High-dimensional model representation for structural reliability analysis: Authors' reply to comments by S. Rahman and H. Xu
High dimensional model representation based higher order limit state function for reliability analysis
A new response surface method for predicting the failure probability of structural or mechanical systems subjected to random loads and material properties is presented in this paper. The method involves high dimensional model representation (HDMR) technique in conjunction with moving least squares to approximate the original implicit performance function with an explicit performance function. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Results of seven numerical examples involving mathematical functions and structural mechanics problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure.
High-dimensional model representation for structural reliability analysis
This paper presents a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry. The method involves high-dimensional model representation (HDMR) that facilitates lower-dimensional approximation of the original high-dimensional implicit limit state/ performance function, response surface generation of HDMR component functions, and Monte Carlo simulation. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by statistical simulation. Results of nine numerical examples involving mathematical functions and structural mechanics problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure. Copyright © 2008 John Wiley & Sons, Ltd.
Hybrid High Dimensional Model Representation for reliability analysis
This paper presents a new probabilistic method based on Hybrid High Dimensional Model Representation (HHDMR) for predicting the failure probability of structural and mechanical systems subject to random loads, material properties, and geometry. High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable cooperative effects are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then all right hand side components of HDMR must be used to be able to obtain the best result. However, if HDMR requires all components, which means 2N number of components, to get a desired accuracy, making the method very expensive in practice, then Factorized HDMR (FHDMR) can be used. But in most cases the limit state/performance function has neither additive nor multiplicative nature, rather it has an intermediate nature. This paper presents a new HHDMR-based approximation for the limit state/performance functions having an intermediate nature. The proposed approximation of an implicit limit state/performance function includes both HDMR and FHDMR expansions through a hybridity parameter. As an alternative to the conventional methods for reliability analysis which are very computationally demanding, when applied in conjunction with complex finite element models, this study aims to assess how accurately and efficiently HHDMR technique can capture complex model output uncertainty. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by statistical simulation. Results of six numerical examples involving mathematical functions and structural/solid-mechanics problems indicate that the failure probability obtained using HHDMR-based approximation of an implicit limit state/performance function provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. © 2008 Elsevier B.V. All rights reserved.
Reliability analysis of 500MWe PHWR inner containment using high-dimensional model representation
In this paper, uncertainty analysis of Indian 500. MWe Pressurized Heavy Water Reactor (PHWR) subjected to an accidental pressure is carried out using a computational tool based on High Dimensional Model Representation (HDMR) that facilitates lower dimensional approximation of the original high dimensional implicit limit state/performance function. The method involves response surface generation of HDMR component functions, and Monte Carlo simulation. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by first few lower-order terms. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by statistical simulation. Reliability estimates of PHWR inner containment subjected to an internal pressure exceeding the design pressure, considering three stages of progressive failure prior to collapse are presented. © 2010 Elsevier Ltd.
Multicut high dimensional model representation for reliability analysis
This paper presents a novel method for predicting the failure probability of structural or mechanical systems subjected to random loads and material properties involving multiple design points. The method involves Multicut High Dimensional Model Representation (Multicut-HDMR) technique in conjunction with moving least squares to approximate the original implicit limit state/performance function with an explicit function. Depending on the order chosen sometimes truncated Cut-HDMR expansion is unable to approximate the original implicit limit state/performance function when multiple design points exist on the limit state/performance function or when the problem domain is large. Multicut-HDMR addresses this problem by using multiple reference points to improve accuracy of the approximate limit state/performance function. Numerical examples show the accuracy and efficiency of the proposed approach in estimating the failure probability.
Hybrid high dimensional model representation for failure probability estimation
This work presents a new probabilistic method based on Hybrid High Dimensional Model Representation (HHDMR) for predicting the failure probability of randomly parametered structural/mechanical system. High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable cooperative effects are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then Factorized HDMR (FHDMR) must be used, to get a desired accuracy with least number of numerical calculations. But in most cases the nature of the limit state/performance function has neither additive nor multiplicative nature. Rather it has an intermediate nature. This paper presents a new HHDMR based approximation of an implicit limit state/performance function has neither additive nor multiplicative nature but rather an intermediate nature. The proposed approximation of an implicit limit state/performance function includes both HDMR and FHDMR expansions through a hybridity parameter. Results of six numerical examples involving elementary mathematical functions and structural/solid-mechanics problems indicate that the failure probability obtained using HHDMR approximation of an implicit limit state/performance function provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.
Assessment of high dimensional model representation techniques for reliability analysis
This paper presents an assessment of efficient response surface techniques based on the High Dimensional Model Representation (HDMR) and the Factorized High Dimensional Model Representation (FHDMR). The HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of an additive nature, allowing the physical model to be captured by the first few lower order terms. But, if the multiplicative nature of the response function is dominant, then all the right hand side components of the HDMR must be used to be able to obtain the best result. However, if the HDMR requires all components, which means 2N of them, to get a desired accuracy, making the method very expensive in practice, then the FHDMR can be used. The component functions of the FHDMR are determined by using the component functions of the HDMR. This paper presents the formulation of the FHDMR based response surface approximation of a limit state/performance function which is dominantly multiplicative in nature. It is a given that conventional methods for reliability analysis are computationally very demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based response surface techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of the HDMR, which is recently applied to reliability analysis, is also demonstrated. The response surface is constructed using the moving least squares interpolation formula by including constant, first-order, and second-order terms of the HDMR and the FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Results of seven numerical examples involving structural/solid-mechanics/geo-technical engineering problems indicate that the failure probability obtained using the FHDMR based response surface method for a limit state/performance function that is dominantly multiplicative in nature, provides a significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. © 2008 Elsevier Ltd. All rights reserved.
Enhanced high-dimensional model representation for reliability analysis
This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high-dimensional model representation (HDMR) generated from low-order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the lower-order terms and facilitating lower-dimensional approximation of the original high-dimensional implicit limit state/ performance function. When first-order HDMR approximation of the original high-dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher-order terms of HDMR expansion by expressions similar to the lower-order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input-output samples without directly invoking the determination of second- and higher-order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high-dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least-squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/ solid-mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/ performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.
Probabilistic analysis using high dimensional model representation and fast fourier transform
This paper presents an efficient probabilistic analysis method for predicting component reliability of structural/mechanical systems subject to random loads, material properties, and geometry. The proposed method involves High Dimensional Model Representation (HDMR) for the limit state/performance function approximation and fast Fourier transform for solving the convolution integral. The limit state/performance function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full-scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through five examples involving explicit/implicit performance functions.