Johannes Blühdorn, TU Kaiserslautern
Efficient Solution of the Unit Cell Problem
For materials with complex microstructure, analytical descriptions of the effective (macroscopic) behaviour are usually not available. In the context of mathematical homogenization, the effective behaviour is approximated based on a simulation of a material unit cell on the microscopic scale. There, one has to solve an elliptic PDE. The best approximation is usually obtained with periodic boundary conditions. While the unit cell problem can traditionally be solved by finite element methods, Moulinec and Suquet proposed a more efficient algorithm that needs no meshing and can work directly on data obtained from CT images. It relies upon the reformulation of the original problem in terms of the periodic Lippmann-Schwinger equation. The solution to this integral equation is approximated in the space of trigonometric polynomials by means of a fixed point iteration. In each iteration, a PDE with constant coefficients must be solved, which is done efficiently in Fourier space. In this talk, we derive the method for linear elasticity at small deformations. We outline how it extends to nonlinear material laws and large deformations and discuss its advantages, limitations and more recent adaptions.