Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.

翻译：数学模型对于理解和预测自然与工程系统中的现象至关重要。然而，数学模型是对真实现象的简化，因此预测结果存在不确定性。因此，量化不确定性的能力对任何建模框架都至关重要，它使用户能够评估特定参数对关注量的重要性，并通过提供对不确定性的严格理解来控制模型输出的质量。近场动力学模型是一类特殊的数学模型，已被证明在解决大量材料失效问题上具有显著的准确性和鲁棒性。然而，近场动力学模型的高计算成本仍是主要限制，阻碍了需要大量模拟的外循环应用，例如不确定性量化。本文提出了一种使此类计算可行的框架。通过采用多级蒙特卡洛（MLMC）框架（其中大部分模拟使用粗网格进行，仅用细网格进行相对较少的模拟），可以实现计算成本的显著降低，并估计结构失效的统计特性。结果表明，与标准蒙特卡洛估计器相比，加速比达到16倍，从而能够在计算成本高昂的近场动力学模型中实现不确定参数的前向传播。此外，多级方法同时提供了离散误差和采样误差的估计，从而提高了数值预测的置信度。通过对准脆性材料中统计尺寸效应的检验，证明了该方法的有效性。