We present a new iterative technique based on radial basis function (RBF) interpolation and smoot... more We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.
High-order, curvilinear meshes have recently become popular due to their ability to conform to th... more High-order, curvilinear meshes have recently become popular due to their ability to conform to the geometry of the domain. Curvilinear 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 algorithms 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.
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Papers by Vidhi Zala