Efficient finite element quadrature on immersed voxel data

Abstract

Abstract High-resolution computed tomography (CT) scans provide voxel-wise information on geometry and material properties. Direct numerical simulations on CT data in which each voxel is considered a finite element can yield billions of unknowns, a resolution that is often excessive. Coarsening the finite element mesh reduces the number of unknowns, but accurate quadrature then becomes a challenge. Composed integration by distributing Gauss-Legendre points on each voxel is computationally expensive due to the resulting large number of integration points. The standard remedy is to pre-integrate the stiffness contribution of each voxel in reference coordinates. However, this approach does not directly extend to local enrichment of the finite element space. For example, under hierarchical octree-based refinement the basis functions vary from element to element. For such cases, we develop an efficient voxel quadrature scheme based on a modified moment-fitting approach. The method computes a custom integration rule for each element which absorbs voxel information directly into the integration weights. This enables exact integration of the stiffness matrix of finite elements with any polynomial degree while limiting the number of quadrature points during assembly. In contrast to pre-integration, our approach decouples the material resolution from the finite element basis and is even faster for coarse finite element resolutions, while remaining far cheaper than composed Gauss-Legendre quadrature. We demonstrate voxel-based moment fitting on three examples: a scaling analysis for heterogeneous voxel data, a voxelized L-shaped domain with adaptivity driven by a residual-based error estimator, and a locally refined phase-field fracture simulation.