Multilevel Monte Carlo for uncertainty quantification in structural engineering

TL;DR

This paper applies Multilevel Monte Carlo combined with Finite Element methods to efficiently quantify uncertainty in structural engineering problems involving material variability, demonstrating significant computational speedups over traditional Monte Carlo methods.

Contribution

It introduces a novel application of MLMC with FEM for uncertainty quantification in structural responses with material randomness, showing substantial efficiency improvements.

Findings

MLMC achieves several orders of magnitude faster computations than classical Monte Carlo.

The method effectively handles both elastic and elastoplastic responses.

Significant reduction in computational costs for uncertainty quantification in structural models.

Abstract

Practical structural engineering problems often exhibit a significant degree of uncertainty in the material properties being used, the dimensions of the modeled structures, etc. In this paper, we consider a cantilever beam and a beam clamped at both ends, both subjected to a static and a dynamic load. The material uncertainty resides in the Young's modulus, which is modeled by means of one random variable, sampled from a univariate Gamma distribution, or with multiple random variables, sampled from a Gamma random field. Three different responses are considered: the static elastic, the dynamic elastic and the static elastoplastic response. In the first two cases, we simulate the spatial displacement of a concrete beam and its frequency response in the elastic domain. The third case simulates the spatial displacement of a steel beam in the elastoplastic domain. In order to compute the statistical quantities of the static deflection and frequency response function, Multilevel Monte Carlo (MLMC) is combined with a Finite Element solver. In this paper, the computational costs and run times of the MLMC method are compared with those of the classical Monte Carlo method, demonstrating a significant speedup of up to several orders of magnitude for the studied cases.