Empirical Material Sampling and Linearisation -- A Simple and Efficient Strain-Space Model Order Reduction Approach for Computational Homogenisation in Large-Deformation Hyperelasticity

TL;DR

This paper introduces EMSL, a simple and efficient strain-space model order reduction method for hyperelastic RVEs, enabling rapid, accurate simulations in large-deformation hyperelasticity without iterative solvers.

Contribution

EMSL combines empirical sampling with linearisation to improve efficiency and accuracy in hyperelastic model reduction, outperforming existing methods in speed and precision.

Findings

EMSL reduces computational time compared to traditional approaches.

It achieves higher accuracy in stress response predictions.

No Newton iterations are needed in the online phase.

Abstract

In this article, we propose a simple and efficient hyperreduced strain-space model order reduction (MOR) approach for hyperelastic representative volume elements (RVEs), called Empirical Material Sampling and Linearisation (EMSL). The approach is conceptually motivated by the Empirically Corrected Cluster Cubature (E3C) of Wulfinghoff and Hauck [36], but also draws on ideas from previous work on incremental variational structure-preserving strain-space model order reduction techniques to achieve rapid evaluations in the online phase. As in E3C, we group the material domain into regions of similar behaviour, and query the material routine at one reference strain value per region. However, we sample these strains only once per load increment, at empirically estimated expected strain values. We use the reference material tangent and strain modes obtained via the Proper Orthogonal Decomposition (POD) to compute a linearised estimate of the stress response in the remainder of the material cluster. In contrast to E3C, which approximately integrates the exact material law, EMSL could therefore be said to exactly integrate an approximation of the material behaviour. The resulting reduced problem is affine in each load step, allowing for integration over the entire computational domain via operations which can readily be preprocessed in the offline phase. Since a linear equation system is obtained in each load increment, no Newton iterations are required in the online phase. For benchmark comparisons, we pose a variant of two popular reduced cubature schemes in strain space and recall the E3C algorithm proposed by Wulfinghoff et al. On an example hyperelastic RVE problem with a porous geometry, we show that EMSL Pareto-dominates competing strain-space approaches in terms of the tradeoff between accuracy and runtime.