Reduced basis hybrid computational homogenization based on a mixed incremental formulation

A new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed. Key features of the Nonuniform Transformation Field Analysis (NTFA) [1,2] are taken as a point of departure. A short-coming of the NTFA method is its limitation to simple constitutive models on the microscale. A second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension. Both issues are related to the simple evolution law for the mode activity coefficients, which are the new macroscopic internal variables of the homogenized material. In the present contribution a generalization of the NTFA is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation. The derivation is based on purely micro-mechanical considerations. The modification allows for arbitrary Generalized Standard Materials [3] on the microscopic scale including, e.g., crystal visco-plasticity. Additionally, the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field. Numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi-scale method.