assumed to be equal to the displacement computed by WAMIT. Note that the displacement is about 4% greater than for
Test Run 2, due to the more accurate description of the columns. (The panel vertices are
deﬁned to lie on the exact circular cylinder surface, hence the ﬂat panels deﬁne a surface
with less displaced volume than the exact body.)
Comparisons should be made with the output ﬁles from TEST06 to judge the convergence of the results with increasing numbers of panels.

A–18

TEST07.GDF: (lines 1-8 only):
TEST07.GDF ISSC TLP -- ILOWHI=0, fine discretization
43.125
9.806650
1
1
1012
49.09267
37.15733
0.00000
49.09267
37.15733
-0.33626
50.43388
38.90522
-0.33626
50.43388
38.90522
0.00000
TEST07.POT:
TEST07.POT ISSC TLP -- ILOWHI=0, fine discretization
450. 0. 0. 0. 0.
0 0
1 0 1 0 1 0
3
5. 10. 15.
1
0.
TEST07.FRC:
TEST07.FRC ISSC TLP -- ILOWHI=0, fine discretization -- IALTFRC=2
1 1 1 -2 0 0 0 0 0 IOPTN (IOPTN(4)<0 signifies fixed modes)
6
NDFR
1 1 0 0 0 1
IMODE
1.
RHO
0. 0. 3.0
XCG
1
IMASS
53066.4
0.
0.
0.
159199.2
0.
0.
53066.4
0. -159199.2
0.
0.
0.
0.
53066.4
0.
0.
0.
0. -159199.2
0. 8.0201552E7
0.
0.
159199.2
0.
0.
0.
8.0201552E7
0.
0.
0.
0.
0.
0. 9.54906731E7
0
IDAMP
0
ISTIFF
0
NBETAH
0
NFIELD

A–19

test07.cfg:
maxscr=1024
ISOR=0
ISOLVE=4
ISCATT=1
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
MONITR=0
NUMHDR=1
IALTFRC=2
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–20

A.8 ELASTIC COLUMN WITH GENERALIZED MODES – TEST08
This test run evaluates the force coeﬃcients and RAO’s for a bottom-mounted vertical
cylinder of circular cross-section, with four bending modes deﬁned by shifted Jacobi polynomials. The hydroelastic analysis of these bending modes is analyzed using the generalized
body mode option described in Chapter 8. Further details are given in Reference [13]. The
cylinder extends from the free surface, where it is free, down to the bottom, at a depth of
200m, where it is clamped. The cylinder radius is 10m. Since the cylinder is clamped at
the bottom the six rigid-body modes are all ﬁxed, and speciﬁed by the values MODE(j)=0
in the POT ﬁle.
External mass and stiﬀness matrices are deﬁned in the (Alternative 2) FRC ﬁle. The
cylinder is considered to have a constant distributed mass equal to half of the displaced
mass of ﬂuid, and also a concentrated mass at the free surface equal to the displaced
mass. The stiﬀness factor EI for the beam equation is assumed constant with the value
0.41m0 h3 , where m0 is the concentrated mass and h is the ﬂuid depth. No matrix elements
are required for the square submatrix (i, j) ≤ 6 since the body is ﬁxed in these modes.
Further details for this case are given in [13].
The cylinder geometry is deﬁned with two planes of symmetry and 512 panels on one
quadrant. The length scale ULEN is speciﬁed as 1.0 to simplify the deﬁnitions of modes and
output quantities. The generalized modes are deﬁned in the subroutine DEFMOD.FOR,
which is distributed to licensed users with the test ﬁles on the CD-ROM disk. The use of
DEFMOD is described in Chapter 8. The output ﬁle from DEFMOD, TEST08.MOD, is
included with the test ﬁles so that this test can be run without prior use of DEFMOD.
Only one wave period is considered here, which coincides with resonant bending motion of
the cylinder.

A–21

TEST08.GDF: (lines 1-8 only):
TEST08.GDF vertical cylinder, 16*32, cosine spacing at free surface
1.0000
9.80665
1
1
512
10.0000
0.000000
-200.000
9.95185
0.980171
-200.000
9.95185
0.980171
-190.186
10.0000
0.000000
-190.186
TEST08.POT:
TEST08.POT -- bending of vertical column at resonance, 200m depth
200.0 0.0 0.0 0.0 0.0
0 0 0
0
0
0
0
0
0
0
0
1
6.5
1
0.0
TEST08.FRC:
TEST08.FRC file,
1
1
1.0
.0000000
1
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0
1
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

vertical column with 4 bending modes
1
1
0
0
0
0
0
.0000000

1.000000

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
69115.
62832.
62832.
62832.

0.
0.
0.
0.
0.
0.
62832.
67320.
62832.
62832.

0.
0.
0.
0.
0.
0.
62832.
62832.
66323.
62832.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

A–22

0.
0.
0.
0.
0.
0.
62832.
62832.
62832.
65688.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0
0

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

test08.cfg:
MAXSCR=1024
NUMHDR=1
NUMNAM=0
ISOR=0
IRR=0
MONITR=0
newmds=4
IALTPOT=1
IALTFRC=2
USERID_PATH=\WAMITv6

0.
0.
0.
0.
0.

0.
0.
0.
0.
0.

0.
0.
0.
0.
103044.
412177. 824354. 1339575.
412177. 4430902. 9789203. 16487078.
824354. 9789203. 37899671. 64382041.
1339575. 16487078. 64382041. 162406554.

(directory for *.exe, *.dll, and userid.wam)

A–23

A.9 SPAR WITH THREE STRAKES – TEST09
This test run analyzes a circular cylinder with three spiral strakes. The strakes are modeled
as zero-thickness dipole panels, following the method described in Section 5.4. The radius
of the cylinder is 18m and the draft is 200m. The strake width is 3.7m. There are no planes
of symmetry, due to the twist of the strakes. A total of 960 panels are used, including 672
on the cylinder plus 288 on the strakes. The excerpts from the GDF ﬁle include the ﬁrst
body panel and also the ﬁrst dipole panel. A perspective view of the discretized structure
is shown in the ﬁgure.
The FORCE run includes all options which can be evaluated without using the source
formulation (ISOR=1), since the latter option cannot be used with dipole panels. The
body pressure ﬁle TEST09.5p includes the pressure on the body panels, and the pressure
jump on the dipole panels. The corresponding panel centroids are listed in the output ﬁle
TEST09.PNL.

A–24

TEST09.GDF: (lines 1-8 and the
SPAR R, D, W, T, NS, TWIST=
18.00000
9.806650
0
0
672
18.00000
0.0000000E+00
17.38667
4.658743
15.58846
9.000000
17.38667
4.658743
.
.
.
288
18.00000
0.0000000E+00
17.38667
4.658743
18.28015
4.898150
18.92500
0.0000000E+00
.
.
.

first 5 lines for dipole panels)
18.00 200.00
3.70 0.000000

0.0000000E+00
-8.333333
-8.333333
0.0000000E+00

0.0000000E+00
-8.333333
-8.333333
0.0000000E+00

TEST09.POT:
TEST09.POT (Spar with three strakes)
-1. 0. 0. 0. 0.
HBOT, XBODY(1-4)
1
1
IRAD, IDIFF
1 1 1 1 1 1
IMODE(1-6)
3
NPER (array PER follows)
0.1 0.5 1.0
2
NBETA (array BETA follows)
0.0 45.
(end of file)
TEST09.FRC:
TEST09.FRC (Spar with three strakes)
1
1
1
1
1
1
1
2
0.000000
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
0
2
23. 0. 0.
15. 15. -0.5

A–25

0

IOPTN(1-9)
VCG
XPRDCT
NBETAH
NFIELD
(end of file)

3

1.000

test09.cfg:
MAXSCR=1000
ISOR=0
ISOLVE=0
ISCATT=0
IQUAD=0
ILOG=0
IDIAG=0
IRR=0
MONITR=0
NUMHDR=1
IPERIO=3
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–26

A.11 CIRCULAR CYLINDER – TEST11
The same cylinder used for the low-order TEST01 is used here with the higher-order option
(ILOWHI=1). Two alternatives are used for the geometry.
In TEST11 the geometry is deﬁned by B-splines (IGDEF=1). The parameters, knot
vectors, and coeﬃcients for each patch are contained in the ﬁle TEST11.GDF. It should
be noted that the circular patches and boundaries cannot be ﬁt exactly with B-splines;
however the geometric errors are generally much smaller in this case, compared to the ﬂatpanel representation in TEST01. For example the maximum error of any point output in
the data ﬁle test11.pnl is less than 3E-5, and the maximum error in the computed volume
is 1E-5. By comparison, using the ﬂat-panel discretization in TEST01, the maximum
error in the computed volume is 3E-3. Thus, when the higher-order method is used, the
principal errors in the results should be associated with the approximation of the potential
by B-splines, as opposed to the representation of the geometry. This approximation can be
systematically reﬁned by increasing the number of panels, or by using the PANEL-SIZE
option in the CONFIG.WAM (or CFG) ﬁle and reducing the value of this parameter.
In TEST11a the geometry is deﬁned analytically by the GEOMXACT.F subroutine
CIRCCYL (IGDEF=-1). The radius and draft of the cylinder are input in TEST11a.GDF.
The parameter INONUMAP=0 speciﬁes uniform mapping. Comparison of the output ﬁles
with TEST01 and TEST11 conﬁrms the statements above regarding accuracy. Most of
the output data from TEST11 and TEST11a agree to at least ﬁve decimals, except for the
third wave period which coincides with an irregular frequency.
In TEST11b the geometry is deﬁned analytically, in the same manner as for TEST11a,
except that nonuniform mapping is speciﬁed by the parameter INONUMAP=1 as explained
in Section 6.8. This modiﬁcation gives a more accurate solution near the corner and waterline, which is particularly beneﬁcial for the pressure drift force evaluation. Comparison
between the outputs for the momentum and pressure drift force shows that the results are
more consistent in this case, compared to the use of uniform mapping in TEST11a. More
extensive comparisons for the same geometry are included in Reference 24.
TEST11c illustrates the use of the option IGDEF=2, where the geometry is described
by MultiSurf (see Section 6.7 and Appendix 2). In this case the same nonuniform mapping
is used as in TEST11b, using the relabeling technique in MultiSurf. Comparison of the
results with TEST11b indicates that they are practically identical.

A–27

TEST11.GDF: (lines 1-8 only):
TEST11 cylinder R=1 T=0.5 defined by B-splines (IGDEF=1)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
2 1
NPATCH, IGDEF
4
2
4
4
-1.00000000000000
-1.00000000000000
-1.00000000000000
-1.00000000000000
-0.500000000000000
0.000000000000000E+000
TEST11.SPL:
TEST11 cylinder R=1 T=0.5 defined by B-splines (IGDEF=1)
4 2
NU NV (Patch 1, side
u azimuthal v vertical)
4 2
NU NV (Parch 2, bottom u azimuthal v radial)
TEST11.POT:
TEST11.POT Cylinder R=1, T=0.5, igdef=1
-1.
1
1
IRAD, IDIFF

A–28

3
8.971402 2.006403 1.003033
1
0.
1
test11.gdf
0. 0. 0. 0.
1 1 1 1 1 1
0

NPER (array PER follows)
NBETA (array BETA follows)
NBODY
XBODY
IMODE(1-6)
NEWMDS

TEST11.FRC:
TEST11.FRC Cylinder R=1, T=0.5, igdef=1
1
1
1
1
3
1 1 2 2
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test11.cfg:
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
MONITR=0
NOOUT= 1 1 1 1 0 1 1 1 1
NUMHDR=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–29

TEST11A.GDF: (lines 1-8 only):
TEST11a cylinder R=1 T=0.5 -- analytic geometry, uniform mapping
1. 9.80665 ULEN GRAV
1 1
ISX ISY
2 -1
NPATCH IGDEF
2
NLINES
1.0 0.5
RADIUS, DRAFT
0
INONUMAP (uniform mapping)
TEST11A.SPL:
TEST11a.spl - cylinder R=1 T=0.5 -- analytic geometry (npatch=2)
4 2
NU NV (Patch 1, side
u azimuthal v vertical)
4 2
NU NV (Parch 2, bottom u azimuthal v radial)
TEST11A.POT:
TEST11A.POT Cylinder R=1, T=0.5, igdef=-1
-1.
1
1
IRAD, IDIFF
2
NPER (array PER follows)
8.971402 2.006403
2
NBETA (array BETA follows)
0. 45.
1
NBODY
test11a.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST11A.FRC:
TEST11a.FRC Cylinder R=1, T=0.5, igdef=-1
1
1
1
1
3
1 1 2 2
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test11a.cfg:
ILOWHI=1
IALTPOT=2

A–30

IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–31

TEST11B.GDF: (lines 1-8 only):
TEST11a cylinder R=1 T=0.5 -- analytic geometry, nonuniform mapping
1. 9.80665 ULEN GRAV
1 1
ISX ISY
2 -1
NPATCH IGDEF
2
NLINES
1.0 0.5
RADIUS, DRAFT
1
INONUMAP (nonuniform mapping)
TEST11B.POT:
TEST11B.POT Cylinder R=1, T=0.5, igdef=-1
-1.
1
1
IRAD, IDIFF
2
NPER (array PER follows)
8.971402 2.006403
2
NBETA (array BETA follows)
0. 45.
1
NBODY
test11b.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST11B.SPL:
TEST11b.spl - cylinder R=1 T=0.5 -- analytic geometry (npatch=2)
4 2
NU NV (Patch 1, side
u azimuthal v vertical)
4 2
NU NV (Parch 2, bottom u azimuthal v radial)
TEST11B.FRC:
TEST11B.FRC Cylinder R=1, T=0.5, igdef=-1
1
1
1
1
3
1 1 2 2
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test11B.cfg:
ILOWHI=1
IALTPOT=2
IRR=0

A–32

ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–33

TEST11C.GDF: (lines 1-8 only):
TEST11 cylinder R=1 T=0.5 -- MultiSurf .ms2 input, nonuniform mapping
1. 9.80665 ULEN GRAV
1 1
ISX ISY
2 2
NPATCH IGDEF
3
NLINES
TEST11C.MS2
(name of .ms2 file)
*
default wetted surface (use all patches)
0 0 0
default settings: FAST, DivMult, outward normal
TEST11C.POT:
TEST11C.POT Cylinder R=1, T=0.5, igdef=-1
-1.
1
1
IRAD, IDIFF
2
NPER (array PER follows)
8.971402 2.006403
2
NBETA (array BETA follows)
0. 45.
1
NBODY
test11c.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST11C.SPL:
TEST11c.spl - cylinder R=1 T=0.5 -- analytic geometry (npatch=2)
4 2
NU NV (Patch 1, side
u azimuthal v vertical)
4 2
NU NV (Parch 2, bottom u azimuthal v radial)
TEST11C.MS2:
MultiSurf 1.23
// Truncated cylinder for base HIPAN example
// Full cosine spacing on side_surf using type-3 BLoftSurf (analytic)
// Half cosine spacing on bottom
Units: m kg
Symmetry: x y
Extents: -1.000 -1.000 -0.500 1.000 1.000 0.000
View: -20.00 120.00 0
Places: 3
Layers: FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
DivMult: 1
BeginModel;
AbsPoint top 14 1 / 0.000 0.000 0.000 ;
EulerFrame F0 15 1 / top * 0.0000 0.0000 0.0000 ;

A–34

FrameAbsPt draft 15 1 / F0 0.000 0.000 -0.500 ;
FrameRelPt radius 15 1 / F0 draft 1.000 0.000 0.000 ;
Line axis 6 1 1x1 / * draft top ;
Helix chine 11 1 8x4 / * radius axis 0.000000 90.0000 ;
Plane2 top_plane 7 1 / top draft ;
ProjCurve top_edge 11 1 8x4 / * chine top_plane ;
ObjectList wetted_surfs /
{ side_surf bottom_surf } ;
BLoftSurf bottom_surf 6 3 8x4 4x4 1 / * 2
{ chine chine draft } ;
BLoftSurf side_surf 2 3 8x4 4x4 1 / * 3
{ top_edge top_edge chine chine } ;
EndModel;
TEST11C.FRC:
TEST11C.FRC Cylinder R=1, T=0.5, igdef=-1
1
1
1
1
3
1 1 2 2
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test11C.cfg:
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–35

A.12 IRREGULAR-FREQUENCY REMOVAL – TEST12
TEST12 is the higher-order analog of TEST02, intended to illustrate the removal of
irregular-frequency eﬀects using the higher-order method. As in TEST11a, the geometry
is deﬁned analytically (IGDEF=-1) and the dimensions are input in the ﬁle TEST12.GDF.
In this case NPATCH=3 is speciﬁed, where the additional patch corresponds to the interior
free surface as required for the irregular-frequency option. The ﬁgures below illustrate the
extra patch and panels.

A–36

TEST12.GDF:
TEST12 cylinder R=1 T=0.5 -- analytic geometry (npatch=3)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
3 -1
NPATCH IGDEF
2
NLINES
1.0 0.5
RADIUS, DRAFT
0
INONUMAP (uniform mapping)
TEST12.SPL:
TEST12.spl - cylinder R=1 T=0.5 -- analytic geometry (npatch=3)
4 2
NU NV (Patch 1, side
u azimuthal v vertical)
4 2
NU NV (Parch 2, bottom u azimuthal v radial)
4 2
NU NV (Parch 3, interior free surface)
TEST12.POT:
TEST12.POT Cylinder R=1, T=0.5, igdef=-1, npatch=3 (IRR=1)
-1.
1
1
IRAD, IDIFF
3
NPER (array PER follows)
8.971402 2.006403 1.003033
2
NBETA (array BETA follows)
0. 45.
1
NBODY
test12.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST12.FRC:
TEST12.FRC Cylinder R=1, T=0.5, igdef=-1 (irr=1)
1
1
1
1
3
1 1 2 2
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1.5 0. 0.
1.5 0. -0.5
(end of file)
test12.cfg:
ILOWHI=1
IALTPOT=2

A–37

IRR=1
ILOG=1
ISOLVE=1
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–38

A.13 MULTIPLE BODIES – TEST13
This test uses the same cylinder and spheroid as in the low-order TEST05. IALTPOT=2
is used to specify the separate input ﬁles TEST13C.GDF and TEST13S.GDF. TEST13C
uses IGDEF=-1 as in TEST11a. TEST13S uses the ELLIPSOID subroutine (IGDEF=-4)
with the semi-axes (2.0, 0.25, 0.25) speciﬁed in TEST13S.GDF. The same separate FRC
ﬁles TEST05C, TEST05S are used with IALTFRC=3. The vector IALTFRCN is included
in TEST13.CFG to indicate that IALTFRC=1 in the separate FRC ﬁles for each body.
(Normally it is necessary to duplicate the FRC ﬁles for analogous runs, as for example in
TEST01 and TEST11, since the output ﬁlenames are assigned based on the FRC ﬁlename.
This is not necessary for individual FRC ﬁles for each body when multiple bodies are
analyzed, since these do not aﬀect the output ﬁlenames.)
The option is used to evaluate the mean drift force and moment using a control surface,
following the instructions in Chapter 14. The parameter ICTRSURF=1 is assigned in
the CFG ﬁle. The control surfaces surrounding the cylinder and spheroid are deﬁned by
the input ﬁles TEST13c.csf and TEST13s.csf. These control surfaces are generated by
the subroutines CIRCYL CS and ELLIPSOID CS in the GEOMXACT DLL library. The
surfaces generated by these subroutines include the portion of the free surface between the
body and outer control surface. The corresponding output for the mean drift force and
moment is contained in the ﬁle TEST13.9c.

A–39

TEST13c.GDF:
TEST13C cylinder R=1 T=2 -- analytic geometry (npatch=2)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
2 -1
NPATCH IGDEF
2
NLINES
1.0 2.0
RADIUS, DRAFT
0
INONUMAP (uniform mapping)
TEST13s.GDF:
TEST13S spheroid a=2, b=c=0.25 -- igdef=-4
1. 9.80665 ULEN GRAV
1 1
ISX ISY
1 -4
NPATCH IGDEF
1
NLINES
2.0 0.25 0.25
A, B, C

A–40

TEST13C.SPL:
TEST13C cylinder R=1 T=2 -- analytic geometry (npatch=2)
4 4
NU NV (side)
4 2
NU NV (bottom)
TEST13S.SPL:
TEST13S spheroid A=2 B=C=.25 -- analytic geometry (npatch=1)
4 2
NU NV
TEST13.POT:
TEST13.POT -- Cylinder + spheroid, ILOWHI=1
-1.
1
1
IRAD, IDIFF
2
NPER (array PER follows)
1.5 2.0
1
NBETA (array BETA follows)
0.
2
NBODY
test13c.gdf
1.25 0.0 0.0 0.0
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
test13s.gdf
-0.5 0.0 0.0 90.0
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST13.FRC:
TEST13.FRC -- Cylinder + spheroid, ILOWHI=1
1 1 1 1 0 1 1 1 1
1.0
test05c.frc
test05s.frc
0
1
0. 0. 0.
TEST05c.FRC:
CYL.FRC
0
0
0
0.000000
1.000000
.0000000
.0000000

0

0

.0000000
1.000000
.0000000

0

0

0

.0000000
.0000000
1.000000

A–41

0

0
0
TEST05s.FRC:
SPD.FRC
0
0
0
0.000000
1.000000
.0000000
.0000000
0
0

0

0

.0000000
1.000000
.0000000

0

0

0

0

.0000000
.0000000
1.000000

TEST13c.csf:
cylinder R=1.2 T=2.2 -- analytic CONTROL SURFACE (npatch=3)
1
ILOWHICSF
1 1
ISX ISY
3 -1001 1. NPATCH ICDEF PSZCSF
2
1.2 2.2 1.0
RADIUS, DRAFT, Inner radius
0
UNIFORM MAPPING
TEST13s.csf:
ELLIPSOID CONTROL SURFACE defined by subroutine ELLIPSOID_CS
1
ILOWHICSF
1 1
ISX ISY
2 -1003 1. NPATCH IGDEF PSZCSF
2
NLINES
2.2 0.3 0.3
A, B, C (semi-axes of outer control surface)
2.0 0.25
(semi-axes of body waterline)
test13.cfg:
IPLTDAT=4
maxscr=1024
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
NUMHDR=1
NOOUT=0 1 1 1 0 1 1 1 1

A–42

IALTFRC = 3
! Alternative Form 3 FRC
IALTFRCN= 1 1
ICTRSURF=1
! Evaluate control surface drift forces
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

A–43

A.14 ISSC TLP – TEST14
The subroutine TLP (IGDEF=-9) is used to generate the ISSC TLP with the dimensions
speciﬁed in TEST14.GDF. Except for the geometry, the inputs correspond to the low-order
test runs TEST06 and TEST07.
For TEST14A, TEST14A.CFG TEST14A.POT and TEST14A.FRC are used to output
data to be used as input to F2T. TEST14.GDF and TEST14.SPL are used without change.
TEST14A.POT has zero and inﬁnite frequencies and 98 uniformly spaced frequencies. In
CFG, IPERIO=2 is speciﬁed. In FRC, IOPTN(4) is set to output RAOs.

A–44

TEST14.GDF:
TEST14 -- ISSC TLP (ILOWHI=1)
43.125 9.80665
ULEN GRAV
1 1
ISX ISY
12 -9
NPATCH, IGDEF
2
NLINES
8.435 35.
43.125
RADIUS DRAFT HSPACE
7.5
10.5
WIDTH HEIGHT
TEST14.SPL:
TEST14 -- ISSC
1 2
NU
1 2
NU
1 2
NU
1 2
NU
1 2
NU
1 2
NU
1 2
NU
1 2
NU
4 1
NU
4 2
NU
4 1
NU
1 1
NU

TLP (ILOWHI=1)
NV patch 1
NV patch 2
NV patch 3
NV patch 4
NV patch 5
NV patch 6
NV patch 7
NV patch 8
NV patch 9
NV patch 10
NV patch 11
NV patch 12

TEST14.POT:
TEST14 -- ISSC TLP (ILOWHI=1)
450
0
0
IRAD, IDIFF
3
NPER (array PER follows)
5. 10. 15.
1
NBETA (array BETA follows)
0.
1
NBODY
test14.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
0
NEWMDS
TEST14.FRC:
TEST14 -- ISSC TLP (ILOWHI=1, IALTFRC=2)
1 1 1 -2 0 0 0 0 0 IOPTN (IOPTN(4)<0 signifies fixed modes)
6
NDFR
1 1 0 0 0 1
IMODE
1.
RHO

A–45

0. 0. 3.0
1
53066.4
0.
0.
53066.4
0.
0.
0. -159199.2
159199.2
0.
0.
0.
0
0
0
0
test14.cfg:
MAXSCR=1024
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=1
IQUADI=4
IQUADO=3
KSPLIN=3
NUMHDR=1
IALTFRC=2
USERID_PATH=\WAMITv6

XCG
IMASS
0.
0.
159199.2
0.
0. -159199.2
0.
0.
53066.4
0.
0.
0.
0. 8.0201552E7
0.
0.
0.
0.
8.0201552E7
0.
0.
0.
0. 9.54906731E7
IDAMP
ISTIFF
NBETAH
NFIELD

(directory for *.exe, *.dll, and userid.wam)

A–46

TEST14A.POT:
TEST14A -- ISSC TLP (ILOWHI=1)
450
0
0
IRAD, IDIFF
-101
NPER (array PER follows)
-0.05 0.05
1
NBETA (array BETA follows)
0.
1
NBODY
test14.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
0
NEWMDS
TEST14A.FRC:
TEST14A -- ISSC TLP (ILOWHI=1, IALTFRC=2)
1 1 1 1 0 0 0 0 0 IOPTN
1.
RHO
0. 0. 3.0
XCG
1
IMASS
53066.4
0.
0.
0.
159199.2
0.
0.
53066.4
0. -159199.2
0.
0.
0.
0.
53066.4
0.
0.
0.
0. -159199.2
0. 8.0201552E7
0.
0.
159199.2
0.
0.
0.
8.0201552E7
0.
0.
0.
0.
0.
0. 9.54906731E7
0
IDAMP
0
ISTIFF
0
NBETAH
0
NFIELD
TEST14A.CFG:
MAXSCR=1024
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=1
IQUADI=4
IQUADO=3
KSPLIN=3
NUMHDR=1
IALTFRC=2
IPERIO=2
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–47

A.15 SEMI-SUB – TEST15
The subroutine SEMISUB (IGDEF=-10) is used to generate a Semi-submersible with the
dimensions speciﬁed in TEST15.GDF. There are ﬁve columns on each pontoon, as shown
in the ﬁgures below. For this structure a total of 10 patches are required. If NPATCH=11,
extensions of the pontoons can be included as explained in the subroutine comments.

A–48

TEST15.GDF:
TEST15 Semi-sub, NCOL=5, IGDEF=-10
1. 9.80665 ULEN GRAV
1
1
ISX ISY
10 -10
NPATCH IGDEF
2
NLINES
260. 20. 40. -30. -20.
XL, Y1, Y2, Z1, Z2
60. 8. 5
DCOL, RCOL, NCOL
TEST15.SPL:
TEST15 Semi-sub, NCOL=5, IGDEF=-10
9 2
NU NV
(patch
32 1
NU NV
(patch
2 2
NU NV
(patch
2 1
NU NV
(patch
5 2
NU NV
(patch
4 2
NU NV
(patch
4 1
NU NV
(patch
5 2
NU NV
(patch
4 2
NU NV
(patch
4 1
NU NV
(patch

10+32 - pontoon bottom)
9+32 - pontoon side)
1+32 - column 3)
2+32 - annulus 3)
3+32 - between annulus 3&4)
4+32 - column 4)
5+32 - annulus 4)
6+32 - between annulus 4&5)
7+32 - column 5)
8+32 - annulus 5)

TEST15.POT:
TEST15 Semi-sub with five columns on each pontoon
-1.
0
0
IRAD, IDIFF
1
NPER (array PER follows)
18.0
1
NBETA (array BETA follows)
180.
1
NBODY
test15.gdf
0. 0. 0. 0.
XBODY
0 0 1 0 1 0
IMODE(1-6)
0
NEWMDS
TEST15.FRC:
TEST15 Semi-sub
1
0.0000
20.0
0.
0
0

with five columns on each pontoon
1
1
1
0
0
0
0
0.0
60.0
0.

0.0
0.0
60.0

A–49

0

0
test15.cfg:
ISOLVE=1
IPERIO=1
NUMHDR=1
KSPLIN=3
IQUADI=4
IQUADO=3
ILOWHI=1
IALTPOT=2
IRR=0
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–50

A.16 BARGE WITH BENDING MODES – TEST16
The test runs TEST16 and TEST16a analyze the structural response of a rectangular
barge with total length 80m, beam 10m, and draft 5m. Eight free-free beam modes are
included to analyze the elastic deformation of the barge. These mode shapes are deﬁned
in the subroutines NEWMODES and DEFINE, as explained in Chapter 8. The response
amplitude of each mode is included in the RAO’s.
In TEST16 the subroutine BARGE (IGDEF=-5) is used. The half-length, half-beam
and draft are speciﬁed in TEST16.GDF. In TEST16a the option IGDEF=0 is used, with
the vertices of the patches speciﬁed in TEST16a.GDF in the same format as for low-order
panels.

A–51

TEST16.GDF:
TEST16 elastic barge
1. 9.80665 ULEN GRAV
1 1
ISX ISY
3 -5
NPATCH IGDEF
1
NLINES
40.0
5.0 5.0
half-length, half-beam, draft
TEST16.SPL:
TEST16 elastic barge
3 3
NU NV
(end)
4 4
KU KV
5 2
(side)
4 4
5 2
(bottom)
4 4
IQUO IQVO are not specified IQUADO=3 in config.wam
IQUI IQVI are not specified IQUADI=4 in config.wam
TEST16.POT:
TEST16 elastic barge with 8 beam modes
-1.
0
0
IRAD, IDIFF
2
NPER (array PER follows)
7. 8.
1
NBETA (array BETA follows)
180.
1
NBODY
test16.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
8
NEWMDS
TEST16.FRC:
TEST16 elastic barge with 8 beam modes
1 1 1 1 0 0 0 0 0
1000.
0. 0. 0.
1
4.00000E+06 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 4.E+06 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 2.13333E+09 0. 0. 0. 0. 0. 0. 0. 0. 0.

A–52

0
1

0
0

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0. 0. 0. 0. 0. 0. 0. 0.
1.E+06 0. 0. 0. 0. 0. 0. 0.
0. 1.E+06 0. 0. 0. 0. 0. 0.
0. 0. 1.E+06 0. 0. 0. 0. 0.
0. 0. 0. 1.E+06 0. 0. 0. 0.
0. 0. 0. 0. 1.E+06 0. 0. 0.
0. 0. 0. 0. 0. 1.E+06 0. 0.
0. 0. 0. 0. 0. 0. 1.E+06 0.
0. 0. 0. 0. 0. 0. 0. 1.E+06

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
6.25705E+06 0. 0. 0. 0. 0. 0. 0.
0. 4.75441E+07 0. 0. 0. 0. 0. 0.
0. 0. 1.82720E+08 0. 0. 0. 0. 0.
0. 0. 0. 4.99297E+08 0. 0. 0. 0.
0. 0. 0. 0. 1.11419E+09 0. 0. 0.
0. 0. 0. 0. 0. 2.17352E+09 0. 0.
0. 0. 0. 0. 0. 0. 3.85260E+09 0.
0. 0. 0. 0. 0. 0. 0. 6.35602E+09

test16.cfg:
ILOWHI=1
IALTPOT=2
IALTFRC=2
IRR=0
ISOLVE=1
IQUADI=5
IQUADO=4
IPERIO=1
MONITR=0
NUMHDR=1
ILOG=0
IGENMDS=16
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–53

TEST16A.GDF: (lines 1-8 only):
TEST16a elastic barge with 8 beam modes - igdef=0 (3 flat panels)
1.0000
9.80665
1
1
3
0
NPATCH, IGDEF
40.0000
0.000000
-5.00000
40.0000
5.00000
-5.00000
40.0000
5.00000
0.000000
40.0000
0.000000
0.000000 (end)
TEST16A.SPL:
TEST16A elastic barge with igdef=0 (patches defined by flat panels)
3 3
NU NV
end
4 4
KU KV
5 2
side
4 4
5 2
bottom
4 4
TEST16A.POT:
TEST16a elastic barge with 8 beam modes - igdef=0
-1.
0
0
IRAD, IDIFF
2
NPER (array PER follows)
7. 8.
1
NBETA (array BETA follows)
180.
1
NBODY
test16a.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
8
NEWMDS
TEST16A.FRC:
TEST16a elastic barge with 8 beam modes (igdef=0)
1 1 1 1 0 0 0 0 0
1000.
0. 0. 0.
1
4.00000E+06 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 4.E+06 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 2.13333E+09 0. 0. 0. 0. 0. 0. 0. 0. 0.

A–54

0
1

0
0

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.

0. 0. 0. 0. 0. 0. 0. 0.
1.E+06 0. 0. 0. 0. 0. 0. 0.
0. 1.E+06 0. 0. 0. 0. 0. 0.
0. 0. 1.E+06 0. 0. 0. 0. 0.
0. 0. 0. 1.E+06 0. 0. 0. 0.
0. 0. 0. 0. 1.E+06 0. 0. 0.
0. 0. 0. 0. 0. 1.E+06 0. 0.
0. 0. 0. 0. 0. 0. 1.E+06 0.
0. 0. 0. 0. 0. 0. 0. 1.E+06

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0.
6.25705E+06 0. 0. 0. 0. 0. 0. 0.
0. 4.75441E+07 0. 0. 0. 0. 0. 0.
0. 0. 1.82720E+08 0. 0. 0. 0. 0.
0. 0. 0. 4.99297E+08 0. 0. 0. 0.
0. 0. 0. 0. 1.11419E+09 0. 0. 0.
0. 0. 0. 0. 0. 2.17352E+09 0. 0.
0. 0. 0. 0. 0. 0. 3.85260E+09 0.
0. 0. 0. 0. 0. 0. 0. 6.35602E+09

test16A.cfg:
ILOWHI=1
IALTPOT=2
IALTFRC=2
IRR=0
ISOLVE=1
IQUADI=5
IQUADO=4
IPERIO=1
MONITR=0
NUMHDR=1
ILOG=0
IGENMDS=16
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–55

A.17 CYLINDER WITH MOONPOOL – TEST17
This test run illustrates two alternative methods for analyzing bodies with moonpools.
The geometry used is the circular cylinder with a concentric ﬂuid chamber, as shown in
the ﬁgure. The inner chamber of ﬂuid, referred to as a ‘moonpool’, is open at the bottom
of the cylinder to the external ﬂuid domain. The top of the moonpool is a free surface
with atmospheric pressure. One of the practical aspects of this problem is the existence of
highly tuned resonant frequencies of the motion at the moonpool free surface. If the draft is
comparable or large compared to the horizontal dimensions of the moonpool, the principal
resonance is a ‘pumping mode’ which occurs when KT , the product of the wavenumber K
and draft T is slightly less than one. Additional resonances occur in ‘sloshing modes’ at
higher frequencies, corresponding approximately to standing waves inside the moonpool.
A cylinder with draft 1m is used, with the outer radius RADIUS=0.5m and the inner radius RADMP=0.25m. The geometry in all cases is represented analytically by the
subroutine CYLMP (IGDEF=-7). To clarify the behavior near resonance, the wavenumber K is input in the POT ﬁle with the corresponding option IPERIO=3 speciﬁed in the
TEST17.cfg ﬁle. Seven values of K are input in the range 0.7 ≤ K ≤ 1.0 to focus on the
regime including the pumping mode. The computed hydrodynamic parameters include
the force coeﬃcients, RAO’s, and the elevation of the free surface at the center of the
moonpool.
In TEST17 three patches are used to represent the outer surface r=RADIUS, the annular bottom z=-DRAFT, and the inner vertical surface r=RADMP. The free surface inside
the moonpool is part of the physical free surface, and the appropriate free-surface boundary condition is satisﬁed automatically by the Green function as described in Chapter 12.
The outputs display singular features characteristic of the resonant pumping mode. This
includes large amplitudes of the heave damping and exciting force, and negative added
mass in the vicinity of KT = 0.85. The heave RAO exceeds 8, and the free-surface elevation in the moonpool exceeds 6, relative to the incident wave amplitude. These very large
responses are non-physical, and their existence in the computations can be attributed primarily to the neglect of viscous damping associated with ﬂow separation at the outer and
inner corners of the cylinder. This damping is only important when the vertical motions
of the cylinder and/or moonpool are large. Experiments suggest that typical resonant
amplitudes are on the order of 2 or 3 times the incident wave amplitude.
In order to damp the moonpool response and heave motions separately, a diﬀerent
physical problem is considered where a ‘lid’ is placed on the free surface of the moonpool. This lid is considered to be an extension of the body surface, and represented by
an additional patch. Thus NPATCH=4 is assigned in TEST17a.GDF (and used also in
TEST17b) and the subroutine CYLMP assigns the patch number 4 to be the circular disc
of radius RADMP in the plane Z=0. However allowance must be made for the motions of
the actual free surface relative to the body. This is done by deﬁning appropriate generalized modes, which are nonzero only on patch 4. The most important mode is a vertical
translation, assigned here in the subroutine ﬁle NEWMODES.F with the index j = 7. A
more complete expansion can be introduced, but at the wavenumbers considered here and

A–56

for head-sea incidence angle it suﬃces to consider only a pitch rotation of the lid (j = 8).
These two generalized modes, physically analogous to pitch and heave but deﬁned relative
to the body, are introduced via the subroutine MOONPOOL FS in NEWMODES.F.
In test run TEST17a, the lid is assumed to be free with no external force or moment
acting on it. The IALTFRC=2 option is employed, and the only external force matrix that
is included in TEST17.FRC is the mass matrix of the body. This mass matrix is equivalent
to the radii of gyration speciﬁed in TEST17.FRC. It can be conﬁrmed by comparison of the
outputs that the motions of the body (RAO) are virtually identical to TEST17, conﬁrming
that the representation of the moonpool free surface in this manner is legitimate. A
comparison can also be made between the moonpool free surface elevation (numeric output
ﬁle TEST17.6) and the response of the lid in mode 7 (RAO(7) in the numeric output ﬁle
TEST17a.4), but in this comparison account must be made for the fact that RAO(7)
is relative to the body motions, and thus it is necessary to compare the (complex) sum
RAO(3)+RAO(7) in TEST17a with the moonpool free surface elevation in TEST17.
Finally, in TEST17b, empirical damping is introduced via the external damping matrix
in TEST17b.FRC. Since this is the only diﬀerence between TEST17a and TEST17b, it
is not necessary to re-run POTEN and the same TEST17a.P2F ﬁle is used for TEST17b.
Thus TEST17a.pot is speciﬁed in FNAMES.17b, and IPOTEN=0 in the TEST17b.cfg ﬁle.
The only nonzero elements of the external damping matrix are for heave (j = 3) and the lid
vertical motion (j = 7). With these empirical damping coeﬃcients added, more appropriate
RAO’s are obtained. This general approach can be reﬁned based on experimental data.
Experience with similar problems suggests that relatively crude estimates based on the
observed response at resonance are suﬃcient to correct the response over a broad range of
wave periods.

A–57

TEST17.GDF:
TEST17 cylinder with moonpool
1. 9.80665 ULEN GRAV
1 1
ISX ISY
3 -7
NPATCH IGDEF
1
NLINES
0.5 1.0 0.25
radius, draft, moonpool radius
TEST17.POT:
TEST17 cylinder with moonpool, NPATCH=3
-1.
0
0
IRAD, IDIFF
7
NPER (array PER follows)
.7 .75 .8 .85 .9 .95 1.0
1
NBETA (array BETA follows)
180.
1
NBODY
test17.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
0
NEWMDS
TEST17.FRC:
TEST17.FRC Cylinder with moonpool
1
1
1
1
0
1 0 0 0
0.000000
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
0
1
0.0 0.0 0.0
test17.cfg:
ILOWHI=1
IALTPOT=2
IALTFRC=1
IRR=0
ISOLVE=1
PANEL_SIZE = 0.2
IPERIO = 3
NUMHDR=1
USERID_PATH=\WAMITv6

IOPTN(1-9)
VCG
XPRDCT
NBETAH
NFIELD

(use default .spl parameters)
(input wavenumber)
(directory for *.exe, *.dll, and userid.wam)

A–58

TEST17A.GDF: (lines 1-8 only):
TEST17a cylinder with moonpool -- undamped patch on free surface
1. 9.80665 ULEN GRAV
1 1
ISX ISY
4 -7
NPATCH IGDEF
1
NLINES
0.5 1.0 0.25
radius, draft, moonpool radius
TEST17A.POT:
TEST17 cylinder with moonpool, NPATCH=3
-1.
0
0
IRAD, IDIFF
7
NPER (array PER follows)
.7 .75 .8 .85 .9 .95 1.0
1
NBETA (array BETA follows)
180.
1
NBODY
test17a.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
2
NEWMDS
TEST17A.FRC:
TEST17a moonpool with generalized modes
1 1 1 1 0 0 0 0 0
1.
0. 0. 0.
1 imass (mass matrix of body)
0.589049 0.0
0.0
0.0
0.0
0.589049 0.0
0.0
0.0
0.0
0.589049 0.0
0.0
0.0
0.0
0.589
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0 idamp
0 istif
0
0
test17A.cfg:
ILOWHI=1
IALTPOT=2

A–59

for free surface - no damping

0.0
0.0
0.0
0.0
0.589
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.589
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

IALTFRC=2
ISOLVE=1
PANEL_SIZE = 0.2
IPERIO = 3
IRR=0
ILOG=1
NUMHDR=1
IGENMDS=17
USERID_PATH=\WAMITv6

(use default .spl parameters)
(input wavenumber)

(directory for *.exe, *.dll, and userid.wam)

A–60

TEST17B.FRC:
TEST17b cylinder+moonpool, generalized
1 1 1 1 0 0 0 0 0
1.
0. 0. 0.
1 imass (mass matrix of body)
0.589049 0.0
0.0
0.0
0.0
0.589049 0.0
0.0
0.0
0.0
0.589049 0.0
0.0
0.0
0.0
0.589
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1 idamp
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0 istif
0
0
test17B.cfg:
ILOWHI=1
IPOTEN=0
IALTPOT=2
IALTFRC=2
ISOLVE=1
PANEL_SIZE = 0.2
IPERIO = 3
IRR=0
ILOG=1
NUMHDR=1
IGENMDS=17
USERID_PATH=\WAMITv6

modes, damping b33=.4, b77=.1

0.0
0.0
0.0
0.0
0.589
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.589
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

(skip POTEN subprogram, use TEST17a.p2f)

(use default .spl parameters)
(input wavenumber)

(directory for *.exe, *.dll, and userid.wam)

A–61

A.18 ELASTIC COLUMN – TEST18
The same inputs are used as in the low-order test run TEST08, except for the GDF ﬁle.
The circular column is represented by the subroutine CIRCCYL (IGDEF=-1). Since the
cylinder is bottom-mounted, NPATCH=1 and the patch on the bottom of the cylinder is
omitted. The draft is set equal to the ﬂuid depth. IGENMDS=18 is assigned in CFG
ﬁle, and in NEWMODES this results in a call to subroutine DEFINE for the four shifted
Jacobi polynomials.

A–62

TEST18.GDF:
TEST18.GDF vertical cylinder, bottom mounted
1.0000 9.80665
ulen, grav
1
1
isx,isy
1 -1
npatch, igdef
2
nlines
10.0000 200.000
radius, draft
0
INONUMAP (uniform mapping)
TEST18.SPL:
TEST18.spl - bottom-mounted cylinder R=10 T=200 -- (npatch=1)
4 8
NU NV (Patch 1, side
u azimuthal v vertical)
TEST18.POT:
TEST18.POT -- bending of vertical column at resonance, 200m depth
200.0 0.0 0.0 0.0 0.0
0 0 0
0
0
0
0
0
0
0
0
1
6.5
1
0.0
TEST18.FRC:
TEST08.FRC file,
1
1
1.0
.0000000
1
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0
1
0.
0.
0.
0.
0.
0.

vertical column with 4 bending modes
1
1
0
0
0
0
0
.0000000

1.000000

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
69115.
62832.
62832.
62832.

0.
0.
0.
0.
0.
0.
62832.
67320.
62832.
62832.

0.
0.
0.
0.
0.
0.
62832.
62832.
66323.
62832.

0.
0.

0.
0.

0.
0.

0.
0.

0.
0.

0.
0.

A–63

0.
0.
0.
0.
0.
0.
62832.
62832.
62832.
65688.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.

0
0

0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.

test18.cfg:
MAXSCR=1024
ISOLVE=1
NUMHDR=1
NUMNAM=0
ISOR=0
IRR=0
NEWMDS=4
ILOWHI=1
IALTPOT=1
IALTFRC=2
KSPLIN=3
IQUADO=3
IQUADI=4
IGENMDS=18
USERID_PATH=\WAMITv6

0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
103044.
412177. 824354. 1339575.
412177. 4430902. 9789203. 16487078.
824354. 9789203. 37899671. 64382041.
1339575. 16487078. 64382041. 162406554.

(directory for *.exe, *.dll, and userid.wam)

A–64

A.19 CATAMARAN BARGE – TEST19
The geometrical conﬁguration is the same as the barge near a wall (TEST04). Since only
head seas are considered, the hydrodynamic outputs correspond to TEST04 except for the
diﬀerent deﬁnition of the incident wave amplitude which applies for a body near a wall.
In TEST19 IGDEF=0 is used, with four patches speciﬁed in the GDF ﬁle corresponding
to one quadrant of the catamaran conﬁguration. Since there are two hulls in this case,
the forces acting on both hulls are two times the corresponding forces in TEST04, but
since the incident wave amplitude in TEST04 is increased by a factor of two, the exciting
force coeﬃcients and RAO’s are the same in both test runs, except for small diﬀerences in
accuracy. Note that in TEST19 two planes of symmetry can be utilized, unlike TEST04
where reﬂection about the plane x = 0 is required by the program. The comparisons of
cross-coupling coeﬃcients and Haskind/Diﬀraction exciting forces implies that the results
of TEST19 are more accurate, with less computational cost.

A–65

Generalized modes can be used to extend the analysis of this conﬁguration to include two
independent bodies. In this case each of the rigid-body modes of the catamaran must be
supplemented by a corresponding generalized mode which has the same normal velocity on
one barge, and the opposite phase on the other. The separate modes of each independent
body are then evaluated by combining the corresponding symmetric and antisymmetric
modes for the catamaran. It is simpler to use the option NBODY=2 for this purpose, but
the number of unknowns is increased by a factor of four, resulting in a substantial increase
of the run time. For the more eﬃcient approach used in TEST19 it is necessary to represent
the entire forward half of one barge, as shown in the patch ﬁgure. The subroutine BARGE
(IGDEF=-5) is not suitable, since this only represents one quadrant of one barge. On the
other hand, BARGE can be used in the alternative NBODY=2 approach.

A–66

TEST19.GDF:
TEST19 one quadrant of catamaran barge configuration
40. 9.80665 ULEN GRAV
1 1
ISX ISY
4 0
NPATCH IGDEF
40.0000
2.000000
-10.00000
40.0000
22.00000
-10.00000
40.0000
22.00000
0.000000
40.0000
2.000000
0.000000 (end)
40.0000
22.00000
-10.00000
0.000000
22.00000
-10.00000
0.000000
22.00000
0.000000
40.0000
22.00000
0.000000 (outside)
40.0000
2.000000
-10.00000
0.000000
2.000000
-10.00000
0.000000
22.00000
-10.00000
40.0000
22.00000
-10.00000 (bottom)
40.0000
2.00000
0.000000
0.000000
2.00000
0.000000
0.000000
2.00000
-10.00000
40.0000
2.00000
-10.00000 (inside)
TEST19.SPL:
TEST19 catamaran barge
2 2
NU NV
(end)
4 4
KU KV
5 2
(outside)
4 4
5 2
(bottom)
4 4
5 2
(inside)
4 4
IQUO IQVO are not specified IQUADO=3 in config.wam
IQUI IQVI are not specified IQUADI=4 in config.wam
TEST19.POT:
TEST19.POT -- Catamaran barge, same geometry as TEST04
-1. 0. 0. 0. 0.
0 0
IQUAD,IDIAG must be read in IALTPOT, not used
0 0
IRAD,IDIFF
1 0 1 0 1 0
3
6. 7. 8.
1

A–67

0.0
TEST19.FRC:
TEST19.FRC -- Catamarn barge, ILOWHI=1 (same as TEST04.FRC)
1
1
1
1
0
0
0
0
1
3.0
20.00000
0.000000
0.000000
0.000000
5.000000
0.000000
0.000000
0.000000
20.00000
0
0
test19.cfg:
ILOWHI=1
IALTPOT=1
IALTFRC=1
IRR=0
ISOLVE=1
IQUADI=5
IQUADO=4
IPERIO=1
MONITR=0
NUMHDR=1
ILOG=0
isor=0
USERID_PATH=\WAMITv6

(must include since ialtpot=1)
(directory for *.exe, *.dll, and userid.wam)

A–68

A.20 MULTISURF BARGE – TEST20
This example illustrates the use of a MultiSurf geometry representation with IGDEF=2.
The barge has a length of 100m, beam 20m, and draft 4.8m with one plane of symmetry
(y = 0). The origin of the body coordinate system is at the intersection of the baseline and
midship section. Thus XBODY=(0.0, 0.0, -4.8, 0.0) is input in the POT ﬁle. The format of
the GDF ﬁle is as explained in Section 6.7 and Appendix 2. Four patches are used on one
side of the body to represent the forebody, parallel middlebody, afterbody, and transom.
Reference 24 includes results for a multiple-body conﬁguration including two barge hulls
identical to this model.
The body pressure is evaluated at the points speciﬁed in the input ﬁle test20.bpi, as
explained in Section 4.11. The parameter IPNLBPT=1 in the .cfg ﬁle is used to specify
this option with the input points speciﬁed in the body coordinate system.

A–69

test20.pot:
single barge based on MultiSurf model (igdef=2)
-1.
1
1
IRAD, IDIFF
3
NPER (array PER follows)
6.0 9.0 12.0
3
NBETA (array BETA follows)
180. 135. 90.
1
NBODY
test20.gdf
0. 0. -4.8 0.0
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
test20.gdf:
Test run for barge modelled with MultiSurf
1.000000 9.806600 ULEN, GRAV
0 1 ISX, ISY
4
2 NPATCH, IGDEF
3 NLINES
test20.ms2
wetted_surfs
0 0 0 FAST,DivMult, outward normals
test20.ms2: (lines 1-8 only)
MultiSurf 1.23
// Barge model for WAMIT
// J. S. Letcher, AeroHydro, Inc. 1/10/2002
Units: m MT
Symmetry: y
Extents: -50.000 -10.000 0.000 50.000 10.000 8.000
View: -30.00 60.00 0
Places: 3
test20.frc:
test20.frc igdef=2
1
1
1
1
1
0.0
10.00000
.0000000
.0000000
25.00000
.0000000
.0000000
0
0

0

0

1 1

.0000000
.0000000
25.00000

A–70

IOPTN(1-9)
VCG
XPRDCT
NBETAH
NFIELD

test20.bpi: (lines 1-8 only)
bpi input file for test20, body pressure points for MultiSurf barge
556
-44.7760
0.0000
3.7926
-42.9236
0.0000
2.9037
-45.0675
0.2891
3.9523
-43.5608
0.6874
3.1868
-45.3541
0.5548
4.1152
-44.1782
1.2902
3.4831
test20.cfg:
MAXSCR=1024
ILOWHI=1
IALTFRC=1
IALTPOT=2
IRR=0
ISOLVE=1
NUMHDR=1
USERID_PATH=\WAMITv6
panel_size= 10.
IPNLBPT=1

(directory for *.exe, *.dll, and userid.wam)

A–71

A.21 SPAR WITH THREE STRAKES – TEST21
The subroutine SPAR (IGDEF=-12) is used to generate the SPAR with three strakes with
the dimensions speciﬁed in TEST21.GDF. Except for the geometry, the inputs correspond
to the low-order test runs TEST09.

A–72

TEST21.GDF:
TEST21 SPAR2 with three strakes IGDEF=-12
18. 9.80665 ULEN GRAV
0 0
ISX ISY
7 -12
NPATCH IGDEF
npatch_dipole = 3
ipatch_dipole = 2 4 6
5
18. 200.
RADIUS, DRAFT
3.7 0. 1. 3
WIDTH, THICKNESS, TWIST, NSTRAKE
0
IRRFRQ
0 0.
IMOONPOOL, RADIUSMP
0
IMPGEN
TEST21.POT:
TEST21.POT SPAR with three strakes igdef=-12 (TEST21.GDF)
-1.
1
1
IRAD, IDIFF
3
NPER (array PER follows)
0.1 0.5 1.
2
NBETA (array BETA follows)
0. 120.
1
NBODY
test21.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST21.FRC:
TEST21.FRC SPAR with three strakes igdef=-12
1 1 1 1 0 1 1 2 0
IOPTN(1-9)
0.000000
VCG
100.000000
.0000000
.0000000
.0000000
100.000000
.0000000
.0000000
.0000000
10.000000
XPRDCT
0
NBETAH
2
NFIELD
23. 0. 0.
15. 15. -0.5
(end of file)
TEST21.CFG:
ipltdat=4

A–73

ilowgdf=4
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=1
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=3
MONITR=0
NUMHDR=1
NOOUT= 1 1 1 1 0 1 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)
PANEL_SIZE=18
IPOTEN=1
ILOG=1

A–74

A.22 FPSO WITH TWO INTERNAL TANKS – TEST22
The subroutine FPSOINT (IGDEF=-21) is used to generate the FPSO with two internal
tanks with the dimensions speciﬁed in TEST22.GDF. One plane of symmetry is speciﬁed,
about y = 0. The tanks are rectangular, and the vertices of each patch are speciﬁed in
TEST22.GDF. Both tanks have the same length (2m), breadth (4.2m), and depth (1.1m).
The aft side of tank 1 and the forward side of tank 2 are in the same plane x = 0.0. The
free surface of tank 1 is at z = 1, 1m above the plane of the exterior free surface. The
free surface of tank 2 is at z = 0.0. The ﬁrst and last patches of each tank are assigned
by the parameter NPTANK. Both tanks contain ﬂuid of relative density 1.0, as speciﬁed
in TEST22.CFG. The parameter ITANKFPT=1 is used so that the ﬁeld points can be
assigned in each tank, on the last two lines of TEST22.FRC.
The option is used to evaluate the mean drift force and moment using a control surface,
following the procedure described in Chapter 14. The parameter ICTRSURF=1 is assigned in the CFG ﬁle. The control surface surrounding the FPSO is a simple rectangular
box, deﬁned by the input ﬁle TEST22.csf. This control surface is generated using three
rectangular patches (ICDEF=0) with the vertices speciﬁed in the CSF ﬁle, and with two
planes of symmetry. (Note that the control surface is symmetric about the plane x = 0,
although the body is not symmetric about this plane.) The parameter PSZCSF is negative,
indicating that the subdivision of the control surface is determined by the parameters in
the ﬁle TEST22.CSP.
Since the portion of the free surface between the body and outer control surface is
not included, the results for the horizontal drift force and vertical moment are correctly
evaluated but the vertical force and horizontal components of the moment are not correct.
The corresponding output for the mean drift force and moment is contained in the ﬁle
TEST13.9c. Comparison of the results for the mean drift force in the sway direction from
the ﬁles TEST22.9 (direct pressure integration) and TEST22.9c (control surface), with the
momentum drift force data in TEST22.8, conﬁrms that the control surface gives a more
accurate result compared to direct pressure integration for this body.

A–75

TEST22.GDF (complete file):
TEST22.GDF -- fpso with 2 tanks, one raised, joined at x=0
1. 9.80665 ULEN GRAV
0 1
ISX ISY
15 -21
NPATCH IGDEF
36
NLINES 4+16*2
3.
15.
2.
XBOW, XMID, XAFT
2.2
1.2
HBEAM, HTRANSOM
1.2 0.6
DRAFT, DTRANSOM
0 2 0
INONUMAP, NTANKS, IRR
2.000000
0.0000000E+00 1.0000000E+00
2.000000
2.100000
1.0000000E+00
2.000000
2.100000
-0.100000
2.000000
0.0000000E+00 -0.100000
2.000000
2.100000
1.0000000E+00
0.000000E+00 2.100000
1.0000000E+00

A–76

0.000000E+00
2.000000
0.000000E+00
2.000000
2.000000
-0.000000E+00
-0.000000
-0.000000
-0.000000
-0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-2.000000E+00
-2.000000E+00
0.000000
-2.000000E+00
0.000000
0.000000
-2.000000E+00
-2.000000
-2.000000
-2.000000
-2.000000

2.100000
2.100000
0.0000000E+00
0.0000000E+00
2.100000
2.100000
0.0000000E+00
2.100000
2.100000
0.0000000E+00
0.0000000E+00
2.100000
2.100000
0.0000000E+00
2.100000
2.100000
2.100000
2.100000
0.0000000E+00
0.0000000E+00
2.100000
2.100000
0.0000000E+00
2.100000
2.100000
0.0000000E+00

-0.100000
-0.100000
-0.100000
-0.100000
-0.100000
-0.100000
-0.100000
-0.100000
1.0000000E+00
1.0000000E+00
0.0000000E+00
0.0000000E+00
-1.100000
-1.100000
0.0000000E+00
0.0000000E+00
-1.100000
-1.100000
-1.100000
-1.100000
-1.100000
-1.100000
-1.100000
-1.100000
0.0000000E+00
0.0000000E+00

TEST22.POT:
TEST22.POT fpso with 2 interior tanks
-1.0
1
1
IRAD, IDIFF
3
2.0 2.5 3.0
1
NBETA (array BETA follows)
90.
1
NBODY
test22.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS

A–77

TEST22.SPL:
TEST22.SPL FPSO with two tanks
4
3
4
2
8
3
8
2
2
1
3
3
3
2
3
2
3
2
3
3
3
2
3
2
3
2
3
3
3
2
TEST22.FRC:
TEST22.FRC fpso with 2 tanks, one field point on free surface in each tank
1
1
1
1
0
1 1 1 1
IOPTN(1-9)
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
2
NFIELD
1 1.0 1.0 1.0
2 -1.0 1.0 0.0
TEST22.CSF:
Test22 control surface box
1
ILOWHICSF
1 1
3 0 -2.
12. 3. 0.
12. 3. -2. 0. 3. -2. 0. 3. 0.
12. 0. 0.
12. 0. -2.
12. 3. -2.
12. 3.
12. 0. -2. 0. 0. -2. 0. 3. -2.
12. 3. -2.
TEST22.CSP:
Test22.CSP control surface spline file
1 6
1 2
6 2

A–78

0.

TEST22.CFG:
ipltdat=4
ILOWHI=1
IALTPOT=2
IRR=0
ILOG=1
ISOLVE=1
KSPLIN=3
IQUADO=3
IQUADI=4
MONITR=0
NUMHDR=1
NOOUT= 0 0 0 0 0 0 0 0 0
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)
NPTANK=(8-11) (12-15)
RHOTANK= 1.0 1.0
(relative densities of tank fluids)
ITANKFPT=1
(tank field points are in .frc file)
ICTRSURF=1
Evaluate control surface drift forces

A–79

A.23 RADIATED WAVE FIELD FROM A BANK OF WAVEMAKERS – TEST23
The option ISOLVE=-1 is used to compute the radiated waves from a bank of ‘paddle’
wavemakers, following the procedure in Section 10.8. The wavemakers are in the plane
x = 0 of a rectangular tank, as shown below. The tank has a reﬂecting wall at y = 0.
The tank depth is 4m. Each wavemaker is represented by one rectangular patch, using
IGDEF=0, with the vertices listed in TEST23.GDF. The motion of each wavemaker is
rotational about its lower edge, at the same depth below the free surface, represented
by a generalized mode with the same distribution of normal velocity and with symmetry
prescribed about the walls x = 0 and y = 0. These generalized modes are deﬁned in the
subroutine WAVEMAKER, in the DLL ﬁle NEWMODES.F, designated by the parameter
IGENMDS=21 in TEST23.CFG. This subroutine reads the depth of the lower edge of the
wavemaker, ZHINGE=-2m, from the ﬁle WAVEMAKER DEPTH.DAT. Wave elevations
are evaluated at a square array of 64 ﬁeld points deﬁned in TEST23.FRC, using the uniform
ﬁeld point array option in Section 3.10.

A–80

test23.cfg:
ILOWHI=1
IALTPOT=2
IALTFRC=2
IRR=0
ISOLVE=-1
MONITR=0
NUMHDR=1
USERID_PATH=\WAMITv6
IGENMDS=21
PANEL_SIZE=1.
INUMOPT6=1
IFIELD_ARRAYS=1

(skip POTEN solutions for wavemakers in walls)
(directory for *.exe, *.dll, and userid.wam)
(use NEWMODES subroutine WAVEMAKER)
(use default .spl parameters)
(output separate radiation modes in .6 file)
(field points input in array format in .frc file)

test23.pot:
TEST23.POT -- 8 wavemaker segments in wall x=0
4. fluid depth
0
-1
IRAD, IDIFF
2
NPER (array PER follows)
2. 4.
1
NBETA (array BETA follows)
0.0
1
NBODY
test23.gdf
0. 0. 0. 0.
XBODY
0 0 0 0 0 0
IMODE(1-6)
8
NEWMDS
test23.gdf:
TEST23.GDF wavemaker, 8 segments in wall x=0 2 0. Four patches are required for these three
elements plus the spheroidal end. The total number of segments, speciﬁed in TEST24.GDF,
is used to read the x-coordinates of the boundaries between adjacent segments and also the
end of the vessel. The total number of segments is equal to seven, including ﬁve cylinders
plus two spheroids. Only the boundaries with coordinates x>0 are included in the last line
of the GDF ﬁle, since ISX=1.
The generalized modes which represent the deﬂection of the hinges are deﬁned in the
subroutine HINGE MODES in the DLL ﬁle NEWMODES.F, designated by the parameter IGENMDS=22 in TEST24.CFG. This subroutine reads the appropriate input data
from the ﬁle XHINGE.DAT, which is shown below. This input ﬁle also speciﬁes the xcoordinates of the hinges. The last cylinder and the spheroidal end are considered to be
rigidly joined. Thus there are ﬁve ‘active’ segments corresponding to the parameter NSEG
in the XHINGE.DAT ﬁle, and NEWMDS=4 is assigned in the TEST24.POT ﬁle.
In the TEST24.FRC ﬁle, the 10×10 matrix of inertia coeﬃcients is speciﬁed. No external
damping or stiﬀness matrices are input, corresponding to the situation where the hinges
are ideal without friction or other mechanical constraints.
Further information can be found in the headers and comments of the subroutines which
are used to generate the geometry and to represent the hinge modes.

A–83

TEST24.GDF:
TEST24 segmented vessel with 7 segments
1. 9.80665 ULEN GRAV
1 1
ISX ISY
4 -32
NPATCH IGDEF
3
NLINES
7
Nsegments
1.
Radius
2. 6. 8. 10. xseg
TEST24.SPL:
TEST24 elastic barge
TEST24 segmented vessel with 7 segments
4 2
NU NV
mid cylinder
4 4
next cylinder
4 2
outer cylinder
4 4
spheroidal end

A–84

TEST24.POT:
TEST24 segmented vessel with 7 segments and 4 hinge modes
-1.
0
0
IRAD, IDIFF
2
NPER (array PER follows)
3. 5.
1
NBETA (array BETA follows)
180.
1
NBODY
test24.gdf
0. 0. 0. 0.
XBODY
1 0 1 0 1 0
IMODE(1-6)
4
NEWMDS
XHINGE.DAT file for hinge coordinates of hinged body used in TEST24
1 5
ISX, NSEG
2. 6. 10. XHINGE(0:NSEG)
TEST24.FRC:
TEST24 segmented vessel with 7
1 1 1 1 0 0 0 0 0
1000.
0. 0. 0.
1
29321.5
0.
0.
0.
0.
29321.5
0.
0.
0.
0. 29321.5 0.
0.
0.
0. 1.5E4
0.
0.
0.
0.
0.
0.
0.
0.
0.
0. 12000.
0.
0.
0.
0.
0.
0.
0. 12000.
0.
0.
0.
0.
0.
0
0
0
0

segments and 4 hinge modes

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0. 12000.
0. 12000.
0.
0.
0.
0.
0.
0.
0.
7.33E5
0.
0. 24000.
0. 72000.
0. 7.33E5
0.
0.
0.
0.
0.
0. 10000.
0. 2000.
0.
24000.
0.
0. 6000.
0. 2000.
0.
0. 2000.
0. 8000.
0.
72000.
0.
0. 2000.
0. 8000.

A–85

TEST24.cfg:
ipltdat=5
ILOWHI=1
IALTPOT=2
IALTFRC=2
IRR=0
ILOG=0
ISOLVE=1
IQUADI=4
IQUADO=3
KSPLIN=3
IPERIO=1
MONITR=0
NUMHDR=1
IGENMDS=22
USERID_PATH=\WAMITv6

(directory for *.exe, *.dll, and userid.wam)

A–86

Appendix B

DESCRIPTION OF
SECOND-ORDER TEST RUNS
WAMIT V6S includes additional 8 standard test runs illustrating the evaluation of various
second-order quantities. It includes 4 low-order and 4 higher-order applications.
The following table gives relevant features of each test run. In this table the ﬁrst
column tst denotes the name of the test run. All of the corresponding input/output ﬁles
(except FNAMES.WAM ) are assigned the ﬁlenames TESTtst. (For example, the input
POT ﬁle for the ﬁrst test run listed below is TEST101.POT.) tst contains three digits
starting with 1 for the second-order applications, the second digit is 0 for low-order test
runs (ILOWHI=0), and 1 for higher-order test runs (ILOWHI=1). Test runs which are
identical except for diﬀerent input options are assigned the same number with a letter
suﬃx. Thus TEST103(TEST113) and TEST103a(TEST113a) describe the same problem
using diﬀerent options to evaluate the second-order solution.
tst
geometry
ILOWHI other parameters/comments
101 Bottom mounted cylinder
0
102 Truncated cylinder
0
103 Cylinder & spheroid
0
Free surface forcing ignored
103a Cylinder & spheroid
0
111 Bottom mounted cylinder
1
112 Truncated cylinder
1
113 Cylinder & spheroid
1
Free surface forcing ignored
113a Cylinder & spheroid
1

B–1

B.1 BOTTOM MOUNTED CYLINDER – TEST101
The ﬁrst-order quantities including exciting forces, pressures and ﬂuid velocities on the
body, ﬁeld pressures and wave elevation, ﬁeld velocities, drift forces are evaluated. In
addtion, the second-order quantities including quadratic forces, complete forces by direct
and indirect methods, pressures on the body, ﬁeld pressures, wave elevations are evaluated
for a bottom mounted vertical circular cylinder of radius 1 meter and draft 1 meter, in
ﬁnite water depth of 1 meter for two wave periods and two wave headings. Two second
harmonic of unidirectional waves, a mean and a diﬀerence frequency of bidirectional waves
are considered in the second-order computation (see PT2 ﬁle).
Using two planes of symmetry, only the ﬁrst quadrant of the surface of the cylinder
is discretized with 100 panels. 10 panels are distributed in the azimuthal and vertical
directions with double cosine spacing in the latter. The characteristic length (ULEN) is
set equal to the radius of the cylinder.
The free surface inside the inner circle of radius 3 meters is discretized automatically with
252 panels on the ﬁrst quadrant. No intermediate region is considered and the partition
radius is the same as the radius of inner circle. Perspective view of the body and the inner
region of the free surface is shown below.

B–2

TEST101.GDF:
TEST101 Bottom mounted cylinder R=1 T=1 Cosine spacing
1.000000
9.806650
1
1
100
0.9876884
0.1564345
0.0000000E+00
1.000000
0.0000000E+00 0.0000000E+00
1.000000
0.0000000E+00 -2.7999997E-02
0.9876884
0.1564345
-2.7999997E-02
TEST101.POT:
TEST101.POT Bottom mounted Cylinder R=1, T=1, igdef=-1
1.
-1
1
IRAD, IDIFF
2
NPER (array PER follows)
0.4
0.8
2
NBETA (array BETA follows)
0. 90.
1
NBODY
test101.gdf
0. 0. 0. 0.
XBODY
1 1 1 1 1 1
IMODE(1-6
0
NEWMDS
TEST101.FRC:
TEST101.FRC Bottom mounted Cylinder R=1, T=1, igdef=-1
0
0
1
0
3
2
2
2
2
1
1
1
1
1
1
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
1
1.044325
8.2290111E-02 0.0000000E+00

0

(IOPTN 1-16)

TEST101.PT2:
TEST101.PT2 -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
1 1
2
1 1 1
1 1
2 2 1

B–3

1
2
1
1
1
1

1
1 1
1
2 1
2

TEST101.FDF:
TEST101.FDF
3.0000E+00
-1.
1.
0

Inner radius (=Partition radius)
Auto Discretization, Scale=1.
No intermediate region

TEST101.CFG:
IPLTDAT=1
ILOWGDF=1
ILOWHI=0
ISOR=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=3
MONITR=0
NUMHDR=1
I2ND=1
ILOG=1
NOOUT= 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–4

B.2 TRUNCATED CIRCULAR CYLINDER – TEST102
The ﬁrst-order exciting forces, drift forces and the second-order quadratic forces and complete second-order forces by direct and indirect methods are evaluated for a vertical circular
cylinder of radius 8.435 meters and draft 35 meters, in inﬁnite water depth for two wave
periods and two wave headings. All sum- and diﬀerence frequency pairs and wave heading
pairs are considered for the second-order forces.
Using two planes of symmetry, only the ﬁrst quadrant of the surface of the cylinder
is discretized with 320 panels. 8, 28, 12 panels are distributed in the azimuthal, vertical
and radial directions with double cosine spacing on the side and cosine spacing toward the
corner at the bottom. The characteristic length (ULEN) is set equal to the radius of the
cylinder.
The free surface inside the inner circle of radius 25.305 meters, three times of the cylinder
radius, is automatically discretized with 159 panels on the ﬁrst quadrant. The intermediate
region consists of two annuli of width 16.87 meters, two times the radius of the cylinder.
On each annulus, 24 + 1 nodes and 8 nodes Gauss quadratures are applied in azimuthal
and radial direction, respectively. The partition circle coincides with the outside circle of
the second annulus and its radius is 58.45 meters. Perspective view of the body and the
inner region of the free surface is shown below.

B–5

TEST102.GDF:
TEST102.GDF -- Cylinder R=8.435 T=35 meters, Cosine spacing at corner and waterline
8.435000
9.806650
1
1
320
8.272924
1.645587
0.0000000E+00
8.435000
0.0000000E+00 0.0000000E+00
8.435000
0.0000000E+00 -0.1307399
8.272924
1.645587
-0.1307399
TEST102.POT:
TEST102 (Cylinder R=8.435 T=35 in infinite depth water)
-1. 0. 0. 0. 0.
-1 1
1 1 1 1 1 1
2
7.8934 5.1207
2
0. 90.
TEST102.FRC:
TEST102 -- Linear, mean and second-order exciting forces
0
0
1
0
0
0
0
2
2
1
1
1
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
0
0

0

0

0

TEST102.PT2:
TEST102 (All frequency and wave heading combinations)
-1 1
IRAD2=-1 IDIF2=1 (Exciting force only)
1 1 1 1 1 1
2 2
TEST102.FDF:
TEST102 Single cylinder R=8.435, Inner radius= 3*R, partition radius= 7*R
25.30500
-1
1.0
2
16.87
4
8
TEST102.CFG:
IPLTDAT=1

B–6

0

MAXSCR=1024
ISOR=1
IQUAD=0
ILOG=1
IDIAG=0
IRR=0
NUMHDR=1
IALTPOT = 1
! Alternative Form 1 POT/CFG
IALTFRC = 1
! Alternative Form 1 FRC
I2ND=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–7

B.3 MULTIPLE BODIES (APPROXIMATION) – TEST103
The NBODY option is illustrated in this test run. The bodies are the same as TEST05.
Body one is a circular cylinder of radius 1 meter and draft 2 meters. Body 2 is a spheroid
of length 4 meters and maximum radius 0.25 meters. The gap between these two bodies is
set equal to the beam of the spheroid (0.5 meters) and the origin of the global coordinate
system is located at the mid-point of this gap. The relative locations of the two bodies
and the orientation of the spheroid are speciﬁed in the GGDF ﬁle. One quadrant of the
cylinder is discretized with 336 panels. 12,12 and 16 panels are distributed in the azimuthal,
radial, and vertical directions using cosine spacing in radial and vertical directions. One
quadrant of the spheroid is discretized with 128 panels. 16 and 8 panels are distributed in
the longitudinal, and transverse directions.
The Alternative 1 input format is used for the POTEN subprogram and the Alternative
3 input format is used for FORCE. The vector IALTFRCN is included in TEST103.CFG
to indicate that IALTFRC(1)=1 for the cylinder and IALTFRC(2)=2 for the spheroid in
the separate FRC ﬁles.
The cylinder is ﬁxed and the spheroid is free to move. The added-mass and damping
coeﬃcients and the ﬁrst-order exciting forces, motions, wave elevations, ﬁeld pressures,
ﬁeld velocities and mean drift forces are evaluated in inﬁnite water depth for two wave
periods and one wave heading (POT ﬁle). The second-order forces, wave elevations and
ﬁeld pressures are evaluated for all diﬀerence frequency pairs (PT2 ﬁle).
The forcing on the body due to the second-order incident wave and due to the ﬁrst-order
body motion of the spheriod are considered but the forcing on the free surface is ignored
(FDF ﬁle).

B–8

TEST103.GDF:
TEST103.GDF -- GGDF file for use with IALTPOT=1 (Cylinder + Spheroid)
-1. 9.80665
2
test103c.gdf
1.25 0.0 0.0 0.0
0
0
0
0
0
0
fixed
test103s.gdf
-0.5 0.0 0.0 90.0
1
1
1
1
1
1
all modes are free
TEST103C.GDF: (first 8 lines only)
TEST103 & TEST103A
Cylinder R=1 T=2 Double cosines side, cosine bottom
1.000000
9.806650
1
1
336
0.0000000E+00 0.0000000E+00 -2.000000
0.0000000E+00 0.0000000E+00 -2.000000
0.1294095
1.7037088E-02 -2.000000
0.1305262
0.0000000E+00 -2.000000
TEST103S.GDF: (first 8 lines only)
TEST103 TEST103A Spheriod L=2 R1=R2=0.25
1.000000
9.806650
1
1
128
1.990369
2.4504285E-02 -1.0927847E-08
2.000000
0.0000000E+00 -1.0927847E-08
1.961571
0.0000000E+00 -4.8772592E-02
1.952125
2.4033442E-02 -4.8772592E-02
TEST103.POT:
TEST103.POT -- CYLINDER + SPHEROID, ILOWHI=0
-1.0 0.0 0.0 0.0 0.0
0
1
Dummy (check TEST103.gdf)
0 0 0 0 0 0
Dummy
2
8.971402 2.006403
1
0.0
TEST103.FRC:
TEST103.FRC -- Cylinder SPHEROID , ILOWHI=0
1 1 1 1 0 1 1 1 1 1 1 1 1 1 1

B–9

0

1.0
test103C.frc
test103S.frc
0
1
0.0001 0.0001 0.
TEST103C.FRC:
TEST 103 Cylinder
0 0 0 0 0 0 0 0 0
0.000000
1.000000
.0000000
.0000000
1.000000
.0000000
.0000000
0
0

0

0

0

0

0

0

0

.0000000
.0000000
1.000000

TEST103S.FRC:
TEST103S Floating Spheroid
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.
rho (water density)
0. 0. 0. XCG (center of grativity w.r.t. body coordinates)
1
IMASS (radii of gyration are assumed 1)
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621
0
IDAMP (if damped change it to 1 and add 6 x 6 damping forces)
0
ISITF (if restoring force applied change it to 1 and add 6 x 6 elements)
0
0
TEST103.PT2:
TEST103 -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
0 2
(all difference frequency pairs)
TEST103.FDF:
TEST103 -- Free suface forcing not included (NPF,NTCL,NAL=0)

B–10

5.
0
0

PARTR (irrelavant because no foricing considered)
0
NPF, NTCL
NAL

TEST103.CFG
IPLTDAT=1
MAXSCR=2000
ISOR=1
IQUAD=0
ILOG=1
IDIAG=0
IRR=0
NUMHDR=1
NOOUT=0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0
IALTPOT = 1
! GDF names in GGDF file
IALTFRC = 3
! Alternative Form 3 FRC for GFRC TEST33.FRC
IALTFRCN= 1 2 ! form 1 TEST103c.FRC and Form 2 spheriod TEST103s.FRC
I2ND=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–11

B.3A MULTIPLE BODIES – TEST103A
This test is the same as TEST103 except the free surface forcing is not ignored in the
second-order computation.
The cylinder is ﬁxed and the spheroid is free to move. The added-mass and damping
coeﬃcients and the ﬁrst-order exciting forces, motions, wave elevations, ﬁeld pressures,
ﬁeld velocities and mean drift forces are evaluated in inﬁnite water depth for two wave
periods and one wave heading.
The second-order forces, wave elevations and ﬁeld pressures are evaluated for all diﬀerence frequency pairs (see PT2 ﬁle).
The free surface inside the inner circle of radius 5 meters is discretized automatically
with 2052 panels over the entire inner region. No intermediate region is considered and
the partition radius is the same as the radius of inner circle. Perspective view of the body
and the inner region of the free surface is shown below.

B–12

TEST103A.GDF:
TEST103A.GDF -- GGDF file for use with IALTPOT=0 (Cylinder + Spheroid)
-1. 9.80665
2
test103c.gdf
1.25 0.0 0.0 0.0
0
0
0
0
0
0
fixed
test103s.gdf
-0.5 0.0 0.0 90.0
1
1
1
1
1
1
all modes are free
TEST103C.GDF: (first 8 lines only)
TEST103 & TEST103A
Cylinder R=1 T=2 Double cosines side, cosine bottom
1.000000
9.806650
1
1
336
0.0000000E+00 0.0000000E+00 -2.000000
0.0000000E+00 0.0000000E+00 -2.000000
0.1294095
1.7037088E-02 -2.000000
0.1305262
0.0000000E+00 -2.000000
TEST103S.GDF: (first 8 lines only)
TEST103 TEST103A Spheriod L=2 R1=R2=0.25
1.000000
9.806650
1
1
128
1.990369
2.4504285E-02 -1.0927847E-08
2.000000
0.0000000E+00 -1.0927847E-08
1.961571
0.0000000E+00 -4.8772592E-02
1.952125
2.4033442E-02 -4.8772592E-02
TEST103A.POT:
TEST103A.POT -- CYLINDER + SPHEROID, ILOWHI=0
-1.0 0.0 0.0 0.0 0.0
0
1
Dummy (check TEST103.gdf)
0 0 0 0 0 0
Dummy
2
8.971402 2.006403
1
0.0
TEST103A.FRC:
TEST103A.FRC -- Cylinder SPHEROID , ILOWHI=0
1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0

B–13

1.0
test103C.frc
test103S.frc
0
1
0.0001 0.0001 0.
TEST103C.FRC:
TEST 103 Cylinder
0 0 0 0 0 0 0 0 0
0.000000
1.000000
.0000000
.0000000
1.000000
.0000000
.0000000
0
0

0

0

0

0

0

0

0

.0000000
.0000000
1.000000

TEST103S.FRC:
TEST103S Floating Spheroid
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.
rho (water density)
0. 0. 0. XCG (center of grativity w.r.t. body coordinates)
1
IMASS (radii of gyration are assumed 1)
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621
0
IDAMP (if damped change it to 1 and add 6 x 6 damping forces)
0
ISITF (if restoring force applied change it to 1 and add 6 x 6 elements)
0
0
TEST103A.PT2:
TEST103 -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
0 2
(all difference frequency pairs)
TEST103A.FDF:
TEST103A AND 113A

B–14

5.
-1
0

PARTR
1.5

AUTO-DISCRETIZATION, SCALE=1.5
NAL

TEST103A.CFG
IPLTDAT=1
MAXSCR=2000
ISOR=1
IQUAD=0
ILOG=1
IDIAG=0
IRR=0
NUMHDR=1
NOOUT=0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0
IALTPOT = 1
! GDF names in GGDF file
IALTFRC = 3
! Alternative Form 3 FRC for GFRC TEST33.FRC
IALTFRCN= 1 2 ! form 1 TEST103c.FRC and Form 2 spheriod TEST103s.FRC
I2ND=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–15

B.11 BOTTOM MOUNTED CYLINDER – TEST111
The test uses the same cylinder as in the low-order TEST101. The ﬁrst-order quantities
including exciting forces, pressures and ﬂuid velocities on the body, ﬁeld pressures and wave
elevation, ﬁeld velocities, drift forces are evaluated. In addtion, the second-order quantities
including quadratic forces, complete forces by direct and indirect methods, pressures on
the body, ﬁeld pressures, wave elevations are evaluated for a bottom mounted vertical
circular cylinder of radius 1 meter and draft 1 meter, in ﬁnite water depth of 1 meter for
two wave periods and two wave headings. Two second harmonic of unidirectional waves, a
mean and a diﬀerence frequency of bidirectional waves are considered in the second-order
computation (see PT2 ﬁle).
Using two planes of symmetry, only the ﬁrst quadrant of the surface of the cylinder is
divided into 4 higher-order panels as shown below. The order of B-spline is set KSPLIN=3
in CFG ﬁle. Uniform mapping is used as the pressure and velocity are ﬁnite on the body
surface.
The free surface inside the inner circle of radius 3 meters is discretized automatically with
159 panels on the ﬁrst quadrant. No intermediate region is considered and the partition
radius is the same as the radius of inner circle. Perspective view of the body and the inner
region of the free surface is shown below.

B–16

TEST111.GDF:
TEST111 Bottom mounted cylinder R=1 T=1 -- analytic geometry (npatch=1)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
1 -1
NPATCH IGDEF
2
1.0 1.
RADIUS, DRAFT
0
Uniform mapping
TEST111.SPL:
TEST111.spl - cylinder R=1 T=1. -- analytic geometry (npatch=1)
2 2
NU NV (Patch 1, side
u azimuthal v vertical)
TEST111.POT
TEST111.POT Cylinder R=1, T=1, igdef=-1
1.
-1
1
IRAD, IDIFF
2
NPER (array PER follows)
0.4 0.8
2
NBETA (array BETA follows)
0. 90.
1
NBODY
test111.gdf
0. 0. 0 0.
XBODY
1 0 0 0 0 0
IMODE(1-6)
0
NEWMDS
TEST111.FRC:
TEST111.FRC Bottom mounted Cylinder R=1, T=1.0, igdef=-1
0
0
1
0
3
1
1
2
2
1
1
1
1
1
1
0.000000
VCG
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
XPRDCT
0
NBETAH
1
NFIELD
1.044325
8.2290111E-02 0.0000000E+00 XFILED

0

IOPTN(1-16)

TEST111.PT2:
TEST111.PT2 -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
1 1

B–17

2
1
1
2
1
2
1
1
1
1

1 1
1
2 1
1
1 1
1
2 1
2

TEST111.FDF:
TEST111.FDF
3.0000E+00
-1.
1.
0

Inner radius (=Partition radius)
Auto Fdf, Scale=1.
No intermediate region

TEST111.CFG:
IPLTDAT=4
ILOWGDF=4
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=1
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=3
MONITR=0
NUMHDR=1
I2ND=1
IPOTEN=1
ILOG=1
NOOUT= 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–18

B.12 TRUNCATED CIRCULAR CYLINDER – TEST112
The test uses the same cylinder as in the low-order TEST102. The ﬁrst-order exciting
forces, drift forces and the second-order quadratic forces and complete second-order forces
by direct and indirect methods are evaluated for a vertical circular cylinder of radius 8.435
meters and draft 35 meters, in inﬁnite water depth for two wave periods and two wave
headings. All sum- and diﬀerence frequency pairs and wave heading pairs are considered
for the second-order forces.
Using two planes of symmetry, the ﬁrst quadrant of the surface of the cylinder is represented with 4 patches including two narrow patches near the corner, one on the side and
the other on the bottom. The width of these patches is 1 meter. Nonuniform mapping
corresponding to the corner ﬂow is used on these patches. The rest of each surface on the
side and bottom is represented be a patch with unform mapping. 20 higher-order panels
are speciﬁed on the ﬁrst quadrant of the surface as shown in the following ﬁgure.
The free surface inside the inner circle of radius 25.305 meters, three times of the
cylinder radius, is automatically discretized with 159 panels on the ﬁrst quadrant. The
intermediate region consists of two annuli with width of 16.87 meters, two times the radius
of the cylinder. On each annulus, 24 + 1 nodes and 8 nodes Gauss quadratures are applied
in azimuthal and radial directions, respectively. The partition circle coincides with the
outside circle of the second annulus and its radius is 58.45 meters. Perspective view of the
body and the inner region of the free surface is shown below.

B–19

TEST112.GDF:
TEST112.GDF -- Cylinder R=8.435m D=35m, nonuniform mapping 1m from corner
8.435
9.80665
1
1
4 -101
1
8.435 35. 1. RADIUS, DRAFT, WIDTH OF CORNER PATCHES
TEST112.SPL:
TEST112.SPL -- 2 azimuthal, 6
2 6
Side above corner patch
2 1
Side corner patch
2 2
Bottom inside patch
2 1
Bottom corner patch

side vertical, 2 bottom radial, 1 corner patches
(2 azimutal, 6 vertical)
(2
1 vertical)
(2
2 radial )
(2
1 radial

TEST112.POT:
TEST112.POT infinite depth, diffraction only
-1 0. 0. 0. 0.
-1 1
1 1 1 1 1 1
2
7.8934 5.1207
2
0. 90.
TEST112.FRC:
TEST112.FRC -- Linear, mean and second-order exciting forces
0
0
1
0
0
0
0
2
2
1
1
1
0
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
.0000000
.0000000
.0000000
1.000000
0
0

TEST112.PT2:
TEST112 (All frequency and wave heading combinations)
-1 1
IRAD2=-1 IDIF2=1 (Exciting force only)
1 1 1 1 1 1
2 2

B–20

0

0

0

TEST112.FDF
TEST112 Single cylinder R=8.435, Inner radius= 3*R, partition radius= 7*R
25.30500
-1
1.0
2
16.87
4
8
TEST112.CFG:
MAXSCR=1000
IPLTDAT=4
IQUAD=0
ILOG=1
IDIAG=0
IRR=0
ISOR=0
NUMHDR=1
IALTPOT = 1
! Alternative Form 1 POT/CFG
IALTFRC = 1
! Alternative Form 1 FRC
I2ND=1
ILOWHI=1
KSPLIN=3
IQUADO=3
IQUADI=4
ISOLVE=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–21

B.13 MULTIPLE BODIES (APPROXIMATION) – TEST113
The bodies are the same as TEST103. One quadrant of the cylinder is represented with
4 patches. Nonuniform mapping corresponding to corner ﬂows is used on the patches of
width 0.1 meter at the corner. Uniform mapping is used on the other patches on the side
and bottom. The ﬁrst quadrant of the cylinder is represented by 21 higher order panels
and that of the spheroid by 8 panels.
The Alternative 2 input format is used for the POTEN subprogram and the Alternative
3 input format is used for FORCE. The vector IALTFRCN is included in TEST113.CFG
to indicate that IALTFRC(1)=1 for the cylinder and IALTFRC(2)=2 for the spheroid in
the separate FRC ﬁles.
The cylinder is ﬁxed and the spheroid is free to move. The added-mass and damping
coeﬃcients and the ﬁrst-order exciting forces, motions, wave elevations, ﬁeld pressures,
ﬁeld velocities and mean drift forces are evaluated in inﬁnite water depth for two wave
periods and one wave heading.
The second-order forces, wave elevations and ﬁeld pressures are evaluated for all diﬀerence frequency pairs (see PT2 ﬁle). The forcing on the free surface is ignored. The forcing
on the body due to the second-order incident wave and due to the ﬁrst-order body motion
of the spheriod are considered but that on the free surface is ignored (see FDF ﬁle).
Perspective view of the body and the inner region of the free surface is shown below.

B–22

TEST113C.GDF:
TEST113C cylinder R=1 T=2 -- analytic geometry (npatch=4)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
4 -13
NPATCH IGDEF
1
NLINES
1.0 2. 0.1
RADIUS, DRAFT

TEST113S.GDF:
TEST113S spheroid a=2, b=c=0.25 -- igdef=-4
1. 9.80665 ULEN GRAV
1 1
ISX ISY
1 -4
NPATCH IGDEF
1
2.0 0.25 0.25
A, B, C
TEST113.POT:
TEST113.POT -- Cylinder + spheroid, ILOWHI=1
-1.
0
1
IRAD, IDIFF
2
NPER (array PER follows)
8.971402 2.006403
1
NBETA (array BETA follows)
0.
2
NBODY
test113c.gdf
1.25 0.0 0.0 0.0
XBODY
0 0 0 0 0 0
IMODE(1-6)
0
NEWMDS
test113s.gdf
-0.5 0.0 0.0 90.0
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST113.FRC:
TEST113.FRC -- Cylinder SPHEROID
1 1 1 1 0 1 1 1 1 1 1
1.0
test113C.frc
test113S.frc
0
1

1

1

B–23

1

1

0

0.0001 0.0001 0.
TEST113C.FRC:
TESTS 103 and
0 0 0 0 0
0.000000
1.000000
.0000000
.0000000
0
0

113 Fixed Cylinder
0 0 0 0 0 0 0
.0000000
1.000000
.0000000

0

0

0

0

.0000000
.0000000
1.000000

TEST113S.FRC:
TESTS 103 and 113 Floating Spheroid
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.
rho (water density)
0. 0. 0. XCG (center of grativity w.r.t. body coordinates)
1
IMASS (radii of gyration are assumed 1)
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621
0
IDAMP (if damped change it to 1 and add 6 x 6 damping forces)
0
ISITF (if restoring force applied change it to 1 and add 6 x 6 elements)
0
0
TEST113.PT2:
TEST103 AND TEST 113 -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
0 2
(all difference frequency pairs)
TEST113.FDF:
TEST103 AND 113 -- Free suface forcing not included (NPF,NTCL,NAL=0)
5.
PARTR (irrelavant because no foricing considered)
0
0
NPF, NTCL
0
NAL
TEST113.CFG:

B–24

ILOWGDF=4
IPLTDAT=4
MAXSCR=1000
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
NUMHDR=1
NOOUT= 0 1 1 1 0 1 1 1 1 1 1
1 0 1 1 0
IALTFRC = 3
! Alternative Form 3 FRC
IALTFRCN= 1 2
I2ND=1
ILOG=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–25

B.13A MULTIPLE BODIES – TEST113A
This test is the same as TEST113 except the free surface forcing is not ignored in the
second-order computation.
The cylinder is ﬁxed and the spheroid is free to move. The added-mass and damping
coeﬃcients and the ﬁrst-order exciting forces, motions, wave elevations, ﬁeld pressures,
ﬁeld velocities and mean drift forces are evaluated in inﬁnite water depth for two wave
periods and one wave heading.
The second-order forces, wave elevations and ﬁeld pressures are evaluated for all diﬀerence frequency pairs (see PT2 ﬁle).
The free surface inside the inner circle of radius 5 meters is discretized automatically
with 2052 panels over the entire inner region. No intermediate region is considered and
the partition radius is the same as the radius of inner circle. Perspective view of the body
and the inner region of the free surface is shown below.

B–26

TEST113C.GDF:
TEST113C cylinder R=1 T=2 -- analytic geometry (npatch=4)
1. 9.80665 ULEN GRAV
1 1
ISX ISY
4 -13
NPATCH IGDEF
1
NLINES
1.0 2. 0.1
RADIUS, DRAFT, WIDTH OF CONNER PATCHES

TEST113S.GDF:
TEST113S spheroid a=2, b=c=0.25 -- igdef=-4
1. 9.80665 ULEN GRAV
1 1
ISX ISY
1 -4
NPATCH IGDEF
1
2.0 0.25 0.25
A, B, C
TEST113A.POT:
TEST113A.POT -- Cylinder + spheroid, ILOWHI=1
-1.
0
1
IRAD, IDIFF
2
NPER (array PER follows)
8.971402 2.006403
1
NBETA (array BETA follows)
0.
2
NBODY
test113c.gdf
1.25 0.0 0.0 0.0
XBODY
0 0 0 0 0 0
IMODE(1-6)
0
NEWMDS
test113s.gdf
-0.5 0.0 0.0 90.0
XBODY
1 1 1 1 1 1
IMODE(1-6)
0
NEWMDS
TEST113A.FRC:
TEST113A.FRC -- Cylinder SPHEROID
1 1 1 1 0 1 1 1 1 1 0 1
1.0
test113C.frc
test113S.frc
0
1

0

B–27

1

1

0

0.0001 0.0001 0.
TEST113C.FRC:
TESTS 103 and
0 0 0 0 0
0.000000
1.000000
.0000000
.0000000
0
0

113 Fixed Cylinder
0 0 0 0 0 0 0
.0000000
1.000000
.0000000

0

0

0

0

.0000000
.0000000
1.000000

TEST113S.FRC:
TESTS 103 and 113 Floating Spheroid
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.
rho (water density)
0. 0. 0. XCG (center of grativity w.r.t. body coordinates)
1
IMASS (radii of gyration are assumed 1)
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621 0.
0.
0.
0.
0.
0.
0.257621
0
IDAMP (if damped change it to 1 and add 6 x 6 damping forces)
0
ISITF (if restoring force applied change it to 1 and add 6 x 6 elements)
0
0
TEST113A.PT2:
TEST103/TEST103A AND TEST113/TEST113A -- PT2
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
-1 1
(diffraction for second-order force for all modes)
1 1 1 1 1 1
0 2
(all difference frequency pairs)
TEST113A.FDF:
TEST103A AND 113A -- Free suface forcing not included (NPF,NTCL,NAL=0)
5.
PARTR
-1
1
NPF, NTCL
0
NAL
TEST113.CFG:

B–28

ILOWGDF=4
IPLTDAT=4
MAXSCR=1000
ILOWHI=1
IALTPOT=2
IRR=0
ISOLVE=2
KSPLIN=3
IQUADO=3
IQUADI=4
IPERIO=1
NUMHDR=1
NOOUT= 0 1 1 1 0 1 1 1 1 1 1
1 0 1 1 0
IALTFRC = 3
! Alternative Form 3 FRC
IALTFRCN= 1 2
I2ND=1
ILOG=1
USERID_PATH=\WAMITv6 (directory for *.exe, *.dll, and userid.wam)

B–29

Using the WAMIT-RGKernel Interface
J. S. Letcher, Jr.
AeroHydro, Inc.
Southwest Harbor, Maine 04679 USA
207-244-4100
Sept. 8, 2002; revised June 21, 2006

Contents
Using the WAMIT-RGKernel Interface ..................................................................................... 1
1. Summary.................................................................................................................................... 1
2. Supported features and options ............................................................................................. 2
3. Required files, versions and file locations............................................................................. 3
4. MultiSurf modeling considerations ....................................................................................... 4
4.1 TrimSurfs are excluded ................................................................................................................... 4
4.4 Surface normal orientations............................................................................................................. 5
4.5 Base plane and waterline ................................................................................................................. 6
4.6 Mirror symmetry ............................................................................................................................. 7
4.7 Rotational symmetry........................................................................................................................ 7
4.8 “Fast” vs. “Accurate” evaluation..................................................................................................... 8
4.9 Divisions and Subdivisions.............................................................................................................. 9
4.10 Irregular frequency removal .......................................................................................................... 9
4.11 Coordinate singularities ................................................................................................................. 9
4.12 Breaklines in surfaces.................................................................................................................. 10
4.13 “Cosine spacing” ......................................................................................................................... 10
4.14 Parameters ................................................................................................................................... 11

5. GDF file format ....................................................................................................................... 11
6. Log file RGKLOG.TXT........................................................................................................... 13
7. Synopsis of operation............................................................................................................. 14
8. Error Conditions ..................................................................................................................... 14

1. Summary
A joint development effort between AeroHydro, Inc. and WAMIT, Inc. has forged an
intimate connection between MultiSurf and WAMIT. As a result, model geometry
developed in MultiSurf’s “relational geometry” (RG) framework can be directly accessed
by WAMIT for all setup and analysis purposes within WAMIT’s “higher-order” solution
method. This feature is available in WAMIT versions 6.1 and higher.
RGKernel is the C library of mathematical, geometric and data storage functions that
supports MultiSurf, SurfaceWorks and other RG applications. It is compiled as a DLL. In
this project a second DLL was developed to provide the interface between WAMIT’s
Fortran code and RGKernel’s C procedures.

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The advantages of this integration are:
(1) Accuracy. RG geometry definitions are fundamentally exact, and in practice can be
evaluated to a high degree of precision. This is in contrast to faceted and NURBSbased geometric representations, whose accuracy is commonly compromised by
approximations. Accuracy of surface-surface junctions is critical in WAMIT’s highorder analysis method.
(2) Convenience. When MultiSurf is used to develop WAMIT models for this interface,
the user avoids all further effort to develop panel files for the low-order method, or
B-spline approximation files as have been needed for the high-order method. The
Geometric Data File (GDF) is usually 8 lines for each body defined by this method.
The purpose of this document is to provide instructions for use of this new WAMIT
feature. The topics include:
Supported features and options
Required files, versions and locations
MultiSurf modeling considerations
GDF file format
Log file RGKLOG.TXT
Error conditions

2. Supported features and options
The WAMIT-RGKernel interface supports all current features of the higher-order
solution option.
Multiple bodies. Each body is associated with a GDF file in the POT or FNAMES file.
IGDEF = 2 is reserved to mean “geometry specified by an .MS2 file.” The different
bodies can have various IGDEF types.
XBODY positioning. Each body has an associated XBODY transformation to position it
in the global coordinate system. The MultiSurf model of a body can be constructed in
any convenient position, e.g. aligned and centered in the body coordinate system.
Alternatively, one MultiSurf model can be constructed with two or more bodies in their
final global positions; this allows visualization of the bodies’ relative and absolute
positioning.
Mirror symmetry. If the body has mirror symmetry with respect to one vertical plane,
only one half of the body needs to be constructed. If the body has mirror symmetry with
respect to two orthogonal vertical planes, only one quarter needs to be constructed.
These symmetry options are represented similarly in MultiSurf and WAMIT.
Rotational symmetry. Alternatively, rotational symmetry is supported. A MultiSurf
model can have rotational symmetry about any one of the coordinate axes, with any
number of copies (NCOPIES), at an angular spacing of 360/NCOPIES degrees. This

C-2

feature greatly simplifies modeling structures that have rotational symmetry; only
1/NCOPIES of the total model needs to be explicitly constructed.
Units. WAMIT is flexible with regard to units, but some of its inputs are quantities with
units of length, and the global length unit is implicit in the GRAV parameter. MultiSurf
supports meters, centimeters, millimeters, feet and inches as length units, and this choice
is specified in the model file. The interface code compares the model and global units,
and transparently performs any needed units conversions.
Parameters. The GDF file permits overriding floating-point values in the definition of
any object in the model. This allows a single model (parametrically constructed) to be
analyzed in a wide variety of configurations without opening the model and making
any changes in MultiSurf.
Variables and formulas. Use of numeric variables and formulas in RG models is an
advanced feature supported by MultiSurf versions 4.8 and higher. This represents a
further powerful capability for constructing parametric model families. A variable can
serve as a parameter in the GDF file.
Irregular frequency removal. Interior free-surface patches can be constructed in
MultiSurf, and included in the data available to WAMIT for use with its IRR = 1 option.

3. Required files, versions and file locations
RG2WAMIT.DLL contains the Fortran-C interface code.
RGKERNEL.DLL is the relational geometry C library. RGKERNEL.DLL is a component
of any MultiSurf or SurfaceWorks installation, however please NOTE: The WAMITRGKernel interface may use a different RGKernel version from MultiSurf. These DLL
versions are not necessarily interchangeable, and should be kept in separate directories.
Although they have the same filename, different DLL versions can be distinguished by
different sizes and different dates.
(This said, it has been our uniform experience to date that a newer version of
RGKERNEL.DLL can be safely substituted for an older one; RG2WAMIT.DLL does NOT
have to be recompiled in this event, and new modeling features incorporated in the
newer RGKERNEL.DLL become available for use with WAMIT.)
RG2WAMIT.DLL and RGKERNEL.DLL must be located in the same directory as
WAMIT.EXE. Also, the GDF files and any model files named in IGDEF=2 GDF’s must be
located in this same directory.
MultiSurf does not need to be running when the WAMIT run takes place. WAMIT does
not need to be running when the model is constructed in MultiSurf. Communication
between MultiSurf and WAMIT is solely through the .MS2 model file created in
MultiSurf and accessed by RG2WAMIT.DLL.

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4. MultiSurf modeling considerations
4.1 TrimSurfs are excluded
Trimmed surfaces (TrimSurf and TrimSurf2 entities) cannot be handled by the
RG2WAMIT facility. This is because WAMIT needs a complete parametric square over
which to distribute its tensor-product description of the unknown potential, while a
trimmed surface is by definition an arbitrary bounded portion of a parametric surface.
RG2WAMIT checks for the presence of TrimSurfs in the wetted-surface ObjectList, and
returns an error when one is encountered.
SubSurfs, on the other hand, are perfectly legal for RG2WAMIT, because they have (like
all other RG surface types except TrimSurf) 4-sided topology and a full [0,1]x[0,1]
parameter space. One or more SubSurfs can always cover a complex-shaped region of a
base surface, where a single TrimSurf might otherwise be used.
4.2 Optional wetted-surface ObjectList (Entity List)
In order to be read and utilized through the WAMIT-RGKernel interface, an MS2 file
(“model file”) may include an ObjectList naming the wetted surfaces. When this option
is used, WAMIT needs to be given the name of this ObjectList (in the GDF file, see
below).
Note: In MultiSurf 6, an ObjectList is called an "Entity List". This is consistent with the
following general terminology conventions:
MultiSurf 4 & 5
Surface Works and MultiSurf 6
"object"
=
"entity"
"entity"
=
"entity type"
"support"
=
"parent"
"dependent" =
"child"
A simple alternative is to specify ‘*’ for the ObjectList. This will signify “all visible
surfaces in the model”. To be visible, a surface has to have a positive visibility index, and
be on an enabled layer. Surfaces that are not to be included in the hydrodynamic
analysis can either be hidden (by editing their visibility attribute), or located on a layer
that is disabled in the MS2 file (unchecked in Settings/Layers dialog).
ObjectList is an entity type that has been supported since MultiSurf 4.0. It is just a list of
other objects, of any type. It is a non-graphical entity; you may or may not be able to see
the objects in the list – that depends on whether each object is made from a graphical
entity type, whether it is visible, and whether it is on a turned-on layer. But, the
ObjectList itself is never drawn – it is just a list.
In MultiSurf 5, to create an ObjectList, use Create/ObjectList. (In MultiSurf 6, Insert/
Entity List.) You can preselect the “supports”, i.e. the objects to be in the list. You can
add or subtract objects just as you edit the multiple supports of any other object. Select/

C-4

By Name is often the best way to select an ObjectList – or find it in the Selection Set view
Availables list, . (In MultiSurf 6, find it in the Entities Manager, under Entity Lists.)
This list must contain each wetted surface, and only the wetted surfaces, in your model
(or in the half, quarter or sector you are modeling explicitly when using symmetry
image options). If you leave a wetted surface out of the list, include a surface twice, or
include a construction surface that should not be included in the hydrodynamic
solution, you will very likely find volume discrepancies in WAMIT. The interface does
not currently produce a warning in these situations.
You can have two or more ObjectLists in your model. They (like all other objects) will be
distinguished by their unique names. By changing the ObjectList name specified in the
GDF file, you can conveniently switch from one ObjectList to another without touching
the model in MultiSurf. For example, you may have two sets of wetted surfaces
representing two different drafts. Some surfaces can be common between these lists
(e.g., surfaces that are fully wetted at the lightest draft), or the lists might be completely
disjoint. (Note that in this scenario, you will need to also change XBODY when the draft
changes.)
4.3 Order of patches
The ordering of patches for the WAMIT solution will often be immaterial, but in some
situations it matters. For example, if you wish to use the NPATCH parameter in the GDF
file to exclude one or more patches (such as a free surface patch, in order to compare
results with and without irregular frequency removal), then you have to be sure that the
patches to be excluded are the last in patch order.
There are two rules governing the order of patches, corresponding to the two
alternatives for the ObjectList:
(1) If you use an explicit ObjectList, the patch order will be the order in which the
surfaces are listed in the ObjectList.
(2) If you use the default identifier ‘*’ for the ObjectList, the patch order will be the
order of surfaces in the MS2 file. You can find out this order by searching for “Surf”
in Edit/ Model File or Notepad. In MultiSurf another alternative is to turn on only
the Surfaces filter; set Select/ Visible to “on”; View/ Selection Set; and read the order
in the “Availables” list box.
4.4 Surface normal orientations
For a proper solution of the potential boundary-value problem, WAMIT requires that
the positive normal to each surface be directed into the body. A patch closing the top of
the body for irregular frequency removal must have its normal directed downward
(again, into the body). The positive normal is calculated as the cross product ∂x/∂u x
∂x/∂v, normalized to unit length.
If normals are backwards on some surfaces, WAMIT’s calculation of volumes will very
likely show discrepancies between the X, Y and Z directions. If all normals are reversed,

C-5

the volumes will agree but will be negative. And the hydrodynamic solution probably
won’t make much sense.
When making a surface in MultiSurf, the normal orientation is determined by:
(1) the orientations of supporting objects;
(2) the order in which the supports are used, and
(3) the value of the “orientation” flag.
The orientation flag is handy, because if a surface comes out with its normal opposite to
what you want, you don’t have to rebuild the surface another way, you just have to flip
its “orientation” switch.
In MultiSurf, we visualize the normal orientation by drawing an arrow at the parametric
center of the surface (u=0.5, v=0.5), in the positive normal direction. This arrow is drawn
whenever a surface is selected. To see normal arrows simultaneously on all wetted
surfaces, just select all the wetted surfaces; for example, select the ObjectList, then Select
Expand ObjectLists/ First Generation. (The orientation of non-wetted construction
surfaces doesn’t matter, as these are not included in the surface definitions accessed by
WAMIT.)
It is generally easier to see normals that point out of the geometry than those pointing
inward. For this reason, our convention when making panel models for low-order
WAMIT, and B-spline surface models for HIPAN and high-order WAMIT has always
been to require outward normals on all wetted surfaces, and then to reverse all normals
automatically in the conversion utility.
In the WAMIT-RGKernel interface, we have made the global reversal of normals
optional. The choice is signaled (on a per-body basis) by the inward-normal flag in the
GDF file (see below). This results in two options for the MultiSurf user who is building a
model for the WAMIT-RGKernel interface:
(1) Build the model with all unit normals pointing inward (WAMIT’s convention), and
set the inward-normal flag to 1; or
(2) Build the model with all unit normals pointing outward, and set the inward-normal
flag to 0.
We prefer option (2), because the outward normals are easier to see. In either case, the
orientation flags of the individual surfaces can be freely used in achieving the desired
consistent orientations.
4.5 Base plane and waterline
In MultiSurf, there are no restrictions on the location and orientation of geometry, but
there are some sensible choices that will make the process easier.
Vertical position: Most models for WAMIT analysis are built in one of three vertical
positions. Either

C-6

(1) The model is built above a base plane, so the lowest Z coordinates are zero, and the
design waterline is at some positive Z (call it Zwl > 0); or
(2) The model is built with Z=0 as the design waterline, so the lowest parts of the wetted
surface are at a negative Z (call it Zdraft < 0); or
(3) The model is built with the point representing its center of gravity (C.G.) at the
origin.
Any of these approaches works fine. In the first case, you have to specify XBODY(3) = Zwl to correctly position the body in WAMIT; in the second case, XBODY(3) will be 0; in
the third case, XBODY(3) will be the height of the C.G. above or below the waterline.
The WAMIT-RGKernel interface tests the “wetted surfaces” (i.e., the surfaces in the
named ObjectList, or when * is used for the ObjectList, the visible surfaces) during its
initialization, and reports (in RGKLOG.TXT) any case where part of a “wetted surface”
extends above the global waterplane. This usually indicates an error, either in the
surfaces specified as wetted, or in XBODY(3).
4.6 Mirror symmetry
If the body has one or two planes of complete mirror symmetry, it is very advantageous
to model only the independent half or quarter, and use mirror symmetry options to get
the rest.
The WAMIT-RGKernel interface checks the model’s symmetry flags against the ISX, ISY
flags specified in the GDF file, and issues a warning if there is any discrepancy.
Sometimes this can be harmless, but you should check your model out to be sure you
haven’t either left surfaces out, or covered some areas twice.
4.7 Rotational symmetry
An object has N-fold rotational symmetry with respect to a particular axis if it is
congruent to itself after a rotation of 360 / N degrees about that axis.
Many offshore structures have complete rotational symmetry about a vertical axis. For
example, three-leg TLP’s may have 3-fold rotational symmetry; spars and buoys often
have rotational symmetry of some order; the Hibernia bottom-mounted platform is a 16pointed symmetric star. (Four-leg TLP’s also usually have 4-fold rotational symmetry,
but this case is best treated as combined X and Y mirror symmetry, because of the large
efficiencies that result from exploiting mirror symmetry during the WAMIT solution.)
In MultiSurf, when the object being modeled has complete rotational symmetry, it is
very beneficial to take advantage of it, as only a fraction (1/N) of the total structure then
needs to be explicitly modeled. Rotational symmetry is a model-level property. Z-axis
rotational symmetry is only allowed when X- and Y-mirror symmetry flags are OFF. In
the Settings/Model dialog, check X-, Y- and Z-mirror symmetry OFF, Z-axis rotational
symmetry ON, and enter the number of copies, e.g. 3 for a 3-leg TLP. Then you can
model only the “active” or “independent” sector of the model (e.g. 1 leg and 2 half-

C-7

pontoons of a 3-leg TLP), and the copies will be present implicitly. The symmetry
images will be shown automatically in Render view, and can be toggled on and off in
Wireframe view with the  key.
In the GDF file for a body using rotational symmetry about the Z-axis, ISX and ISY will
be 0, and NPATCH needs to include the symmetry images. For example, if a 3-leg TLP is
modeled with 11 wetted surfaces in the explicitly modeled 120-degree sector, NPATCH
will be 3 x 11 = 33.
Rotational symmetry about the X- or Y-axis is also supported – but only one axis at a
time. (In WAMIT, this would have to be a completely submerged body, positioned
vertically with XBODY(3).)
Rotational symmetry that is incomplete precludes the use of model-level rotational
symmetry. In this situation, the entire geometry has to be built explicitly. But note, the
RotatSurf and CopySurf entity type often provide a highly efficient way to construct the
portions that are rotationally symmetric.
4.8 “Fast” vs. “Accurate” evaluation
RGKernel has two evaluation modes, denoted “Fast” and “Accurate”. In “Accurate”
evaluation, all curves and surfaces are evaluated recursively, using the actual math
functions that provide their definitions. In “Fast” evaluation, curves and surfaces are
evaluated approximately by interpolation in a stored tabulation of the curve or surface;
the fineness of this tabulation is controlled by the divisions x subdivisions products.
Fast vs. accurate evaluation is specified in the GDF file for each body (see below).
Accurate evaluation is typically slower -- in some cases, orders of magnitude slower; this
especially occurs when there are deep levels of dependency in the model, and/or the
model involves entities requiring iterative solutions, such as ProjSnakes and IntSnakes.
The divisions and subdivisions should have no effect on the accuracy under “accurate”
evaluation, since the table is not involved in the calculations.
Fast evaluation is usually much faster, and always more predictable in terms of
performance, but involves some compromise of accuracy due to interpolation errors.
Our general experience is that with the default 8x4 divisions x subdivisions, most curves
and surfaces evaluate to near single-precision accuracy under fast evaluation. The
accuracy should improve as the fourth power of the divisions, provided the divisions on
all supporting curves and surfaces are increased in constant proportion. The “divisions
multiplier” is a simple way to make this increase uniformly across the model.
The GDF file format provides for overriding the model’s divisions multiplier (see
below).
Pending further investigation of the accuracy of the integrated system, our current
recommendation is to use fast evaluation mode, with default divisions and a modest

C-8

divisions multiplier of 2, 3, or 4. This should have little impact on performance, and
should provide WAMIT with full single-precision accuracy for the geometry.
4.9 Divisions and Subdivisions
In MultiSurf, each surface has “division” and “subdivision” attributes that control how
the surface is subdivided for tabulation and display. For low-order WAMIT
panelizations, we use the divisions x subdivisions to determine the mesh density, and
some care with divisions was required to make neat, watertight junctions between the
panels on adjacent surfaces.
Since the surface geometry is more directly accessed in the WAMIT-RGKernel interface,
divisions and subdivisions are less important in general. In “Accurate” mode evaluation,
they should have no bearing at all. However, in “Fast” mode, there are still accuracy
benefits in coordinating divisions between adjacent surfaces: for example, use the same
number of (divisions x subdivisions) on a surface and its supporting curves; where
possible, use matching divisions on surfaces that share a common edge.
4.10 Irregular frequency removal
In the WAMIT 6.1 higher-order solution, irregular frequency removal is effected by
providing additional surface patches that cover any “interior” portions of the plane of
the free surface (IRR = 1).
Such interior free surface patches can be part of the MultiSurf model. They must be
oriented with their positive normal direction consistent with the wetted patches; i.e., if
your model has all outward normals, an interior free surface patch must have its normal
upward. Interior free surface patches must also be included in the ObjectList of wetted
surfaces, and in NPATCH. If you put them last in the ObjectList, you can easily run
WAMIT with and without irregular frequency removal just by setting NPATCH
appropriately in the GDF file.
4.11 Coordinate singularities
A coordinate singularity is a place on a parametric surface where the cross product
∂x/∂u x ∂x/∂v vanishes. This can occur because either ∂x/∂u or ∂x/∂v vanishes, or
because they have the same directions. The most commonly encountered coordinate
singularity is a “pole”, where one or the other of ∂x/∂u, ∂x/∂v vanishes along one whole
edge of the parameter space, e.g. the type of singularity at the north and south poles of a
sphere that is parameterized by latitude and longitude. There are many ways to create
surfaces with poles and other coordinate singularities in MultiSurf; for example when a
RuledSurf is made between a curve and a point, a pole occurs at the point.
Some kinds of coordinate singularities are associated with numerical ill-conditioning in
WAMIT, and so should be avoided as much as possible in modeling for RG2WAMIT.
There are situations where coordinate singularities are apparently harmless, e.g. the
south pole of a half-submerged sphere, or the “south pole” of a truncated vertical

C-9

cylinder. There are other situations where a coordinate singularity is distinctly
beneficial. (See the remarks below under “Cosine spacing”.)
WAMIT, Inc. is recommending especially against poles on the free surface or at
“chines”, i.e. sharp edges that protrude into the water (such as the junction between side
and bottom surfaces of a truncated vertical cylinder).
4.12 Breaklines in surfaces
The RGKernel versions used in RG2WAMIT and MultiSurf 5 and 6 recognize the
presence of “breakpoints” in curves and “breaklines” in surfaces. A breakpoint is a place
where a curve has a discontinuous derivative; if this is the first derivative (a “degree-1
breakpoint") the curve typically has a knuckle. A “breakline” is a u- or v-constant line
across which the surface has a discontinuous derivative; if this is the first derivative, the
surface typically has a knuckle line at this parameter value.
Thus one reasonable way to build a truncated vertical cylinder is to construct a meridian
curve with a breakpoint at the “chine” (e.g., a type-1 BCurve or CCurve), and revolve it
90 degrees about a vertical axis, creating a single RevSurf covering both the bottom and
sides. MultiSurf will be aware of the degree-1 breakline along the chine and will display
the sharp edge correctly.
However, WAMIT will not be aware of the breakline and will be modeling a smooth,
continuous distribution of potential across the chine.
Therefore, we recommend in general that breaklines not be used on surfaces intended
for WAMIT consumption. In this example, modeling the side and bottom surfaces as
two separate surfaces (rather than one surface with a breakline) will make it much easier
to utilize cosine spacing, to better resolve the rapidly changing potential around the
chine.
4.13 “Cosine spacing”
The phrase “cosine spacing” refers to systematic refinement of the mesh in regions
where gradients are high, to provide better resolution in these areas. Although we don’t
have a mesh anymore in the usual sense of a panel file, the concept of cosine spacing is
in fact still quite pertinent, and the techniques for achieving it are highly similar to those
used for the low-order method.
Exterior corners (“chines”) are the typical places where cosine spacing improves
solution accuracy. Also, accuracy benefits are experienced with cosine spacing near the
free surface. In MultiSurf there are two common techniques for achieving mesh
concentration:
(1) Relabeling of lines, curves, snakes and surfaces
(2) Type-2 B-spline lofted surfaces (BLoftSurfs) with duplicate master curves at the
ends; also (usually for flat rectangular patches) B-spline surfaces (BSurfs) with
duplicate control points at corners and/or edges.

C-10

4.14 Parameters
As described below, the GDF file can contain any number of “parameter lines”, which
override specified floating-point data values in the model. The format of a parameter
line is just:
object-name float-index value
For example, if AbsPoint ‘draft_pt’ is a point whose Z coordinate (its third floating-point
data value) controls the draft of the model, the parameter line:
draft_pt 3 -35.5
sets the draft to 35.5.
Obviously, carelessly setting parameters can “break” the MultiSurf geometry. For
example, the leg radius and pontoon radius for a TLP might be separate parameters. As
long as the pontoon radius is smaller than the leg radius, the geometry works; but if
pontoon radius is set larger than the leg radius, a projection will fail, and the pontoon
surface can’t be evaluated. This failure will be reported in RGKLOG.TXT. If you try the
same settings in MultiSurf, you can see what goes wrong.
Even when all surfaces evaluate geometrically, there can be modeling problems that
affect the WAMIT solution. When the pontoon radius is somewhat smaller than the leg
radius, it’s possible for the two pontoons arriving at the leg to intersect each other,
which may or may not cause a recognizable error condition in WAMIT. It is thus a good
idea to check all parameter combinations that you plan to use visually in MultiSurf,
before starting the WAMIT solutions.

5. GDF file format
A GDF file represents geometry for a single body. WAMIT’s GDF file has a new format
option, with IGDEF = 2 specifying “geometry to be obtained from an MS2 model file”.
The first four lines of the GDF are the same as in other optional GDF formats:
header
ULEN GRAV
ISX ISY
NPATCH IGDEF
Where:
header is an identifying character string, up to 72 chars.
ULEN is a characteristic length
GRAV is acceleration due to gravity
ISX, ISY are symmetry flags (0 or 1) for X and Y mirror symmetry
NPATCH = number of patches in this body (not counting X and Y mirror symmetry
images, but counting Z-axis rotational symmetry images)
IGDEF = 2, for using geometry from an .MS2 file

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In an IGDEF=2 GDF file, there are always 4 more lines:
Line 5: NLINES = no. of lines to follow this line (3 plus number of parameter lines)
Line 6: Model filename, including extension .MS2 (no embedded spaces)
Line 7: Object name of ObjectList to be present in the named MS2 file, listing the wetted
surfaces. (Alternative: use ‘*’ to signify “all visible surfaces” – and make sure only the
wetted surfaces are visible in your model when the .MS2 file is saved.)
Line 8: 3 integers:
Fast/accurate flag (0 = fast, 1 = accurate);
Divisions multiplier override (0 to 10). A 0 for DivMult will mean “use the value
in the model file”.
Inward-normal flag (0 = outward normals, 1 = inward normals)
Then there are (NLINES – 3) optional parameter lines, each with 3 items:
object-name float-index value
object-name is the name of an object in the model that has one or more floating-point
numbers in its object data. For example, an AbsPoint has 3 floats -- its X, Y and Z
coordinates --; an AbsBead has just one, its t parameter.
float-index is an integer from 1 to the number of floats the object has, e.g. 1 to 3 for an
AbsPoint; 1 to 1 for an AbsBead.
value is the floating-point value to substitute.

Note re symmetry: The actual number of patches produced internally in WAMIT by this
GDF is:
(number of wetted surfaces) x (ISX+1) x (ISY+1) x N
The interface code does not need to count the X- and Y-symmetry images in indexing the
patches. All mirror-image and XBODY transformations, and mirror-image indexing, are
performed inside WAMIT. However, NPATCH does need to count implicit rotational
images produced by Z-axis rotational symmetry.
Note re Fortran READS: Fortran will read this GDF with (5+NLINES) READ
statements, each reading a complete record. This means any of the records can have
other text following the required entries, separated only by one or more spaces. This
additional data has no effect in Fortran, and is likewise being ignored by C.
Example GDF:
Tension-leg platform example
120. 32.2 ULEN, GRAV
1 1 ISX,ISY
16 2
NPATCH, IGDEF
3
NLINES
TLP4H.MS2
wetted_surfs

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0

2

0 Fast; DivMult; outward normals

Explanation: The first line is an identifying message. ULEN is 120 and GRAV is 32.2.
This value of GRAV implies that the length unit for the global WAMIT model is feet. (If
this MS2 file uses meters for its units, conversion from meters to feet will be performed
in RGKINIT.) The model has mirror symmetry with respect to its X and Y planes (only
one quadrant is explicitly modeled). NPATCH is 16, the number of patches in one
quadrant of the body. IGDEF is 2 to signify geometry from an MS2 file. 3 for NLINES
indicates 3 lines to follow. The MS2 file is TLP4H.MS2. This model file must contain an
ObjectList named ‘wetted_surfs’ which names the wetted surfaces. The last line specifies
Fast evaluation; a divisions multiplier of 2, overriding any divisions multiplier in the
model file; and the model is constructed with outward normals. There are no parameter
lines in this example.

6. Log file RGKLOG.TXT
During operation of the WAMIT-RGKernel interface, a log file is opened and written to
in the same directory as WAMIT.EXE. This file, named RGKLOG.TXT, contains an echo
of XBODY and GDF data, a reflection of some information extracted from the MS2
model file(s), brief reports on some aspects of interface initialization and utilization, and
possible warning and error messages.
RGKLOG.TXT also includes a summary tabulation of surfaces vs. global, body and
patch indices, which will give the correct order of surfaces for constructing an SPL file, if
desired.
Example:
RGKLOG.TXT -- log file for RG2WAMIT.DLL
=====================================================
Starting procedure RGKINIT at 25-Nov-2001 13:59:49
NBODY: 1
----------------------------------------------------Data for body no. 1
XBODY:
0.000
0.000
0.000
0.000
Truncated cylinder example 1.0 m radius, 0.5 m draft
ULEN:
1.000 GRAV: 9.80660
Global length unit (deduced from GRAV) is m.
ISX: 1 ISY: 1
NPATCH: 2 IGDEF: 2
MS2 filename: TRCYL2.MS2
ObjectList of wetted surfaces: wetted_surfs
Model will be evaluated in FAST mode.
Unit normals are outward.
Opened RG model file: TRCYL2.MS2
Body 1 patch 1 surface name: side_surf
Body 1 patch 2 surface name: bottom_surf
----------------------------------------------------index body igdef patch ior surface name
1
1
2
1
1 side_surf

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2
1
2
2
1 bottom_surf
----------------------------------------------------Exiting RGKINIT at 25-Nov-2001 13:59:49, err = 0
=====================================================
There were 1 RGKEVAL calls.
Executing RGKDONE at 25-Nov-2001 13:59:49

7. Synopsis of operation
The user generally will not need to know more about the functioning of the RG2WAMIT
interface than has been explained in the operating instructions above. However, a little
more perspective may be useful in case error conditions are encountered or suspected.
The sequence of communication between the programs is as follows:
1. During WAMIT’s initialization process, it opens and writes to an ASCII text file
named RGKINIT.TXT. This file is located in the same directory as WAMIT.EXE. It is
created for each WAMIT run, whether or not the run uses IGDEF = 2 GDF’s.
RGKINIT.TXT contains, for each body, XBODY(1:4) plus the first 4 lines of any GDF;
in addition it contains the remainder of the GDF file whenever IGDEF = 2.
2. WAMIT calls the procedure RGKINIT in RG2WAMIT.DLL. This initializes the
interface, by using information read from RGKINIT.TXT. All specified model files
are opened and prepared for evaluation calls.
3. WAMIT makes as many calls as it needs, to the procedure RGKEVAL in
RG2WAMIT.DLL. Each such call evaluates one 3-D point at a specified u,v
parameter location on one of the wetted surfaces furnished by the RG model.
4. When finished with its RGKEVAL calls, WAMIT makes one call to the procedure
RGKDONE in RG2WAMIT.DLL. This frees all memory that was allocated during the
RGKINIT procedure, storing data about the open models.

8. Error Conditions
There are a number of error conditions that can occur during the initialization phase of
RG2WAMIT operation. If one of these occurs, there will be a message at the end of
RGKLOG.TXT, and the WAMIT run will abort. The contents of RGKLOG.TXT will often
be helpful in identifying the problem.
(Certain errors are also trapped during the WAMIT equation setup phase, but this error
handling is primarily for debugging purposes; in our opinion such errors should not
occur once the initialization phase has completed successfully.)

RGKINIT error codes
0. No error
1. Can’t open RGKINIT.TXT.

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This probably indicates that RG2WAMIT.DLL and/or RGKERNEL.DLL are in the
wrong directories. They must both be installed in the same directory as
WAMIT.EXE.
2. Unexpected end-of-file in RGKINIT.TXT.
The initialization file did not contain all required information. RGKINIT.TXT is
written by WAMIT, and should not be modified by the user. If the file is incomplete,
there’s likely a bug in WAMIT. If the file appears complete, there’s likely a bug in
RG2WAMIT.DLL.
3. Can’t open a specified model file.
A model file named in an IGDEF=2 GDF file was not found. Check to see that the file
exists in the same directory as WAMIT.EXE. Check to see if the file name was spelled
correctly in the GDF. Blanks are not allowed in filenames for RG2WAMIT. (NOTE:
Filenames are not case sensitive in Windows.)
4. The specified ObjectList was not found in the model file.
An ObjectList of wetted surfaces may be specified by name in the GDF file. Object
names in RG are case sensitive and must use an underscore ‘_’ in place of a space.
Check to see that the ObjectList name is correctly spelled and correctly capitalized in
the GDF. Open the MS2 file in either MultiSurf or Notepad and see that the
ObjectList is there, and you have given its correct name in the GDF. You can find the
ObjectList in the Selection Set view (), Availables list, or use Select/ By Name.
5. ObjectList has an error
The RGKernel error code will be given. This is most likely to occur because one of
the surfaces failed to evaluate, and the ObjectList got an error 284 (support failed) as
a result. If the .MS2 opens in MultiSurf without error or warnings, it is very likely to
open and evaluate successfully in RG2WAMIT. However, the two RGKernel.dll
versions will generally be different and will have some incompatibilities, so it is
possible for this error to occur through no fault of the user. This situation should be
investigated by AeroHydro.
6. Failed to build surface table
Open the MS2 file in MultiSurf and confirm that the surface in question evaluates
without error. If the surface evaluates without error in MultiSurf, it is very unlikely
to fail in RG2WAMIT. However, because of some incompatibilities between
RGKernel.dll versions, this error might occur through no fault of the user.
AeroHydro should investigate. (Note: a table is built for each surface in the model
even when “Accurate” evaluation is specified.)
7. Unexpected end-of-line in RGKINIT.TXT
A file record in RGKINIT.TXT did not contain all required data. For example, if the
file line giving fast/accurate, divisions multiplier, and inward-normal settings has
only one or two tokens, this error will occur. Check that your GDF file contains all
required information, and that NLINES is correct.

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8. ObjectList has fewer than NPATCH surfaces
The GDF file specifies NPATCH, the number of patches to use in the solution for a
given body; the ObjectList does not contain the full complement. Check that
NPATCH is correctly specified in the GDF. Open the MS2 file in either MultiSurf or
Notepad and check that all wetted surfaces and/or interior free surface patches are
in the ObjectList. If the number of surfaces is large, making counting difficult, select
the ObjectList in MultiSurf (usually Select/ By Name is the easiest way), edit its
attributes, edit the supports (“Objects”), and the number of them will be indicated in
the Selection Set view (F6); cancel the edit if no changes are needed. Visually, you
can select the ObjectList and Select Support/ First Generation; all surfaces in the list
will then be highlighted.
9. ObjectList contains a non-surface object.
The wetted-surface ObjectList must contain all the wetted patches and/or interior
free-surface patches, and no other objects. (An ObjectList, in general, can legally
contain objects of any type.) Check the contents of the ObjectList and see that they
are correct in number and identity.
10. ObjectList includes a TrimSurf.
TrimSurfs are not eligible for use in WAMIT models, because they do not provide a
complete square parameter space for distribution of the unknown potential. You can
replace a TrimSurf with one or more SubSurfs.
11. Surface object is in error.
The RGKernel error code will be given. This error code is compatible with the
MultiSurf error system, so can be looked up in the MultiSurf reference manual or
help system under “Error codes”. The same error should occur when the model is
opened in MultiSurf; however, this will not always be true, because of the presence
of incompatibilities between the two RGKernel.dll versions. See also remarks for
error 6.

Using WAMIT-RGKernel Interface1.doc

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