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

Abstract: 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.