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

Monolithic Hyper ROM FE$^2$ Method with Clustered Training at Finite Deformations