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.

翻译：本文提出了一种面向超弹性代表体积单元(RVE)的简单高效超降阶应变空间模型降阶方法(MOR)，称为"实证材料采样与线性化"(EMSL)。该方法在概念上受Wulfinghoff与Hauck[36]提出的实证校正聚类积分(E3C)启发，同时借鉴了先前关于增量变分保结构应变空间模型降阶技术的研究成果，以实现在线阶段的高效评估。与E3C类似，我们将材料域划分为行为相似的区域，并在每个区域的单个参考应变值处查询材料本构程序。然而，我们仅在每个载荷增量步中通过经验估计的预期应变值进行一次采样，利用通过本征正交分解(POD)获得的参考材料切线模量与应变模态，对材料簇其余部分的本构响应进行线性化估计。与近似积分精确材料律的E3C不同，EMSL可视为对材料行为近似值的精确积分。所得到的降阶问题在每一载荷步中均为仿射形式，使得可通过离线阶段预处理的运算对整个计算域进行积分。由于每个载荷增量步中仅需求解线性方程组，在线阶段无需进行牛顿迭代。为进行基准比较，我们在应变空间中构建了两种主流降阶积分格式的变体，并复现了Wulfinghoff等人提出的E3C算法。在含孔洞几何的超弹性RVE算例中，我们展示了EMSL在精度与运行时间权衡方面对竞争性应变空间方法的帕累托优势。