Finite elements for Mat\'ern-type random fields: Uncertainty in computational mechanics and design optimization

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.