Log In Sign Up

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

by   Philippe Blondeel, et al.

Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior. The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Loève (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to 1/epsilon under certain conditions, where epsilon is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions.


page 1

page 2

page 3

page 4


Multilevel Monte Carlo for uncertainty quantification in structural engineering

Practical structural engineering problems often exhibit a significant de...

On the Selection of Random Field Evaluation Points in the p-MLQMC Method

Engineering problems are often characterized by significant uncertainty ...

Multilevel Quasi-Monte Carlo for Optimization under Uncertainty

This paper considers the problem of optimizing the average tracking erro...

Quantifying uncertainties on excursion sets under a Gaussian random field prior

We focus on the problem of estimating and quantifying uncertainties on t...

Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty

We interpret uncertainty in the parameters of a model for seismic wave p...

Effect of an attached end mass in the dynamics of uncertainty nonlinear continuous random system

This work studies the dynamics of a one dimensional elastic bar with ran...

Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes

This paper deals with the applications of stochastic spectral methods fo...