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Practical structural engineering problems are often characterized bysignificant uncertainties. Historically, one of the prevalent methods toaccount for this uncertainty has been the standard Monte Carlo (MC) method.Recently, improved sampling methods have been proposed, based on the idea ofvariance reduction by employing a hierarchy of mesh refinements. We combine anh- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) andMultilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicabilityof these novel combination methods on three structural engineering problems,for which the uncertainty resides in the Young's modulus: the static responseof a cantilever beam with elastic material behavior, its static response withelastoplastic behavior, and its dynamic response with elastic behavior. Theuncertainty is either modeled by means of one random variable sampled from aunivariate Gamma distribution or with multiple random variables sampled from agamma random field. This random field results from a truncated Karhunen-Lo\`eve(KL) expansion. In this paper, we compare the computational costs of theseMonte Carlo methods. We demonstrate that MQLMC and MLMC have a significantspeedup with respect to MC, regardless of the mesh refinement hierarchy used.We empirically demonstrate that the MLQMC cost is optimally proportional to1/epsilon under certain conditions, where epsilon is the tolerance on theroot-mean-square error (RMSE). In addition, we show that, when the uncertaintyis modeled as a random field, the multilevel methods combined with p-refinementhave a significant lower computation cost than their counterparts based onh-refinement. We also illustrate the effect the uncertainty models have on theuncertainty bounds in the solutions.

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