Iterated two-scale asymptotic method and numerical algorithm for the elastic structures of composite materials ☆

In this paper, we shall discuss the elasto-static problem of composite materials with two or more different length scales, and shall present iterated two-scale asymptotic expansion of the solution for considering problem. The mathematical proofs of some convergence theorems will be given under some assumptions. Finally, some numerical results are reported.

[1]  J. Bourgat Numerical experiments of the homogenization method , 1979 .

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

[3]  U. Hornung Homogenization and porous media , 1996 .

[4]  P. Grisvard,et al.  BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN , 1976 .

[5]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1966 .

[6]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[7]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[8]  David Gilbarg,et al.  Intermediate Schauder estimates , 1980 .

[9]  Grégoire Allaire,et al.  Multiscale convergence and reiterated homogenisation , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  M. Avellaneda Iterated homogenization, differential effective medium theory and applications , 1987 .

[11]  A. Norris A differential scheme for the effective moduli of composites , 1985 .

[12]  Jacques-Louis Lions,et al.  Some Methods in the Mathematical Analysis of Systems and Their Control , 1981 .

[13]  Willi Jäger,et al.  Multiscale problems in science and technology : challenges to mathematical analysis and perspectives : proceedings of the conference on multiscale problems in science and technology, Dubrovnik, Croatia, 3-9 September 2000 , 2002 .

[14]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[15]  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.

[16]  J. L. Lions,et al.  Asymptotic expansions in perforated media with a periodic structure , 1980 .

[17]  Doina Cioranescu,et al.  Homogenization of Reticulated Structures , 1999 .

[18]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[19]  H. Kardestuncer,et al.  Finite element handbook , 1987 .

[20]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[21]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .