E3C hyper-reduction for computational homogenization in nonlinear mechanics

Abstract In computational homogenization, a fast solution of the microscopic problem can be achieved by model order reduction in combination with hyper-reduction. Such a technique, which has recently been proposed in the context of magnetostatics, is applied to nonlinear mechanics in this work. The method is called ’Empirically Corrected Cluster Cubature’ (E3C), as it combines clustering techniques with an empirical correction step to compute a novel type of integration points, which does not form a subset of the finite element integration points. The method is adopted to the challenges arising in nonlinear mechanics and is tested in plane strain for different microstructures (porous and reinforced) in dependence of the material nonlinearity. The results show that hyper-reduction errors $$\lesssim 1$$ % can be achieved with a comparably small number of integration points, which is in the order of the number of modes. A two-scale example is provided and the research code can be downloaded.