Non-intrusive Uncertainty Quantification with Sparse Grids for Multivariate Peridynamic Simulations

Abstract

Peridynamics is an accepted method in engineering for modeling crack propagation on a macroscopic scale. However, the sensitivity of the method to two important model parameters – elasticity and the particle density – has not yet been studied. Motivated by Silling and Askari (Comput Struct 83(17–18):1526–1535, 2005) and Kidane et al. (J Mech Phys Solids 60(5):983–1001, 2012) we use Peridynamics to simulate a high-speed projectile impacting a plate and study the overall damage on the plate. We have extended the setting by the magnitude of the force of the indenter and selected the parameter range such that a sharp transition in the response function occurs.We describe the simulation setting as an uncertainty quantification problem and use a non-intrusive stochastic collocation method based on spatially adaptive sparse grids to propagate the uncertainty. We show first convincing results of its successful application to Peridynamics and compare to Monte Carlo sampling.If the magnitude of the force is deterministic, a strong sensitivity of the damage in the plate with respect to the elasticity factor can be shown for the 2-dimensional setting. If it is non-deterministic, it dominates the simulation and explains most of the variance of the solution. The error of the expectation value estimation reaches an early saturation point for the studied collocation methods: We found parameter ranges where the quantity of interest oscillates. Moreover, faster convergence and higher robustness than for the Monte Carlo method can be observed.