FFT-based homogenization on periodic anisotropic translation invariant spaces
For researchers in computational homogenization, this provides a theoretical unification of existing methods, but the contribution is primarily mathematical and incremental.
This paper unifies Fourier-based and finite element methods for quasi-static elasticity homogenization within a common mathematical framework using anisotropic translation invariant spaces, deriving a generalized Lippmann-Schwinger equation and characterizing the periodized Green operator. Numerical examples with de la Vallée Poussin means and Box splines demonstrate the framework's flexibility.
In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann-Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann-Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.