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