The hole-filling method and the multiscale computation for the wave equations in perforated domains

In this paper we discuss the multiscale computation of the wave equations with rapidly oscillating coefficients in a perforated domain. The hole-filling method through filling all holes with a very compliant material is introduced. As the material parameters of the weak phase go to zero, the proof of this limiting process is derived. Based on the above hole-filling method, the multiscale asymptotic method and the convergence results for the associated multiphase problem in a domain without holes are presented. The numerical experiments are carried out to validate the theoretical results.

[1]  Gilles A. Francfort,et al.  Correctors for the homogenization of the wave and heat equations , 1992 .

[2]  Hung Nguyen-Xuan,et al.  High-order B-splines based finite elements for delamination analysis of laminated composites , 2013 .

[3]  J. Cui,et al.  Error estimate of the homogenization solution for elliptic problems with small periodic coefficients on L∞(Ω) , 2010 .

[4]  P. Donato,et al.  An introduction to homogenization , 2000 .

[5]  Hung Nguyen-Xuan,et al.  An isogeometric analysis for elliptic homogenization problems , 2013, Comput. Math. Appl..

[6]  Li-Qun Cao,et al.  Multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains , 2014, Appl. Math. Comput..

[7]  M. Vanninathan.  Homogenization of eigenvalue problems in perforated domains , 1981 .

[8]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[9]  Qiao-Li Dong,et al.  Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients , 2009 .

[10]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[11]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[12]  W. Chew.  Waves and Fields in Inhomogeneous Media , 1990 .

[13]  Jun-zhi Cui,et al.  Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains , 2004, Numerische Mathematik.

[14]  Liqun Cao,et al.  The hole-filling method and the uniform multiscale computation of the elastic equations in perforated domains , 2008 .

[15]  Li-qun Cao,et al.  Multiscale numerical algorithm for the elliptic eigenvalue problem with the mixed boundary in perforated domains , 2008 .

[16]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[17]  Marcus J. Grote,et al.  Finite Element Heterogeneous Multiscale Method for the Wave Equation , 2011, Multiscale Model. Simul..

[18]  E Weinan,et al.  Numerical methods for multiscale transport equations and application to two-phase porous media flow , 2005 .

[19]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[20]  O. Oleinik,et al.  Mathematical Problems in Elasticity and Homogenization , 2012 .