# DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior

## TL;DR

This paper introduces DBFFT, a displacement-based FFT homogenization method that improves efficiency and robustness for non-linear mechanical behavior modeling, avoiding redundant information and rank-deficiency issues of previous methods.

## Contribution

The paper presents a novel displacement-based FFT framework that handles general non-linear constitutive behaviors and improves computational efficiency over existing methods.

## Key findings

- Achieved around 30% reduction in computational cost compared to Galerkin-FFT.
- The method is robust for elastic, hyperelastic, and viscoplastic materials.
- The linear system in the linear case has a full rank Hermitian matrix, enabling efficient iterative solutions.

## Abstract

Most of the FFT methods available for homogenization of the mechanical response use the strain/deformation gradient as unknown, imposing their compatibility using Green's functions or projection operators. This implies the allocation of redundant information and, when the method is based in solving a linear equation, the rank-deficiency of the resulting system. In this work we propose a fast, robust and memory-efficient FFT homogenization framework in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. The framework allows any general non-linear constitutive behavior for the phases and direct strain, stress and mixed control of the macroscopic load. In the linear case, the method results in a linear system defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples including elastic, hyperelastic and viscoplastic materials are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.