A boundary element method for homogenization of periodic structures

Abstract

Homogenized coefficients of periodic structures are calculated via an auxiliary partial differential equation in the periodic cell. Typically, a volume finite element discretization is employed for the numerical solution. In this paper, we reformulate the problem as a boundary integral equation using Steklov-Poincaré operators. The resulting boundary element method only discretizes the boundary of the periodic cell and the interface between the materials within the cell. We prove that the homogenized coefficients converge super-linearly with the mesh size, and we support the theory with examples in two and three dimensions. KEYWORDS boundary element method, homogenization MSC CLASSIFICATION 65N38; 65N12; 35B27 | INTRODUCTIONSolving a boundary value problem, which involves materials with composite microstructure, is computationally demanding. Therefore, we look for homogeneous (constant) material coefficients imitating the original microstructure so that the solution to the original problem with a highly oscillating material function is in a sense close to the solution of a related problem with a constant material function. There are two well-known approaches to homogenization. In both approaches, auxiliary boundary value problems in the so-called representative volume element (RVE) are solved. In the first approach, the geometry of the microstructure does not need to be periodic, and a large portion of the domain has to be covered in the auxiliary problems. In such cases, energy methods 1 are often employed. The methods provide effective coefficients that preserve the energy of the solution for a fixed type of boundary conditions. The second approach, which our paper actually deals with, assumes a periodic microstructure. As stated in Allaire, 2 energy methods do not take full advantage of the periodicity. The homogenized coefficients are instead determined by solutions to auxiliary problems with periodic boundary conditions. For a presentation of mathematical theory and methods for the periodic case, we refer to Cioranescu and Donato. 3 As far as numerical methods for the solution to the auxiliary problems are concerned, volume discretization techniques such as finite element methods prevail in literature, cf Rohan and Lukeš. 4 In this paper, we consider boundary

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