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：本研究提出了一种基于有限元的科学计算工作流中引入实际不确定性的方法，重点面向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（GRFs）来建模随机不确定性，包括环境影响、材料属性变异及几何模糊性。研究重点在于提供能够精确捕捉缺陷与变异的实用GRF实现，并理解它们如何影响计算模型的预测结果及优化设计的拓扑构型。我们描述了一种基于求解广义SPDE的数值算法，用于在任意网格化域上采样GRF。该算法融合成熟技术，可与开源有限元库MFEM及相关科学计算工作流（如工业与国家实验室环境中的典型工作流）无缝集成。我们的求解器在大规模问题上具有高效可扩展性，并支持多种域类型，包括曲面与嵌入流形。我们通过生物力学与拓扑优化应用展示了其多功能性。基于SPDE的GRF生成方法兼具灵活性与高效性，使我们能够在二维和三维域上运行大规模优化问题（包括在嵌入曲面上寻找优化设计），并生成超越传统技术能力范围的拓扑构型。此外，这些功能使我们能够建模重建子流形（如脑动脉瘤表面）的几何不确定性。除了在特定领域带来优势外，所提出的技术超越了具体应用范畴，可推广至涉及有限元的不确定性量化中的任意正反问题。