A mesh warping algorithm based on weighted Laplacian smoothing

Proceedings, 12th International Meshing Roundtable, Sandia National Laboratories, pp.147-158, Sept. 2003

MESHING
RESEARCH
CORNER

12th International Meshing Roundtable
September 14-17, 2003
Santa Fe, New Mexico, U.S.A.

Center for Applied Mathematics
Cornell University, Ithaca, NY 14853
shontz@cam.cornell.edu

Department of Computer Science
Cornell University, Ithaca, NY 14853
vavasis@cs.cornell.edu

Abstract
We present a new mesh warping algorithm for tetrahedral meshes based upon weighted laplacian smoothing. We start with a 3D domain which is bounded by a triangulated surface mesh and has a tetrahedral volume mesh as its interior. We then suppose that a movement of the surface mesh is prescribed and use our mesh warping algorithm to update the nodes of the volume mesh. Our method determines a set of local weights for each interior node which describe the relative distances of the node to its neighbors. After a boundary transformation is applied, the method solves a system of linear equations based upon the weights to determine the ?nal position of the interior nodes. We study mesh invertibility and prove a theorem which gives su?cient conditions for a mesh to resist inversion by a transformation. We prove that our algorithm yields exact results for a?ne mappings and state a conjecture for more general mappings. In addition, we prove that our algorithm converges to the same point as both the local weighted laplacian smoothing algorithm and the Gauss-Seidel algorithm for linear systems. We test our algorithm’s robustness and present some numerical results. Finally, we use our algorithm to study the movement of the canine heart.

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