Metamodel-based importance sampling for structural reliability analysis (1105.0562v2)

Abstract: Structural reliability methods aim at computing the probability of failure of systems with respect to some prescribed performance functions. In modern engineering such functions usually resort to running an expensive-to-evaluate computational model (e.g. a finite element model). In this respect simulation methods, which may require $10^{3-6}$ runs cannot be used directly. Surrogate models such as quadratic response surfaces, polynomial chaos expansions or kriging (which are built from a limited number of runs of the original model) are then introduced as a substitute of the original model to cope with the computational cost. In practice it is almost impossible to quantify the error made by this substitution though. In this paper we propose to use a kriging surrogate of the performance function as a means to build a quasi-optimal importance sampling density. The probability of failure is eventually obtained as the product of an augmented probability computed by substituting the meta-model for the original performance function and a correction term which ensures that there is no bias in the estimation even if the meta-model is not fully accurate. The approach is applied to analytical and finite element reliability problems and proves efficient up to 100 random variables.

• The paper presents a method integrating kriging surrogates with importance sampling to efficiently estimate failure probabilities in engineering systems.
• It introduces an adaptive refinement strategy that selectively enriches design experiments to optimize reliability assessments.
• The approach significantly reduces computational effort compared to Monte Carlo methods, making it suitable for complex high-dimensional problems.

Metamodel-based Importance Sampling for Structural Reliability Analysis

The paper presents a method to enhance structural reliability analysis by incorporating metamodel-based importance sampling. Structural reliability analysis aims to estimate failure probabilities of engineering systems, often necessitating expensive computational models like finite element models. Traditional methods like Monte Carlo simulation are computationally prohibitive when the required number of model evaluations is exorbitant. The proposed approach employs kriging surrogates to build quasi-optimal importance sampling densities, significantly reducing computational costs.

Core Contributions

1. Kriging Surrogates: The paper utilizes kriging metamodels to approximate the performance function, enabling efficient evaluations. This surrogate model is used to approximate the probability of failure and construct an importance sampling density that emphasizes regions close to the failure domain.
2. Importance Sampling Framework: The method reformulates failure probability estimation by leveraging the probabilistic predictions from kriging. A novel estimator combines an augmented failure probability with a correction factor, ensuring unbiasedness even when the metamodel is not entirely accurate.
3. Adaptive Refinement: An adaptive strategy is included to refine the probabilistic classification function, optimizing the instrumental density iteratively. The approach selectively enriches the design of experiments, focusing computational effort on significant regions.
4. Algorithmic Implementation: The paper outlines an efficient algorithm, detailing initialization, refinement, and parallel estimation steps. The implementation ensures practical applicability to high-dimensional and complex reliability problems.

Numerical Results and Implications

The methodology is applied to both analytical and finite element problems, demonstrating efficiency with up to 100 random variables. The strong numerical results indicate significant reductions in computational efforts compared to traditional Monte Carlo methods. The approach aligns well with industrial applications where computational resources are a critical constraint.

Theoretical and Practical Implications

Theoretically, the work advances the integration of metamodeling and variance reduction techniques, offering a robust framework for reliability analysis. Practically, it allows for a feasible evaluation of complex systems under uncertainty without excessive computational burdens. The method holds potential for widespread use in engineering fields requiring reliability assessments, such as aerospace, civil, and mechanical engineering.

Future Directions

This research opens avenues for further developments, including enhancements to adaptive refinement strategies and exploration of alternative metamodels. Additionally, integrating the method within reliability-based design optimization frameworks could provide comprehensive solutions to engineering design challenges.

In conclusion, the paper provides a significant contribution to structural reliability analysis, leveraging the strengths of metamodels and importance sampling. It addresses key challenges in computational cost and accuracy, laying the groundwork for future advancements in the field.