{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:07Z","timestamp":1760148967714,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,22]],"date-time":"2023-06-22T00:00:00Z","timestamp":1687392000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key of R&amp;D Program of China","award":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"],"award-info":[{"award-number":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"]}]},{"name":"National Natural Science Foundation of China","award":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"],"award-info":[{"award-number":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"]}]},{"DOI":"10.13039\/501100018542","name":"Sichuan Natural Science Foundation","doi-asserted-by":"publisher","award":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"],"award-info":[{"award-number":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"publisher"}]},{"name":"State Key Laboratory of Science and Engineering Computing and the Fundamental Research Funds for the Central Universities","award":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"],"award-info":[{"award-number":["2019YFA040520X","11801387","11971336","11971337","2022NSFSC0322","YJ201811"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.<\/jats:p>","DOI":"10.3390\/sym15071299","type":"journal-article","created":{"date-parts":[[2023,6,23]],"date-time":"2023-06-23T01:54:56Z","timestamp":1687485296000},"page":"1299","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multiscale Asymptotic Computations for the Elastic Quadratic Eigenvalue Problem in Periodically Composite Structure"],"prefix":"10.3390","volume":"15","author":[{"given":"Qiang","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics, Sichuan University, Chengdu 610041, China"}]},{"given":"Hongyu","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sichuan University, Chengdu 610041, China"}]},{"given":"Lin","family":"Bi","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research and Development Center, Mianyang 621000, China"}]},{"given":"Junzhi","family":"Cui","sequence":"additional","affiliation":[{"name":"LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing 100190, China"}]},{"given":"Tingyan","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sichuan University, Chengdu 610041, China"}]},{"given":"Yuting","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sichuan University, Chengdu 610041, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/0022-247X(68)90115-7","article-title":"Quadratic eigenvalue problems","volume":"23","author":"Eisenfeld","year":"1968","journal-title":"J. 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