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Mesh Smoothing Challenges in the Industry

• Smooth mixed meshes
• Move nodes in all dimensions (coupled/uncoupled)
• Offer enough user-control
• Obtain higher mesh quality
• Test for invalid mesh
• Handle structured/mapped mesh regions

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What is Variational Smoothing ?

• This is a shell mesh smoothing technique in 3D
  space that combines a variety of conventional
  smoothing methods in an effort to reap the best
  benefits and prevent irreversible mesh
  distortions.
• The variational algorithm smoothes each node
  according to a specific smoothing technique
  defined by variational rules.
• The smoothing method selection depends on the
  mesh unit connected to a node.
• It is a hybrid and heuristic approach

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Problem Statement

• Smoothes shell meshes in 1D/2D/3D space
• Is iterative/ almost as efficient as Laplace
• Gives several controls to the user
• Tries to preserve mapped/structured meshes or
  mesh regions
• Works better than most smoothers in concave
  domains
• Rarely creates inverted elements
• Improves element included angles, average element
  skew and hence mesh quality
• Smoothed mesh may/may not be projected back to
  surface

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VARIATIONAL SMOOTHING MODEL

• The governing equation
• N
• Pi' ? Fn(C,V) ?n (C,V)
• n 1
• where Pi' New position of node i,
• Fn Variational weight factor
  for n-th element
• ?n Positional function for
  n-th element
• C Connectivity pattern of
  the node,
• V Nodal valency

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What is a Mesh Unit ?

• Mesh Unit Vtq
• V nodal valency
• q no of quads
• t no of triangles
• A mesh unit is defined by a node
• Number of elements converging at the node
• The topology of connecting elements

• Mesh Unit 303

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SMOOTHING Schemes Incenter Smoothing

• N
• Pi' Pi ? Wn(Pn - Pi)
• n 1
• Pi (x, y, z) is the position vector of node i
• Pn(x, y, z) is the incenter vector of element n
• N No. of elements at node i
• Initial Mesh After
  Laplacian Smoothing After Incenter
  smoothing

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SMOOTHING Schemes Isoparametric-Laplace

• 1 N
• Pi' ------------ ? Wn (Pnj Pnl - wPnk)
• N(2 w) n 1
• N no. of elements at node i
• w coupling factor, 0.0 - Laplace
• 1.0 -
  Isoparametric
• 0.5 - Iso-Laplace
• Laplace
  Isoparametric Isoparametric-Laplac
  e

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SMOOTHING Schemes Equipotential/Winslow Smoothing

• The governing equation for equipotential
  (Winslow) smoothing can written for node i as
• ?Pi?? - 2?Pi?? ?Pi?? 0
• where ?,? are logical variables that are harmonic
  in nature, while ?, ?, ? are constant
  coefficients that depend on the problem.
• The weighing factors of the 8 neighboring nodes
  are given by
• W1-?/2,W2?,W3?/2,W4?,W5-?/2,W6?,W7?/2,W8?
  
• where
• ? xp2 yp2 zp2
• ? xpxq ypyq zpzq
• ? xq2 yq2 zq2
• xp (x2 -x6)/2, yp (y2 - y6)/2, zp (z2 -
  z6)/2
• xq (x8 -x4)/2, yq (y8 - y4)/2, zq (z8 -
  z4)/2

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SMOOTHING Schemes Equipotential Smoothing

• Original Mesh
• 
  After Laplacian
• 
  smoothing
• After Winslow smoothing

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SMOOTHING Schemes Equipotential Smoothing

• Original mapped mesh Mesh
  after tangling

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SMOOTHING Schemes Equipotential Smoothing

• After Laplace Smoothing After
  Winslow Smoothing
• 
  
• 
  Initial tangled mesh

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Mesh Units All-Quad

• Mesh Unit 303
• Isoparametric-Laplace smoothing
• Length/angle-weighted Laplace
• Mesh Unit 404
• Equi-potential smoothing
• Mesh Unit 505
• Isoparametric-Laplace smoothing

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Mesh Units All Triangular

• Mesh Unit 660
• Incenter/AngleZhou Shimada/Laplace smoothing
• Mesh Unit 770
• Incenter/Laplace/Angle smoothing Zhou Shimada
  
• Mesh Unit 880
• Equi-potential smoothing

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Mesh Units Mixed

• Mesh Unit 413
• Incenter-Iso-Laplace smoothing

• Mesh Unit 514
• Incenter/Iso-Laplace
• Angle smoothing
• Mesh Unit 624
• Incenter/Iso-Laplace smoothing

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SMOOTHING Schemes Handling Bivalent nodes

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Smart Smoothing Constraints

• Angle check
• Check element included angles during smoothing
• Region Check
• Keep node inside the bounding box formed by the
  barycenters of the connected element

• Constrained node movement

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Smoothing Boundary-Morphed Orphaned Shell Meshes

• Mesh Quality No is defined as
• N
• MQ No (?Ei)1/N
• i1
• where Ei is the element quality number for
  element i
• It measures element skew, warp, stretch, aspect
  ratio and Jacobian
• Ei is non-dimensional and varies from 0 and 1.

• Shift this
  hole Shrink this hole

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Smoothing Boundary-Morphed Orphaned Shell Meshes

• Bad Mesh - MQN lt 0.4
• OK Mesh - 0.5 lt MQN gt 0.4
• Good Mesh - 0.6 lt MQN gt0.5
• Excellent Mesh - MQN gt 0.6
• Perfect Mesh - MQN 1.0
• After Variational
• smoothing
• Mesh Quality No.
• 0.504

• After Laplace smoothing
• Mesh Quality No. Invalid/Unsolvable mesh

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Morphing And Remeshing On Legacy FEM How
Variational Smoothing Helps

• Steps to morph
• shift hole
• gouge hole out
• stretch ends
• bend tail
• add new tail cut-outs for wiring access

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Morphing And Remeshing On Legacy FEM After
preliminary Morphing Steps

• After preliminary morphing
• Original mesh

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Morphing And Remeshing On Legacy FEM 3D
smoothing the morphed mesh

• After Laplace smoothing
• Mesh Quality No. 0.440

• After variational smoothing
• Mesh Quality No. 0.653

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Morphing And Remeshing On Legacy FEM
Refeaturing steps

• B-Rep is added to the
• raw morphed mesh
• New features (cut-outs for wiring access) are
  added

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Morphing And Remeshing On Legacy FEM 2D
smoothing during remesh

• Remeshed, re-smoothed morphed legacy FEM with
  new features
• - After Length-weighted smart Laplacian smoothing
• Mesh Quality No. 0.507
• -After Variational smoothing
• Mesh Quality No. 0.759

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Global Smoothing

• Mesh Quality before global smoothing (3700
  elements) 0.45 393 elements fail different
  element quality checks
• After length-weighted smart Laplacian smoothing
  Mesh Quality No. - 0.44 379 elements fail
• After variational smoothing Mesh Quality No. -
  0.59 210 elements fail

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Acknowledgements

• Jean Cabello
• Michael Hancock