Multilevel sampling with Monte Carlo and Quasi-Monte Carlo methods for uncertainty quantification in structural engineering

Practical structural engineering applications tend to exhibit a certain degree of uncertainty in their material parameters, loading forces and so forth. As such, the accurate quantification of the effect of those uncertainties is of capital importance. The standard Monte Carlo method is one of the most common sampling methods used to compute this effect. In this paper we compare two extensions of the standard Monte Carlo method: the Multilevel Monte Carlo (MLMC) and the Multilevel Quasi-Monte Carlo (MLQMC) method. These two methods are tested on a structural engineering problem: a cantilever steel beam clamped at both sides and loaded in the middle, with an uncertain Young's modulus. A Gamma random field is used to model the uncertainty. For the response of the beam we consider its spatial displacement in the elasto-plastic domain. Our aim is to demonstrate the effectiveness and versatility of both MLMC and MLQMC by coupling them with this Finite Element code. We show that MLQMC has a lower computational cost than MLMC for a desired tolerance on the root mean square error. Furthermore both methods are significantly faster than a standard Monte Carlo method.