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 Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials Ma, Qiang1; Ye, Shuyu1; Cui, Junzhi2; Yang, Zhiqiang3; Jiang, Xue4; Li, Zhihui5,6 2021-04-01 Source Publication APPLIED MATHEMATICAL MODELLING ISSN 0307-904X Volume 92Pages:565-593 Abstract 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. Keyword 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 DOI 10.1016/j.apm.2020.11.028 Indexed By SCI Language 英语 Funding Project 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 Research Area Engineering ; Mathematics ; Mechanics WOS Subject Engineering, Multidisciplinary ; Mathematics, Interdisciplinary Applications ; Mechanics WOS ID WOS:000617130300005 Publisher ELSEVIER SCIENCE INC Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/58201 Collection 中国科学院数学与系统科学研究院 Corresponding Author Yang, Zhiqiang Affiliation 1.Sichuan Univ, Sch Math, Chengdu 610041, Peoples R China2.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China3.Harbin Inst Technol, Dept Astronaut Sci & Mech, Harbin 150001, Peoples R China4.Beijing Univ Technol, Fac Sci, Sch Math, Beijing 100124, Peoples R China5.China Aerodynam Res & Dev Ctr, Hypervelocity Aerodynam Inst, Mianyang 621000, Sichuan, Peoples R China6.BUAA, Natl Lab Computat Fluid Dynam, Beijing 100191, Peoples R China Recommended CitationGB/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.
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