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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 PublicationCOMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
ISSN0045-7825
Volume340Pages:340-365
AbstractA 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.
KeywordHomogenization method Multi-scale asymptotic expansion Eigenvalue problems Finite element approximation Coordinate transformation
DOI10.1016/j.cma.2018.05.035
Language英语
Funding ProjectNational 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 AreaEngineering ; Mathematics ; Mechanics
WOS SubjectEngineering, Multidisciplinary ; Mathematics, Interdisciplinary Applications ; Mechanics
WOS IDWOS:000442385600016
PublisherELSEVIER SCIENCE SA
Citation statistics
Cited Times:1[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/31087
Collection计算数学与科学工程计算研究所
Affiliation1.Sichuan Univ, Coll Math, Chengdu 610043, Sichuan, Peoples R China
2.China Aerodynam Res & Dev Ctr, Hyperveloc Aerodynam Inst, Mianyang 621000, Peoples R China
3.BUAA, Natl Lab Computat Fluid Dynam, Beijing 100191, Peoples R China
4.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
Recommended Citation
GB/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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