Model order reduction of nonlinear homogenization problems using a Hashin-Shtrikman type finite element method

摘要

AbstractThis work presents a computational nonlinear homogenization approach, the starting point of which is a model order reduction method based on data-clustering. To this end, the micromechanical data from numerical experiments (snapshots) is analyzed in order to identify characteristic microstructural deformation patterns. These describe how the macroscopic strain typically localizes within the microstructure. The outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points. Within each cluster the strain is then approximated as being constant.The mechanical problem is formulated in terms of a three-field Hashin–Shtrikman type variational formulation which is based on the introduction of a linear-elastic reference medium. After discretization, most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns. The resulting homogenization scheme includes, as special cases, the finite element method as well as Hashin–Shtrikman and Talbot–Willis type homogenization approaches with phase-wise constant trial fields (as well as related bounds). The limit case ’finite element method’ allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material. The method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy. The results are shown to be in good agreement with full-field FE-simulations. Furthermore, the method is used to compute a refined upper bound of the Talbot–Willis type (compared to phase-wise constant trial fields), which converges to the finite element solution with increasingly refined discretization.