Aggressive Tetrahedral Mesh Improvement

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Aggressive Tetrahedral Mesh Improvement

• Bryan Matthew Klinger and Jonathan Richard
  Shewchuk
• UC Berkeley
•  
• Presented by Jacob Johnson

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Aggressive optimization

• aggressively optimize the worst tetrahedra
• most methods get stuck in local optima
• those methods are weak
• these methods are strong
• uses three techniques together
• smoothing
• topological transformations
• vertex insertion
• gets dihedral angles to 31o to 149o or 23o to
  136o (depending on the objective function)

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Overview

• Past methods
• The problem
• Mesh quality measures
• Mesh operations
• The scheduling of mesh operations
• Results

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Introduction

• The quality of a mesh is limited by the quality
  of its worst elements
• Current methods improve the overall quality of a
  mesh, but can't salvage bad elements
• The paper tries to improve the quality of the
  worst elements aggressively and at the expense of
  speed

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Existing methods

• Hill-climbing
• consider an individual operation
• will it improve the quality of the mesh?
• will always result in an improvement in mesh
  quality
• Smoothing
• moving vertices without changing the topology of
  the mesh
• numerical optimization

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Existing methods

• Topological transformations
• include 2-3, 3-2, 4-4, and 2-2 flips
• changing the connectivity of the mesh without
  moving vertices
• combinatorial optimization

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Existing methods

• Freitag and Ollivier-Gooch combine smoothing,
  flips, and edge removal with great success

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The problem

• Objective functions have many local optima, and
  most methods can only find these
• The authors find it unlikely that any method will
  ever come close to finding a global optimum
• However, their method aggressively chases a local
  optimum with no low-quality tetrahedra

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Mesh quality

• Large dihedral angles reduce the accuracy of the
  finite element method
• Small dihedral angles degrade the condition of
  the stiffness matrices
• A single element with a poor angle can ruin the
  simulation
• "a large dihedral angle can engender a huge
  spurious strain in the discretized solution of a
  mechanical system"

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Mesh quality

• Only isotropic simulations are considered, and
  all input tetrahedra are assumed to be
  non-inverted
• The desired metrics are normalized so
• larger values indicate better tets
• positive values indicate correct orientation
• scores range from -1 to 1

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Mesh quality

• Three quality measures are used
• minimum sine measure - the minimum sine of the
  tetrahedron's six dihedral angles
• biased minimum sine measure - minimum sine with
  obtuse angles' scores multiplied by 0.7
• these miss skinny tetrahedra
• volume-length measure - V/lrms3 - volume divided
  by the cube of the root-mean-square edge length
• radius ratio - radius of inscribed sphere divided
  by radius of circumscribing sphere

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Mesh quality

• Quality vector
• a vector of tet qualities, from worst to best
• the objective function
• Imagine an N-digit number where the worst tets
  lower the most significant digits and the best
  tets raise the least significant digits
• 1122234455677777788
• 1222234455677777788
• When a subset of the mesh is changed, only the
  quality vectors of the affected tetrahedra need
  be compared

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Mesh operations

• Smoothing
• Freitag and Ollivier-Gooch use a nonsmooth
  optimization algorithm which optimizes the worst
  tetrahedron in a group
• This algorithm is used with a small change
  vertices on boundaries can be smoothed as well,
  but their movement is constrained to the plane or
  line they lie on
• Smoothing is only ever done if it improves the
  quality of the worst tetrahedron

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Mesh operations

• Edge Removal
• Replaces m tets with 2m - 4 tets
• Previous work has demonstrated its effectiveness
• Seven tets are replaced
• by 27 - 4 10 tets

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Mesh operations

• Edge Removal
• The trick to edge removal is to find the
  triangulation of T that maximizes the quality of
  the worst tet
• The problem was solved in 1980 in an unrelated
  paper by Klincsek
• The algorithm is O(m3), but m is always fairly
  small

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Mesh operations

• Multi-face Removal
• The inverse of edge removal replaces 2m tets
  with m 2 tets
• A face f is sandwiched between a and b if the two
  tets that share f consist of the points in f and
  a or b
• The algorithm used is Shewchuck's own, which
  finds the optimal operation and avoids
  tetrahedral inversion

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Mesh operations

• Vertex Insertion
• The real innovation of the paper
• Delaunay triangulations are often improved by
  inserting vertices, but the analogue is not seen
  in tetrahedralizations
• The method used maximizes the quality of the
  worst new tetrahedron
• Compound vertex insertion is effective at getting
  a mesh over a optimization valley

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Mesh operations

• Given a tetrahedron to insert a vertex into, try
  to
• insert the new point at the barycenter of any
  boundary face
• insert the new point at the baryceter of the
  tetrahedron
• insert the new point at the midpoint of any
  boundary edge
• If all of these fail to improve the quality of
  the mesh, give up

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Mesh operations

• The mesh is represented as a directed graph
• Each node represents a tetrahedron
• Each edge represents a shared face
• The edge is directed towards the tetrahedron
  whose "view" of the new vertex is blocked by the
  other tetrahedron
• Coplanar edges are directed arbitrarily

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Mesh operations

• View the graph as a tree, with the tetrahedra
  containing the new vertex at the root
• A subgraph G is examined
• consists of nodes six or fewer levels from the
  root
• G typically has 5-100 tetrahedra
• Nodes on the boundary are linked to "ghost nodes
  across their unshared faces, so all
  leaves are ghost nodes

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Mesh operations

• Finding the Optimal Cut
• Vertex insertion carves out a cavity around the
  new vertex and links each face of the cavity to
  the vertex to create the new tetrahedra
• Carving out the cavity means finding a cut in the
  graph of G

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Mesh operations

• The cut should maximize the quality of the worst
  new tetrahedron
• To this end, each edge is labeled with the
  quality of the tetrahedron that will be created
  if its origin is cut by its destination remains
• The algorithm must therefore maximize the
  weight of the smallest edge cut

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Mesh operations

• Given that
• the root must be inside the cut
• the ghost nodes must be outside the cut
• the algorithm goes through the list of edges in
  order from worst to best, and greedily blocks bad
  cuts

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Mesh operations

• This will create the optimal cut, which typically
  entails replacing 5-15 tetrahedra
• Any vertex inside the cavity (except the inserted
  vertex) will be deleted, so this operation can
  reduce the number of vertices in the mesh

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Mesh operations

• Composite vertex insertion
• Vertex insertion by itself doesn't often improve
  the quality vector of the mesh
• So vertex insertion is followed by smoothing and
  topological transformations
• vertex insertion algorithm is biased to give
  far-away edges better qualities
• If these methods do not improve the quality
  vector, the mesh is rolled back to the way it was
  before insertion

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Scheduling the operations

• The scheduling of the operations is shown in
  pseudocode in figure 4, and is ordered basically
  as follows
• Smoothing pass
• Topological pass, which consists of
• Edge removal
• Multi-face removal or 2-3 flips
• Smoothing pass
• if this improves the quality of the mesh, goto 3
• Topological pass
• if this improves the quality of the mesh, goto 3

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Scheduling the operations

• Vertex insertion
• if we're seeing no improvement, attempt insertion
  in all tets with a dihedral angle 140o
• otherwise, attempt insertion in the worst 3.5 of
  tets
• if this improves the quality of the mesh, goto 3
• If the mesh goes through steps 3-5 three times in
  a row without seeing significant improvement, the
  algorithm terminates

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Scheduling the operations

• Significant improvement means 
• improvement in the quality of the worst element
  in the mesh, or
• slight improvement in a threshold mean
• A threshold mean is the mean of all qualities,
  with qualities above a threshold d reduced to d

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Results

• Tested on a dozen meshes, including
• RAND1 and RAND2, which have horrible quality
• DRAGON, COW, STGALLEN, and STAYPUFT, which have
  curved boundaries
• Minimum sine objective improved dihedral angles
  to 31o to 149o
• Volume-length measure improved dihedral angles to
  23o to 136o
• Poor-quality input meshes significantly reduced
  speed
• Composite vertex insertion accounted for up to
  90 of runtime
• Size of output meshes varied greatly
• Maximum increase of 41 of elements
• Some meshes shrank

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Results

• Using individual methods
• Smoothing most valuable, followed by vertex
  insertion
• Edge removal had little effect
• 4-4 flip is most effective common topological
  transformation
• Removing individual methods
• Removing smoothing or vertex insertion
  significantly reduced the quality of the output
  mesh

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Results

• Metrics
• Minimum sine measure
• Created skinny tets
• Optimized smallest dihedral angle best
• Radius ratio
• Created "rounded" tets
• Volume-length measure
• Created "rounded" tets
• Optimized radius ratio better than radius ratio
  does
• Best overall

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Conclusions

• The implementation can be improved by
• Allowing variable spacing, affecting vertex
  insertion
• Reducing the ridiculous run time
• The authors feel that traditional mesh
  improvement methods should integrated into mesh
  generation