Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction

In the present work, we propose a method combining a multi-scale approach and a model reduction technique based on proper orthogonal decomposition (POD) to solve highly nonlinear conduction problems in structures made of periodic heterogeneous materials. Following classical computational homogenization schemes, a representative volume element is associated with each integration point of the macrostructure. The local macroscopic response is computed directly on the RVE through solving an incremental problem with appropriate boundary and initial conditions. In the proposed method, the equations of the linearized micro problem are projected on the reduced basis, which is obtained using POD via preliminary computations. The set of unknowns and Lagrange multipliers associated with periodic boundary conditions is largely reduced. The technique called reduced model multi-scale method (R3M) lowers the computational costs. Both accuracy and efficiency are examined through numerical tests involving thermal and electric nonlinear conduction problems.