Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales
Tóm tắt
A family of effective equations for wave propagation in periodic media for arbitrary timescales
$$\mathcal {O}(\varepsilon ^{-\alpha })$$
, where
$$\varepsilon \ll 1$$
is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017,
arXiv:1701.08600
; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018,
arXiv:1803.09455
). The effective solutions in the family are proved to be
$$\varepsilon $$
close to the original wave in a norm equivalent to the
$${\mathrm {L}^{\infty }}(0,\varepsilon ^{-\alpha }T;{{\mathrm {L}^{2}}(\varOmega )})$$
norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.
Tài liệu tham khảo
Abdulle, A., Grote, M.J.: Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul. 9(2), 766–792 (2011)
Abdulle, A., Grote, M.J., Stohrer, C.: Finite element heterogeneous multiscale method for the wave equation: long-time effects. Multiscale Model. Simul. 12(3), 1230–1257 (2014). https://doi.org/10.1137/13094195X.
Abdulle, A., Henning, P.: Chapter 20 - multiscale methods for wave problems in heterogeneous media 18, 545–576 (2017). https://doi.org/10.1016/bs.hna.2016.10.007.
Abdulle, A., Henning, P.: Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comp. 86(304), 549–587 (2017). https://doi.org/10.1090/mcom/3114.
Abdulle, A., Pouchon, T.: Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization. Math. Models Methods Appl. Sci. 26(14), 2651–2684 (2016). https://doi.org/10.1142/S0218202516500627.
Abdulle, A., Pouchon, T.: A priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time. SIAM J. Numer. Anal. 54(3), 1507–1534 (2016)
Abdulle, A., Pouchon, T.: Effective models for long time wave propagation in locally periodic media. SIAM J. Numer. Anal. 56(5), 2701–2730 (2018)
Allaire, G., Briane, M., Vanninathan, M.: A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures. SeMA J. 73(3), 237–259 (2016)
Allaire, G., Lamacz, A., Rauch, J.: Crime pays; homogenized wave equations for long times (2018). ArXiv preprint arXiv:1803.09455
Arjmand, D., Runborg, O.: Analysis of heterogeneous multiscale methods for long time wave propagation problems. Multiscale Model. Simul. 12(3), 1135–1166 (2014)
Arjmand, D., Runborg, O.: Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media. Multiscale Model. Simul. 15(2), 948–976 (2017)
Bakhvalov, N.S., Panasenko, G.P.: Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36. Kluwer Academic Publishers Group, Dordrecht (1989). https://doi.org/10.1007/978-94-009-2247-1. Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. Leĭtes
Benoit, A., Gloria, A.: Long-time homogenization and asymptotic ballistic transport of classical waves (2017). Preprint arXiv:1701.08600
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978)
Brahim-Otsmane, S., Francfort, G.A., Murat, F.: Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71(3), 197–231 (1992)
Christov, C., Maugin, G., Velarde, M.: Well-posed boussinesq paradigm with purely spatial higher-order derivatives. Physical Review E 54(4), 3621 (1996)
Cioranescu, D., Donato, P.: An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17. Oxford University Press, New York (1999)
Conca, C., Orive, R., Vanninathan, M.: On burnett coefficients in periodic media. J. Math. Phys. 47(3), 032902 (2006)
De Giorgi, E., Spagnolo, S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 4(8), 391–411 (1973)
Dohnal, T., Lamacz, A., Schweizer, B.: Bloch-wave homogenization on large time scales and dispersive effective wave equations. Multiscale Model. Simul. 12(2), 488–513 (2014)
Dohnal, T., Lamacz, A., Schweizer, B.: Dispersive homogenized models and coefficient formulas for waves in general periodic media. Asymptot. Anal. 93(1-2), 21–49 (2015)
Duerinckx, M., Otto, F.: Higher-order pathwise theory of fluctuations in stochastic homogenization. Stochastics and Partial Differential Equations: Analysis and Computations pp. 1–68 (2019)
Engquist, B., Holst, H., Runborg, O.: Multi-scale methods for wave propagation in heterogeneous media. Commun. Math. Sci. 9(1), 33–56 (2011). http://projecteuclid.org/euclid.cms/1294170324
Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
Fish, J., Chen, W., Nagai, G.: Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Internat. J. Numer. Methods Engrg. 54(3), 331–346 (2002)
Jiang, L., Efendiev, Y.: A priori estimates for two multiscale finite element methods using multiple global fields to wave equations. Numer. Methods Partial Differential Equations 28(6), 1869–1892 (2012). https://doi.org/10.1002/num.20706.
Jiang, L., Efendiev, Y., Ginting, V.: Analysis of global multiscale finite element methods for wave equations with continuum spatial scales. Appl. Numer. Math. 60(8), 862–876 (2010). https://doi.org/10.1016/j.apnum.2010.04.011.
Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, Heidelberg (1994)
Lamacz, A.: Dispersive effective models for waves in heterogeneous media. Math. Models Methods Appl. Sci. 21(9), 1871–1899 (2011)
Lamacz, A.: Waves in heterogeneous media: Long time behavior and dispersive models. Ph.D. thesis, TU Dortmund (2011)
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, Travaux et recherches mathématiques, vol. 1. Dunod, Paris (1968)
Murat, F., Tartar, L.: \(H\)-convergence. In: Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, pp. 21–43. Birkhäuser Boston, Boston, MA (1997)
Owhadi, H., Zhang, L.: Numerical homogenization of the acoustic wave equations with a continuum of scales. Comput. Methods Appl. Mech. Engrg. 198(3), 397–406 (2008)
Owhadi, H., Zhang, L.: Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast. Multiscale Model. Simul. 9(4), 1373–1398 (2011). https://doi.org/10.1137/100813968
Pouchon, T.: Effective models and numerical homogenization methods for long time wave propagation in heterogeneous media. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Lausanne (2017)
Sánchez-Palencia, E.: Nonhomogeneous media and vibration theory, Lecture Notes in Phys., vol. 127. Springer-Verlag, Berlin-New York (1980)
Santosa, F., Symes, W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51(4), 984–1005 (1991). https://doi.org/10.1137/0151049.
Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Super. Pisa Cl. Sci. 22(4), 571–597 (1968)