Factorized high dimensional model representation for structural reliability analysis
Purpose: To develop a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry. Design/methodology/approach: 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 correlations 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. The component functions of FHDMR are determined by using the component functions of HDMR. This paper presents the formulation of FHDMR approximation of a multivariate limit state/performance function, which is dominantly of multiplicative nature. Given that conventional methods for reliability analysis are very computationally demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based approximation techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of HDMR, which is recently applied to reliability analysis, is also demonstrated. Response surface is constructed using moving least squares interpolation formula by including constant, first-order and second-order terms of HDMR and FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Findings: Results of five numerical examples involving structural/solid-mechanics/geo-technical engineering problems indicate that the failure probability obtained using FHDMR approximation for the limit state/performance function of dominantly multiplicative in nature, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Originality/value: This is the first time where application of FHDMR concepts is explored in the field of reliability and system safety. Present computational approach is valuable to the practical modeling and design community, where user often suffers from the curse of dimensionality. © Emerald Group Publishing Limited.
Hysteresis model for RC structural element accounting bi-directional lateral load interaction
Experimental investigations on cyclic behaviour of reinforced concrete (RC) structures have demonstrated that, bidirectional flexural deformation is much more, compared to unidirectional responses separately. Biaxial lateral load interaction, therefore, has a significant role on the dynamic responses of structure. The present paper projected a new formulation of the biaxial forcedeformation model, accounting strength and stiffness degradation in a rather simplistic manner. Also the modelling scheme dose not requires much computational rigour. The present effort is based on yield surface approach of biaxial modelling scheme. The validity of the proposed model is appraised through available biaxial test of RC column. Compared to other existing models, present model requires simple input parameters and the prediction of hysteretic responses are quite faithful. In the RC containment structures, which are the prime concern to the safety of nuclear power plant, this model may pertain to evaluate the hysteretic responses with fair accuracy during seismic loading. Copyright © 2005 by ASME.
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.
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.
Probabilistic characterization of AHWR Inner Containment using High Dimensional Model Representation
In this paper, uncertainty analysis of Advanced Heavy Water Reactor (AHWR) 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 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. Reliability estimates of AHWR Inner Containment subjected to an internal pressure exceeding the design pressure, considering four stages of progressive failure prior to collapse are presented. © 2009 Elsevier B.V. All rights reserved.
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 piece-wise continuous function approximation
High dimensional model representation (HDMR) approximates multivariate functions in such a way that the component functions of the approximation are ordered starting from a constant and gradually approaching to multivariance as we proceed along the terms like first-order, second-order and so on. Until now HDMR applications include construction of a computational model directly from laboratory/field data, creating an efficient fully equivalent operational model to replace an existing time-consuming mathematical model, identification of key model variables, global uncertainty assessments, efficient quantitative risk assessments, etc. In this paper, the potential of HDMR for tackling univariate and multivariate piece-wise continuous functions is explored. Eight numerical examples are presented to illustrate the performance of HDMR for approximating a univariate or a multivariate piece-wise continuous function with an equivalent continuous function. Copyright © 2007 John Wiley & Sons, Ltd.
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.
Probabilistic stability assessment of slopes using high dimensional model representation
This paper presents a new computational tool for probabilistic stability assessment of earth slopes/embankments. The method involves high dimensional model representation (HDMR) that facilitates lower dimensional approximation of the original limit state, 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 limit state is defined, the failure probability can be obtained by statistical simulation. Results of four numerical examples indicate that the proposed method provides accurate and computationally efficient estimates of the failure probability of earth slopes/embankments. © 2010 Elsevier Ltd.