Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes

Abstract

Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson’s equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 24 in the L2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O(h3). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O(h4). An optimal reduction of 24 is observed when the functions on the C1 geometry are relaxed to be C0.