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：本文提出一种简单高效的超降阶应变空间模型降阶方法（MOR），用于超弹性代表性体积单元（RVE），称为经验性材料采样与线性化（EMSL）。该方法在概念上受Wulfinghoff与Hauck[36]提出的经验校正聚类积分（E3C）启发，同时借鉴了以往关于增量变分结构保持应变空间模型降阶技术的研究成果，以实现在线阶段的快速评估。与E3C类似，我们将材料域划分为行为相似的区域，并针对每个区域在单一参考应变值处查询材料本构例程。然而，我们仅在每个载荷增量中基于经验估计的期望应变值对这些应变进行一次采样。利用通过本征正交分解（POD）获得的参考材料切线及应变模态，计算材料簇其余部分应力响应的线性化估计。与近似精确积分材料本构律的E3C不同，EMSL可视为精确积分材料行为近似的一种方法。所得降阶问题在每个载荷步中具有仿射形式，这使得可通过离线阶段即可预处理的运算对整个计算域进行积分。由于每个载荷增量中均获得线性方程组，在线阶段无需进行牛顿迭代。为进行基准对比，我们构建了应变空间中两种流行降阶积分格式的变体，并回顾了Wulfinghoff等人提出的E3C算法。通过一个含孔洞几何的超弹性RVE算例，我们证明EMSL在精度与运行时间的权衡上帕累托优于竞争性的应变空间方法。