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 2N2N 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.