An adaptive polynomial chaos expansion for high-dimensional reliability analysis

Efficiency is greatly concerned in reliability analysis community, especially for the problems with high-dimensional input random variables, because the computation cost of common reliability analysis methods may increase sharply with respect to the dimension of the problem. This paper proposes a novel meta-model based on the concepts of polynomial chaos expansion (PCE), dimension-reduction method (DRM), and information-theoretic entropy. Firstly, a PCE method based on DRM is developed to approximate the original function by a series of PCEs of univariate components. Compared with the PCE of the original function, the DRM-based PCE can reduce the computational cost. Before constructing the meta-model, a prior of the degree of the PCE is required, which determines the accuracy and efficiency of the PCE. However, the prior is usually determined by experience. According to the maximum entropy principle, this paper proposes an adaptive method for the selection of the polynomial chaos basis efficiently. With the adaptive PCE method based on DRM, a novel meta-model method is proposed, with which the reliability analysis can be achieved by Monte Carlo simulation efficiently. In order to verify the performance of the proposed method, three numerical examples and one structural dynamics engineering example are tested, with good accuracy and efficiency.

[1]  Zhenzhou Lu,et al.  Reliability sensitivity method by line sampling , 2008 .

[2]  R. Rackwitz,et al.  A benchmark study on importance sampling techniques in structural reliability , 1993 .

[3]  Geoffrey T. Parks,et al.  Multi-fidelity non-intrusive polynomial chaos based on regression , 2016 .

[4]  Byeng D. Youn,et al.  Performance Moment Integration (PMI) Method for Quality Assessment in Reliability-Based Robust Design Optimization , 2005 .

[5]  Zhenzhou Lu,et al.  Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation , 2018, Structural and Multidisciplinary Optimization.

[6]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[7]  M. Lemaire,et al.  Stochastic finite element: a non intrusive approach by regression , 2006 .

[8]  B. Youn,et al.  Reliability-based robust design optimization using the eigenvector dimension reduction (EDR) method , 2009 .

[9]  Olivier P. Le Maître,et al.  Polynomial chaos expansion for sensitivity analysis , 2009, Reliab. Eng. Syst. Saf..

[10]  Jun Xu,et al.  A new unequal-weighted sampling method for efficient reliability analysis , 2018, Reliab. Eng. Syst. Saf..

[11]  Adam Hapij,et al.  Refined Stratified Sampling for efficient Monte Carlo based uncertainty quantification , 2015, Reliab. Eng. Syst. Saf..

[12]  Ding Wang,et al.  Structural reliability analysis based on polynomial chaos, Voronoi cells and dimension reduction technique , 2019, Reliab. Eng. Syst. Saf..

[13]  Chao Hu,et al.  A comparative study of probability estimation methods for reliability analysis , 2012 .

[14]  Pol D. Spanos,et al.  Stochastic Finite Element Method: Response Statistics , 1991 .

[15]  Wei Chen,et al.  A non‐stationary covariance‐based Kriging method for metamodelling in engineering design , 2007 .

[16]  Xiaoping Du,et al.  Reliability Analysis With Monte Carlo Simulation and Dependent Kriging Predictions , 2016 .

[17]  Hao Zhang,et al.  A Multiwavelet Neural Network‐Based Response Surface Method for Structural Reliability Analysis , 2015, Comput. Aided Civ. Infrastructure Eng..

[18]  A. Doostan,et al.  Least squares polynomial chaos expansion: A review of sampling strategies , 2017, 1706.07564.

[19]  Xiaoping Du,et al.  Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design , 2004, DAC 2002.

[20]  B. Youn,et al.  Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis , 2008 .

[21]  S. Rahman,et al.  A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics , 2004 .

[22]  A. Sudjianto,et al.  First-order saddlepoint approximation for reliability analysis , 2004 .

[23]  Meng Li,et al.  A High-Dimensional Reliability Analysis Method for Simulation-Based Design under Uncertainty , 2018 .

[24]  Wei Chen,et al.  A Most Probable Point-Based Method for Efficient Uncertainty Analysis , 2001 .

[25]  Chao Dang,et al.  Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods , 2017 .

[26]  Georgios Karagiannis,et al.  Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs , 2014, J. Comput. Phys..

[27]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[28]  Chao Hu,et al.  A Copula-based sampling method for data-driven prognostics and health management , 2013, 2013 IEEE Conference on Prognostics and Health Management (PHM).

[29]  M. Pandey,et al.  Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method , 2013 .

[30]  Francesco Contino,et al.  A robust and efficient stepwise regression method for building sparse polynomial chaos expansions , 2017, J. Comput. Phys..

[31]  Jun Xu,et al.  A cubature collocation based sparse polynomial chaos expansion for efficient structural reliability analysis , 2018, Structural Safety.

[32]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[33]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[34]  Lei Wang,et al.  REIF: A novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis , 2019, Reliab. Eng. Syst. Saf..

[35]  Gang Li,et al.  Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method , 2019, Journal of Mechanical Design.

[36]  John Dalsgaard Sørensen,et al.  AKOIS: An adaptive Kriging oriented importance sampling method for structural system reliability analysis , 2020 .

[37]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[38]  N. Wiener.  The Homogeneous Chaos , 1938 .

[39]  Zhenzhou Lu,et al.  Sparse polynomial chaos expansion based on D-MORPH regression , 2018, Appl. Math. Comput..

[40]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[41]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[42]  Jianbin Guo,et al.  Structural reliability analysis based on analytical maximum entropy method using polynomial chaos expansion , 2018, Structural and Multidisciplinary Optimization.

[43]  Xu Han,et al.  Kinematic Reliability Analysis of Robotic Manipulator , 2020, Journal of Mechanical Design.

[44]  Pan Wang,et al.  Efficient structural reliability analysis method based on advanced Kriging model , 2015 .

[45]  Xu Han,et al.  A Moment Approach to Positioning Accuracy Reliability Analysis for Industrial Robots , 2020, IEEE Transactions on Reliability.

[46]  Zhenzhou Lu,et al.  Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression , 2018 .

[47]  Thierry A. Mara,et al.  Bayesian sparse polynomial chaos expansion for global sensitivity analysis , 2017 .

[48]  Jie Liu,et al.  Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions , 2018, Computer Methods in Applied Mechanics and Engineering.

[49]  Bruno Sudret,et al.  Efficient computation of global sensitivity indices using sparse polynomial chaos expansions , 2010, Reliab. Eng. Syst. Saf..

[50]  Tao Zhou,et al.  A Christoffel function weighted least squares algorithm for collocation approximations , 2014, Math. Comput..

[51]  Wei Chen,et al.  Confidence-based adaptive extreme response surface for time-variant reliability analysis under random excitation , 2017 .

[52]  Gang Li,et al.  An improved maximum entropy method via fractional moments with Laplace transform for reliability analysis , 2018, Structural and Multidisciplinary Optimization.

[53]  Xiaoping Du,et al.  Uncertainty Analysis by Dimension Reduction Integration and Saddlepoint Approximations , 2005, DAC 2005.

[54]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[55]  Zeng Meng,et al.  A novel experimental data-driven exponential convex model for reliability assessment with uncertain-but-bounded parameters , 2020 .

[56]  Han Wang,et al.  Evaluating Influence of Variable Renewable Energy Generation on Islanded Microgrid Power Flow , 2018, IEEE Access.

[57]  Alireza Doostan,et al.  Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression , 2014, 1410.1931.

[58]  Jie Liu,et al.  Time-Variant Reliability Analysis through Response Surface Method , 2017 .

[59]  Masoud Rais-Rohani,et al.  Reliability estimation using univariate dimension reduction and extended generalised lambda distribution , 2010 .

[60]  Hao Hu,et al.  Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization , 2018, Applied Mathematical Modelling.

[61]  Gang Li,et al.  An active weight learning method for efficient reliability assessment with small failure probability , 2020 .

[62]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[63]  Zeng Meng,et al.  Adaptive conjugate single-loop method for efficient reliability-based design and topology optimization , 2019, Computer Methods in Applied Mechanics and Engineering.

[64]  Zhenzhou Lu,et al.  Multi-level multi-fidelity sparse polynomial chaos expansion based on Gaussian process regression , 2019, Computer Methods in Applied Mechanics and Engineering.

[65]  Xufang Zhang,et al.  Efficient Computational Methods for Structural Reliability and Global Sensitivity Analyses , 2013 .

[66]  Yan Zeng,et al.  A novel structural reliability analysis method via improved maximum entropy method based on nonlinear mapping and sparse grid numerical integration , 2019, Mechanical Systems and Signal Processing.

[67]  K. Rasmussen,et al.  Wavelet density-based adaptive importance sampling method , 2015 .

[68]  I. Sobol,et al.  About the use of rank transformation in sensitivity analysis of model output , 1995 .