Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a twoor three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the nodes of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior node that describes each interior node in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior nodes. Our main concern is when this algorithm reverses elements. We analyze the causes for this undesirable behavior and propose techniques to make the method more robust against reversals. Among the methods includes combining FEMWARP with an optimization-based untangler. Finally, we use FEMWARP to study the movement of the canine heart.