# An algorithm for stress and mixed control in Galerkin based FFT homogenization

## TL;DR

This paper introduces a novel algorithm for imposing stress or mixed control in FFT-based homogenization, enhancing the flexibility and efficiency of non-linear multiscale simulations without additional computational cost.

## Contribution

It develops a modified projection operator enabling stress or strain control in Galerkin FFT homogenization, maintaining the original variational framework without extra iterations.

## Key findings

- Efficient stress/mixed control achieved without additional iterations.
- Validated on polycrystal and hyperelastic material examples.
- Demonstrates improved flexibility in multiscale material modeling.

## Abstract

A new algorithm is proposed to impose a macroscopic stress or mixed stress/deformation gradient history in the context of non-linear Galerkin based FFT homogenization. The method proposed is based in the definition of a modified projection operator in which the null frequencies enforce the type of control (stress or strain) for each component of either the macroscopic first Piola stress or the deformation gradient. The resulting problem is solved exactly as the original variational method and it does not require additional iterations compared to the strain control version, neither in the linear iterative solver, nor in the Newton scheme. The efficiency of the method proposed is demonstrated with a series of numerical examples, including a polycrystal and a particle reinforced hyperelastic material.