Mesh Smoothing and Untangling

1
Mesh Smoothing and Untangling

• Optimization of vertex locations in simplicial
  meshes.
• Sean Mauch
• Caltech
• October, 2003

2
Optimizing Vertex Locations

• One can improve the quality of the simplices in a
  triangle or tetrahedral mesh by moving the
  vertices. (We consider moving only the internal
  vertices.)
• We iterate over the vertices and consider the
  local mesh of the simplices incident to a single
  vertex.
• In this Gauss-Seidel iteration, we move each
  vertex to optimize the local mesh quality.
• One can use geometric quality measures, like the
  minimum dihedral angle, in the optimization.
  Lori Freitag implemented this approach with the
  Opt-MS package. One can also use algebraic
  quality measures defined in terms of the Jacobian
  matrix of the transformation from the equilateral
  simplex. (Simultaneous untangling and smoothing
  of tetrahedral meshes by Escobar et. al.) We
  have implemented the latter approach.

3
An Example of Smoothing
Initial mesh.
Tangled and distorted mesh.
One smoothing iteration.
Two smoothing iterations.
Three smoothing iterations.
4
Preliminary Results

• We have implemented mesh smoothing with algebraic
  quality measures as a C class library.
• The problem dimension is a template parameter.
  The code will smooth 2-D triangle meshes, 3-D
  tetrahedral meshes and higher dimensional
  simplicial meshes.
• Initial tests indicate that the approach may be
  useful in optimizing input meshes and in
  repairing meshes during the course of a
  simulation.

5
Future Work

• We may implement additional quality measures and
  compare their performance.
• We will implement constrained optimization of the
  vertices on the boundary.
• If the boundary faces have poor quality, so will
  some of the simplices.
• Move the boundary vertices to optimize the
  quality of the simplices subject to maintaining
  the shape of the boundary.
• We will also implement topological optimization.
  Flip edges/faces to increase simplex quality.