A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

doi:10.1007/s10483-020-2604-6

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (5): 769-784.doi: https://doi.org/10.1007/s10483-020-2604-6

A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

Zigang LI¹, Jun JIANG², Ling HONG², J. Q. SUN³

1. Department of Mechanics, School of Science, Xi'an University of Science and Technology, Xi'an 710054, China;
2. State Key Laboratory for Strength and Vibration, Xi'an Jiaotong University, Xi'an 710049, China;
3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A.

收稿日期:2020-01-02 修回日期:2020-02-19 发布日期:2020-04-20
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. 11702213, 11772243, 11572215, and 11332008) and the Natural Science Foundation of Shaanxi Province of China (No. 2018JQ1061)

A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

Zigang LI¹, Jun JIANG², Ling HONG², J. Q. SUN³

1. Department of Mechanics, School of Science, Xi'an University of Science and Technology, Xi'an 710054, China;
2. State Key Laboratory for Strength and Vibration, Xi'an Jiaotong University, Xi'an 710049, China;
3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A.

Received:2020-01-02 Revised:2020-02-19 Published:2020-04-20
Contact: J. Q. SUN E-mail:jqsun@ucmerced.edu
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11702213, 11772243, 11572215, and 11332008) and the Natural Science Foundation of Shaanxi Province of China (No. 2018JQ1061)

摘要/Abstract

摘要： A subspace expanding technique (SET) is proposed to efficiently discover and find all zeros of nonlinear functions in multi-degree-of-freedom (MDOF) engineering systems by discretizing the space into smaller subdomains, which are called cells. The covering set of the cells is identified by parallel calculations with the root bracketing method. The covering set can be found first in a low-dimensional subspace, and then gradually extended to higher dimensional spaces with the introduction of more equations and variables into the calculations. The results show that the proposed SET is highlyefficient for finding zeros in high-dimensional spaces. The subdivision technique of the cell mapping method is further used to refine the covering set, and the obtained numerical results of zeros are accurate. Three examples are further carried out to verify the applicability of the proposed method, and very good results are achieved. It is believed that the proposed method will significantly enhance the ability to study the stability, bifurcation, and optimization problems in complex MDOF nonlinear dynamic systems.

关键词: spatial discretization, subspace expanding technique (SET), parallel computing, subdivision, global zero finding

Abstract: A subspace expanding technique (SET) is proposed to efficiently discover and find all zeros of nonlinear functions in multi-degree-of-freedom (MDOF) engineering systems by discretizing the space into smaller subdomains, which are called cells. The covering set of the cells is identified by parallel calculations with the root bracketing method. The covering set can be found first in a low-dimensional subspace, and then gradually extended to higher dimensional spaces with the introduction of more equations and variables into the calculations. The results show that the proposed SET is highlyefficient for finding zeros in high-dimensional spaces. The subdivision technique of the cell mapping method is further used to refine the covering set, and the obtained numerical results of zeros are accurate. Three examples are further carried out to verify the applicability of the proposed method, and very good results are achieved. It is believed that the proposed method will significantly enhance the ability to study the stability, bifurcation, and optimization problems in complex MDOF nonlinear dynamic systems.

Key words: spatial discretization, subspace expanding technique (SET), parallel computing, subdivision, global zero finding

中图分类号:

Zigang LI, Jun JIANG, Ling HONG, J. Q. SUN. A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(5): 769-784.

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A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems

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