From CAD surface models to quality meshes. Patrick LAUG. Projet GAMMA. INRIA Rocquencourt. Outline

Transcription

1 From CAD surface models to quality meshes Patrick LAUG Projet GAMMA INRIA Rocquencourt Tetrahedron II Oct Outline 1. Introduction B-Rep, patches 2. CAD repair ant topology recovery 3. Discretization of the boundary curves 4. Patch meshing 5. Mesh simplification 6. Application eamples 7. Conclusion and future work Tetrahedron II Oct

2 1. Introduction Tetrahedron II Oct Solid model in CAD systems Two major approaches to solid modeling: CSG(constructive solid geometry) The solid is defined using set operations on elementary blocks (cube, cylinder, sphere, ) B-Rep (boundary representation) The solid is defined by its boundary composed of patches topology and geometry of the curves and the surfaces CSG B-Rep Tetrahedron II Oct

3 Objective From a CSG or B-Rep representation, automatically generate a mesh for finite element computation Two cases CSG:voelization + marching cube B-Rep : direct method, or indirect method via the parameter domains Tetrahedron II Oct General scheme 1. CAD repair 2. Discretization of the boundary curves (in particular the interface curves) 3. Surface meshing of the patches 4. Volume meshing of the solid 5. Numerical simulation For an evolutive phenomenon and/or adaptive meshing, reiterate steps 2 5. Tetrahedron II Oct

4 2. CAD repair and topology recovery Tetrahedron II Oct CAD repair Overlapping or intersecting patches: Compute the intersection curves, report in parametric domains and rectify Gaps between neighboring patches: Build patches to fill the gaps Simple method for repairing if the defaults are small sized Tetrahedron II Oct

5 References of points and curves Initial numbering of points and curves Numbers reflecting the topology of conforming patches project points split curves Tetrahedron II Oct Conforming mesh Tetrahedron II Oct

6 3. Discretization of the boundary curves Tetrahedron II Oct Formal definition of the problem ( t) γ : I = [ a, b] Γ, t a γ ( t) = y( t) (in R 3 ) z( t) Leibniz formula : ds = dγ equivalent to : t a s( t) = γ& ( u) du Method for discretization (uniform in this eample) 1. Compute the total length L = s(b) of curve Γ 2. Determine parameters t i such that s(t i ) = i (L / n), i = 0,, n Tetrahedron II Oct

7 General method Given: uniform, isotropic or anisotropic specifications Discretization without or with preliminary construction of a polygonal support such that linear interpolations are valid Report into the parametric domains by linear interpolation Tetrahedron II Oct Application DASSAULT AVIATION Falcon business jet: ε 1 = 2 Tetrahedron II Oct

8 4. Patch meshing Tetrahedron II Oct Model for each patch Parameterized surface v (u,v) z y u Tetrahedron II Oct

9 Problem to solve Generate a surface mesh respecting isotropic or anisotropic specifications Isotropic triangle in 3D Proposed solution Idea: mesh the parametric domain ( fast and robust algorithms) Utilization of metrics Anisotropic triangle in 2D Tetrahedron II Oct Some more details Formal definition of a surface : Ω Σ, ( u, v) u, v a ( u, v) = y( u, v) z( u, v) From a 3D metric defined on Σ, a 2D metric defined on Ω can be deduced using the first derivatives u and v A planar mesh is then constructed with a combined advancig front / Delaunay approach in a Riemannian contet Tetrahedron II Oct

10 Tetrahedron II Oct Combined approach: Advancing front and Delaunay Tetrahedron II Oct Metric induced in the parametric domain ( ) v u T v T u M M 3 ~ = Anisotropic case Isotropic case ( ) v u T v T u I P h M 3 2 ) ( 1 ~ = M P h P h M v v u v v u u u ) ( 1,,,, ) ( 1 ~ 2 2 = = =

11 5. Mesh simplification Tetrahedron II Oct Mesh simplification Global elimination of small edges (w.r.t. the size specifications) Geometry preserved using Hausdorff distances and angle deviations Patch dependent or patch independent meshing Tetrahedron II Oct

12 6. Application eamples Surface mesher: BLSURF Tetrahedron II Oct Application eample: Falcon DASSAULT AVIATION Tetrahedron II Oct

13 Application eample: Falcon DASSAULT AVIATION Uniform mesh Tetrahedron II Oct Application eample: Falcon DASSAULT AVIATION Geometric mesh Tetrahedron II Oct

14 Application eample: Falcon DASSAULT AVIATION Geometric mesh with gradation Tetrahedron II Oct CAD integration BLSURF plugged in TopSolid, Catia, Open Cascade, Import several file formats: IGES, STEP, Tetrahedron II Oct

15 CAD integration Tetrahedron II Oct CAD integration Tetrahedron II Oct

16 Car decklid Tetrahedron II Oct Car decklid Tetrahedron II Oct

17 Molecular surfaces Tetrahedron II Oct Conclusion and future work Tetrahedron II Oct

18 Conclusion and future work A methodology for meshing a surface composed of parameterized patches has been presented Various applications have proven the efficiency of this method Future work: difficult surfaces including thin or neighboring patches, degenerated or unsteady parameterizations, very large meshes, etc. Tetrahedron II Oct