Second-order three-scale asymptotic analysis and algorithms for Steklov eigenvalue problems in composite domain with hierarchical cavities. (English) Zbl 1539.65163
The authors develop a second-order three-scale expansion to analyze Steklov eigenvalue problems for the hierarchically perforated structure. By considering three types of perforated domains, it is demonstrated numerically by a three-scale finite element algorithm that the three-scale asymptotic solutions corresponding to both the simple and multiple eigenvalues can exhibit correct distributions in each scale. The main difficulty lies in dealing with the coupling of the corresponding parameters due to the eigenvalue distribution so that new cell functions depending on two consecutive scales should be defined.
Reviewer: Zhiming Chen (Beijing)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
Steklov eigenvalue problems; composite perforated materials; three-scale asymptotic expansion; second-order expansion terms; multi-scale finite element algorithmReferences:
| [1] | Stekloff, M.: Sur les problmes fondamentaux de la physique math \(\grave{e}\) matique. Ann. Sci. \( \grave{E}\) c. Norm. Sup \(\grave{e}\) r. 19(1902), 455-490 (in French) · JFM 33.0800.01 |
| [2] | Hadjesfandiari, AR; Dargush, GF, Theory of boundary eigensolutions in engineering mechanics, J. Appl. Mech., 68, 101-108 (2001) · Zbl 1110.74466 · doi:10.1115/1.1331059 |
| [3] | Hadjesfandiari, AR; Dargush, GF, Boundary eigensolutions in elasticity. I. Theoretical development, Int. J. Solids Struct., 38, 6589-6625 (2001) · Zbl 1013.74006 · doi:10.1016/S0020-7683(01)00028-2 |
| [4] | Hadjesfandiari, AR; Dargush, GF, Boundary eigensolutions in elasticity. II. Application to computational mechanics, Int. J. Solids Struct., 40, 1001-1031 (2003) · Zbl 1087.74637 · doi:10.1016/S0020-7683(02)00586-3 |
| [5] | Bermudez, A.; Rodriguez, R.; Santamarina, D., A finite element solution of an added mass formulation for coupled fluid-solid vibrations, Numer. Math., 87, 201-227 (2000) · Zbl 0998.76046 · doi:10.1007/s002110000175 |
| [6] | Doumate, J.; Leadi, L.; Marcos, A., Asymmetric Steklov problems with sign-changing weights, J. Math. Anal. Appl., 425, 1004-1038 (2015) · Zbl 1433.35148 · doi:10.1016/j.jmaa.2015.01.002 |
| [7] | Andreev, AB; Todorov, TD, Isoparametric finite element approximation of a Steklov eigenvalue problem, IMA J. Numer. Anal., 24, 309-322 (2004) · Zbl 1069.65120 · doi:10.1093/imanum/24.2.309 |
| [8] | Armentano, MG; Padra, C., A posteriori error estimates for the Steklov eigenvalue problem, Appl. Numer. Math., 58, 593-601 (2008) · Zbl 1140.65078 · doi:10.1016/j.apnum.2007.01.011 |
| [9] | Bi, H.; Yang, YD, A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem, Appl. Math. Comput., 217, 23, 9669-9678 (2011) · Zbl 1222.65121 |
| [10] | Ma, YY; Sun, JG, Integral equation method for a Non-Selfadjoint Steklov Eigenvalue Problem, Commun. Comput. Phys., 31, 1546-1560 (2022) · Zbl 1486.65234 · doi:10.4208/cicp.OA-2022-0016 |
| [11] | Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North - Holland, Amsterdam (1978) · Zbl 0404.35001 |
| [12] | Bourget, J.F., Iria-Laboria. Numerical Experiments of the Homogenization Method for Operators with Periodic Coefficients. Springer-Verlag, Heidelberg (1979) |
| [13] | Lions, JL, Some Methods in the Mathematical Analysis of Systems and their Control (1981), Gordon and Breach, Beijing: Science Press, Gordon and Breach, Beijing · Zbl 0542.93034 |
| [14] | Cioranescu, D.; Donato, P., An Introduction to Homogenization (1999), New York: Oxford University Press, New York · Zbl 0939.35001 · doi:10.1093/oso/9780198565543.001.0001 |
| [15] | Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. North - Holland, Amsterdam (1992) · Zbl 0768.73003 |
| [16] | Allaire, G., Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 6, 1482-1518 (1992) · Zbl 0770.35005 · doi:10.1137/0523084 |
| [17] | Hou, TY; Wu, XH, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 1, 169-189 (1997) · Zbl 0880.73065 · doi:10.1006/jcph.1997.5682 |
| [18] | Abdulle, A., E, W.N., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1-87 (2012) · Zbl 1255.65224 |
| [19] | E, W.N.: Principles of Multiscale Modeling. Science Press, Beijing (2012) |
| [20] | Cui, JZ; Cao, LQ, Finite element method based on two-scale asymptotic analysis, Math. Numer. Sin., 1, 89-102 (1998) · Zbl 0910.65076 |
| [21] | Feng, YP; Cui, JZ, Multi-scale analysis and FE computation for the structure of composite materials with small periodicity configuration under condition of coupled thermo-elasticity, Int. J. Numer. Methods Eng., 60, 241-269 (2004) · Zbl 1060.74626 · doi:10.1002/nme.1029 |
| [22] | Yang, ZH; Cui, JZ, The statistical second-order two-scale analysis for dynamic thermo-mechanical performances of the composite structure with consistent random distribution of particles, Comput. Mater. Sci., 69, 359-373 (2013) · doi:10.1016/j.commatsci.2012.12.011 |
| [23] | Yang, ZH; Cui, JZ; Wu, YT; Wang, ZQ; Wan, JJ, Second-order two-scale analysis method for dynamic thermo-mechanical problems in periodic structure, Int. J. Numer. Anal. Model., 12, 1, 144-161 (2015) · Zbl 1332.35356 |
| [24] | Yang, ZQ; Cui, JZ; Sun, Y.; Liang, J.; Yang, ZH, Multiscale analysis method for thermo-mechanical performance of periodic porous materials with interior surface radiation, Int. J. Numer. Methods Eng., 105, 323-350 (2016) · Zbl 07868644 · doi:10.1002/nme.4964 |
| [25] | Yang, ZH; Huang, JZ; Feng, XB; Guan, XF, An efficient multi-modes Monte Carlo homogenization method for random materials, SIAM J. Sci. Comput., 44, 3, A1752-A1774 (2022) · Zbl 1492.65334 · doi:10.1137/21M1454237 |
| [26] | Yang, ZH; Wang, XT; Guan, XF; Huang, JZ; Wu, XX, A normalizing field flow induced two-stage stochastic homogenization method for random composite materials, Commun. Comput. Phys., 34, 3, 787-812 (2023) · Zbl 1538.65404 · doi:10.4208/cicp.OA-2023-0007 |
| [27] | Zhang, S.; Yang, ZH; Guan, XF, Multi-modal multiscale method for heat conduction problem in heterogeneous solids with uncertain material parameters, Adv. Appl. Math. Mech., 15, 1, 69-93 (2023) · Zbl 1513.65474 · doi:10.4208/aamm.OA-2022-0048 |
| [28] | Su, F.; Cui, JZ, A second-order and two-scale analysis method for the quasi-periodic structure of composite materials, Finite Elem. Anal. Des., 46, 320-327 (2010) · doi:10.1016/j.finel.2009.11.004 |
| [29] | Allaire, G.; Habibi, Z., Second order corrector in the homogenization of a conductive-radiative heat transfer problem, Discrete Contin. Dyn. Syst. Ser. B, 18, 1-36 (2013) · Zbl 1270.35053 |
| [30] | Yang, ZH; Zhang, Y.; Dong, H., High-order three-scale method for mechanical behavior analysis of composite structures with multiple periodic configurations, Compos. Sci. Technol., 152, 198-210 (2012) · doi:10.1016/j.compscitech.2017.09.031 |
| [31] | Yang, ZH; Guan, XF; Cui, JZ, Stochastic multiscale heat transfer analysis of heterogeneous materials with multiple random configurations, Commun. Comput. Phys., 22, 2, 431-459 (2020) · Zbl 1476.35025 · doi:10.4208/cicp.OA-2018-0311 |
| [32] | Dong, H.; Yang, ZH; Guan, XF; Cui, JZ, Stochastic higher-order three-scale strength prediction model for composite structures with micromechanical analysis, J. Comput. Phys., 465, 111352 (2022) · Zbl 07561046 · doi:10.1016/j.jcp.2022.111352 |
| [33] | Dong, H., Computationally efficient higher-order three-scale method for nonlocal gradient elasticity problems of heterogeneous structures with multiple spatial scales, Appl. Math. Model., 109, 426-454 (2022) · Zbl 1505.74028 · doi:10.1016/j.apm.2022.05.010 |
| [34] | Kesavan, S., Homogenization of elliptic eigenvalue problems: part I, Appl. Math. Optim., 5, 1, 153-167 (1979) · Zbl 0415.35061 · doi:10.1007/BF01442551 |
| [35] | Kesavan, S., Homogenization of elliptic eigenvalue problems: part II, Appl. Math. Optim., 5, 1, 197-216 (1979) · Zbl 0428.35062 · doi:10.1007/BF01442554 |
| [36] | Nankakumar, AK, Homogenization of eigenvalue problems of elasticity in perforated domains, Asymptot. Anal., 9, 337-358 (1994) · Zbl 0814.35135 |
| [37] | Vanninathan, M., Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci., 90, 3, 239-271 (1981) · Zbl 0486.35063 · doi:10.1007/BF02838079 |
| [38] | Cao, LQ; Cui, JZ; Zhu, DC, Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general domains, SIAM J. Numer. Anal., 40, 2, 543-577 (2002) · Zbl 1019.65085 · doi:10.1137/S0036142900376110 |
| [39] | Cao, LQ; Cui, JZ, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problems for second order elliptic equations in perforated domains, Numer. Math., 96, 528-581 (2004) · Zbl 1049.65126 |
| [40] | Allaire, G.; Piatnitski, A., Homogenization of the Schrodinger equation and effective mass theorem, Comm. Math. Phys., 258, 1-22 (2005) · Zbl 1081.35092 · doi:10.1007/s00220-005-1329-2 |
| [41] | Zhang, L.; Cao, LQ; Wang, X., Multiscale finite element algorithm of the eigenvalue problems for the elastic equations in composite materials, Comput. Methods Appl. Mech. Eng., 198, 2539-2554 (2009) · Zbl 1228.74099 · doi:10.1016/j.cma.2009.03.015 |
| [42] | Craster, R.V., Kaplunov, J., Pichugin, A.V.: High frequency homogenization for periodic media. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466, 2341-2362 (2010) · Zbl 1196.35038 |
| [43] | Piat, VC; Nazarov, SS; Piatniski, AL, Steklov problems in perforated domains with a coefficient of indefinite sign, Netw. Heterog. Media, 7, 1, 151-178 (2012) · Zbl 1262.35025 · doi:10.3934/nhm.2012.7.151 |
| [44] | Douanla, H., Homogenization of Steklov Spectral problems with indefinite density function in perforated domains, Acta Appl. Math., 123, 261-284 (2013) · Zbl 1263.35022 · doi:10.1007/s10440-012-9765-4 |
| [45] | Cao, LQ; Zhang, L.; Allegretto, W.; Lin, YP, Multiscale computation of a Steklov eigenvalue problem with rapidly oscillating coefficients, Int. J. Numer. Anal. Model., 10, 1, 42-73 (2013) · Zbl 1290.65106 |
| [46] | Cao, LQ; Zhang, L.; Allegretto, W.; Lin, YP, Multiscale asymptotic method for Steklov eigenvalue equations in composite media, SIAM J. Numer. Anal., 51, 1, 273-296 (2013) · Zbl 1267.65172 · doi:10.1137/110850876 |
| [47] | Li, ZH; Ma, Q.; Cui, JZ, Multi-scale modal analysis for axisymmetric and spherical symmetric structures with periodic configurations, Comput. Methods Appl. Mech. Eng., 317, 1068-1101 (2017) · Zbl 1439.74324 · doi:10.1016/j.cma.2017.01.013 |
| [48] | Ma, Q.; Li, ZH; Cui, JZ, Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates, Comput. Methods Appl. Mech. Eng., 340, 340-365 (2018) · Zbl 1440.65165 · doi:10.1016/j.cma.2018.05.035 |
| [49] | Ye, S.Y., Ma, Q., Hu, B., Cui, J.Z., Jiang, X.: Multiscale asymptotic analysis and computations for steklov eigenvalue problem in periodically perforated domain. Math. Meth. Appl. Sci. 1-21 (2021) · Zbl 1479.35066 |
| [50] | Allaire, G.; Briane, M., Multiscale convergence and reiterated homogenization, Proc. R. Soc. Edinb. Sect. A Math., 126, 2, 297-342 (1996) · Zbl 0866.35017 · doi:10.1017/S0308210500022757 |
| [51] | Trucu, D.; Chaplain, MAJ; Marciniak-Czochra, A., Three-scale convergence for processes in heterogeneous media, Appl. Anal., 91, 7, 1351-1373 (2012) · Zbl 1252.35038 · doi:10.1080/00036811.2011.569498 |
| [52] | Telega, JJ; Galka, A.; Tokarzewski, S., Application of the reiterated homogenization to determination of effective moduli of a compact bone, J. Theor. Appl. Mech., 37, 3, 687-706 (1999) · Zbl 0963.74037 |
| [53] | Ramirez-Torres, A.; Penta, R.; Rogriguez-Ramos, R.; Merodio, J.; Sabina, FJ; Bravo-Castillero, J.; Guinovart-Diaz, R.; Preziisi, L.; Grillo, A., Three scales asymptotic homogenization and its application to layered hierarchical hard tissues, Int. J. Solids Struct., 130-131, 190-198 (2018) · doi:10.1016/j.ijsolstr.2017.09.035 |
| [54] | Yang, ZQ; Sun, Y.; Liu, YZ; Guan, TY; Dong, H., A three-scale asymptotic analysis for ageing linear viscoelastic problems of composites with multiple configurations, Appl. Math. Model., 71, 223-242 (2019) · Zbl 1481.74638 · doi:10.1016/j.apm.2019.02.021 |
| [55] | Dong, H.; Zheng, XJ; Cui, JZ; Nie, YF; Yang, ZQ; Yang, ZH, High-order three-scale computational method for dynamic thermo-mechanical problems of composite structures with multiple spatial scales, Int. J. Solids Struct., 169, 95-121 (2019) · doi:10.1016/j.ijsolstr.2019.04.017 |
| [56] | Yang, ZQ; Sun, Y.; Cui, JZ; Ge, JG, A three-scale asymptotic expansion for predicting viscoelastic properties of composites with multiple configuration, Eur. J. Mech. A-Solid., 76, 235-246 (2019) · Zbl 1470.74020 · doi:10.1016/j.euromechsol.2019.04.016 |
| [57] | Dong, H.; Cui, JZ; Nie, YF; Yang, ZH; Wang, ZQ, High-order three-scale computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales, Adv. Eng. Softw., 121, 1-12 (2018) · doi:10.1016/j.advengsoft.2018.03.005 |
| [58] | Ma, Q.; Cui, JZ; Yang, Z.; Yang, ZQ; Jiang, X.; Li, ZH, Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials, Appl. Math. Model., 92, 6, 565-593 (2020) · Zbl 1481.35361 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
