Xiangrong Li, Mark S. Shephard

1
Accounting for Curved Domains in Mesh Refinement

• Xiang-rong Li, Mark S. Shephard
• Scientific Computation Research Center
• Rensselaer Polytechnic Institute
• and
• Mark W. Beall
• Simmetrix Inc.

• Outline
• Complexity of mesh generation for curved domains
• Single approach to support accounting for curved
  geometry in adaptively modified meshes
• Technical aspects of procedure
• Results

2
Mesh Modification Involving Geometry Changes

• Curved domain problems introduce complexities
  into the application of mesh modification
  procedures since the geometry of the mesh must
  also change
• Key tools where this is critical
• Adaptive mesh refinement/coarsening
• Curving initially straight-edged mesh entities
  and needed by higher-order elements

3
Dealing with Mesh Modification for Curved Domains

• Refinement vertices classified on curved
  boundaries need to be snapped to the boundary
• Mesh entity curving for p-version analysis has
  same complexities

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3
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1
2
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Overall procedure for vertex snapping

• Input a list of mesh vertices to be snapped
• Two parts of the procedure
• Part I Process all that snap directly or require
  local modification only
• While any mesh vertex in the list still can
  be moved ahead
• determine target move for next vertex
  in list
• if the current vertex can move to
  target location without a problem then
• move to that location and remove vertex
  from list
• else
• determine the first problem preventing
  motion
• define a move that ensures the
  first problem is passed
• analyze current situation to determine
  best local modification
• if there is an acceptable local
  modification then
• perform selected local modification
  and remove from list if at target
• end if
• end if
• end while

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Overall procedure for vertex snapping

• Part II Ensure to eliminate remaining vertices
  in the list using cavity re-triangulation
• Traverse remaining vertices in the list
  once
• perform local cavity construction
• apply cavity mesher, remove form list add
  any new ones created into list
• end traverse
• New boundary mesh vertices may be created during
  cavity re-triangulation. But, this is not common,
  and they are typically closer to boundary

Yes
Done
If any created vertices need to be snapped
If list is empty
Vertex list to be snapped
No
No
Part II
Part I
Done
Yes
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Outline of technical aspects

• Snap vertices by performing local modification
• Analyze current situation
• Requirement on where to move past first problem
  plane
• Evaluation and selection of most appropriate mesh
  modification operator with respect to local
  element shape and mesh size
• Snap vertices using local cavity re-triangulation
• Handling of new boundary vertices created

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Analyze current situation

• Definition of first problem plane
• The plane containing a mesh face opposite to
  current snapping vertex
• The first plane the snapping vertex runs into
  while moving toward target
• Other information includes
• Key mesh entities to cause mesh invalid
• Problem mesh faces on the first problem plane

the mesh vertex classified on the model
boundary not snapped
B-C-D a mesh face opposite to
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Need to Move Past First Problem Plane

• Vertex may be snapped incrementally along
  different path
• Any intermediate target must at least pass first
  problem plane

Green line model Black line mesh Red arrow
snap direction Cyan arrow possible snap path
Pass first problem plane
Stop before reaching first problem plane
(3) move to vertex 2, which gets onto the first
plane and typically a better solution
(1) original mesh
(2) create poorly-shaped element not desirable
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Mesh Modification Operators

• Single step operators

Edge swapping
Face swapping
Face splitting
Edge splitting
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Mesh Modification Operators

• Edge collapsing
• Region collapsing

Green model edge Black mesh edge
Split edge 2-4 1-3, and collapse the diagonal
edge
Remove the tetrahedron, face 1-2-4 2-3-4, edge
2-3 and re-classify face 1-2-3 1-3-4, edge 1-3
Remove the tetrahedron, face 1-2-4 and
re-classify all other interior bounding mesh
entities
Remove the tetrahedron, face 1-2-3, 2-4-3
3-4-1, edge 1-3, 2-3 3-4 and reclassify face
1-2-4
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Mesh Modification Operators

• Compound operators modify two or more mesh
  entities
• Examples of two step compound operators
• Split a mesh entity, and then move the new vertex
• Split a mesh entity, and then collapse the new
  vertex
• Swap entity A, and then collapse entity B
• Collapse entity A, and then collapse entity B
• Etc.

Green line model edge Black line mesh
edge Arrow snap direction
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Selection of most appropriate local modifications

• Collapse to existing vertices of problem
  face(s)
• This is useful way to pass multiple problem
  faces
• Modification operators
• Single step edge collapse,
• Compound swap(s)edge collapse,
  collapse(s)edge collapse
• Modifications to move to its projection on
  first problem plane along snap direction.
  Modification operators include
• Collapse flat region(s) created by the move
• Swap key mesh entities
• Split key mesh entities, followed by processing
  new vertices created

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Cavity Creation and Meshing in Vertex Snapping

• In some cases the available combinations of mesh
  modification operators does not succeed or
  produce poor element shapes
• In those cases a local remesh that ensures the
  current vertex is eliminated is applied
• Terms used
• minimum loop(s) of boundary mesh
  edges encircling
• a triangulation of
  which has classified on the
  model boundary with parameter values
  interior to

model
original mesh
(example to define terms only -
algorithm not needed in this case)
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Cavity Creation and Meshing in Vertex Snapping

• Step1. Generate
• Step 2. Perform swaps to minimize the
  intersection of with existing mesh
  entities
• Step 3. Determine mesh entities that interact
  with and delete the associated mesh
  regionsStep 4. Determine resulting cavity (or
  cavities), send to cavity mesher
  (see Karamete,Beall Shephard, IJNME, 2000)
• Step 5. Add any created in Steps 4
  classified on the boundary to the
  list to be snapped

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2-D Example of Cavity Creation

Green line model edge Black line mesh edge Red
dash line Blue dot
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Handling of new boundary vertices created

• This procedure may create new vertices that need
  to be snapped not common and typically close
  to its classified model boundary
• These new vertices will be added into the vertex
  list to be processed by the overall procedure

Blue dot A new boundary vertex introduced
during re-triangulation The new vertex
is easier to be snapped since it is close to
classified model boundary
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Examples - torus with holes

• The model, a coarse initial mesh and the mesh
  after multiple refinement/coarsening
• The spatial field of desired size
• Number of vertices snapped in refinement/coarsenin
  g iterations

z
x
y
This example did not require local
re-triangulation
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Examples - sleeve of ball bearing

• The model, a coarse initial mesh and the mesh
  after multiple refinement/coarsening
• The spatial field of desired size
• of vertices snapped in refinement/coarsening
  iterations

x
y
z
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Closing Remarks

• The key complexity of mesh generation and mesh
  adaptation is with dealing with curved geometry
• Properties that hold for polygonal domains do not
  provide complete set needed for curved domains
• Key features of presented approach
• Vertex can be snapped incrementally. Each
  intermediate target must pass at least first
  problem plane
• Boundary cavity construction and re-triangulation
  algorithm ensures the snapping of a given vertex
• Potential challenge
• New vertices are possibly introduced during
  re-triangulation
• To date, it has not been a problem in examples
  tested
• Efforts to complete qualification remain