A Mesh Warping Algorithm Based on Weighted Laplacian Smoothing.
Suzanne Michelle Shontz, Stephen A. Vavasis · 2003
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 final 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.