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Mesh Smoothing Challenges in the Industry

Mesh Smoothing Challenges in the Industry. S mooth mixed meshes M ove nodes in all dimensions (coupled/uncoupled) O ffer enough user-control O btain higher mesh quality T est for invalid mesh H andle structured/mapped mesh regions. What is Variational Smoothing ?.

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Mesh Smoothing Challenges in the Industry

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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 What is a Mesh Unit ?

  6. 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

  7. 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-Laplace

  8. 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

  9. SMOOTHING Schemes: Equipotential Smoothing Original Mesh After Laplacian smoothing After Winslow smoothing

  10. SMOOTHING Schemes: Equipotential Smoothing Original mapped mesh Mesh after tangling

  11. SMOOTHING Schemes: Equipotential Smoothing After Laplace Smoothing After Winslow Smoothing Initial tangled mesh

  12. 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

  13. Mesh 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

  14. Mesh Unit = 514 Incenter/Iso-Laplace Angle smoothing Mesh Unit = 624 Incenter/Iso-Laplace smoothing Mesh Unit = 413 Incenter-Iso-Laplace smoothing Mesh Units: Mixed

  15. SMOOTHING Schemes: Handling Bivalent nodes

  16. 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 Smart Smoothing Constraints

  17. 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

  18. 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 Smoothing Boundary-Morphed Orphaned Shell Meshes

  19. 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

  20. Morphing And Remeshing On Legacy FEM : After preliminary Morphing Steps After preliminary morphing Original mesh

  21. After Laplace smoothing Mesh Quality No.= 0.440 After variational smoothing Mesh Quality No. = 0.653 Morphing And Remeshing On Legacy FEM : 3D smoothing the morphed mesh

  22. 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

  23. 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

  24. 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

  25. Acknowledgements • Jean Cabello • Michael Hancock

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