The usage of numerical homogenization to obtain structure-property relations using the finite element method at both the micro and macroscale has gained much interest in the research community. However the computational cost of this so called FE$^2$ method is so high that algorithmic modifications and reduction methods are essential. Currently the authors proposed a monolithic algorithm. Now this algorithm is combined with ROM and ECM hyper integration, applied at finite deformations and complemented by a clustered training strategy, which lowers the training effort and the number of necessary ROM modes immensely. The applied methods are modularly combinable as aimed in finite element approaches. An implementation in terms of an extension for the already established MonolithFE$^2$ code is provided. Numerical examples show the efficiency and accuracy of the monolithic hyper ROM FE$^2$ method and the advantages of the clustered training strategy. Online times of below $1\%$ of the conventional FE$^2$ method could be gained. In addition the training stage requires around $3\%$ of that time, meaning that no extremely expensive offline stage is necessary as in many Neural Network approaches, which only pay off when a lot of online simulations will be conducted.

翻译：采用数值均匀化方法，通过宏微观尺度有限元分析获取材料结构-性能关系的研究已引起学界广泛关注。然而，这种称为FE$^2$方法的计算成本极高，必须引入算法改进与降阶处理。本文作者团队此前已提出单片算法，现将其与ROM及ECM超积分相结合，在有限变形条件下应用，并补充了聚类训练策略，从而大幅降低训练开销与所需ROM模态数量。所采用的方法遵循有限元分析的模块化组合原则。本文提供了基于已有MonolithFE$^2$代码的扩展实现方案。数值算例表明，单片超ROM FE$^2$方法兼具高效性与准确性，且聚类训练策略优势显著。在线计算时间可降至传统FE$^2$方法的$1\%$以下，同时训练阶段仅需该时间的$3\%$，这意味着无需像许多神经网络方法那样进行极其昂贵的离线预训练（后者仅在大量在线仿真场景下才具成本效益）。