
ThinPlateSpline Curvilinear Meshing on a CalculusofVariations Framework
Sastry, Shankar P., Vidhi Zala, Robert M. Kirby
24th International Meshing Roundtable, Elsevier Ltd., October 1214 2015

IMR PROCEEDINGS

24th International Meshing Roundtable
Austin, TX
October 1214,2014
Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, U.S.A.
Email: sastry@sci.utah.edu
Abstract
Highorder, 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 straightsided 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 straightsided 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 RBFbased techniques have always yielded higherquality
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 secondorder 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 firstorder 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 higherquality meshes than prior algorithms.
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