On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers

Heterogeneous structures like composites often need a fine-scale resolution of micro-effects which influence the macroscopic overall response. This is of particular relevance in the fully non-linear range of large strains and inelastic material response of the constituents. Suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales. For big differences of micro- and macro-scales, the argument of scale separation induces the application of homogenization methods. Such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine-scale micro-structure as well as the macroscopic boundary-value problem. In contrast, small-scale differences require full resolution of the heterogeneous structure. Effective solution methods for the resulting large-scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques, which may be considered as numerical multiscale approaches. In both scenarios, a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales. The paper outlines new mesh-bridging techniques in a deformationdriven context for fully non-linear response, which exploit in a non-trivial manner weak constraints on the average deformation in typical finite element patches. The framework is based on an incremental variational structure of finite inelasticity. The proposed new formulations provide variational-based homogenization algorithms for physical multiscale scenarios and problem-dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials. Copyright q 2007 John Wiley & Sons, Ltd.