Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization

TL;DR

This work presents a scalable SPDE-based framework to generate Matérn-type Gaussian random fields on arbitrary domains and manifolds for uncertainty quantification in finite-element workflows. It delivers a practical, open-source MFEM implementation capable of sampling complex uncertainties, including anisotropy and spatial inhomogeneity, and demonstrates its impact on biomechanics-inspired geometry and topology optimization problems. The methodology enables forward and inverse analyses on 2D and 3D domains, including embedded surfaces, and reveals novel design features inaccessible to traditional discretizations. The study also discusses symmetry enforcement, robust formulations, and computational considerations, highlighting the SPDE approach as a versatile tool for uncertainty quantification in computational mechanics and design optimization.

Abstract

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 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 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.

Figures (78)

• Figure 1: Isotropic, homogeneous, Matérn-type Gaussian random fields of different smoothness and correlation length generated with the SPDE method. The smoothness and correlation-length increase along their respective axis. The examples are generated with homogeneous Neumann boundary conditions on $D=(0,1)^2$ using identical white noise realizations for all samples.
• Figure 2: The finite element assembly procedure in a typical finite element code reduces to matrix multiplication. For the mass matrix, we multiply through the factorization $\mathbf{M} = \mathbf{P}^\top \mathbf{G}^\top \mathbf{B}^\top \mathbf{D} \mathbf{B} \mathbf{G} \mathbf{P}$ depicted above. Figure modified from Anderson2021andrej2024mfem.
• Figure 4: Matérn random fields on sphere, mobius strip, and Klein bottle geometries.
• Figure 5: The synthetic aneurysms are obtained from a reference geometry (grey mesh) yang2020intra via the SPDE method and shifting the respective vertices in normal direction proportionally to the random field values (red-blue color bar) multiplied by a scaling parameter $\alpha_S > 0$. Regions with positive random field values are lifted while negative values are lowered. Matérn-type GRF parameters: $l =5.0$, $\nu=2.0$, and $\alpha_S = 0.2$. We choose a larger scaling parameter $\alpha_S$ than in Table \ref{['tab:aneurysm-uq-param']} so that the geometric variations are easier to observe.
• Figure 8: Effects of geometric uncertainties for three realizations on a fluid flow throughout the aneurysm. The streamlines are colored by the pressure in the fluid using the color scale shown in the figure.