Investigations on the restrictions of stochastic collocation methods for high dimensional and nonlinear engineering applications

Abstract

Sophisticated sampling techniques used for solving stochastic partial differential equations efficiently and robustly are still in a state of development. It is known in the scientific community that global stochastic collocation methods using isotropic sparse grids are very efficient for simple problems but can become computationally expensive or even unstable for non-linear cases. The aim of this paper is to test the limits of these methods outside of a basic framework to provide a better understanding of their possible application in terms of engineering practices. Specifically, the stochastic collocation method using the Smolyak algorithm is applied to finite element problems with advanced features, such as high stochastic dimensions and non-linear material behaviour. We compare the efficiency and accuracy of different unbounded sparse grids (Gauss–Hermite, Gauss–Leja and Kronrod–Patterson) with Monte Carlo simulations. The sparse grids are constructed using an open source toolbox provided by Tamellini et al., ©2009–2018, https://sites.google.com/view/sparse-grids-kit, while Abaqus is used as a finite element solver.