Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials

doi:10.1016/j.apm.2020.11.028

	Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials
	Ma, Qiang 1; Ye, Shuyu 1; Cui, Junzhi 6; Yang, Zhiqiang 5; Jiang, Xue 4; Li, Zhihui 2,3
刊名	APPLIED MATHEMATICAL MODELLING
	2021-04-01
卷号	92 页码:565-593
关键词	Hierarchically perforated structure Successive two-scale asymptotic analysis Neumann-type elliptic eigenvalue problem Second-order two-scale and three-scale asymptotic computations Finite element simulation
ISSN号	0307-904X
DOI	10.1016/j.apm.2020.11.028
英文摘要	A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the firstand second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of "corrector equation", the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells. (c) 2020 Elsevier Inc. All rights reserved.
资助项目	National Natural Science Foundation of China[11801387] ; National Natural Science Foundation of China[11971336] ; National Natural Science Foundation of China[11971337] ; National Natural Science Foundation of China[11701123] ; National Natural Science Foundation of China[11771057] ; National Natural Science Foundation of China[11671052] ; State Key Laboratory of Science and Engineering Computing ; Fundamental Research Funds for the Central Universities
WOS研究方向	Engineering ; Mathematics ; Mechanics
语种	英语
出版者	ELSEVIER SCIENCE INC
WOS记录号	WOS:000617130300005
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/58200]
专题	中国科学院数学与系统科学研究院
通讯作者	Yang, Zhiqiang
作者单位	1.Sichuan Univ, Sch Math, Chengdu 610041, Peoples R China 2.BUAA, Natl Lab Computat Fluid Dynam, Beijing 100191, Peoples R China 3.China Aerodynam Res & Dev Ctr, Hypervelocity Aerodynam Inst, Mianyang 621000, Sichuan, Peoples R China 4.Beijing Univ Technol, Fac Sci, Sch Math, Beijing 100124, Peoples R China 5.Harbin Inst Technol, Dept Astronaut Sci & Mech, Harbin 150001, Peoples R China 6.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Ma, Qiang,Ye, Shuyu,Cui, Junzhi,et al. Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials[J]. APPLIED MATHEMATICAL MODELLING,2021,92:565-593.
APA	Ma, Qiang,Ye, Shuyu,Cui, Junzhi,Yang, Zhiqiang,Jiang, Xue,&Li, Zhihui.(2021).Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials.APPLIED MATHEMATICAL MODELLING,92,565-593.
MLA	Ma, Qiang,et al."Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials".APPLIED MATHEMATICAL MODELLING 92(2021):565-593.