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 Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates Ma, Qiang1; Li, Zhihui2,3; Cui, Junzhi4 2018-10-01 Source Publication COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING ISSN 0045-7825 Volume 340Pages:340-365 Abstract A novel second-order two-scale asymptotic method is presented for the eigenvalue problems of the second-order elliptic operator in the general composite domain. The eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation, and by applying the asymptotic expansion technique, the eigenfunctions of the system are expanded to the second-order terms. Using the argument of the so-called "corrector equations", the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain. The feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates. Various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations, and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated. The finite element algorithm is developed and three numerical experiments are carried out, which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures. It is also indicated that the second-order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions. By the coordinate transformation, the asymptotic analysis method can be generalized to more general composite domain with quasi-periodic and non-periodic configurations. (C) 2018 Elsevier B.V. All rights reserved. Keyword Homogenization method Multi-scale asymptotic expansion Eigenvalue problems Finite element approximation Coordinate transformation DOI 10.1016/j.cma.2018.05.035 Language 英语 Funding Project National Key Basic Research and Development Program[2014CB744100] ; National Nature Science Foundation of China[11325212] ; National Nature Science Foundation of China[91530319] ; China Postdoctoral Science Foundation[2016T91019] ; Fundamental Research Funds for the central Universities WOS Research Area Engineering ; Mathematics ; Mechanics WOS Subject Engineering, Multidisciplinary ; Mathematics, Interdisciplinary Applications ; Mechanics WOS ID WOS:000442385600016 Publisher ELSEVIER SCIENCE SA Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/31087 Collection 计算数学与科学工程计算研究所 Corresponding Author Li, Zhihui Affiliation 1.Sichuan Univ, Coll Math, Chengdu 610043, Sichuan, Peoples R China2.China Aerodynam Res & Dev Ctr, Hyperveloc Aerodynam Inst, Mianyang 621000, Peoples R China3.BUAA, Natl Lab Computat Fluid Dynam, Beijing 100191, Peoples R China4.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China Recommended CitationGB/T 7714 Ma, Qiang,Li, Zhihui,Cui, Junzhi. Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates[J]. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING,2018,340:340-365. APA Ma, Qiang,Li, Zhihui,&Cui, Junzhi.(2018).Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates.COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING,340,340-365. MLA Ma, Qiang,et al."Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates".COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 340(2018):340-365.
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