FFT-based method for homogenization of periodic media: Theory and applications

Abstract

This dissertation is devoted to an FFT-based homogenization scheme, a numerical method for the evaluation of the effective (homogenized) matrix of periodic linear heterogeneous materials. A problem is explained and demonstrated on a scalar problem modeling electric conduction, heat conduction, or diffusion. Originally, FFT-based homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from Lippmann-Schwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of Lippmann-Schwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce Moulinec-Suquet algorithm; trigonometric polynomials are taken as the trial space. Convergence of approximate solutions to the solution of weak formulation is provided using a standard finite element approach together with approximation properties of trigonometric polynomials. Then, the solution of assembled non-symmetric linear system by Conjugate gradients, proposed by Zeman et al., is clarified. Next, we study arbitrary accurate guaranteed bounds of homogenized matrix introduced by Dvořák for a scalar problem and later independently by Wieckowski for linear elasticity. This approach is also applicable for FFT-based homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upper-lower bound structure. Dual formulation is employed to obtain lower bounds - for odd number of discretization points, the solution of dual formulation can be avoided. A general number of discretization points leads to a more complicated theory in both discretization and numerical treatment. Finally, applications of FFT-based homogenization to real-world problems is demonstrated. The method is used to calculate homogenized matrix for cement paste, gypsum and aluminum alloy with local data obtained from nanoindentation. Next, it is employed as a part of two-step homogenization for a highly porous aluminium foam.

This dissertation is devoted to an FFT-based homogenization scheme, a numerical method for the evaluation of the effective (homogenized) matrix of periodic linear heterogeneous materials. A problem is explained and demonstrated on a scalar problem modeling electric conduction, heat conduction, or diffusion. Originally, FFT-based homogenization, that was proposed by Moulinec and Suquet, is a numerical algorithm derived from Lippmann-Schwinger equation. Its equivalence to a corresponding weak formulation is shown; it eliminates a reference homogeneous material, a parameter of Lippmann-Schwinger equation. Next, Galerkin approximation with numerical integration is introduced to produce Moulinec-Suquet algorithm; trigonometric polynomials are taken as the trial space. Convergence of approximate solutions to the solution of weak formulation is provided using a standard finite element approach together with approximation properties of trigonometric polynomials. Then, the solution of assembled non-symmetric linear system by Conjugate gradients, proposed by Zeman et al., is clarified. Next, we study arbitrary accurate guaranteed bounds of homogenized matrix introduced by Dvořák for a scalar problem and later independently by Wieckowski for linear elasticity. This approach is also applicable for FFT-based homogenization. A general technique is proposed to allow for efficient calculation by FFT algorithm and to maintain the upper-lower bound structure. Dual formulation is employed to obtain lower bounds - for odd number of discretization points, the solution of dual formulation can be avoided. A general number of discretization points leads to a more complicated theory in both discretization and numerical treatment. Finally, applications of FFT-based homogenization to real-world problems is demonstrated. The method is used to calculate homogenized matrix for cement paste, gypsum and aluminum alloy with local data obtained from nanoindentation. Next, it is employed as a part of two-step homogenization for a highly porous aluminium foam.