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Mathematical models are essential for understanding and making predictionsabout systems arising in nature and engineering. Yet, mathematical models are asimplification of true phenomena, thus making predictions subject touncertainty. Hence, the ability to quantify uncertainties is essential to anymodelling framework, enabling the user to assess the importance of certainparameters on quantities of interest and have control over the quality of themodel output by providing a rigorous understanding of uncertainty. Peridynamicmodels are a particular class of mathematical models that have proven to beremarkably accurate and robust for a large class of material failure problems.However, the high computational expense of peridynamic models remains a majorlimitation, hindering outer-loop applications that require a large number ofsimulations, for example, uncertainty quantification. This contributionprovides a framework to make such computations feasible. By employing aMultilevel Monte Carlo (MLMC) framework, where the majority of simulations areperformed using a coarse mesh, and performing relatively few simulations usinga fine mesh, a significant reduction in computational cost can be realised, andstatistics of structural failure can be estimated. The results show a speed-upfactor of 16x over a standard Monte Carlo estimator, enabling the forwardpropagation of uncertain parameters in a computationally expensive peridynamicmodel. Furthermore, the multilevel method provides an estimate of both thediscretisation error and sampling error, thus improving the confidence innumerical predictions. The performance of the approach is demonstrated throughan examination of the statistical size effect in quasi-brittle materials.

A probabilistic peridynamic framework with an application to the study of the statistical size effect