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Petra-M: Physics Equation Translator for MFEM
S. Shiraiwa
November 12, 2017

Chapter 1

Introduction
Petra-M (Physics Equation Translator for MFEM) is a tool to build a finite
element simulation model using MFEM. MFEM is a scalable finite element
library built at LLNL. In MFEM, a user has to fill the linear system matrix and the right hand side by adding mfem::BilinearFormIntegrator and
mfem::LinearFormIntegrator to mfem::BilinearForm and mfem::LinearForm.
While a variety of integrators are already defined in MFEM, translating a
physics equation to a weakform and choosing a proper integrator for each
case is an error-prone process. Another practical issue is to assign domain
and boundary conditions for a particular element. A real world 3D geometry
could contain several handreds or even more surfaces and domains. Without an interactive interface, it is difficult to perform this step in a reliable
manner.
Using Petra-M and πScope, a user can build physics simulation model
quickly and reliably. Petra-M currently support only frequency domain
Maxwell problems and simple thermal diffusion model. However, its low
level engine is design to be flexible to expand in future.
Goal of this report is to describe how Petra-M construct a linear system.
An emphasis is on showing the weakform equation systems used for each
physics module. Note that these equations are mostly well established and
found in various literature, and therefore, giving detailed derivation is not
the point of this report.

1

Chapter 2

Linear system construction
strategy

2

Chapter 3

EM Modules
3.1
3.1.1

3D frequency domain(EM3D)
Weakform of the Maxwell’s equation

This module uses the Cartesian coordinate system (x, y, z), and solves the
following weakform of Maxwell equations.
1
(∇ × F, ∇ × E) − (F, (ω 2  + iωσ)E) + hF, Qi
µ
−γhF, n × n × Ei = iω(F, Jext ),
(3.1)
1
n × ( ∇ × E) + γn × n × E = Q on ∂Ω2
(3.2)
µ
R
where (A, B) Ris the domain integral ( Ω ABdxdydz) and hA, Bi is the boundary integral ( ∂Ω2 ABdxdydz).

3

3.1.2

Anisotropic media

3.1.3

External current source

3.1.4

PMC (perfect magnetic conductor)

3.1.5

PEC (perfect electric conductor/electric field BC

3.1.6

Surface Current

3.1.7

Port

3.1.8

Periodic (Floquet-Bloch) BC

3.2
3.2.1

Axisymmetric frequency domain(EM2Da)
Weakform of the Maxwell’s equation

This module uses the cylindrical coordinate system (r, φ, z). Physics quantities are supposed to have a periodic dependence to φ direction (∼ emφ ),
where m is the out-of-plane mode number. First, we write the curl operator
in the following form
∂Eφ
∂Er ∂Ez
1 ∂
im
im
Ez −
)er + (
−
)eφ + (
(rEφ ) −
Er )ez
r
∂z
∂z
∂r
r ∂r
r
eφ
× (imEt − ∇t (rEφ )),
(3.3)
=∇t × Et +
r

∇ × E =(

∂ ∂
where Et = (Er , Ez ) and ∇t = ( ∂r
, ∂z ) are 2D vectors on r, z plane.
Then, we exapned the weakform for Maxwell equations (Eq. 3.1) using
Et , Eφ , and ∇t so that a direct one-by-one connection to mfem::LinearFormIntegrator
and mfem::BilinearFormIntegrator becomes clear. For a scolar µ, , and σ,
it can be written as

1
1
1
1
1
r(∇t × Ft , ∇t ×Et ) + m2 (Ft , Et ) + (∇t (rFφ ), ∇t (rEφ ))
µ
r
µ
r
µ
1
1
1
im
Et )
+ (imFt , ∇t (rEφ )) − (∇t (rFφ ),
r
µ
r
µ
1
−(ω 2  + iωσ)(r(Ft , Et ) + (rFφ , rEφ ))
r
+rhFt , Qt i + hrFφ , Qphi i − rγhFt , Et i
=iωr(Ft , Jt ) + iω(rFφ , Jφ ),
(3.4)
R
where (A,RB) is the domain integral ( Ω ABdrdz) and hA, Bi is the boundary
integral ( ∂Ω ABdrdz). The module uses the H(curl) element for Et and the
H1 element for rEφ . Note that the integration does not consider 2πr and r
is included in the coefficient of the linear/bilinear forms.

4

3.2.2

Anisotropic media

3.2.3

External current source

This domain condition is implemented using the last line of Eq. 3.4.

3.2.4

PMC (perfect magnetic conductor)

3.2.5

PEC (perfect electric conductor/electric field BC

5

Chapter 4

Thermal Modules
4.1

3D static (TH3Ds)

6

Chapter 5

Cross-Physics Coupling

7



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