# Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

## TL;DR

This paper introduces a reduced basis surrogate model for efficient finite strain homogenization of hyperelastic solids, significantly speeding up computations while maintaining accuracy and robustness in multiscale simulations.

## Contribution

It presents a novel snapshot POD approach on deformation gradient fluctuations and an efficient sampling method using Hencky strain for hyperelastic homogenization.

## Key findings

- Speed-up factors of 5-100 times achieved.
- Enhanced robustness with maintained accuracy.
- Open-source demonstrator tool with 50 lines of code.

## Abstract

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5-100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.