A numerical approximation method for nonlinear dynamic systems based on radial basis functions

LI Yan-ting; XU Xi-bin; ZHOU Shi-liang; XU Ji-qing

doi:10.3879/j.issn.1000-0887.2016.03.009

Volume 37 Issue 3

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LI Yan-ting, XU Xi-bin, ZHOU Shi-liang, XU Ji-qing. A numerical approximation method for nonlinear dynamic systems based on radial basis functions[J]. Applied Mathematics and Mechanics, 2016, 37(3): 311-318. doi: 10.3879/j.issn.1000-0887.2016.03.009

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A numerical approximation method for nonlinear dynamic systems based on radial basis functions

doi: 10.3879/j.issn.1000-0887.2016.03.009

School of River & Ocean Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R.China;National Engineering Research Center for Inland Waterway Ragulation, Key Laboratory of Hydraulic & Waterway Engineering of the Ministry of Education, Chongqing Jiaotong University, Chongqing 400074, P.R.China

Received Date: 2015-10-12
Rev Recd Date: 2015-11-22
Publish Date: 2016-03-15

Abstract

Abstract

The radial basis functions have the advantages of simple forms and isotropy. A new numerical method for solving the initial-value problems of nonlinear dynamic systems was constructed through combination of the idea of the radial basis function approximation and the weighted residual collocation point method. The advantages and disadvantages of several methods for the numerical solution of nonlinear dynamic systems were analyzed. Some practical numerical examples were given to compare the proposed method with the existing methods. The results show that the present method is easily applicable with good convergence and high accuracy.
- nonlinear dynamic system,
- initial-value problem,
- radial basis function,
- weighted residual collocation point method

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References(13)

References

[1]	胡海岩. 应用非线性动力学[M]. 北京: 航空工业出版社, 2000.（HU Hai-yan. Applied Nonlinear Transient Dynamical [M]. Beijing: Aviation Industry Press, 2000.（in Chinese））
[2]	凌复华. 非线性动力学系统的数值研究[M]. 上海: 上海交通大学出版社, 1989.（LING Fu-hua. Numerische Untersuchung Uichtlinearer Dynamischer System [M]. Shanghai: Shanghai Jiao Tong University Press, 1989.（in Chinese））
[3]	刘向军, 石磊, 徐旭常. 稠密气固两相流欧拉-拉格朗日法的研究现状[J]. 计算力学学报, 2007,24(2): 166-172.（LIU Xiang-jun, SHI Lei, XU Xu-chang. Activities of dense particle-gas two-phase flow modeling in Eulerian-Lagrangian approach[J]. Chinese Journal of Computational Mechanics,2007,24(2):166-172. （in Chinese））
[4]	刘石, 陈德祥, 冯永新, 徐自力, 郑李坤. 等几何分析的多重网格共轭梯度法[J]. 应用数学和力学, 2014,35(6): 630-639.（LIU Shi, CEHN De-xiang, FENG Yong-xin, XU Zi-li, ZHENG Li-kun. A multigrid preconditioned conjugate method for isogeometric analysis[J]. Applied Mathematics and Mechanics,2014,35(6): 630-639.（in Chinese））
[5]	陈全发, 肖爱国. Runge-Kutta-Nystrom方法的若干新性质[J]. 计算数学, 2008,30(2): 201-212. （CHEN Quan-fa, XIAO Ai-guo. Some new properties of Runge-Kutta-Nystrom methods[J]. Mathematic Numeric Sinica,2008,30(2): 201-212.（in Chinese））
[6]	樊文欣, 杨桂通, 岳文忠. 基于ADAMS的发动机动力学通用分析模型[J]. 应用基础与工程科学学报, 2009,17(S1): 172-178.（FAN Wen-xin, YANG Gui-tong, YUE Wen-zhong. The dynamic unicersal analysis model of engine based on ADAMS[J]. Journal of Basic Science and Engineering,2009,17(S1): 172-178.（in Chinese））
[7]	徐次达, 陈学潮, 郑瑞芬. 新计算力学加权残值法——原理、方法及应用[M]. 上海: 同济大学出版社, 1997.（XU Ci-da, CHEN Xue-chao, ZHENG Rui-fen. A New Computational Mechanics of Weighted Residual Method—Principle, Method and Application [M]. Shanghai: Tongji University Press, 1997.（in Chinese））
[8]	李鹏柱, 李风军, 李星, 周跃亭. 基于三次样条插值函数的非线性动力系统数值求解[J]. 应用数学和力学, 2015,36(8): 887-896.（LI Peng-zhu, LI Feng-jun, LI Xing, ZHOU Yue-ting. A numerical method for the solution to nonlinear dynamic systems based on cubic spline[J].Applied Mathematics and Mechanics,2015,36(8): 887-896.（in Chinese））
[9]	吴宗敏. 径向基函数、散乱数据拟合与无网格偏微分方程数值解[J]. 工程数学学报, 2002,19(2): 1-12.（WU Zong-min. Radial basis function scattered data interpolation and the meshless method of numerical solution of PDEs[J].Journal of Engineering Mathematics,2002,19(2): 1-12.（in Chinese））
[10]	马利敏. 径向基函数逼近中的若干理论、方法及其应用[D]. 博士学位论文. 上海: 复旦大学, 2009.（MA Li-min. Some theory, methods and application in RBF approaching[D]. PhD Thesis. Shanghai: Fudan University, 2009.（in Chinese））
[11]	陈文, 傅卓佳, 魏星. 科学与工程计算中的径向基函数方法[M]. 北京: 科学出版社, 2014.（CHEN Wen, FU Zhuo-jia, WEI Xing. The Radial Basis Function Methods in Science and Engineering Mathematics [M]. Beijing: Science Press, 2014.（in Chinese））
[12]	张雄, 刘岩. 无网格法[M]. 北京: 清华大学出版社, 2004.（ZHANG Xiong, LIU Yan. Meshless Method [M]. Beijing: Tsinghua University Press, 2004.（in Chinese））
[13]	徐绩青, 李正良, 吴林键. 基于径向基函数逼近的结构动力响应计算方法[J]. 应用数学和力学, 2014,35(5): 533-541.（XU Ji-qing, LI Zheng-liang, WU Lin-jian. A calculation method for structural dynamic responses based on the approximation theory of radial basis function[J]. Applied Mathematics and Mechanics,2014,35(5): 533-541.（in Chinese））