A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

doi:10.1007/s10483-020-2614-7

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (6): 967-982.doi: https://doi.org/10.1007/s10483-020-2614-7

• 论文 • 上一篇

A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Lincong CHEN^1,2, J. Q. SUN³

1. College of Civil Engineering, Huaqiao University, Xiamen 361021, Fujian Province, China;
2. Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province, Jimei Avenue 668, Xiamen 361021, Fujian Province, China;
3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A.

收稿日期:2020-02-02 修回日期:2020-03-09 发布日期:2020-06-08
通讯作者: J. Q. SUN E-mail:jqsun@ucmerced.edu
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11672111, 11332008, 11572215, and 11602089), the Program for New Century Excellent Talents in Fujian Province's University, the Natural Science Foundation of Fujian Province of China (No. 2019J01049), and the Scholarship for Overseas Studies from Fujian Province of China

A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Lincong CHEN^1,2, J. Q. SUN³

1. College of Civil Engineering, Huaqiao University, Xiamen 361021, Fujian Province, China;
2. Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province, Jimei Avenue 668, Xiamen 361021, Fujian Province, China;
3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A.

Received:2020-02-02 Revised:2020-03-09 Published:2020-06-08
Contact: J. Q. SUN E-mail:jqsun@ucmerced.edu
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11672111, 11332008, 11572215, and 11602089), the Program for New Century Excellent Talents in Fujian Province's University, the Natural Science Foundation of Fujian Province of China (No. 2019J01049), and the Scholarship for Overseas Studies from Fujian Province of China

摘要/Abstract

摘要： Analytical and numerical studies of multi-degree-of-freedom (MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge. This paper presents a highly-efficient method for determining the stationary probability density functions (PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov (FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in a higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation (MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS.

关键词: stationary response, multi-degree-of-freedom (MDOF) nonlinear system, Fokker-Planck-Kolmogorov (FPK) equation, least square method

Abstract: Analytical and numerical studies of multi-degree-of-freedom (MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge. This paper presents a highly-efficient method for determining the stationary probability density functions (PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov (FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in a higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation (MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS.

Key words: stationary response, multi-degree-of-freedom (MDOF) nonlinear system, Fokker-Planck-Kolmogorov (FPK) equation, least square method

中图分类号:

Lincong CHEN, J. Q. SUN. A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(6): 967-982.

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A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

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