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

Suzanne M. Shontz, Stephen A. Vavasis

TL;DR

This work analyzes FEMWARP, a finite-element-based mesh-warping method that updates interior vertex positions by solving a sparse linear system derived from a discrete Laplacian with Dirichlet boundary data. It proves exactness for affine boundary deformations and identifies element reversal as the main practical failure mode, classifying reversals into three types. To enhance robustness, the authors propose and evaluate strategies including small-step (infinitesimal-step) deformations, mesh refinement, and a hybrid FEMWARP/Opt-MS untangling approach, as well as comparing with a mean-value weight map. Across 2D and 3D tests, small-step FEMWARP and the FEMWARP/Opt-MS hybrid generally outperform plain FEMWARP, though guarantees on mesh quality and trajectory continuity are not always available, pointing to future work on quality guarantees and symmetry-based formulations.

Abstract

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or 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 vertices 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 vertex that describes each interior vertex 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 vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.

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

TL;DR

This work analyzes FEMWARP, a finite-element-based mesh-warping method that updates interior vertex positions by solving a sparse linear system derived from a discrete Laplacian with Dirichlet boundary data. It proves exactness for affine boundary deformations and identifies element reversal as the main practical failure mode, classifying reversals into three types. To enhance robustness, the authors propose and evaluate strategies including small-step (infinitesimal-step) deformations, mesh refinement, and a hybrid FEMWARP/Opt-MS untangling approach, as well as comparing with a mean-value weight map. Across 2D and 3D tests, small-step FEMWARP and the FEMWARP/Opt-MS hybrid generally outperform plain FEMWARP, though guarantees on mesh quality and trajectory continuity are not always available, pointing to future work on quality guarantees and symmetry-based formulations.

Abstract

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or 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 vertices 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 vertex that describes each interior vertex 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 vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.

Paper Structure

This paper contains 8 sections, 2 theorems, 25 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The FEMWARP algorithm
  3. FEMWARP is exact for affine mappings
  4. Element reversal and small-step FEMWARP
  5. Mesh warping and mesh untangling
  6. Three-dimensional tests
  7. Conclusions
  8. Acknowledgements

Key Result

lemma thmcounterlemma

Let $A_B$ and $A_I$ be generated using FEMWARP. Then $A_I$ is nonsingular based upon well-known finite element theory johnsonbook. Let $[\hat{x}_B,\hat{y}_B]$ be the user-specified deformed coordinates of the boundary. Suppose there exists a $2\times 2$ nonsingular matrix $L$ and $2$-vector $v$ such Let $[\hat{x}_I,\hat{y}_I]$ be the deformed interior coordinates computed by the method. Then for e

Figures (3)

  • Figure 1: Translation and rotation of annulus mesh: (a) original mesh; (b)-(d) meshes obtained by translating the inner circle outwards and rotating it counterclockwise by $(1,0^{o}), (2,20^{o}), (3,40^{o})$, and $(3.5,50^{o})$, respectively. Meshes (a) and (b) are valid meshes; mesh (c) is also valid but contains elements that are near reversal; mesh (d) contains reversed elements and is invalid.
  • Figure 2: All meshes in this section were generated by the Triangle mesh generator; this is an example of a mesh used herein.
  • Figure 3: The 3D test meshes used in this section: (a) foam5 pat_mesh, (b) gear pat_mesh, (c) hook pat_mesh, and (d) tire lori_mesh.

Theorems & Definitions (4)

  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof