Algorithms for mesh warping with applications to cardiology

Suzanne Shontz, Cornell University

Thursday, March 4, 16:00, 655 MCB

Abstract:

Moving meshes arise in cardiology, computer graphics, animation, and crash simulation, 
among other applications in science and engineering. With moving meshes, the mesh is 
updated at each step in time due to a moving domain boundary, thus resulting in 
potentially drastically varying mesh quality from step to step.  One problem that can 
occur at each timestep is element inversion.  Our focus is on the development of mesh 
warping algorithms that maintain good mesh quality at each timestep.

We have developed several different algorithms for the warping of tetrahedral meshes.  
These methods are based upon weighted Laplacian smoothing, the finite element method, 
and quadratic programming.  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 any of our mesh warping algorithms 
to update the nodes of the volume mesh.  Each method determines a set of local weights 
for each interior node which describe the relative positions of the node to each of its 
neighbors.  After a boundary transformation is applied, the method solves a system of 
linear equations based upon the weights and the new boundary positions to determine the 
final positions of the interior nodes.

We study mesh invertibility and prove a theorem which gives sufficient conditions for a 
mesh to resist inversion by a transformation.  Our theory shows that two of our methods 
yield exact results for affine mappings, and we state a conjecture for more general mappings.  
In addition, our theory ensures that the same methods yield the mesh to which both the local 
weighted laplacian smoothing algorithm and the Gauss-Seidel algorithm for linear systems converge.  
We test the robustness of our methods and present some numerical results.  Finally, we use our 
algorithm to study the movement of the beating canine heart.

Part of this talk represents joint work with S. Vavasis.