小波Galerkin法在非线性分岔问题求解中的应用

张磊; 唐从刚; 王德全; 刘冰

doi:10.21656/1000-0887.410085

小波Galerkin法在非线性分岔问题求解中的应用

doi: 10.21656/1000-0887.410085

¹国防科技大学空天科学学院, 长沙410073;²湖北三江航天集团红阳机电有限公司, 湖北孝感 432000

基金项目: 湖南省自然科学基金（2019JJ50735）

详细信息

作者简介:
张磊（1989—），男，讲师，博士(通讯作者. E-mail: leizhg2016@163.com).

中图分类号: O302
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出版历程
- 收稿日期: 2020-03-25
- 修回日期: 2020-05-06
- 刊出日期: 2021-01-01

Application of the Wavelet Galerkin Method to Solution of Nonlinear Bifurcation Problems

¹College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, P.R.China;²Hubei Sanjiang Aerospace Hongyang Electromechanical Co. Ltd., Xiaogan, Hubei 432000, P.R.China

摘要

摘要: 通过一个典型的Bratu问题，研究了小波Galerkin法（WGM）在非线性分岔问题求解方面的应用.首先，利用基于Coiflet的小波Galerkin法，对一维和二维Bratu方程进行离散；然后针对单参数问题，推导了追踪解曲线的伪弧长格式和直接计算极值型分岔点的扩展方程；针对双参数问题，推导了追踪稳定边界的伪弧长格式和求解尖点型分岔点的扩展方程.数值结果表明，基于小波Galerkin法的非线性分岔计算不仅具有更高的计算精度，而且能够有效地捕捉双参数分岔问题的折迭线和尖点突变曲面.该算例展示了基于小波Galerkin法的数值分岔计算的具体过程及其求解多参数分岔问题复杂行为的应用潜力.
- 小波Galerkin法 /
- 分岔计算 /
- 双参数问题 /
- 尖点突变
Abstract: Application of the wavelet Galerkin method to solution of nonlinear bifurcation problems was studied through a typical Bratu problem. Firstly, 1D and 2D Bratu equations were discretized with the Coiflet based wavelet Galerkin method, then both the pseudo arclength scheme for tracing solution curves and the extended equations for calculating limit bifurcation points were derived in the case of 1parameter Bratu problems, similarly both the pseudo arclength scheme for tracing solution surfaces and the extended equations for solving cusp bifurcation points were also derived in the case of 2parameter Bratu problems. Numerical results show that, the wavelet Galerkin method not only has higher accuracy during bifurcation point calculation, but also is capable of capturing fold lines and cusp catastrophe quantitatively in the case of 2parameter bifurcation problems. This example exhibits the specific procedure of numerical bifurcation analysis based on the wavelet Galerkin method and demonstrates its potential for capturing complex bifurcation behaviors of multiparameter problems.
- wavelet Galerkin method /
- bifurcation computation /
- 2parameter problem /
- cusp catastrophe

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参考文献(20)

[1]	陈予恕, 唐云. 非线性动力学中的现代分析方法[M]. 北京: 科学出版社, 1992.(CHEN Yushu, TANG Yun. Modern Analytical Methods in Nonlinear Dynamics [M]. Beijing: Science Press, 1992.(in Chinese))
[2]	SEYDEL R. Practical Bifurcation and Stability Analysis [M]. New York: Springer, 2009.
[3]	季海波, 武际可. 分叉问题及其数值方法[J]. 力学进展, 1993,23(4): 493-502.(JI Haibo, WU Jike. Bifurcation and its numerical methods[J]. Advances in Mechancis,1993,23(4): 493-502.(in Chinese))
[4]	ROOSE D, PIESSENS R. Numerical computation of non-simple turning points and cusps[J]. Numerische Mathematik,1985,46(2): 189-211.
[5]	陈仲英, 巫斌. 小波分析[M]. 北京: 科学出版社, 2007.(CHEN Zhongying, WU Bin. Wavelet Analysis [M]. Beijing: Science Press, 2007.(in Chinese))
[6]	KRISHNAN J, RUNBORG O, KEVREKIDIS I G. Bifurcation analysis of nonlinear reaction-diffusion problems using wavelet: based reduction techniques[J]. Computers & Chemical Engineering,2004,28(4): 557-574.
[7]	张磊. 高精度小波数值方法及其在结构非线性分析中的应用[D]. 博士学位论文. 兰州: 兰州大学, 2016.(ZHANG Lei. High-precision wavelet numerical methods and applications to nonlinear structural analysis[D]. PhD Thesis. Lanzhou: Lanzhou University, 2016.(in Chinese))
[8]	刘小靖, 王记增, 周又和. 一种适用于强非线性结构力学问题数值求解的修正小波伽辽金方法[J]. 固体力学学报, 2011,32(3): 249-257.(LIU Xiaojing, WANG Jizeng, ZHOU Youhe. A modified wavelet Galerkin method for computations in structural mechancis with strong nonlinearity[J]. Chinese Journal of Solid Mechanics,2011,32(3): 249-257.(in Chinese))
[9]	LIU X J, ZHOU Y H, WANG X M, et al. A wavelet method for solving a class of nonlinear boundary value problems[J]. Communications in Nonlinear Science and Numerical Simulation,2013,18(8): 1939-1948.
[10]	WANG X M, LIU X J, WANG J Z, et al. A wavelet method for bending of circular plate with large defelction[J]. Acta Mech Solida Sinica,2015,28(1): 83-90.
[11]	ZHANG L, WANG J Z, ZHOU Y H. Wavelet solution for large deflection bending problems of thin rectangular plates[J]. Archive of Applied Mechanics,2015,85(3): 355-365.
[12]	MA X L, WU B, ZHANG J H, et al. A new numerical scheme with wavelet-Galerkin followed by spectral deferred correction for solving string vibration problems[J]. Mechanism and Machine Theory,2019,142(12): 103623.
[13]	ANDERSON C A, ZIENKIEWICZ O C. Spontanenous ignition: finite element solutions for steady state and transient conditions[J]. Journal of Heat Transfer,1974,96(3): 398-404.
[14]	KAPANIA R K. A pseudo-spectral solution of 2-parameter Bratu’s equation[J]. Computational Mechanics,1990,6(1): 55-63.
[15]	KARKOWSKI J. Numerical experiments with the Bratu equation in one, two and three dimensions[J]. Computational and Applied Mathematics,2013,32(1): 231-244.
[16]	洪文强, 徐绩青, 许锡宾, 等. 求解Bratu型方程的径向基函数逼近法[J]. 应用数学和力学, 2016,37(6): 617-625.(HONG Wenqiang, XU Jiqing, XU Xibin, et al. The radial basis function approximation method for solving Bratu-type equations[J]. Applied Mathematics and Mechanics,2016,37(6): 617-625.(in Chinese))
[17]	BODDINGTON T, FENG C G, GRAY P. Thermal explosion and the theory of its initiation by steady intense light[J]. Proceedings of the Royal Society A: Mathematical,1983,390(1799): 265-281.
[18]	GREENWAY P, SPENCE A. Numerical calculation of critical points for a slab with partial insulation[J]. Combust Flame,1985,62(2): 141-156.
[19]	CLIFFE K A, HALL E, HOUSTON P, et al. Adaptivity and a posteriori error control for bifurcation problems Ⅰ: the Bratu problem[J]. Communications in Computational Physics,2010,8(4): 845-865.
[20]	王记增. 正交小波统一理论与方法及其在压电智能结构等力学研究中的应用[D]. 博士学位论文. 兰州: 兰州大学, 2001.(WANG Jizeng. Generalized theory and arithmetic of orthogonal wavelets and applications to researches of mechanics including piezoeiectric smart structures[D]. PhD Thesis. Lanzhou: Lanzhou University, 2001.(in Chinese))