A reduced order variational spectral method for efficient construction of eigenstrain-based reduced order homogenization models

Abstract

Reduced order models (ROMs) are often coupled with concurrent multiscale simulations to mitigate the computational cost of nonlinear computational homogenization methods. Construction (or training) of ROMs typically requires evaluation of a series of linear or nonlinear equilibrium problems, which itself could be a computationally very expensive process. In the eigenstrain-based reduced order homogenization method (EHM), a series of linear elastic microscale equilibrium problems are solved to compute the localization and interaction tensors that are in turn used in the evaluation of the reduced order multiscale system. These microscale equilibrium problems are typically solved using either the finite element method or semi-analytical methods. In the present study, a reduced order variational spectral method is developed for efficient computation of the localization and interaction tensors. The proposed method leads to a small stiffness matrix that scales with the order of the reduced basis rather than the number of degrees of freedom in the finite element mesh. The reduced order variational spectral method maintains high accuracy in the computed response fields. A speedup higher than an order of magnitude can be achieved compared to the finite element method in polycrystalline microstructures. The accuracy and scalability of the method for large polycrystals and increasing phase property contrast are investigated.