Monolithic Hyper ROM FE\textsuperscript{2} Method with Clustered Training at Finite Deformations

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.