Acceleration of the spectral stochastic FEM using POD and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry

Abstract

The spectral stochastic FEM with local basis functions in the stochastic domain (SL-FEM) is one of the most flexible and accurate stochastic methods, however, also the most computationally expensive. These expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re-integrated in every iteration. In this work we incorporate the proper orthogonal decomposition (POD) into the SL-FEM, thus performing a drastic reduction of the stiffness matrix. In order to reduce the integration costs by hyperreduction, a novel element-based modification of the discrete empirical interpolation, the so-called element-based empirical approximation method (EDEAM), is developed and combined with the POD. Particular advantages of the SL-FEM for order reduction and hyperreduction compared to other stochastic techniques are discussed. The new reduced-order SL-FEM is applied to the computational homogenization of materials with random geometry of the microstructure, i.e. to a general class of problems exhibiting strongly nonlinear, non-smooth and sometimes discontinuous dependency of the solution on some random parameters. The reduced-order SL-FEM demonstrates a high accuracy and a high solution speed, whereby the solution time for the reduced-order SL-FEM is comparable to the solution time of only one single Monte-Carlo sample.