This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on applications in computational mechanics and design optimization. We leverage Mat\'ern-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 and the 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. The flexibility and efficiency of SPDE-based GRF generation empower us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model geometric uncertainties of reconstructed submanifolds, such as the surfaces of cerebral aneurysms. 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.

翻译：本文提出了一种基于有限元方法将实际不确定性引入科学计算工作流的策略，重点聚焦于计算力学与设计优化领域的应用。我们利用基于SPDE方法生成的Matérn型高斯随机场来建模随机不确定性，涵盖环境影响、材料属性变异及几何模糊性。研究核心在于生成能够精准捕获瑕疵与变异的实用高斯随机场实现，并探究其对计算模型预测结果及优化设计拓扑结构的影响。我们描述了一种基于广义SPDE求解的数值算法，可在任意网格化域上采样高斯随机场。该算法融合已有成熟技术，能够无缝对接开源有限元库MFEM及相关工业界与国家实验室环境中常见的科学计算工作流。求解器具备大规模问题高效求解能力，支持包括曲面与嵌入流形在内的多种域类型。我们通过生物力学与拓扑优化应用案例展示了其通用性。基于SPDE的高斯随机场生成技术兼具灵活性与高效性，使我们能够在二维/三维域上开展大规模优化问题（包括在嵌入曲面上寻找最优设计），并生成超越传统方法可达范围的拓扑结构。此外，这些能力允许我们对颅内动脉瘤表面等重构子流形的几何不确定性进行建模。所提技术不仅在上述特定领域具有优势，还能推广至涉及有限元的不确定性量化中的任意正反演问题。