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Meta-models for structural reliability and uncertainty quantification

Bruno Sudret

arXiv:1203.2062v1stat.MEstat.AP

TL;DR

Structural reliability often requires costly repeated evaluations of computational models, while direct Monte Carlo simulation can demand about 10^(r+2) simulations for failure probabilities of order 10^-r. The paper reviews polynomial response surfaces, polynomial chaos expansions, and kriging, then proposes kriging-assisted importance sampling to obtain unbiased failure-probability estimates with substantially reduced computational cost.

  • Problem

    Costly computational-model evaluations make precise structural-reliability estimation impractical, and surrogate-based failure probabilities are not guaranteed to be sufficiently accurate or unbiased.

  • Method

    The paper reviews polynomial response surfaces, polynomial chaos expansions, and kriging, and combines kriging with importance sampling so the surrogate builds an instrumental density rather than directly replacing the limit-state function.

  • Results

    The meta-importance-sampling approach reduced computational cost by two orders of magnitude, using 40 limit-state-function runs and 200 true-model runs to compute failure probability within 5% accuracy.

  • Takeaways & Limitations

    Meta-modelling can make real-world structural-reliability problems computationally affordable, while kriging and polynomial chaos expansions show promising performance and accuracy.

  • Takeaways & Limitations

    Quadratic response surfaces assume smooth behavior and a single design point, while meta-importance sampling requires balancing surrogate-construction calls against correction-sample calls.

Abstract

from arXiv · show

A meta-model (or a surrogate model) is the modern name for what was traditionally called a response surface. It is intended to mimic the behaviour of a computational model M (e.g. a finite element model in mechanics) while being inexpensive to evaluate, in contrast to the original model which may take hours or even days of computer processing time. In this paper various types of meta-models that have been used in the last decade in the context of structural reliability are reviewed. More specifically classical polynomial response surfaces, polynomial chaos expansions and kriging are addressed. It is shown how the need for error estimates and adaptivity in their construction has brought this type of approaches to a high level of efficiency. A new technique that solves the problem of the potential biasedness in the estimation of a probability of failure through the use of meta-models is finally presented.

1. Introduction

Structural reliability and uncertainty quantification increasingly rely on repeated evaluations of computational models, making costly models impractical for precise analysis. The paper reviews meta-modelling techniques developed to address this computational burden and improve reliability estimation.

  • 1. Introduction: Structural reliability accounts for uncertainties in models and parameters such as geometry, material properties, and applied forces.The field has developed industrial applications across civil, environmental, mechanical, and aerospace engineering.
  • 1. Introduction: Repeated model evaluations make precise reliability predictions impractical when each computational-model run is costly.Monte Carlo simulation may require many evaluations, which is incompatible with expensive finite-element models.
  • 1. Introduction: Meta-modelling uses inexpensive approximations to support structural reliability analysis involving computational models.The paper presents meta-modelling as a research field focused on addressing the cost of repeated evaluations.
  • 1. Introduction: The paper reviews meta-models including polynomial response surfaces, polynomial chaos expansions, and kriging.Its organization also covers uncertainty quantification, Monte Carlo limitations, unbiased importance sampling, and applications.

2. Problem statement

Structural reliability evaluates failure probabilities for systems governed by computational models and uncertain inputs, but implicit high-dimensional integrals and expensive model runs make direct Monte Carlo estimation costly. The paper frames surrogate models and other reliability methods as ways to reduce this computational burden.

  • 2. Problem statement: A computational model maps input parameters to response quantities and is treated as a deterministic black box evaluated point-by-point.Inputs may describe geometry, materials, and loading, while outputs may include displacements, strains, stresses, and internal variables.
  • 2. Problem statement: Structural failure is represented by a limit-state function whose nonpositive values define the failure domain and whose zero level defines the limit-state surface.The probability of failure is obtained from the uncertain input distribution over this domain.
  • 2. Problem statement: The probability-of-failure integral is difficult because its domain is implicit and its dimension equals the usually large number of uncertain parameters.Monte Carlo estimates are unbiased and mean-square convergent, but their sample requirement can remain prohibitive.
  • 2. Problem statement: Monte Carlo simulation requires about 10^(r+2) simulations for failure probabilities of order 10^-r with coefficient of variation at most 10%.This requirement becomes unaffordable for probabilities around 10^-3 to 10^-6 when each limit-state evaluation is costly.
  • 2. Problem statement: The paper reviews approximation-based and simulation-based reliability methods before focusing on surrogate models and their use in structural reliability.The reviewed methods are intended either to reduce computational cost through approximation or to improve Monte Carlo convergence.

3.1. Introduction

A meta-model is a fast analytical approximation fitted to observations of an expensive computational model. The paper reviews polynomial response surfaces, polynomial chaos expansions, kriging, and related reliability-oriented approaches.

  • 3.1. Introduction: A meta-model is a fast analytical function fitted to input/output observations of the original computational model.It belongs to a selected function class and is characterized by fitted parameters.
  • 3.1. Introduction: The reviewed meta-model classes are linear or quadratic response surfaces, polynomial chaos expansions, and Gaussian-process kriging surrogates.These classes provide different ways to approximate the original model for reliability analysis.
  • 3.1. Introduction: Support vector machines have also been applied to classify safe and failure points and combined with subset simulation for small failure probabilities.This approach comes from statistical learning and is presented as related work rather than among the three principal reviewed classes.

3.2. FOSM and FORM method viewed as linear response surfaces

FOSM and FORM can be interpreted as reliability analyses based on linear response surfaces. FOSM linearizes around the input mean, whereas FORM transforms inputs to standard normal space and linearizes around a design point.

  • 3.2. FOSM and FORM method viewed as linear response surfaces: FOSM implicitly uses a linear response surface constructed around the mean values of the input parameters.Its reliability index uses the mean and standard deviation of the safety margin, with variance obtained from a Taylor expansion.
  • 3.2. FOSM and FORM method viewed as linear response surfaces: FORM transforms inputs into standard normal space and approximates the transformed limit-state surface at the design point.The Hasofer-Lind reliability index is the algebraic distance from the origin to that surface, and failure probability follows from the linearization.

3.3. Quadratic response surfaces

Quadratic polynomial response surfaces approximate the limit-state function by fitting a second-order polynomial to selected model observations, but their smoothness and single-design-point assumptions limit versatility.

  • 3.3. Quadratic response surfaces: Quadratic response surfaces approximate the limit-state function using a fitted second-order polynomial built from selected observations.The coefficients are computed by minimizing least-square error between model responses and the surrogate.
  • 3.3. Quadratic response surfaces: Variants differ in polynomial terms, including whether cross terms x_i x_j are included, and in the experimental design used for fitting.
  • 3.3. Quadratic response surfaces: Quadratic response surfaces have been widely used for reliability analysis over the last 20 years.
  • 3.3. Quadratic response surfaces: Their versatility is limited because second-order polynomials can mimic only smoothly behaving models and generally assume a single design point.The single-design-point assumption is associated with FORM-style approximation and is seldom met in industrial problems.

3.4. Polynomial chaos expansions

Polynomial chaos expansions represent the random margin with orthonormal polynomial bases and can serve as global surrogate models for reliability analysis. Regression-based, adaptive constructions extend beyond quadratic approximations while providing error control.

  • 3.4. Polynomial chaos expansions: Polynomial chaos expansions represent the random margin as a truncated sum of multivariate orthonormal polynomials in the input variables.The basis is constructed from univariate orthogonal polynomials and tensorized across input dimensions.
  • 3.4. Polynomial chaos expansions: p = 3 is required to compute probabilities of failure down to 10−4 with satisfactory accuracy, whereas p = 2 usually suffices for estimating the margin’s mean and variance.
  • 3.4. Polynomial chaos expansions: Regression-based PC expansions fit coefficients from a space-filling experimental design and substitute the resulting global polynomial response surface for the true limit-state function.Latin Hypercube sampling or quasi-random numbers can define the design, with empirical size N=2-3 CardA.
  • 3.4. Polynomial chaos expansions: Unlike quadratic response surfaces, PC expansions allow arbitrarily high polynomial degree and provide error estimates, adaptivity, moments, and sensitivity information.
  • 3.4. Polynomial chaos expansions: Adaptive error estimates select sparse PC representations and increase the maximal polynomial degree until the prescribed admissible error is attained.These procedures address the arbitrary nature of a priori truncation.

3.5. Kriging meta-models

Kriging models a computational response as a Gaussian process, providing interpolating predictions and a pointwise epistemic uncertainty measure. Adaptive enrichment uses this uncertainty and limit-state proximity to refine reliability surrogates.

  • 3.5. Kriging meta-models: Kriging provides a meta-model independent of the probabilistic model for the input random vector X.
  • 3.5. Kriging meta-models: The kriging predictor is a best linear unbiased estimator formed from observed computer-model responses, with a regression mean and covariance-based correction.
  • 3.5. Kriging meta-models: The mean kriging estimator interpolates the observations, and the kriging variance is zero at observed points.
  • 3.5. Kriging meta-models: Kriging variance measures epistemic prediction uncertainty and can guide adaptive enrichment of the experimental design.It is distinct from aleatoric uncertainty in the input vector X.
  • 3.5. Kriging meta-models: Active kriging methods sequentially add points near the surrogate limit state or where prediction uncertainty is high.AK-MCS evaluates a U-function on a large Monte Carlo sample without additional calls to the true limit-state function, then adds the point with minimum U.
  • 3.5. Kriging meta-models: Sequential one-point enrichment is a weakness when several candidate extrema exist and distributed computing could evaluate multiple limit-state points in parallel.

3.6. Conclusion

The paper reviews quadratic response surfaces, polynomial chaos expansions, and kriging for structural reliability, then addresses bias in failure-probability estimation from surrogate substitution. Adaptive kriging illustrates sequential sampling near uncertainty margins.

  • 3.6. Conclusion: Quadratic response surfaces, polynomial chaos expansions, and kriging are reviewed as meta-models for structural reliability analysis.
  • 3.6. Conclusion: Directly evaluating failure probability on a surrogate is not guaranteed to equal or sufficiently approximate the probability from the original limit-state function.
  • 3.6. Conclusion: Adaptive kriging concentrates later sample points within the uncertainty margin defined by kriging variance, using clustering to select enrichment points.

4. Meta-models as a means for importance sampling

Meta-IS combines kriging with importance sampling to avoid directly substituting a surrogate for the true limit-state function, addressing potential bias in failure-probability estimates. The method uses a probabilistic classification function to construct a quasi-optimal instrumental density and applies a correction factor to obtain an unbiased estimator.

  • Meta-IS uses kriging to construct a smart importance-sampling density rather than directly replacing the true limit-state function, addressing potential estimator bias.The surrogate defines the sampling density, not the final failure indicator used by the estimator.
  • Importance sampling concentrates samples in regions relevant to failure, whereas crude Monte Carlo usually draws most samples from the central input-distribution region.The instrumental density is intended to reduce estimator variance by focusing computation on the failure region.
  • The probabilistic classification function smoothly approximates the failure indicator, approaching 0 for safe points and 1 for failure points when kriging is accurate.Its behavior follows the sign of the true limit-state function as surrogate uncertainty becomes small.
  • The meta-IS estimator combines a correction factor based on true failure indicators with a low-cost probabilistic-classification estimate.The correction-factor sum uses Ncorr meta-IS samples, while the classification-based sum uses Nε samples of X.
  • The estimator is proven unbiased, with Nε = 10^5−10^6 affordable because the probabilistic classification function is analytical once kriging is available.Evaluating the true indicator for the correction factor remains costly, so Ncorr is limited to a few hundred in practice.
  • Meta-IS requires balancing surrogate-building calls N against correction-sample calls Ncorr because total cost is N + Ncorr.Cross-validation is proposed to stop surrogate refinement when the estimated failure probability is sufficiently close to Pf.

5. Application examples

Application examples show that polynomial chaos and meta-importance sampling can estimate structural reliability accurately while substantially reducing evaluations of costly models.

  • 5.1. Frame structure - Polynomial chaos expansion (Blatman and Sudret, 2010a): Less than 5% error is achieved for generalized reliability indices up to β = 4 using sparse polynomial chaos expansion.The expansion uses a target approximation error of 10^-3 and estimates are reported for different displacement thresholds.
  • 5.1. Frame structure - Polynomial chaos expansion (Blatman and Sudret, 2010a): The frame-structure analysis requires 450 finite element model calls while also providing response moments and sensitivity results.The polynomial chaos expansion is computed once, enabling threshold-based parametric analysis without additional model evaluations.
  • 5.1. Frame structure - Polynomial chaos expansion (Blatman and Sudret, 2010a): Polynomial chaos accuracy is controlled globally in a least-square sense, so tail accuracy is not perfectly controlled for very small failure probabilities.Reported literature results indicate robustness up to Pf = 10^-4/-5.
  • 5.2. System reliability - meta-IS (Dubourg, 2011): Meta-importance sampling builds a surrogate with 40 limit-state evaluations and then reaches 5% probability-of-failure accuracy with 200 true-model evaluations.The reported computational cost is decreased by two orders of magnitude relative to the reference analysis.

6. Conclusions

The conclusions present meta-modelling as a practical way to make real-world structural reliability problems computationally affordable. They review polynomial chaos expansions and kriging as promising approaches while emphasizing continued engagement with computer science and applied mathematics.

  • 6. Conclusions: Meta-modelling enables real-world engineering reliability problems to be addressed at an affordable computational cost.The paper also interprets FOSM and FORM/SORM as basic meta-models.
  • 6. Conclusions: Polynomial chaos expansions and kriging show promising performance and accuracy among the reviewed advanced meta-modelling techniques.
  • 6. Conclusions: The paper emphasizes that engineers should engage with technologies from computer science, applied mathematics, and statistics to support further innovation.
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