This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on prevalent applications in computational mechanics and design optimization. We leverage Matérn-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models as well as the shape and topology of optimized designs. We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications, emphasizing the potential to influence these domains. The flexibility and efficiency of SPDE-based GRF generation empowers us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate design features and topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model and quantify geometric uncertainties on reconstructed submanifolds, such as the interpolated surfaces of cerebral aneurysms provided by postprocessing CT scans. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.

中文翻译：

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Matérn 型随机场的有限元：计算力学和设计优化中的不确定性


这项工作重点介绍了一种将现实不确定性纳入基于有限元的科学计算工作流程的方法，重点关注计算力学和设计优化中的普遍应用。我们利用 SPDE 方法生成的 Matérn 型高斯随机场 (GRF) 来模拟任意不确定性，包括环境影响、变化的材料属性和几何模糊性。我们的重点在于提供实用的 GRF 实现，准确捕获缺陷和变化，并了解它们如何影响计算模型的预测以及优化设计的形状和拓扑。我们描述了一种基于求解广义 SPDE 来对任意网格域上的 GRF 进行采样的数值算法。该算法利用现有技术，并与开源有限元库 MFEM 和相关的科学计算工作流程（如工业和国家实验室环境中的工作流程）无缝集成。我们的求解器可以有效地解决大规模问题，并支持各种域类型，包括曲面和嵌入式流形。我们通过生物力学和拓扑优化应用展示其多功能性，强调影响这些领域的潜力。基于 SPDE 的 GRF 生成的灵活性和效率使我们能够在 2D 和 3D 域上运行大规模优化问题，包括在嵌入式表面上查找优化设计，并生成超出传统技术范围的设计特征和拓扑。此外，这些功能使我们能够对重建子流形的几何不确定性进行建模和量化，例如后处理 CT 扫描提供的脑动脉瘤的插值表面。 除了在这些特定领域提供好处之外，所提出的技术超越了特定应用，并可推广到涉及有限元的不确定性量化中的任意前向和后向问题。