The usage of numerical homogenization to obtain structure-property relations using the finite element method at both the micro and macro scale has gained much interest in the research community. However the computational cost of this so called FE\textsuperscript{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. An implementation in terms of an extension for the already established MonolithFE\textsuperscript{2} code is provided. Numerical examples show the efficiency and accuracy of the monolithic hyper ROM FE\textsuperscript{2} method and the advantages of the clustered training strategy. The applied methods are modularly combinable as aimed in finite element approaches.

翻译：数值均匀化方法通过在微尺度与宏尺度同时采用有限元法获取结构-性能关系，已在研究领域引起广泛关注。然而，这种所谓FE²方法的计算成本极高，亟需算法改进与降阶处理。作者此前提出了单片算法，现将其与超降阶集成方法和经验正交基降阶模型结合，应用于有限变形问题，并辅以聚类训练策略，从而大幅降低训练成本及所需降阶模态数量。本文还提供了基于现有MonolithFE²代码的扩展实现方案。数值算例验证了单片超ROM FE²方法的效率与精度，以及聚类训练策略的优越性。所采用的方法遵循有限元方法的模块化组合原则。