UQLab User Manual – Structural Reliability (Rare Events Estimation)
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UQL AB U SER M ANUAL S TRUCTURAL R ELIABILITY (R ARE E VENTS E STIMATION ) S. Marelli, R. Schöbi, B. Sudret C HAIR OF R ISK , S AFETY AND U NCERTAINTY Q UANTIFICATION S TEFANO -F RANSCINI -P LATZ 5 CH-8093 Z ÜRICH Risk, Safety & Uncertainty Quantification How to cite UQL AB S. Marelli, and B. Sudret, UQLab: A framework for uncertainty quantification in Matlab, Proc. 2nd Int. Conf. on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, 2014, 2554-2563. How to cite this manual S. Marelli, R. Schöbi and B. Sudret, UQLab user manual – Structural Reliability, Report UQLab-V1.1-107, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich, 2018. BIBTEX entry @TechReport{UQdoc 11 107, author = {Marelli, S. and Schöbi, R. and Sudret, B.}, title = {{UQLab user manual -- Reliability analysis}}, institution = {Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich}, year = {2018}, note = {Report # UQLab-V1.1-107}, } List of contributors: C. Lamas: Original implementation of FORM, SORM, Monte Carlo and importance sampling methods Document Data Sheet Document Ref. Title: Authors: Date: Doc. Version V1.1 UQL AB-V1.1-107 UQL AB User Manual – Structural Reliability (Rare Events Estimation) S. Marelli, R. Schöbi, B. Sudret Chair of Risk. Safety and Uncertainty Quantification, ETH Zurich, Switzerland 01/07/2018 Date 01/07/2018 Comments UQL AB V1.1 release • Updated usage and reference list for AK-MCS V1.0 01/05/2017 UQL AB V1.0 release • added PC-Kriging to AK-MCS • bugfixes and display improvements V0.92 15/02/2016 Initial release Abstract Structural reliability methods aim at the assessment of the probability of failure of complex systems due to uncertainties associated to their design, manifacturing, environmental and operating conditions. The name structural reliability comes from the emergence of such computational methods back in the mid 70’s to evaluate the reliability of civil engineering structures. As these probabilities are usually small (e.g. 10−2 − 10−8 ), this type of problems is also known as rare events estimation in the recent statistics literature. The structural reliability module of UQL AB offers a comprehensive set of techniques for the efficient estimation of the failure probability of a wide range of systems. Classical (crude Monte Carlo simulation, FORM/SORM, Subset Simulation) and state-of-the-art algorithms (AK-MCS) are available and can be easily deployed in association with other UQL AB tools, e.g. surrogate modelling or sensitivity analysis. The structural reliability user manual is divided in three parts: • A short introduction to the main concepts and techniques used to solve structural reliability problems, with a selection of references to the relevant literature • A detailed example-based guide, with the explanation of most of the available options and methods • A comprehensive reference list detailing all the available functionalities in the UQL AB structural reliability module. Keywords: Structural Reliability, FORM, SORM, Importance Sampling, Monte Carlo Simulation, Subset Simulation, AK-MCS, UQL AB, rare event estimation Contents 1 Theory 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Limit-state function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.2 Failure Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Strategies for the estimation of Pf . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 First Order Reliability Method (FORM) . . . . . . . . . . . . . . . . . . 4 1.3.2 Second Order Reliability Method (SORM) . . . . . . . . . . . . . . . . 9 1.3.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5 Subset Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.6 Adaptive Kriging Monte Carlo Simulation . . . . . . . . . . . . . . . . 15 2 Usage 19 2.1 Reference problem: R-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Reliability analysis with different methods . . . . . . . . . . . . . . . . . . . . 20 2.3.1 First Order Reliability Method (FORM) . . . . . . . . . . . . . . . . . . 20 2.3.2 Second Order Reliability Method (SORM) . . . . . . . . . . . . . . . . 23 2.3.3 Monte Carlo Sampling (MCS) . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.4 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.5 Subset Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.6 Adaptive Kriging Monte Carlo Sampling (AK-MCS) . . . . . . . . . . . 32 2.4 Advanced limit-state function options . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Specify failure threshold and failure criterion . . . . . . . . . . . . . . 35 2.4.2 Vector Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Excluding parameters from the analysis . . . . . . . . . . . . . . . . . . . . . 3 Reference List 3.1 Create a reliability analysis 36 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Accessing the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 FORM and SORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Importance sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.4 Subset simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.5 AK-MCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Printing/Visualizing of the results . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 Printing the results: uq print . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Graphically display the results: uq display . . . . . . . . . . . . . . . . 54 Chapter 1 Theory 1.1 Introduction A structural system is defined as a structure required to provide specific functionality under well-defined safety constraints. Such constraints need to be taken into account during the system design phase in view of the expected environmental/operating loads it will be subject to. In the presence of uncertainties in the physical properties of the system (e.g. due to tolerances in the manufacturing), in the environmental loads (e.g. due to exceptional weather conditions), or in the operating conditions (e.g. traffic), it can occur that the structure operates outside of its nominal range. In such cases, the system encounters a failure. Structural reliability analysis deals with the quantitative assessment of the probability of occurrence of such failures (probability of failure), given a model of the uncertainty in the structural, environmental and load parameters. Following the formalism introduced in Sudret (2007), this chapter is intended as a brief theoretical introduction and literature review of the available tools in the structural reliability module of UQL AB. Consistently with the overall design philosophy of UQL AB, all the algorithms presented follow a black-box approach, i.e. they rely on the point-by-point evaluation of a computational model, without knowledge about its inner structure. 1.2 1.2.1 Problem statement Limit-state function A limit state can be defined as a state beyond which a system no longer satisfies some performance measure (ISO Norm 2394). Regardless on the choice of the specific criterion, a state beyond the limit state is classified as a failure of the system. Consider a system whose state is represented by a random vector of variables X ∈ DX ⊂ RM . One can define two domains Ds , Df ⊂ DX that correspond to the safe and failure regions of the state space DX , respectively. In other words, the system is failing if the current state x ∈ Df and it is operating safely if x ∈ Ds . This classification makes it possible to construct a limit-state function g(X) (sometimes also referred to as performance function) that assumes 1 UQL AB User Manual Figure 1: Schematic representation of the safe and failure domains Ds and Df and the corresponding limit-state surface g(x) = 0. positive values in the safe domain and negative values in the failure domain: x ∈ Ds ⇐⇒ g(x) > 0 x ∈ Df ⇐⇒ g(x) ≤ 0 (1.1) The hypersurface in M dimensions defined by g(x) = 0 is known as the limit-state surface, and it represents the boundary between safe and failure domains. A graphical representation of Ds , Df and the corresponding limit-state surface g(x) = 0 is given in Figure 1. 1.2.2 Failure Probability If the random vector of state variables X is described by a joint probability density function (PDF) X ∼ fX (x), then one can define the failure probability Pf as: Pf = P (g(X) ≤ 0) . (1.2) This is the probability that the system is in a failed state given the uncertainties of the state parameters. The failure probability Pf is then calculated as follows: Z Z Pf = fX (x)dx = fX (x)dx. Df (1.3) {x: g(x)≤0} Note that the integration domain in Eq. (1.3) is only implicitly defined by Eq. (1.1), hence making its direct estimation practically impossible in the general case. This limitation can be circumvented by introducing the indicator function of the failure domain, a simple classifier given by: ( 1 1Df (x) = 0 if g(x) ≤ 0 , x ∈ DX . if g(x) > 0 In other words, 1Df (x) = 1 when the input parameters x cause the system to fail and UQL AB-V1.1-107 -2- Structural Reliability (Rare Events Estimation) 1Df (x) = 0 otherwise. This function allows one to cast Eq. (1.3) as follows: Z 1Df (x)fX (x)dx = E 1Df (X) , Pf = (1.4) DX where E [·] is the expectation operator with respect to the PDF fX (x). This reduces the calculation of Pf to the estimation of the expectation value of 1Df (X). 1.3 Strategies for the estimation of Pf From the definition of 1Df (x) in Section 1.2.2 it is clear that determining whether a certain state vector x ∈ DX belongs to Ds or Df requires the evaluation of the limit-state function g(x). In the general case this operation can be computationally expensive, e.g. when it entails the evaluation of a computational model on the vector x. For a detailed overview of standard structural reliability methods and applications, see e.g. Ditlevsen and Madsen (1996); Melchers (1999); Lemaire (2009). In the following, three strategies are discussed for the evaluation of Pf , namely approximation, simulation and adaptive surrogate-modelling-based methods. Approximation methods Approximation methods are based on approximating the limit-state function locally at a reference point (e.g. with a linear or quadratic Taylor expansion). This class of methods can be very efficient (in that only a relatively small number of model evaluations is needed to calculate Pf ), but it tends to become unreliable in the presence of complex, non-linear limit-state functions. Two approximation methods are currently available in UQL AB: • FORM (First Order Reliability Method) – it is based on the combination of an iterative gradient-based search of the so-called design point and a local linear approximation of the limit-state function in a suitably transformed probabilistic space. • SORM (Second Order Reliability Method) – it is a second-order refinement of the solution of FORM. The computational costs associated to this refinement increase rapidly with the number of input random variables M . Simulation methods Simulation methods are based on sampling the joint distribution of the state variables X and using sample-based estimates of the integral in Eq. (1.4). At the cost of being computationally very expensive, they generally have a well-characterized convergence behaviour that can be exploited to calculate confidence bounds on the resulting Pf estimates. Three sampling-based algorithms are available in UQL AB: • Monte Carlo simulation – it is based on the direct sample-based estimation of the expectation value in Eq. (1.4). The total costs increase very rapidly with decreasing values of the probability Pf to be computed. UQL AB-V1.1-107 -3- UQL AB User Manual • Importance Sampling – it is based on improving the efficiency of Monte Carlo simulation by changing the sampling density so as to favour points in the failure domain Df . The choice of the importance sampling (a.k.a. instrumental) density generally uses FORM results. • Subset Simulation – it is based on iteratively solving and combining a sequence of conditional reliability analyses by means of Markov Chain Monte Carlo simulation (MCMC). Metamodel-based adaptive methods Metamodel-based adaptive methods are based on iteratively building surrogate models that approximate the limit-state function in the direct vicinity of the limit-state surface. The metamodels (see e.g. UQL AB User Manual – Polynomial Chaos Expansions and UQL AB User Manual – Kriging (Gaussian process modelling) ) are adaptively refined by adding limit-state function evaluations to their experimental designs until a suitable convergence criterion related to the accuracy of Pf is satisfied. One algorithm is currently available in UQL AB, namely Adaptive Kriging Monte Carlo Sampling (AK-MCS). It is based on building a Kriging (aka Gaussian process regression) surrogate model from a small initial sampling of the input vector X. The surrogate is then iteratively refined close to the currently estimated limit-state surface so as to evaluate accurately the probability of failure. In the following, a detailed description of each of the methods is given. 1.3.1 First Order Reliability Method (FORM) The first order reliability method aims at the approximation of the integral in Eq. (1.3) with a three-step approach: • An isoprobabilistic transform of the input random vector X ∼ fX (x) into a standard normal vector U ∼ N (0, IM ) • A search for the most likely failure point in the standard normal space (SNS), known as the design point U ∗ • A linearization of the limit-state surface at the design point U ∗ and the analytical computation of the resulting approximation of Pf . 1.3.1.1 Isoprobabilistic transform The first step of the FORM method is to transform the input random vector X ∼ fX into a standard normal vector U ∼ N (0, IM ). The corresponding isoprobabilistic transform T reads: X = T −1 (U ) (1.5) For details about the available isoprobabilistic transforms in UQL AB, please refer to the UQL AB User Manual – The INPUT module (Section 1.5). UQL AB-V1.1-107 -4- Structural Reliability (Rare Events Estimation) Figure 2: Graphical representation of the isoprobabilistic transform from physical to standard normal space in Eq. (1.5). From Sudret, 2015: Lectures on structural reliability and risk analysis. This transform can be used to map the integral in Eq. (1.3) from the physical space of X to the standard normal space of U : Z Z Pf = fX (x)dx = Df ϕM (u)du (1.6) {u∈RM : G(u)≤0} where G(u) = g(T −1 (u)) is the limit-state function evaluated in the standard normal space and ϕM (u) is the standard multivariate normal PDF given by: 1 ϕM (u) = (2π)−M/2 exp − (u21 + · · · + u2M ) . 2 (1.7) A graphical illustration of the effects of this transform for a simple 2-dimensional case is given in Figure 2. The advantage of casting the problem in the standard normal space is that it is a probability space equipped with the Gaussian probability measure PG : Z Z PG (U ∈ A) = ϕM (u)du = (2π)−M/2 exp u21 + · · · + u2M du. A (1.8) A This probability measure is spherically symmetric: ϕM (u) only depends on kuk2 and it de cays exponentially as ϕM (u) ∼ exp −kuk2 /2 . Therefore, when evaluating the integral in Eq. (1.6) in the standard normal space, most of the contributions are given by the region closest to the origin. The FORM method capitalizes on this property by linearly approximating the limit-state surface in the region closest to the origin of the standard normal space. UQL AB-V1.1-107 -5- UQL AB User Manual Figure 3: Graphical representation of the linearization of the limit-state function around the design point at the basis of the FORM estimation of Pf . From Sudret, 2015: Lectures on structural reliability and risk analysis. 1.3.1.2 Search for the design point The design point U ∗ is defined as the point in the failure domain closest to the origin of the standard normal space: U ∗ = argmin {kuk, G(u) ≤ 0)} . (1.9) u∈RM Due to the probability measure in Eq. (1.8), U ∗ can be interpreted as the most likely failure point in the standard normal space. The norm of the design point kU ∗ k is an important quantity in structural reliability known as the Hasofer-Lind reliability index (Hasofer and Lind, 1974): βHL = kU ∗ k. (1.10) An important property of the βHL index is that it is directly related to the exact failure probability Pf in the case of linear limit-state function in the standard normal space: Pf = Φ(−βHL ), (1.11) where Φ is the standard normal cumulative density function. The estimation of Pf in the FORM algorithm is based on approximating the limit-state function as the hyperplane tangent to the limit-state function at the design point. Figure 3 illustrates this approximation graphically for the two-dimensional case. In the general non-linear case, Eq. (1.9) may be cast as a constrained optimization problem with Lagrangian: 1 L(u, λ) = kuk2 + λG(u) 2 UQL AB-V1.1-107 (1.12) -6- Structural Reliability (Rare Events Estimation) where λ is the Lagrange multiplier. The related optimality conditions read: ∇u L(U ∗ , λ∗ ) = 0, ∂L ∗ ∗ (U , λ ) = 0, ∂λ (1.13) which can be explicitly written as: G(U ∗ ) = 0, U ∗ + λ∗ ∇G(U ∗ ) = 0. (1.14) The first condition in Eq. (1.14) guarantees that the design point belongs to the limit-state surface. The second condition guarantees that the vector U ∗ is colinear to the limit-state surface normal vector at U ∗ , i.e. ∇G(U ∗ ). The standard iterative approach to solve this nonlinear constrained optimization problem is given by the Rackwitz-Fiessler algorithm (Rackwitz and Fiessler, 1978). Hasofer-Lind - Rackwitz-Fiessler algorithm (HL-RF) The rationale behind the Rackwitz-Fiessler algorithm is to iteratively solve a linearized problem around the current point. Normally, the algorithm is started with U0 = 0. At each iteration, the limit-state function is approximated as: G(U ) ≈ G(Uk ) + ∇G|Uk · (U − Uk ) (1.15) The two optimality conditions in Eq. (1.14) read for each iteration k: ∇G|Uk · (Uk+1 − Uk ) + G(Uk ) = 0 Uk+1 = λ∇G|Uk , (1.16) which after some basic algebra reduce to: Uk+1 = ∇G|Uk · Uk − G(Uk ) ∇G|Uk . k∇G|Uk k2 (1.17) ∇G|Uk , k∇G|Uk k (1.18) By introducing the unit vector: αk = − one finally obtains: Uk+1 G(Uk ) = αk · Uk + αk . k∇G|Uk k (1.19) The associated estimate of the reliability index βk associated to the k-th iteration is then: β k = αk · Uk + G(Uk ) . k∇G|Uk k (1.20) Perfect convergence of the algorithm is obtained when G(U ∗ ) = 0, yielding βHL = α∗ · U ∗ . However, in practice the algorithm is iterated until some stopping criteria are satisfied, i.e., UQL AB-V1.1-107 -7- UQL AB User Manual Table 1: Common stopping criteria for the FORM algorithm and associated description. Criterion |βk+1 − βk | ≤ β kUk+1 − Uk k ≤ U |G(Uk+1 )/G(U0 )| ≤ G Typical value 10−3 10−3 10−6 Description Stability of β between iterations Stability of U between iterations Closeness to the limit-state surface until one or more convergence conditions are verified. The standard stopping criteria used in FORM are reported in Table 1. Note: In UQL AB the gradients ∇G(Uk ) in Eqs. (1.13) to (1.20) are calculated numerically in the standard normal space and not in the physical space. Improved HL-RF algorithm (iHL-RF) The Rackwitz-Fiessler algorithm is a particular case of a wide class of iterative algorithms generically denoted as descent direction algorithms, of the form: Uk+1 = Uk + λk dk , (1.21) where λk is the step size at the k-th iteration and dk is the corresponding descent direction given by: dk = ∇G|Uk · Uk − G(Uk ) ∇G|Uk − Uk . k∇G|Uk k2 (1.22) In the original HL-RF algorithm, λk = 1 ∀k. Zhang and Der Kiureghian (1995) proposed an “improved” version of the same algorithm that takes advantage of a more sophisticated step-size calculation based on the assumption that G(U ) is differentiable everywhere. They introduced the merit function m(U ): 1 m(U ) = kU k + c|G(U )|, 2 (1.23) kU k is a real penalty parameter. This function has its global minimum in the k∇G(U )k same location as the original Eq. (1.9), as well as the same descent direction d. In addition, where c > it allows one to use the Armijo rule (Zhang and Der Kiureghian, 1995) to determine the best step length λk at each iteration as: λk = max{bs | m(Uk + bs dk ) − m(Uk ) ≤ −abs ∇m(Uk ) · dk }, s (1.24) where a, b ∈ (0, 1) are pre-selected parameters, and s ∈ N. 1.3.1.3 FORM results Once the design point U ∗ is identified, it can be used to extract additional important information. According to Eq. (1.20), after the convergence of FORM the Hasofer-Lind index βHL UQL AB-V1.1-107 -8- Structural Reliability (Rare Events Estimation) is given by: βHL = α∗ · U ∗ , (1.25) Pf,FORM = Φ−1 (βHL ) (1.26) with associated failure probability: The local sensitivity indices Si are defined as the fraction of the variance of the safety margin g(X) = G(U ) due to the component of the design vector Ui . It can be demonstrated that they are given by: Si = ∂G ∂ui 2 /k∇G(U ∗ )k2 . (1.27) U∗ From Eq. (1.18) it follows that: Si = αi2 . (1.28) If the input variables are independent, then each coordinate in the SNS Ui corresponds to a single input variable in the physical space Xi . Therefore, the importance factor of each Xi is identified with αi2 . 1.3.2 Second Order Reliability Method (SORM) The second-order reliability method (SORM) is a second-order refinement of the FORM Pf estimate. After the design point U ∗ is identified by FORM, the failure probability is approximated by a tangent hyperparaboloid defined by the second order Taylor expansion of G(U ∗ ) given by: 1 G(U ) ≈ ∇GT|U ∗ · (U − U ∗ ) + (U − U ∗ )T H(U − U ∗ ), 2 where H is the Hessian matrix of the second derivatives of G(U ) evaluated at U ∗ . (1.29) The failure probability in the SORM approximation can be written as a correction factor of the FORM estimate that depends on the curvatures of the hyper-hyperboloid in Eq. (1.29). To estimate the curvatures, the hyperparaboloid in Eq. (1.29) is first cast in canonical form by rotating the coordinates system such that one of its axes is the α vector. Usually the last coordinate is chosen arbitrarily for this purpose. A rotation matrix Q can be built by setting α as its last row and by using the Gram-Schmidt procedure to orthogonalize the remaining components of the basis. Q is a square matrix such that QT Q = I. The resulting vector V satisfies: (1.30) U = QV . In the new coordinates system and after some basic algebra (see e.g. Breitung (1989) and Cai and Elishakoff (1994)), one can rewrite Eq. (1.29) as: 1 G(V ) ≈ k∇G(U ∗ )k(β − VM ) + (V − V ∗ )QHQT (V − V ∗ ) 2 (1.31) def where β is the Hasofer-Lind reliability index calculated by FORM, VM = αT (QT V ) and UQL AB-V1.1-107 -9- UQL AB User Manual Figure 4: Comparison between FORM and SORM approximations of the failure domain for a simple 2-dimensional case. From Sudret, 2015: Lectures on structural reliability and risk analysis. V ∗ = {0, · · · , β}T is the design point in the new coordinates system. By dividing Eq. (1.31) def by the gradient norm k∇G(U ∗ )k and introducing the matrix A = QHQT /k∇G(U ∗ )k, one obtains: 1 G̃(V ) ≈ β − VM + (V − V ∗ )A(V − V ∗ ), 2 (1.32) where G̃(V ) = G(V )/k∇G(U ∗ )k. After neglecting second-order terms in VM and diagonalizating the A matrix via eigenvalue decomposition one can rewrite Eq. (1.32) explicitly in terms of the curvatures κi of an hyper-paraboloid with axis α: G̃(V ) ≈ β − VM + M −1 1 X κi Vi . 2 (1.33) 1 For small curvatures κi < 1, the failure probability Pf can be approximated by the Breitung formula (Breitung, 1989): B Pf,SORM = Φ(−βHL ) M −1 Y 1 (1 + βHL κi )− 2 κi < 1 (1.34) i=1 Note that for small curvatures the Breitung formula approaches the FORM linear limit. The accuracy of Eq. (1.34) decreases for larger values of κi , sometimes even if κi < 1 (Cai and Elishakoff, 1994). A more accurate formula is given by the Hohenbichler formula (Hohenbichler et al., 1987): H Pf,SORM = Φ(−βHL ) M −1 Y i=1 ϕ(βHL ) 1+ κi Φ(−βHL ) − 1 2 . (1.35) Additional methods are available in the literature for the exact computation of the failure probability, e.g.Tvedt (1990). They are, however, outside the scope of this manual. UQL AB-V1.1-107 - 10 - Structural Reliability (Rare Events Estimation) 1.3.3 Monte Carlo Simulation Monte Carlo (MC) simulation is used to directly compute the integral in Eq. (1.4) by sampling the probabilistic input model. Given a sample of size N of the input random vector X, X = x(1) , . . . x(N ) , the unbiased MCS estimator of the expectation value in Eq. (1.4) is given by: Pf,MC N Nf ail 1 X b = Pf = 1Df (x(k) ) = , N N def (1.36) k=1 where Nf ail is the number of samples such that g(x) ≤ 0. In other words, the Monte Carlo estimate of the failure probability is the fraction of samples that belong to the failure domain over the total number of samples. An advantage of Monte Carlo simulation is that it provides an error estimate for Eq. (1.36). Indeed the indicator function 1Df (x) follows by construction a Bernoulli distribution with mean µ1Df = Pf and variance σ12D = Pf (1 − Pf ). For large f enough N it can be approximated by the normal distribution: Pbf ∼ N µ b1Df , σ b1Df , (1.37) q where µ b1Df = Pbf and σ b1Df = Pbf (1 − Pbf ). Hence, the estimator of Pf has a normal distribution with mean Pbf and variance given by: σ bP2 f = σ12D N f = Pbf (1 − Pbf ) . N Confidence intervals on Pbf can therefore be given as follows (Rubinstein, 1981): h i def def Pbf ∈ Pbf− = Pbf + σ bPf Φ−1 (α/2), Pbf+ = Pbf + σ bPf Φ−1 (1 − α/2) , (1.38) (1.39) where Φ(x) is the standard normal CDF and α ∈ [0, 1] is a scalar such that the calculated bounds correspond to a confidence level of 1 − α. An important measure for assessing the convergence of a MCS estimator is given by the coefficient of variation CoV defined as: v u σPbf u 1 − Pbf =t . (1.40) CoV = Pbf N Pbf The coefficient of variation of the MCS estimate of a failure probability therefore decreases √ with N and increases with decreasing Pf . To give an example, to estimate a Pf = 10−3 with 10% accuracy N = 105 samples are needed. The CoV is often used as a convergence criterion to adaptively increase the MC sample size until some desired CoV is reached. An associated generalized reliability index βM CS can be defined as: βMCS = −Φ−1 (Pbf ). (1.41) In analogy, upper and lower confidence bounds on βMCS can be directly inferred from the UQL AB-V1.1-107 - 11 - UQL AB User Manual confidence bounds on Pbf in Eq. (1.39): ± βMCS = −Φ−1 (Pbf± ). (1.42) The MCS method is powerful, when applicable, due to its statistically sound formulation and global convergence. However, its main drawback is the relatively slow converge rate that depends strongly on the probability of failure. 1.3.4 Importance Sampling Importance sampling (IS) is an extension of the FORM and MCS methods that combines the fast convergence of FORM with the robustness of MC. The basic idea is to recast Eq. (1.4) as: Z fX (x) fX (X) 1Df (x) , (1.43) Pf = Ψ(x)dx = EΨ 1Df (X) Ψ(x) Ψ(X) DX where Ψ(X) is an M −dimensional sampling distribution (also referred to as importance distribution) and EΨ denotes the expectation value with respect to the same distribution. The estimate of Pf given a sample X = x(1) , . . . , x(N ) drawn from Ψ is therefore given by: Pf,IS N 1 X fX (x(k) ) = . 1Df (x(k) ) N Ψ(x(k) ) (1.44) k=1 In the standard normal space, Eq. (1.43) can be rewritten as: ϕM (U ) −1 Pf = EΨ 1Df (T (U )) . Ψ(U ) (1.45) When the results from a previous FORM analysis are available, a particularly efficient sampling distribution in the standard normal space is given by (Melchers, 1999): Ψ(u) = ϕM (u − U ∗ ) (1.46) where U ∗ is the estimated design point. Given a sample U = u(1) , . . . , u(N ) of Ψ(u), the estimate of Pf becomes: Pf,IS = N X 1 2 exp −βHL /2 1Df T −1 (u(k) ) exp −u(k) · U ∗ N (1.47) k=1 with corresponding variance: σ bP2 f,IS N ϕ(u(k) ) X 1 1 − Pf,IS = 1Df T −1 (u(k) ) N N −1 Ψ(u(k) ) !2 . (1.48) k=1 ± The coefficient of variation and the confidence bounds Pf,IS can be calculated analogously to Eqs. (1.40) and (1.39), respectively, and can be used as a convergence criterion to adaptively UQL AB-V1.1-107 - 12 - Structural Reliability (Rare Events Estimation) improve the estimation of Pf,IS . The corresponding generalized reliability index reads: βIS = −Φ−1 (Pbf,IS ), (1.49) ± ± βIS = −Φ−1 (Pbf,IS ). (1.50) with upper and lower bounds: Note that exact convergence of FORM is not necessary to obtain accurate results, even an approximate sampling distribution can significantly improve the convergence rate compared to standard MC sampling. 1.3.5 Subset Simulation Monte Carlo simulation may require a large number of limit-state function evaluations to converge with an acceptable level of accuracy when Pf is small (see Eq. (1.40)). Subset simulation is a technique introduced by Au and Beck (2001) that aims at offsetting this limitation by solving a series of simpler reliability problems with intermediate failure thresholds. T Consider a sequence of failure domains D1 ⊃ D2 ⊃ · · · ⊃ Dm = Df such that Df = m k=1 Dk . With the conventional definition of limit-state function in Eq. (1.1), such sequence can be built with a series of decreasing failure thresholds t1 > · · · > tm = 0 and the corresponding intermediate failure domains Dk = {x : g(x) ≤ tk }. One can then combine the probability mass of each intermediate failure region by means of conditional probability. By introducing the notation P (DX ) = P (x ∈ DX ) one can write (Au and Beck, 2001): ! m m−1 \ Y Pf = P (Dm ) = P P (Dk ) = P (D1 ) P (Di+1 |Di ) . (1.51) i=1 k=1 With an appropriate choice of the intermediate thresholds t1 , . . . , tm , Eq. (1.51) can be evaluated as a series of structural reliability problems with relatively high probabilities of failure that are then solved with MC simulation. In practice the intermediate probability thresholds ti are chosen on-the-fly such that they correspond to intermediate values P (Dk ) ≈ 0.1. The convergence of each intermediate estimation is therefore much faster than the direct search for Pf given in Eq. (1.36). 1.3.5.1 Sampling To estimate Pf from Eq. (1.51) one thus needs to estimate the intermediate probabilities P (D1 ) and conditional probabilities {P (Di+1 |Di ) , i = 1, . . . , m − 1}. Given an initial threshold t1 , P (D1 ) can be readily estimated from a sample of size NS of the input distribution X = x(1) , . . . , x(N ) with Eq. (1.36): NS 1 X P (D1 ) ≈ Pb1 = 1D1 (x(k) ). NS (1.52) k=1 The remaining conditional probabilities can be estimated similarly, but an efficient sampling UQL AB-V1.1-107 - 13 - UQL AB User Manual algorithm is needed for the underlying conditional distributions. The latter can be efficiently accomplished by using the modified Metropolis-Hastings Markov Chain Monte Carlo sampling (MCMC) introduced by Au and Beck (2001). 1.3.5.2 Intermediate failure thresholds The efficiency of the subset-simulation method depends on the choice of the intermediate failure thresholds tk . If the thresholds are too large the MCS convergence in each subset would be very good, but the number of subsets needed would increase. Vice-versa, too small intermediate thresholds would correspond to fewer subsets with inaccurate estimates of the underying P (Dk ). A strategy to deal with this problem comes by sampling each subset Dk and determining each threshold tk as the empirical quantile that correspond to a predetermined failure probability, typically P (Dk ) ≈ P0 = 0.1. Note that for practical reasons, P0 is normally limited to 0 < P0 ≤ 0.5. For each subset, the samples falling below the calculated threshold are used as MCS seeds for the next subset (Au and Beck, 2001). 1.3.5.3 Subset simulation algorithm The subset simulation algorithm can be summarized in the following steps: 1. Sample the original space with standard MC sampling (see UQL AB User Manual – The INPUT module for efficient sampling strategies) 2. Calculate the empirical quantile tk in the current subset such that Pbk ≈ P0 3. Using the samples below the identified quantile as the seeds of parallel MCMC chains, sample Dk+1 |Dk until a predetermined number of samples is available 4. Repeat Steps 2 and 3 until the identified quantile tm < 0 5. Calculate the failure probability of the last subset Pbm by setting tm = 0 6. Combine the intermediate calculated failure probabilities into the final estimate of Pbf . The last step of the algorithm consists simply in evaluating Eq. (1.51) with the current estimates of the conditional probabilities Pi : Pbf,SS = m Y Pbi = P0m−1 Pbm . (1.53) i=1 1.3.5.4 Error estimation Due to the intrinsic correlation of the samples drawn from each subset resulting from the MCMC sampling strategy, the estimation of a CoV for the Pf estimate in Eq. (1.53) is nontrivial. Au and Beck (2001) and Papaioannou et al. (2015) derived an estimate for the CoV of Pbf : m X CoVf ≈ δi2 , (1.54) i=1 UQL AB-V1.1-107 - 14 - Structural Reliability (Rare Events Estimation) where m is the number of subsets and δi is defined as: r 1 − Pi δi = (1 + γi ), N Pi (1.55) with N/Ns γi = 2 X k=1 kNs 1− N ρi (k), (1.56) where NS is the number of seeds, Pi is the conditional failure probability of the i−th subset and ρi (k) is the average k-lag auto-correlation coefficient of the Markov Chain samples in ± the i−th subset. By assuming normally distributed errors, confidence bounds Pf,SS can be given on Pf,SS based on the calculated CoVf in analogy with Eq. (1.39). The corresponding generalized reliability index reads: βSS = −Φ−1 (Pbf,SS ), (1.57) ± ± ). βSS = −Φ−1 (Pbf,SS (1.58) with upper and lower bounds: 1.3.6 Adaptive Kriging Monte Carlo Simulation Adaptive Kriging Monte Carlo Simulation (AK-MCS) combines Monte Carlo simulation with adaptively built Kriging (a.k.a. Gaussian process modelling) metamodels. In cases where the evaluation of the limit-state function is costly, Monte Carlo simulation and its variants may become intractable due to the large number of limit-state function evaluations they require. In AK-MCS, a Kriging metamodel surrogates the limit-state function to reduce the total computational costs of the Monte Carlo simulation. Kriging metamodels (see UQL AB User Manual – Kriging (Gaussian process modelling) ) predict the value of the limit-state function most accurately in the vicinity of the experimental design samples X = x(1) , . . . , x(n) . These samples, however, are generally not optimal to estimate the failure probability. Thus, an adaptive experimental design algorithm is introduced to increase the accuracy of the surrogate model in the vicinity the limit-state function. This is achieved by adding carefully selected samples to the experimental design of the Kriging metamodel based on the current estimate of the limit-state surface (g(x) = 0). The adaptive experimental design algorithm is summarized as follows (Echard et al., 2011; Schöbi et al., 2016): 1. Generate a small initial experimental design X = x(1) , . . . , x(N0 ) and evaluate the corresponding limit-state function responses Y = y (1) , . . . , y (N0 ) = g(x(1) ), . . . , g(x(N0 ) ) 2. Train a Kriging metamodel gb based on the experimental design {X , Y} 3. Generate a large set of NM C candidate samples S = s(1) , . . . , s(NM C ) and predict the corresponding metamodel responses gb(s(1) ), . . . , gb(s(NM C ) ) UQL AB-V1.1-107 - 15 - UQL AB User Manual 4. Choose the best next sample s∗ to be added to the experimental design X based on an appropriate learning function 5. Check whether some convergence criterion is met. If it is, skip to Step 7, otherwise continue with Step 6 6. Add s∗ and the corresponding limit-state function response y ∗ = g(s∗ ) to the experimental design of the metamodel. Return to Step 2 7. Estimate the failure probability through Monte Carlo simulation with the final limitstate function surrogate gb(x). 1.3.6.1 Selection of the best next candidate sample A learning function is a measure of the attractiveness of a candidate sample x with respect to improving the estimate of the failure probability when it is added to the experimental design X . A variety of learning functions are available in the literature (Bichon et al., 2008; Dani et al., 2008; Echard et al., 2011; Srinivas et al., 2012; Ginsbourger et al., 2013; Dubourg, 2011), amongst which the U -function (Echard et al., 2011). The U -function is based on the concept of misclassification and is defined for a Gaussian process as follows: U (x) = µgb(x) , σgb(x) (1.59) where µgb(x) and σgb(x) are the prediction mean and standard deviation of gb. A misclassification happens when the sign of the surrogate model and the sign of the underlying limit-state function do not match. The corresponding probability of misclassification is then: Pm (x) = Φ (−U (x)) , where Φ is the CDF of a standard Gaussian variable. The next candidate sample from the set S = s(1) , . . . , s(NM C ) is chosen as the one that maximizes the probability of misclassification or, in other words, as the one most likely to have been misclassified as safe/failed by the surrogate limit-state function gb(x): s∗ = arg min U (s) ≡ arg max Pm (s). s∈S s∈S (1.60) Another popular learning function is the expected feasibility function (EFF) (Bichon et al., 2008): −µgb(x) − − µgb(x) − µgb(x) EF F (x) = µgb(x) 2Φ −Φ −Φ σgb(x) σgb(x) σgb(x) −µgb(x) − − µgb(x) − µgb(x) − σgb(x) 2ϕ −ϕ −ϕ σgb(x) σgb(x) σgb(x) − µgb(x) − − µgb(x) + Φ −Φ , (1.61) σgb(x) σgb(x) UQL AB-V1.1-107 - 16 - Structural Reliability (Rare Events Estimation) where = 2σgb(x) and ϕ is the PDF value of a standard normal Gaussian variable. The next candidate sample is then chosen by: s∗ = arg max EF F (s). s∈S 1.3.6.2 (1.62) Convergence criteria The convergence criterion terminates the addition of samples to the experimental design of the Kriging metamodel and thus terminates the improvement in the accuracy of the failure probability estimate. The standard convergence criterion related to the U -function is defined as follows (Echard et al., 2011): the iterations stop when mini U (s(i) ) > 2 where i = 1, . . . , NM C . Schöbi et al. (2016) demonstrated that this criterion is very conservative and that an alternative stopping criterion, related to the uncertainty in the estimate of the failure probability itself, is often more efficient in the context of structural reliability. It is given by the following condition: Pbf+ − Pbf− ≤ Pbf , Pb0 (1.63) f where Pbf = 10% and the three failure probabilities are defined as: Pbf0 = P µgb(x) ≤ 0 , (1.64) Pbf± = P µgb(x) ∓ kσgb(x) ≤ 0 , (1.65) where k = Φ−1 (1 − α/2) sets the confidence level (1 − α), typically k = Φ−1 (97.5%) = 1.96. A similar convergence criterion can be defined for the reliability index: βb+ − βb− ≤ βb, βb0 (1.66) where the threshold βb = 5% and the three reliability indices correspond to the aforemenf tioned failure probabilities: 1.3.6.3 βb0 = −Φ−1 (Pbf0 ), (1.67) βb± = −Φ−1 (Pbf∓ ). (1.68) AK-MCS with a PC-Kriging metamodel As originally proposed by Echard et al. (2011), AK-MCS uses an ordinary Kriging model for approximating the limit-state function. As demonstrated by Schöbi et al. (2016) replacing the ordinary Kriging metamodel with a Polynomial-Chaos-Kriging (PC-Kriging) one (see also UQL AB User Manual – PC-Kriging ) can significantly improve the convergence of the algorithm. The corresponding reliability methods is called Adaptive PC-Kriging Monte Carlo Simulation (APCK-MCS) and is available in UQL AB as an advanced option of AK-MCS (see Section 2.3.6.2). UQL AB-V1.1-107 - 17 - Chapter 2 Usage In this section, a reference problem will be set up to showcase how each of the techniques in Chapter 1 can be deployed in UQL AB. 2.1 Reference problem: R-S The benchmark of choice to showcase the methods described in Section 1.3 is a basic problem in structural reliability, namely the R-S case. It is one of the simplest possible abstract setting consisting of only two input state variables: a resistance R and a stress S. The system fails when the stress is higher than the resistance, leading to the following limit-state function: X = {R, S} g(X) = R − S; (2.1) The two-dimensional probabilistic input model consists of independent variables distributed according to Table 2. Table 2: Distributions of the input parameters of the R − S model in Eq. (2.1). Name R S Distributions Gaussian Gaussian Parameters [5, 0.8] [2, 0.6] Description Resistance of the system Stress applied to the system An example UQL AB script that showcases how to deploy all of the algorithms available in the structural reliability module can be found in the example file: Examples/Reliability/uq_Examples_Reliability_01_RS.m 2.2 Problem set-up Solving a structural reliability problem in UQL AB requires the definition of three basic components: • a MODEL object that describes the limit-state function • an INPUT object that describes the probabilistic model of the random vector X 19 UQL AB User Manual • a reliability ANALYSIS object. The UQL AB framework is first initialized with the following command: uqlab The model in Eq. (2.1) can be added as a MODEL object directly with a M ATLAB vectorized string as follows: MOpts.mString = 'X(:,1) - X(:,2)'; MOpts.isVectorized = 1; myModel = uq_createModel(MOpts); % R−S For more details on the available options to create a model object in UQL AB, please refer to the UQL AB User Manual – The MODEL module . Correspondingly, an INPUT object with independent Gaussian variables as specified in Table 2 can be created as: IOpts.Marginals(1).Name = 'R'; IOpts.Marginals(1).Type = 'Gaussian'; IOpts.Marginals(1).Moments = [5, 0.8]; IOpts.Marginals(2).Name = 'S'; IOpts.Marginals(2).Type = 'Gaussian'; IOpts.Marginals(2).Moments = [2, 0.6]; % Resistance % Stress myInput = uq_createInput(IOpts); For more details about the configuration options available for an INPUT object, please refer to the UQL AB User Manual – The INPUT module . 2.3 Reliability analysis with different methods This section showcases how all the methods introduced in Section 1.3, can be deployed in UQL AB. In addition, visualization and advanced options are also described in detail. The following methods are showcased in this section: • FORM: Section 2.3.1 • SORM: Section 2.3.2 • Monte Carlo Simulation: Section 2.3.3 • Importance Sampling: Section 2.3.4 • Subset Simulation: Section 2.3.5 • AK-MCS: Section 2.3.6 2.3.1 First Order Reliability Method (FORM) Running a FORM analysis on the specified UQL AB MODEL and INPUT objects does not require any specific configuration. The following minimum syntax is required: UQL AB-V1.1-107 - 20 - Structural Reliability (Rare Events Estimation) FORMOpts.Type = 'Reliability'; FORMOpts.Method = 'FORM'; FORMAnalysis = uq_createAnalysis(FORMOpts); Once the analysis is performed, a report with the FORM results can be printed on screen by: uq_print(FORMAnalysis) which produces the following: ------------------------------------------FORM ------------------------------------------Pf 1.3499e-03 BetaHL 3.0000 ModelEvaluations 8 ------------------------------------------Variables R S Ustar -2.400000 1.800000 Xstar 3.08e+00 3.08e+00 Importance 0.640000 0.360000 ------------------------------------------- The results can be visualized graphically as follows: uq_display(FORMAnalysis) which produces the images in Figure 5. Note that the graphical representation of the FORM iterations (right panel of Figure 5) is only produced for the 2-dimensional case. FORM - Design point, failure plane FORM - Convergence 3.5 Iterations FORM limit state surface 3.5 3 2.5 2.5 2 2 u2 βHL 3 1.5 1.5 1 1 0.5 0.5 0 0 1 2 number of iterations -5 -4 -3 -2 -1 0 u1 Figure 5: Graphical visualization of the results of the FORM analysis in Section 2.3.1. UQL AB-V1.1-107 - 21 - UQL AB User Manual Note: In the preceding example no specifications are provided. If not further specified the FORMruns with the following defaults: – Algorithm to find the design point: 'iHLRF'; – Starting point for the Rackwitz-Fiessler (RW) algorithm: (0, . . . , 0); – Tolerance value for the RW algorithm on the design point: 10−4 ; – Tolerance value for the RW algorithm on the limit-state function: 10−4 ; – Maximum number of iterations for the RW algorithm: 100; – Failure is defined for: limit-state g(x) ≤ 0. Since FORM is a gradient-based method, the gradient of the limit-state function needs to be computed. This is done using finite differences with the following defaults: – Type of finite difference scheme: 'forward'; – Value of the difference scheme: 10−3 . 2.3.1.1 Accessing the results The analysis results can be accessed in the FORMAnalysis.Results structure: FORMAnalysis.Results ans = BetaHL: Pf: ModelEvaluations: Ustar: Xstar: Importance: Iterations: History: 3 0.0013 8 [-2.4000 1.8000] [3.0800 3.0800] [0.6400 0.3600] 2 [1x1 struct] In the Results structure, Pf is the estimate of Pf according to Eq. (1.26), BetaHL the corresponding Hasofer-Lindt reliability index, Ustar the design point U ∗ in the standard normal space, Xstar the correspondingly transformed design point in the original space X ∗ = T −1 (U ∗ ), Importance the importance factors Si in Eq. (1.28), Iterations the number of FORM iterations needed to converge, ModelEvaluations the total number of evaluations of the limit-state function, and History a set of additional information about the convergence behaviour of the algorithm. 2.3.1.2 Advanced options Several advanced options are available for the FORM method to tweak which algorithm is used to calculate the solution. They can be specified by adding a FORM field to the FORM options structure FORMOpts. In the following, the most common advanced options for FORM are specified: UQL AB-V1.1-107 - 22 - Structural Reliability (Rare Events Estimation) • Specify the FORM algorihm: by default, the iHL-RF algorithm is used (see page 8). The original HL-RF algorithm (see page 7) can be enforced by adding: FORMOpts.FORM.Algorithm = 'HLRF'; • Specify a starting point: by default the search of the design point is started in the SNS at U0 = 0. It is possible to specify an alternative starting point (useful, e.g., when multiple design points are expected) as: FORMOpts.FORM.StartingPoint = [u1,...,uM]; where [u1,...,uM] are the desired coordinates of the starting point in the SNS. • Numerical calculation of the gradient: advanced options related to the numerical calculation of the gradient can be specified by using the FORMOpts.Gradient structure. As an example, to specify a gradient relative step-size h = 0.001 one can write: FORMOpts.Gradient.h = 0.001; Details on the gradient computation options are given in Table 7, page 43. For a comprehensive list of the advanced options available to the FORM method, please see Table 6, page 42. 2.3.2 Second Order Reliability Method (SORM) A SORM analysis is set up very similarly to its FORM counterpart: SORMOpts.Type = 'Reliability'; SORMOpts.Method = 'SORM'; SORMAnalysis = uq_createAnalysis(SORMOpts); Once the analysis is performed, a report with the FORM+SORM results can be printed by: uq_print(SORMAnalysis) which produces the following: ------------------------------------------FORM/SORM ------------------------------------------Pf 1.3499e-03 BetaHL 3.0000 PfFORM 1.3499e-03 PfSORM 1.3499e-03 PfSORMBreitung 1.3499e-03 ModelEvaluations 20 ------------------------------------------Variables R S Ustar -2.400000 1.800000 Xstar 3.08e+00 3.08e+00 Importance 0.640000 0.360000 ------------------------------------------- UQL AB-V1.1-107 - 23 - UQL AB User Manual The results can be visualized graphically as follows: uq_display(SORMAnalysis) which produces the same images as in FORM (Figure 5), as SORM is only a refinement of the final FORM Pf estimate. Note: In the preceding example no specifications are provided. If not further specified the SORM runs with the same defaults as FORM. 2.3.2.1 Accessing the results The analysis results can be accessed in the SORMAnalysis.Results structure: SORMAnalysis.Results ans = Pf: 0.0013 BetaHL: 3 ModelEvaluations: 20 PfFORM: 0.0013 PfSORM: 0.0013 PfSORMBreitung: 0.0013 Ustar: [-2.4000 1.8000] Xstar: [3.0800 3.0800] Importance: [0.6400 0.3600] Iterations: 2 History: [1x1 struct] The Results structure contains the same fields as FORM (see Section 2.3.1) (when necessary with a FORM or SORM suffix for clarity). In addition, the two PfSORMBreitung and PfSORM H given in Eqs. (1.34) and (1.35), respectively. fields provide the Pf,SORM and Pf,SORM 2.3.2.2 Advanced options The SORM method shares the same advanced options as the FORM method, described in Section 2.3.1.2. 2.3.3 Monte Carlo Sampling (MCS) The Monte Carlo Sampling algorithm only requires the user to specify the maximum number of limit-state function evaluations, corresponding to N in Eq. (1.36), if different from the default value N = 105 . As an example, to run a reliability analysis with N = 106 samples one can write: MCOpts.Type = 'Reliability'; MCOpts.Method = 'MCS'; MCOpts.Simulation.MaxSampleSize = 1e6; MCAnalysis = uq_createAnalysis(MCOpts); Once the analysis is performed, a report with the Monte-Carlo sampling results can be printed on screen by: UQL AB-V1.1-107 - 24 - Structural Reliability (Rare Events Estimation) uq_print(MCAnalysis) which produces the following: ------------------------------------------Monte Carlo simulation ------------------------------------------Pf 1.3110e-03 Beta 3.0089 CoV 0.0276 ModelEvaluations 1000000 PfCI [1.2401e-03 1.381919e-03] BetaCI [2.9929e+00 3.025751e+00] ------------------------------------------- The results can be visualized graphically as follows: uq_display(MCAnalysis) which produces the convergence images in Figure 6 and the Monte Carlo samples image in Figure 7. Note that in uq_display, the maximum number of samples plotted is n = 105 to limit the size of the figure. Figure 6: Graphical visualization of the convergence of the Monte Carlo Sampling analysis in Section 2.3.3. Note: In the preceding example only the maximum sample size for the analysis is provided. If not further specified the Monte Carlo Sampling runs with the following defaults: – Confidence level: 0.05; – Number of samples evaluated per batch: 104 ; – Failure is defined for: limit-state g(x) ≤ 0. UQL AB-V1.1-107 - 25 - UQL AB User Manual MCS - Samples 5 4 x2 3 2 1 0 g(X) ≤ 0 g(X) > 0 -1 2 4 6 8 x1 Figure 7: Graphical visualization of the samples of the Monte Carlo Sampling analysis in Section 2.3.3. 2.3.3.1 Accessing the results The analysis results can be accessed in the MCAnalysis.Results structure: MCAnalysis.Results ans = Pf: Beta: CoV: ModelEvaluations: PfCI: BetaCI: History: 0.0013 3.0013 0.0273 1000000 [0.0013 0.0014] [2.9855 3.0180] [1x1 struct] The Results structure contains the following fields: Pf, the estimated Pbf,MC as in Eq. (1.36); Beta, the corresponding generalized reliability index in Eq. (1.41); CoV, the coefficient of variation calculated with Eq. (1.40); ModelEvaluations, the total number of limit-state function evaluations; PfCI, the confidence intervals calculated with Eq. (1.39); BetaCI, the corresponding confidence intervals on βHL calculated with Eq. (1.42); History, a structure containing the convergence of Pf , CoV and the corresponding confidence intervals calculated at preset sample batches (by default once every 104 samples). The content of the History structure is used to produce the convergence plot shown in Figure 6. 2.3.3.2 Advanced options The advanced options available in Monte Carlo sampling are related to the convergence criterion of the algorithm and to the definition of the confidence bounds reported in the MCAnalysis.Results structure. In the following, a list of the most commonly used param- eters for a MC analysis are given: • Specify a target CoV and a corresponding batch size: in addition to specifying the UQL AB-V1.1-107 - 26 - Structural Reliability (Rare Events Estimation) MaxSampleSize option, one can specify a target CoV . The algorithm will sequentially add batches of points to the current sample and stop as soon as the current CoV is below the specify threshold. To specify a target CoV = 0.01 and batches of size NB = 104 , one can write: MCOpts.Simulation.TargetCoV = 0.01; MCOpts.Simulation.BatchSize = 1e4; Note that the two options are independent from each other. The BatchSize option is also used to set the breakpoints for the MCAnalysis.Results.History structure. • Specify α for the confidence intervals: the α in Eq. (1.39) can also be specified. To set α = 0.1, one can write: MCOpts.Simulation.Alpha = 0.1; For a comprehensive list of the advanced options available for Monte Carlo simulation, please refer to Table 5, page 41. 2.3.4 Importance Sampling Importance sampling shares configuration options from both the FORM and the Monte Carlo simulation methods. A basic IS analysis can be setup with the following code: ISOpts.Type = 'Reliability'; ISOpts.Method = 'IS'; ISAnalysis = uq_createAnalysis(ISOpts); Using this minimal setup the analysis will run FORM first with the default options as in Section 2.3.1 to determine the design point U ∗ . Then the standard normal importance density centred at the obtained design point is used. Sampling is carried out with the following default options: N = 1000, batch size NB = 100. Once the analysis is performed, a report with the importance sampling results can be printed on screen by: uq_print(ISAnalysis) which produces the following: ------------------------------------------Importance Sampling ------------------------------------------Pf 1.3132e-03 Beta 3.0084 CoV 0.0599 ModelEvaluations 1008 PfCI [1.1591e-03 1.467421e-03] BetaCI [2.9745e+00 3.046121e+00] ------------------------------------------- The results can be visualized graphically as follows: UQL AB-V1.1-107 - 27 - UQL AB User Manual uq_display(ISAnalysis) which produces the convergence image in Figure 8. As with FORM, the second panel in Figure 8 is produced only in the 2D case. IS - FORM Design point and failure plane 5 FORM iterations FORM limit state surface g(X) ≤ 0 g(X) > 0 4 u2 3 2 1 0 -1 -6 -4 -2 0 u1 Figure 8: Graphical visualization of the convergence of the importance sampling analysis in Section 2.3.4. Note: In the preceding example no specifications are provided. The Importance Sampling shares the same defaults values as FORM and MCS. The exceptions are: – Maximum number of evaluated samples: 103 ; – Number of samples evaluated per batch: 102 . 2.3.4.1 Accessing the results The results of the importance sampling analysis can be accessed with the ISAnalysis.Results structure: ISAnalysis.Results ans = Pf: Beta: CoV: ModelEvaluations: PfCI: BetaCI: History: FORM: 0.0013 3.0095 0.0593 100008 [0.0012 0.0015] [2.9759 3.0468] [1x1 struct] [1x1 struct] The basic structure of Results closely resembles that of Monte Carlo sampling (see Section 2.3.3.1). However, an additional structure Results.FORM is available: ISAnalysis.Results.FORM ans = BetaHL: 3 UQL AB-V1.1-107 - 28 - Structural Reliability (Rare Events Estimation) Pf: ModelEvaluations: Ustar: Xstar: Importance: Iterations: History: 0.0013 8 [-2.4000 1.8000] [3.0800 3.0800] [0.6400 0.3600] 2 [1x1 struct] This structure is identical to the FORM results given in Section 2.3.1.1. 2.3.4.2 Advanced options The importance sampling algorithm accepts all of the options specific to both FORM and Monte Carlo sampling, described in Section 2.3.1.2 and Section 2.3.3.2. In addition, two additional options can be specified for importance sampling: • Specify existing FORM results: by default, importance sampling first runs FORM to determine the design point, followed by sampling around this design point to calculate Pf,IS . If the results of a previous FORM analysis are already available, they can be specified so as to avoid running FORM again. If the results are stored in a FORMResults structure with the same format as described in Section 2.3.1.1, one can write: ISOpts.IS.FORM = FORMResults; Alternatively, one can also directly specify a pre-existing UQL AB FORM or SORM analysis, say FORMAnalysis: ISOpts.IS.FORM = FORMAnalysis; • Specify a custom sampling distribution: alternatively, one can directly specify a custom sampling distribution. This can be achieved by providing the marginals and copula structure of the desired distribution, e.g.IOpts in Section 2.2, as follows: ISOpts.IS.Instrumental = IOpts; Alternatively, a pre-existent UQL AB INPUT object, say myISInput, can also be specified: ISOpts.IS.Instrumental = myISInput; In case the model has multiple outputs Nout , it might be desirable to specify a custom sampling distribution for each one of them. This can be done by either providing the IOpts as a 1 × Nout structure or the pre-existing inputs myISInputs as a 1 × Nout uq_input object. Please note, that custom distributions should be specified either for all outputs or for none. For a complete overview of the available options specific to the importance sampling algorithm, see Table 8. UQL AB-V1.1-107 - 29 - UQL AB User Manual 2.3.5 Subset Simulation The subset simulation algorithm can be used with the default options P0 = 0.1 and NS = 103 by specifying: SSOpts.Type = 'Reliability'; SSOpts.Method = 'Subset'; SSimAnalysis = uq_createAnalysis(SSOpts); Once the analysis is performed, a report with the subset-simulation results can be printed on screen by: uq_print(SSimAnalysis) which produces the following: ------------------------------------------Subset simulation ------------------------------------------Pf 1.0600e-03 Beta 3.0729 CoV 0.2433 ModelEvaluations 2680 PfCI [5.5463e-04 1.565373e-03] BetaCI [2.9546e+00 3.261242e+00] ------------------------------------------- The results can be visualized graphically as follows: uq_display(SSimAnalysis) which illustrates the samples of each subset in Figure 9 (applicable only for one and twodimensional problems). SubsetSim - Samples in each subset 5 4 x2 3 2 1 0 0 2 4 6 8 x1 Figure 9: Graphical visualization of the convergence of the subset simulation analysis in Section 2.3.5. UQL AB-V1.1-107 - 30 - Structural Reliability (Rare Events Estimation) Note: In the preceding example no specifications are provided. The Importance Sampling shares the same defaults values as FORM and MCS. Additionally, there are the following default values: – Target conditional failure probability of auxiliary limit-staates: 0.1; – Maximum number of subsets: 20; – Type of the proposal distribution in the Markov Chain: 'uniform'; – Parameter (standard deviation / halfwidth) of the proposal distribution: 1. 2.3.5.1 Accessing the results The results of subset simulation are stored in the SSimAnalysis.Results structure: SSimAnalysis.Results ans = Pf: 0.0016 Beta: 2.9402 CoV: 0.2426 ModelEvaluations: 2597 PfCI: [8.4859e-04 0.0024] BetaCI: [2.8160 3.1387] NumberSubsets: 3 History: [1x1 struct] The fields in the Results structure have the same meaning as their counterparts in importance sampling and Monte-Carlo sampling. Further, the field NumberSubsets denotes the number of subsets. Note that the ModelEvaluations field does not contain exactly the expected N = NS ∗ m ∗ (1 − P0 ) = 27000 limit-state function evaluations, but a slightly smaller N = 26095. This discrepancy is due to the modified Metropolis-Hastings MCMC acceptance criterion described in Au and Beck (2001), which in some uncommon cases can reject samples without the need of evaluating the limit-state function. 2.3.5.2 Advanced options Subset simulation uses the same advanced options as Monte-Carlo sampling described in Section 2.3.3.2, as well as some additional options. The most important are summarized in the following: • Specify P0 : the value of P0 in Eq. (1.53) can be specified in 0 < P0 ≤ 0.5. One can set e.g. P0 = 0.2 as follows: SSOpts.SubsetSim.p0 = 0.2; UQL AB-V1.1-107 - 31 - UQL AB User Manual • Specify the number of samples in each subset: the number of samples in each subset NS can be specified by using the .Simulation.BatchSize field. To set it to NS = 1000 one can write: SSOpts.Simulation.BatchSize = 1000; For a comprehensive overview of the available options specific to subset simulation see Table 9, page 44. 2.3.6 Adaptive Kriging Monte Carlo Sampling (AK-MCS) Adaptive Kriging Monte Carlo Sampling method with default values (see Table 11 and the related linked tables for details on the defaults) can be deployed in UQL AB with the following code: AKOpts.Type = 'Reliability'; AKOpts.Method = 'AKMCS'; AKAnalysis = uq_createAnalysis(AKOpts); Once the analysis is complete, a report with the AK-MCS results can be printed on screen by: uq_print(AKAnalysis) which produces the following: ------------------------------------------AK-MCS ------------------------------------------Pf 1.5200e-03 Beta 2.9637 CoV 0.0810 ModelEvaluations 18 PfCI [1.2785e-03 1.761457e-03] BetaCI [3.0165e+00 2.917994e+00] PfMinus/Plus [1.5200e-03 1.520000e-03] ------------------------------------------- The results can be visualized graphically as follows: uq_display(AKAnalysis) which produces the images in Figure 10. Note that the plot on the right of Figure 10 is only available when the input is two-dimensional. Additionally, if the verbosity is set to .Display ≥ 5 the AK-MCS analysis will plot the convergence of Pf and β while the analysis is running. UQL AB-V1.1-107 - 32 - Structural Reliability (Rare Events Estimation) AK-MCS - Experimental design 4 3.5 3 2.5 2 1.5 2 3 4 5 6 Figure 10: Graphical visualization of the convergence of the AK-MCS analysis in Section 2.3.6. Note: In the preceding example no specifications are provided. If not further specified the Monte Carlo Sampling runs with the following defaults: – Confidence level: 0.05; – Maximum number of evaluated samples: 105 ; – Number of samples evaluated per batch: 104 ; – Failure is defined for: limit-state g(x) ≤ 0; – Type of metamodel: 'Kriging'; – Learning function to determine the best next sample(s): 'U'; – Convergence criterion for the adaptive ED algorithm: 'stopU'; – Number of samples added to the ED for the metamodel: 103 ; – Number of samples in the initial ED: Nini = max(10, 2M ) – Initial ED sampling strategy: 'LHS'. 2.3.6.1 Accessing the results The results from the AK-MCS algorithm are stored in the AKAnalysis.Results structure: AKAnalysis.Results ans = Pf: Beta: CoV: ModelEvaluations: PfCI: BetaCI: Kriging: History: UQL AB-V1.1-107 0.0015 2.9637 0.0810 25 [0.0013 0.0018] [2.9180 3.0165] [1x1 uq_model] [1x1 struct] - 33 - UQL AB User Manual The fields in the Results structure have the same meaning as their counterparts in MonteCarlo sampling. Further, the field Kriging contains the final Kriging metamodel used to estimate the failure probability. This metamodel can be reused within UQL AB for any other purpose, see UQL AB User Manual – Kriging (Gaussian process modelling) for details. 2.3.6.2 Advanced options AK-MCS uses the same advanced options as Monte-Carlo simulation described in Section 2.3.3.2, as well as some additional options. The most important are summarized in the following: • Learning function: The learning function can be set to a custom function handle, which points to an existing M ATLAB function file. For instance, for a learning function called EFF, a function called uq_LF_EFF must be available in the M ATLAB path. It is then used in AK-MCS as: AKOpts.AKMCS.LearningFunction = 'EFF'; • Convergence criterion: There are three different convergence criteria mentioned in Section 1.3.6.2. They are all available and can be specified e.g.criterion on failure probability: AKOpts.AKMCS.Convergence = 'stopPf'; • Specify the Kriging metamodel: The specifications for the Kriging metamodel (see also UQL AB User Manual – Kriging (Gaussian process modelling) ) can be set in the field .AKMCS.Kriging, e.g.for an ordinary Kriging model: AKOpts.AKMCS.Kriging.Trend = 'ordinary'; • Specify the initial experimental design: Apart from specifying a number of points and a sampling strategy, the initial experimental design can be specified by providing X = x(1) , . . . , x(N0 ) in the matrix X and the corresponding limit-state function values g(x(1) , . . . , x(N0 ) in G: AKOpts.AKMCS.IExpDesign.X = X; AKOpts.AKMCS.IExpDesign.G = G; • Specify the number of added experimental design points: The maximum number of samples added to the experimental design of the Kriging metamodel can be specified to e.g.100: AKOpts.AKMCS.MaxAddedED = 100; Note that the total number of runs of the limit-state function then is at most the initial ED size plus the above number. UQL AB-V1.1-107 - 34 - Structural Reliability (Rare Events Estimation) • Use a PC-Kriging metamodel: Instead of Kriging, a PC-Kriging model (see also UQL AB User Manual – PC-Kriging ) can be used as a surrogate model in AK-MCS. AKOpts.AKMCS.MetaModel = 'PCK'; Specific options of the PCK model can be added in the field .AKMCS.PCK. As an example, a Gaussian correlation function in PC-Kriging is set as: AKOpts.AKMCS.PCK.Kriging.Corr.Family = 'Gaussian'; For an overview of the advanced options available for the AK-MCS method, refer to Table 11, page 45. 2.4 Advanced limit-state function options 2.4.1 Specify failure threshold and failure criterion While it is normally good practice to define the limit-state function directly as a UQL AB MODEL object as in Section 2.2, in some cases it can be useful to be able to create one from small modifications of existing MODEL objects. A typical scenario where this is apparent is when the same objective function needs to be tested against a set of different failure thresholds, e.g. for a parametric study. In this case, the limit-state specifications can be modified. As an example, when g(x) ≤ T = 5 defines the failure criterion, one can use the following syntax: MCOpts.LimitState.Threshold = 5; MCOpts.LimitState.CompOp = '<='; UQL AB offers several possibilities to create simple (or arbitrarily complex) objective functions from existing MODEL objects (see also UQL AB User Manual – The INPUT module ). For an overview of the advanced options for the limit-state function, refer to Table 4, page 41. 2.4.2 Vector Output In case the limit-state function g(x) results in a vector rather than a scalar value, the structural reliability module estimates the failure probability for each component independently. Note: There is no system-type reasoning implemented to combine the failure probabilities of each component. However, the implemented methods make use of evaluations of the limit-state function if available, as follows: • Monte Carlo simulation: The enrichment of the sample size is increased until the convergence criteria are fulfilled for all components. UQL AB-V1.1-107 - 35 - UQL AB User Manual • Subset Simulation: The first batch of samples (MCS) is reused for every output component limit-state. • AK-MCS: The initial experimental design for the Kriging model of output component i consists of the final experimental design of component i − 1. 2.5 Excluding parameters from the analysis In various usage scenarios (e.g. parametric studies) one or more input variables may be set to fixed constant values. This can have important consequences for many of the methods available in UQL AB e.g. FORM/SORM and AK-MCS, whose costs increase significantly with the number of input variables. Whenever applicable, UQL AB will appropriately account for the set of constant input parameters and exclude them from the analysis so as to avoid unnecessary costs. This process is transparent to the user as the analysis results will still show the excluded variables, but they will not be included in the calculations. To set a parameter to constant, the following command can be used when the probabilistic input is defined (See UQL AB User Manual – The INPUT module ): inputOpts.Marginals.Type = 'Constant' ; inputOpts.Marginals.Parameters = value; Furthermore, when the standard deviation of a parameter equals zero, UQL AB treats it as a Constant. For example, the following uniformly distributed variable whose upper and lower bounds are identical is automatically set to a constant with value 1: inputOpts.Marginals.Type = 'Uniform' ; inputOpts.Marginals.Parameters = [1 1]; UQL AB-V1.1-107 - 36 - Chapter 3 Reference List How to read the reference list Structures play an important role throughout the UQL AB syntax. They offer a natural way to group configuration options and output quantities semantically. Due to the complexity of the algorithms implemented, it is not uncommon to employ nested structures to fine-tune inputs/outputs. Throughout this reference guide, we adopt a table-based description of the configuration structures. The simplest case is given when a field of the structure is a simple value/array of values: Table X: Input String .Name A description of the field is put here which corresponds to the following syntax Input.Name = 'My Input'; The columns correspond to name, data type and a brief description of each field. At the beginning of each row a symbol is given to inform as to whether the corresponding field is mandatory, optional, mutually exclusive, etc. The comprehensive list of symbols is given in the following table: ⊕ Mandatory Optional Mandatory, mutually exclusive (only one of the fields can be set) Optional, mutually exclusive (one of them can be set, if at least one of the group is set, otherwise none is necessary) When one of the fields of a structure is a nested structure, we provide a link to a table that describes the available options, as in the case of the Options field in the following example: 37 UQL AB User Manual Table X: Input .Name String Description .Options Table Y Description of the Options structure .Field1 String Description of Field1 .Field2 Double Description of Field2 Table Y: Input.Options In some cases an option value gives the possibility to define further options related to that value. The general syntax would be Input.Option1 = 'VALUE1' ; Input.VALUE1.Val1Opt1 = ...; Input.VALUE1.Val1Opt2 = ...; This is illustrated as follows: Table X: Input .Option1 String Short description 'VALUE1' Description of 'VALUE1' 'VALUE2' Description of 'VALUE2' .VALUE1 Table Y Options for 'VALUE1' .VALUE2 Table Z Options for 'VALUE2' Table Y: Input.VALUE1 .Val1Opt1 String Description .Val1Opt2 Double Description Table Z: Input.VALUE2 .Val2Opt1 String Description .Val2Opt2 Double Description Note: In the sequel, double/doubles mean a real number represented in double precision (resp. a set of such real numbers). UQL AB-V1.1-107 - 38 - Structural Reliability (Rare Events Estimation) 3.1 Create a reliability analysis Syntax myAnalysis = uq_createAnalysis(ROpts) Input All the parameters required to determine the analysis are to be given as fields of the structure ROpts. Each method has its own options, that will be reviewed in different tables. The options described in Table 3 are common to all methods. Table 3: ROpts .Type 'uq_reliability' Identifier of the module. The options corresponding to other types are in the corresponding guides. .Method String Type of structural reliability method. The available options are listed below: 'MCS' Monte Carlo simulation. 'FORM', First order reliability method. 'SORM', Second order reliability method. 'IS' Importance sampling. 'Subset' Subset simulation. 'AKMCS' Adaptive Kriging Monte Carlo Simulation (AK-MCS). Name of the module. If not set by the user, a unique string is automatically assigned to it. .Name String .Input INPUT .Model MODEL .LimitState See Table 4 UQL AB-V1.1-107 object object INPUT object used in the analysis. If not specified, the currently selected one is used. MODEL object used in the analysis. If not specified, the currently selected one is used. Specification of the limit-state function. - 39 - UQL AB User Manual .Display String default: 'standard' Level of information displayed by the methods. 'quiet' Minimum display level, displays nothing or very few information. 'standard' Default display level, shows the most important information. 'verbose' Maximum display level, shows all the information on runtime, like updates on iterations, etc. .Simulation See Table 5 Options field for the simulation methods. Only applies when ROpts.Method is 'MCS', 'IS', 'SS', or 'AKMCS'. .FORM See Table 6 Options field for the FORM algorithm methods. Only applies when ROpts.Method is 'FORM', 'SORM', or 'IS'. .Gradient See Table 7 Options field for computing the gradient. It applies to the methods that use FORM, namely, when ROpts.Method is 'FORM', 'SORM', or 'IS'. .IS See Table 8 Options field for importance sampling. It applies only when ROpts.Method is 'IS'. .Subset See Table 9 Options field for subset simulation. It applies only when ROpts.Method is 'Subset'. .AKMCS See Table 11 Options field for the adaptive experimental design algorithm in AK-MCS. This applies when ROpts.Method is 'AKMCS'. .SaveEvaluations Logical default: true Storage or not of performed evaluations of the limit-state function. true Store the evaluations. false Do not store the evaluations. UQL AB-V1.1-107 - 40 - Structural Reliability (Rare Events Estimation) In order to perform a structural reliability analysis, the limit-state function g(x) is compared to a threshold value T (by default T = 0). In analogy with Eq. (1.1), failure is defined as g(x) ≤ T . Alternatively, failure can be specified as g(x) ≥ T by adjustment of the field ROpts.LimitState.CompOp to '>='. The relevant options are summarized in Table 4: Table 4: ROpts.LimitState .Threshold Double default: 0 Threshold T , compared to the limit-state function g(x). .CompOp String default: '<=' Comparison operator for the limit-state function. '<', '<=' Failure is defined by g(x) < T . '>', '>=' Failure is defined by g(x) > T . The available methods to perform structural reliability analysis are Monte Carlo simulation, importance sampling, subset simulation, AK-MCS, FORM, and SORM. The first four methods share the simulation options. FORM and SORM are gradient-based, so they allow the user to specify the finite difference options as well as the algorithm options. In Table 5, the options for the simulation methods (Monte Carlo, importance sampling, subset simulation and AK-MCS) are shown: Table 5: ROpts.Simulation .Alpha Double default: 0.05 Confidence level α. For the Monte Carlo estimators, a confidence interval is constructed with confidence level 1 − α. .MaxSampleSize Integer default: 103 for 'IS'; 105 otherwise Maximum number of samples to be evaluated. If there is no target coefficient of variation (CoV), this is the total number of samples to be evaluated. If the target CoV is present, the method will run until TargetCoV or MaxSampleSize is reached. In this case, the default value of MaxSampleSize, if not specified in the options, is Inf, i.e.the method will run until the target CoV is achieved. UQL AB-V1.1-107 - 41 - UQL AB User Manual .TargetCoV Double Target coefficient of variation. If present, the method will run until the estimate of the CoV (Eq. (1.40)) is below TargetCoV or until MaxSampleSize function evaluations are performed. The value of the coefficient of variation of the estimator is checked after each BatchSize evaluations. By default this option is disabled. Note: this option has no effect in Method = 'Subset' and 'AKMCS'. .BatchSize Integer default: 104 for 'MCS' and 'AKMCS'; 103 for 'Subset'; 102 for 'IS' Number of samples that will be evaluated at once. Note that this option has no effect in Method = 'AKMCS'. Note: In order to use importance sampling after an already computed FORM analysis, one can provide these results to the analysis options in order to avoid repeating FORM. If FORMResults is a structure containing the results of a FORM analysis, the syntax reads: ISOpts.Type = 'Reliability'; ISOpts.Method = 'IS'; ISOpts.FORM = FORMResults; ISAnalysis = uq_createAnalysis(ISOpts); The FORM algorithm has special parameters that can be tuned in the subfield FORM of the options. These parameters also affect the methods that depend on FORM, namely importance sampling and SORM. These are listed in Table 6. Table 6: ROpts.FORM .Algorithm .StartingPoint UQL AB-V1.1-107 String default: 'iHLRF' Algorithm used to find the design point. 'iHLRF' Improved HLRF. 'HLRF' HLRF. 1 × M Double default: zeros(1,M) Starting point for the Rackwitz-Fiessler algorithm. - 42 - Structural Reliability (Rare Events Estimation) .StopU Double default: 10−4 Tolerance value for the Rackwitz-Fiessler algorithm on the design point. The algorithm will stop when |Uk+1 − Uk | < StopU . .StopG Double default: 10−6 Tolerance value for the Rackwitz-Fiessler algorithm on the limit-state function value. The algorithm will stop when k) | G(U G(U0 ) | < StopG. .MaxIterations Integer default: 100 Maximum number of iterations allowed in the Rackwitz-Fiessler algorithm. If this property should be ignored, it can be set to Inf. Since FORM is a gradient-based method, the gradient of the limit-state function needs to be computed. This is done using finite differences. The options for the differentiation are listed in Table 7. Table 7: ROpts.Gradient .h Double default: 10−3 Value of the difference for the scheme. .Method String default: 'forward' Specifies the type of finite differences scheme to be used. 'forward' ∂g Forward finite differences. ∂x is i approximated using g(x) and g(x + hei ). 'backward' ∂g ∂xi is approximated using g(x) and g(x − hei ). 'centered' ∂g ∂xi UQL AB-V1.1-107 is approximated using g(x + hei ) and g(x − hei ). (More accurate and more costly.) - 43 - UQL AB User Manual The options specifically set for the importance sampling are presented in Table 8. Note that the options of .Simulation and .FORM are also processed in the case of importance sampling due to the nature of the MCS, FORM and IS. Table 8: ROpts.IS .Instrumental 1 × Nout INPUT object or Struct Instrumental distribution defined as either a structure of input marginals and copula or an INPUT object (refer to UQL AB User Manual – The INPUT module for details). .FORM FORM ANALYSIS object or FORMAnalysis.Results Struct FORM results computed previously. See Section 2.3.4.2 for details. The options specifically set for subset simulation are presented in Table 9. Note that the options of .Simulation are also processed in the case of subset simulation due to the similar nature of Monte Carlo simulation and subset simulation. Table 9: ROpts.Subset .p0 Double default: 0.1 Target conditional failure probability of auxiliary limit-states (0
Source Exif Data:File Type : PDF File Type Extension : pdf MIME Type : application/pdf PDF Version : 1.5 Linearized : No Page Count : 64 Page Mode : UseOutlines Author : S. Marelli, R. Schöbi, B. Sudret Title : UQLab User Manual – Structural Reliability (Rare Events Estimation) Subject : UQLab Structural Reliability Manual Creator : LaTeX with hyperref package Producer : pdfTeX-1.40.15 Keywords : UQLab;Structural, Reliability;Rare, Events, Estimation;, Uncertainty, Quantification;Monte, Carlo, Sampling;Importance, Sampling;Subset, Simulation;FORM;SORM; Create Date : 2018:07:05 12:17:49+02:00 Modify Date : 2018:07:05 12:17:49+02:00 Trapped : False PTEX Fullbanner : This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) kpathsea version 6.2.1devEXIF Metadata provided by EXIF.tools