A Chaotic Response Surface Method for Non-Gaussian Stochastic Analysis of Structural Responses

LI Zhaoyang; XU Yujiao; YANG Lufeng

doi:10.21656/1000-0887.450093

Volume 46 Issue 7

Jul. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(7): 855-866

LI Zhaoyang, XU Yujiao, YANG Lufeng. A Chaotic Response Surface Method for Non-Gaussian Stochastic Analysis of Structural Responses[J]. Applied Mathematics and Mechanics, 2025, 46(7): 855-866. doi: 10.21656/1000-0887.450093

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A Chaotic Response Surface Method for Non-Gaussian Stochastic Analysis of Structural Responses

doi: 10.21656/1000-0887.450093

LI Zhaoyang^{1, 2
,},
XU Yujiao^1
,,
YANG Lufeng^{1
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,}

1.
School of Civil Engineering and Architecture, Guangxi University, Nanning 530004, P.R.China
2.
Guangxi Polytechnic of Construction, Nanning 530003, P.R.China

Received Date: 2024-04-10
Rev Recd Date: 2024-12-31

Available Online: 2025-07-30

Publish Date: 2025-07-01

Abstract

Abstract

A chaotic response surface method was proposed to improve the computational efficiency and accuracy of the traditional response surface method for stochastic analysis of structural responses involving non-Gaussian random variables. Firstly, the non-Gaussian response variable was expanded by a hybrid generalized polynomial chaos constructed according to the probability distribution function types of the basic random variables. Secondly, the candidate probability collocation points in the non-Gaussian probability space were determined through the combination of the roots of the 1D generalized polynomial chaos with the next higher order, then the probability optimal collocation points in the non-Gaussian probability space were picked out based on the full row rank principle of the coefficient matrix. Finally, the unknown coefficients of the proposed response surface were determined by means of the least squares method. Comparison of examples shows that, the proposed method requires fewer collocation points and lower expansion orders, and achieves higher calculation accuracy and efficiency than those of the traditional response surface methods.
- response surface method,
- generalized polynomial chaos,
- non-Gaussian,
- optimal probability collocation point

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References(28)

References

[1]	ZHAO Z, LU Z H, ZHAO Y G. Simulating multivariate stationary non-Gaussian process based on wavenumber-frequency spectrum and unified Hermite polynomial model[J]. Probabilistic Engineering Mechanics, 2022, 69: 103272.
[2]	GHANEM R G, SPANOS P D. Spectral stochastic finite-element formulation for reliability analysis[J]. Journal of Engineering Mechanics, 1991, 117(10): 2351-2372.
[3]	LU N, LI Y F, HUANG H Z, et al. AGP-MCS+D: an active learning reliability analysis method combining dependent Gaussian process and Monte Carlo simulation[J]. Reliability Engineering & System Safety, 2023, 240: 109541.
[4]	LI X, GONGC L, GU L X, et al. A sequential surrogate method for reliability analysis based on radial basis function[J]. Structural Safety, 2018, 73: 42-53.
[5]	LI J, CHEN J. Stochastic Dynamics of Structures[M]. New York: John Wiley & Sons, 2009.
[6]	CHENG H, YANG D X. Direct probability integral method for stochastic response analysis of static and dynamic structural systems[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 357: 112612.
[7]	BUCHER C G, BOURGUND U. A fast and efficient response surface approach for structural reliability problems[J]. Structural Safety, 1990, 7(1): 57-66.
[8]	ISUKAPALLI S S, ROY A, GEORGOPOULOS P G. Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems[J]. Risk Analysis, 1998, 18(3): 351-363.
[9]	HAMPTON J, DOOSTAN A. Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies[J]. Journal of Computational Physics, 2015, 280: 363-386.
[10]	SU G, YU B, XIAO Y, et al. Gaussian process machine-learning method for structural reliability analysis[J]. Advances in Structural Engineering, 2014, 17(9): 1257-1270.
[11]	DENG J, GU D, LI X, et al. Structural reliability analysis for implicit performance functions using artificial neural network[J]. Structural Safety, 2005, 27(1): 25-48.
[12]	陶然, 周焕林, 孟增, 等. 基于响应面法和改进算术优化算法的抱杆优化设计[J]. 应用数学和力学, 2022, 43(10): 1113-1122. doi: 10.21656/1000-0887.420318 TAO Ran, ZHOU Huanlin, MENG Zeng, et al. Optimization design of holding poles based on the response surface methodology and the improved arithmetic optimization algorithm[J]. Applied Mathematics and Mechanics, 2022, 43(10): 1113-1122. (in Chinese) doi: 10.21656/1000-0887.420318
[13]	王涵, 史治宇. 最优阶次多项式响应面法及其在模型修正中的应用[J]. 机械制造与自动化, 2021, 50(6): 93-98. WANG Han, SHI Zhiyu. Construction of response surface with optimal order polynomial and its application in FEM updating[J]. Machine Building & Automation, 2021, 50(6): 93-98. (in Chinese)
[14]	HUANG S, MAHADEVAN S, REBBA R. Collocation-based stochastic finite element analysis for random field problems[J]. Probabilistic Engineering Mechanics, 2007, 22(2): 194-205.
[15]	LI D Q, CHEN Y F, LU W B, et al. Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables[J]. Computers and Geotechnics, 2011, 38(1): 58-68.
[16]	XIU D, EM KARNIADAKIS G. The Wiener-Askey polynomial chaos for stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2002, 24(2): 619-644.
[17]	JAKEMAN J D, FRANZELIN F, NARAYAN A, et al. Polynomial chaos expansions for dependent random variables[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 351: 643-666.
[18]	姜昌伟, 刘星, 石尔, 等. 基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化[J]. 应用数学和力学, 2021, 42(5): 481-491. doi: 10.21656/1000-0887.410217 JIANG Changwei, LIU Xing, SHI Er, et al. Uncertainty quantification of flow and heat transfer problems with stochastic boundary conditions based on the intrusive polynomial chaos expansion method[J]. Applied Mathematics and Mechanics, 2021, 42(5): 481-491. (in Chinese) doi: 10.21656/1000-0887.410217
[19]	SUDRET B. Global sensitivity analysis using polynomial chaos expansions[J]. Reliability Engineering & System Safety, 2008, 93(7): 964-979.
[20]	CHENG K, LU Z Z. Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression[J]. Computers & Structures, 2018, 194: 86-96.
[21]	吕大刚. 基于线性化Nataf变换的一次可靠度方法[J]. 工程力学, 2007, 24(5): 79-86. LV Dagang. First order reliability method based on linearized Nataf transformation[J]. Engineering Mechanics, 2007, 24(5): 79-86. (in Chinese)
[22]	LU Z H, ZHAO Z, ZHANG X Y, et al. Simulating stationary non-Gaussian processes based on unified Hermite polynomial model[J]. Journal of Engineering Mechanics, 2020, 146(7): 04020067.
[23]	GAUTSCHI W. Orthogonal polynomials (in MATLAB)[J]. Journal of Computational and Applied Mathematics, 2005, 178(1/2): 215-234.
[24]	HADIGOL M, DOOSTAN A. Least squares polynomial chaos expansion: a review of sampling strategies[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 382-407.
[25]	YANG L F, LI Z Y, YU B, et al. Full-space response surface method for analysis of structural reliability[J]. International Journal of Structural Stability and Dynamics, 2020, 20(8): 2050096.
[26]	LI W X, LU Z M, ZHANG D X. Stochastic analysis of unsaturated flow with probabilistic collocation method[J]. Water Resources Research, 2009, 45(8): W08425.
[27]	蒋水华, 李典庆, 周创兵. 随机响应面法最优概率配点数目分析[J]. 计算力学学报, 2012, 29(3): 345-351. JIANG Shuihua, LI Dianqing, ZHOU Chuangbing. Optimal probabilistic collocation points for stochastic response surface method[J]. Chinese Journal of Computational Mechanics, 2012, 29(3): 345-351. (in Chinese)
[28]	杨绿峰, 杨显峰, 余波, 等. 基于逐步回归分析的随机响应面法[J]. 计算力学学报, 2013, 30(1): 88-93. YANG Lufeng, YANG Xianfeng, YU Bo, et al. Improved stochastic response surface method based on stepwise regression analysis[J]. Chinese Journal of Computational Mechanics, 2013, 30(1): 88-93. (in Chinese)