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.