What Makes for a Good Mesh? CS101 – Meshing Winter 2007 1 Mesh Quality What makes a mesh good? � application dependent � some general principles � PL approximation (triangles/tets) � faithful to geometry � values � gradients � simulation: numerical error CS101 – Meshing Winter 2007 2
Judging Quality Approximation � given: some smooth function � best PL approximation � distance: Hausdorff Leif Kobbelt CS101 – Meshing Winter 2007 3 Simplified Setting Approximating a function � over a simplicial complex � values at vertices � how close is linear interpolant to underlying function? � element shape and size CS101 – Meshing Winter 2007 4
Criteria Setup � f given; g is PL interp. � interpolation error Size very important; shape not so much Jonathan Shewchuk � gradient interpolation error Size important; okay bad angles very important � stiffness matrix bad okay CS101 – Meshing Winter 2007 5 Assumptions Curvature (directional) � assumed to be bounded � then over element t weaker but simpler CS101 – Meshing Winter 2007 6
Minimum Containment How to find circle/sphere? � if circumcenter interior to triangle � circumradius � else: half the edge length of edge with circumcenter on “wrong” side � tet: if circumcenter inside: r circ � else: r mc of triangles (1 or 2) with circumcenter on wrong side CS101 – Meshing Winter 2007 7 Gradients Values not enough CS101 – Meshing Winter 2007 8
Gradients Accuracy important as well � discretization error in FEM � mechanics: ∇ f represents strains in circle edge lengths area Jonathan Shewchuk circumcircle CS101 – Meshing Winter 2007 9 Tetrahedra Error bounds � approximation � gradient Jonathan Shewchuk Good Bad CS101 – Meshing Winter 2007 10
Delaunay Optimality For a given set of vertices � many triangulations � for any dimension � Delaunay min. largest r mc � in 2D Delaunay min. largest r circ CS101 – Meshing Winter 2007 11 Stiffness Matrix Condition number � accuracy of linear algebra � iterative solvers are slower � all solvers less accurate � ratio of largest to smallest eigen value � model problem: CS101 – Meshing Winter 2007 12
Conditioning Global stiffness matrix � max eigen value � dominated by single worst element � depends on shape � 2D indep. of size; 3D largest ele. domin. � min eigen value � relatively independent of shape � proportional to areas/volumes of ele. CS101 – Meshing Winter 2007 13 Poisson Equation Specific for each case � max eigen value matters most � prefers equilateral triangles � small angles bad � simpler: CS101 – Meshing Winter 2007 14
3D Case Still Poisson equation � need to solve cubic equation… � � smallest for equilateral tets � dihedral angles relevant � small ones ok if opposing edge long Good Bad Jonathan Shewchuk CS101 – Meshing Winter 2007 15 So Far: Isotropic Adapt to function � when function is known � or anisotropy in PDE � curvature variation CS101 – Meshing Winter 2007 16
Anisotropic Meshes Deform to isotropic space � judge Et � careful with gradients Actually: they are ok, but have to be � big nasty expressions carefully aligned � good element if Et has no large angles CS101 – Meshing Winter 2007 17 Anisotropic Meshes Direction matters now � not-/aligned with principal curvature directions � “equilateral” now with � simplices may not be the best primitives anymore… CS101 – Meshing Winter 2007 18
Quality Measures How to optimize a mesh? � use reciprocal � differentiability matters � don’t want gradient to vanish for the near bad elements… � types of quality measures � long catalogues… CS101 – Meshing Winter 2007 19 Types of Quality Ms. Scale invariant � separate size and shape � can be misleading… � error depends on size too � smaller may be worse shape & OK Size and shape � all effects in one number � more specific to application CS101 – Meshing Winter 2007 20
Error & Quality Mesh refinement � refine element if error too large � Delaunay refinement for bad shape Mesh smoothing � optimize placement of vertices � differentiability matters now CS101 – Meshing Winter 2007 21
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