Thin-plate-Spline Curvilinear Meshing on a Calculus-of-Variations Framework

Abstract

High-order, curvilinear meshes have recently become popular due to their ability to conform to the geometry of the domain. Curvi- linear meshes are generated by first constructing a straight-sided mesh and then curving the boundary elements (and, consequently, some of the interior edges and faces) to respect the geometry of the domain. The locations of the interior vertices can be viewed as an interpolation of a mapping function whose values at the boundary vertices (of the straight-sided mesh) are equal to the vertex locations on the curved domain. We solve this interpolation problem using radial basis functions (RBFs) by extending earlier algo- rithms that were developed for linear mesh deformation. An RBF interpolation technique using a biharmonic kernel is also called a thin plate spline. We analyze the resulting mapping function (the RBF interpolation) in a framework based on calculus of variations and provide a detailed explanation of the reasons the thin plate kernel RBF-based techniques have always yielded higher-quality meshes than other techniques. It is known that the thin plate kernel RBF interpolation minimizes the “bending energy” associated with a function, which depends on its second-order partial derivatives. We show that the minimization of the bending energy attempts to preserve the shape of an element after the transformation. Other techniques minimize either a functional (that depends on the first-order partial derivatives) that attempts to preserve the size of an element, or the bending energy in a smaller subspace of functions. Thus, our experimental results show that our algorithm generates higher-quality meshes than prior algorithms.