Nilanjan Mukherjee
Meshing & Abstraction GroupEDS PLM Solutions, Milford, OH
A Hybrid, Variational 3D Smoother for Orphaned Shell Meshes
Mesh Smoothing Challenges in the IndustryMesh 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
What is Variational Smoothing ?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
Problem StatementProblem 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
VARIATIONAL SMOOTHING MODELVARIATIONAL 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
What is a Mesh Unit ?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
SMOOTHING Schemes: Incenter SmoothingSMOOTHING Schemes: Incenter Smoothing
NPi
' = 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
SMOOTHING Schemes: Isoparametric-LaplaceSMOOTHING 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-Laplace
SMOOTHING Schemes: Equipotential/Winslow SmoothingSMOOTHING Schemes: Equipotential/Winslow Smoothing
• The governing equation for equipotential (Winslow) smoothing can written for node i as
Pi - 2Pi + 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 + yp
2 + zp2
= xpxq + ypyq + zpzq
= xq2 + yq
2 + 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
SMOOTHING Schemes: Equipotential SmoothingSMOOTHING Schemes: Equipotential Smoothing
Original Mesh
After Laplacian
smoothing
After Winslow smoothing
SMOOTHING Schemes: Equipotential SmoothingSMOOTHING Schemes: Equipotential Smoothing
Original mapped mesh Mesh after tangling
SMOOTHING Schemes: Equipotential SmoothingSMOOTHING Schemes: Equipotential Smoothing
After Laplace Smoothing After Winslow Smoothing
Initial tangled mesh
Mesh Units: All-QuadMesh 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
Mesh Units: All TriangularMesh Units: All Triangular
Mesh Unit = 660
Incenter/Angle[Zhou & Shimada]/Laplace smoothing
Mesh Unit = 770
Incenter/Laplace/Angle smoothing[ Zhou & Shimada ]
Mesh Unit = 880
Equi-potential smoothing
Mesh Units: MixedMesh Units: Mixed
Mesh Unit = 514
Incenter/Iso-Laplace
Angle smoothing
Mesh Unit = 624
Incenter/Iso-Laplace smoothing
Mesh Unit = 413
Incenter-Iso-Laplace smoothing
SMOOTHING Schemes: Handling Bivalent nodesSMOOTHING Schemes: Handling Bivalent nodes
Smart Smoothing ConstraintsSmart Smoothing Constraints
• Constrained node movement Angle check
• Check element included angles during smoothing
Region Check
• Keep node inside the bounding box formed by the barycenters of the connected element
Smoothing Boundary-Morphed Orphaned Shell MeshesSmoothing Boundary-Morphed Orphaned Shell Meshes
Shift this hole Shrink this hole
Mesh Quality No is defined as
N
MQ No = (Ei)1/N
i=1
• 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.
Smoothing Boundary-Morphed Orphaned Shell Meshes Smoothing Boundary-Morphed Orphaned Shell Meshes
After Laplace smoothing
Mesh Quality No. = Invalid/Unsolvable mesh
Bad Mesh - MQN <= 0.4
OK Mesh - 0.5 <= MQN > 0.4
Good Mesh - 0.6 <= MQN >0.5
Excellent Mesh - MQN > 0.6
Perfect Mesh - MQN = 1.0
After Variational
smoothing
Mesh Quality No.=
0.504
Morphing And Remeshing On Legacy FEM : How Variational Smoothing HelpsMorphing 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
Morphing And Remeshing On Legacy FEM : After preliminary Morphing StepsMorphing And Remeshing On Legacy FEM : After preliminary Morphing Steps
After preliminary morphing
Original mesh
Morphing And Remeshing On Legacy FEM : 3D smoothing the morphed meshMorphing 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
Morphing And Remeshing On Legacy FEM : Refeaturing stepsMorphing 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
Morphing And Remeshing On Legacy FEM : 2D smoothing during remeshMorphing 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
Global SmoothingGlobal 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
AcknowledgementsAcknowledgements
• Jean Cabello
• Michael Hancock