Two reduction methods for stochastic FEM based homogenization using global basis functions

Abstract

The computationally most expensive part of stochastic FEM based homogenization is the inversion of the stochastic stiffness matrix. Previously we studied different combinations of global and local basis functions in the stochastic domain while keeping the common FEM discretization in the physical domain. Simulation results showed that a local basis (SL-FEM) results in a sparse but bigger stiffness matrix compared to a global basis (SG-FEM). In this work we introduce a SFEM modification involving global basis functions in both the physical and the stochastic domain. We introduce two different strategies for the implementation of the global basis. The first is the application of a trigonometric basis within a global Galerkin framework. The second is the use of vibration modes of some simple deterministic auxiliary model as global basis functions. The second approach in particularly results in a basis reduction technique. However this approach differs markedly from stochastic spectral decomposition which uses the matrix of the original stochastic problem. We address two very important questions, namely the treatment of the Gibbs phenomenon in the case of physical-stochastic trigonometric basis functions and the evaluation of the eigenvectors for the stochastic problem which are suitable for further basis reduction. Convergence studies and detailed comparison with Monte-Carlo results are presented for both methods.