Anisotropic Voronoi diagrams and Delaunay Meshes Mariette Yvinec - PowerPoint PPT Presentation

Anisotropic Voronoi diagrams and Delaunay Meshes Mariette Yvinec INRIA Sophia Antipolis Geometric Computing Workshop Heraklion, January 2013 MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 1 / 49

Introduction Motivation and Definitions The need for anisotropy What is an anisotropic simplicial mesh ? • a mesh with simplicial elements • elongated according to prescribed directions. • Required anisotropy is described at each point using an anisotropic metric • with spatial variations: metric field Why anisotropic meshes ? ◮ reduce interpolation error: anisotropy according to Hessian ◮ adaptative solving of PDE for anisotropic phenomenon ◮ accurate surface discretisation anisotropy according to curvature tensor Images from Adrien Loiselle s Phd MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 2 / 49

Introduction Motivation and Definitions How to prescribe anisotropy Metric field each point z of the domain − → M ( z ) a positive definite quadratic matrix M -distance between two points a and b : � ( a − b ) t M ( a − b ) d M ( a , b ) = ◮ Fonction approximation Metric field related to the Hessian of the function ◮ Approximation of surfaces Metric field related to the normal field and to the second fundamental form (principal curvatures) ◮ Adaptative FEM for PDE Metric field related to the error estimation on previous iteration solution MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 3 / 49

Introduction Previous work Previous Work Many heuristics for anisotropic simplicial 2D or 3D meshes • Ellipses packing [ Li et al. 99 ] , [ Yamakawa Shimada 03 ] • Anisotropic Delaunay refinement [ Borouchaki et al. 97 ] [ Frey Alauzet 04 ] , [ Dobrzynski Frey 08 ] • Continuous mesh [ Loseille Alauzet 09 ] • Anisotropic mesh optimization myciteBossen Heckbert 96, [ Li et al. 05 ] More heuristics for surface meshes • Alliez et al (03), based on prinicipal curvature lines • Jiao et al (06), anisotropic adaptation using Garland Heckbert quadratic error • Azernikov Fischer 05 , grid based MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 4 / 49

Introduction Previous work Previous Work (cont’d) Voronoi approaches • Voronoi diagram on Riemannian manifold [ Leibon Letscher 00 ] , [ Bouglueux et al. 08 ] • Anisotropic Voronoi diagram [ Labelle Shewchuk 03 ] , [ Boissonnat et al. 05 ] [ Du Wang 05 ] [ Cheng at al. 06 ] surface meshes based on 3D AVD Delaunay approach • Anisotropic Delaunay meshes The locally uniform anisotropic Delaunay meshes approach [ Boissonnat et al. 10 ] , [ Boissonnat et al. (to appear) ] MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 5 / 49

Outline Introduction Motivation and Definitions Previous work Anisotropic Voronoi diagrams Meshing Algorithm I Meshing Algorithm II Anisotropic Delaunay meshes Meshing Algorithm Proof of the Algorithm Termination Anisotropic surface meshes MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 6 / 49

Anisotropic Voronoi diagrams Definition of Anisotropic Voronoi Diagrams [Labelle and Shewchuk 03] Let D ⊂ R d be a domain with a metric field defined on D : ∀ p ∈ D − → M p . � ( x − y ) t M p ( x − y ) d p ( x , y ) = Anisotropic Voronoi diagram P a set of sites in D ∀ p ∈ P , Voronoi cell V ( p ) V ( p ) = { x ∈ R d : d p ( p , x ) ≤ d q ( q , x ) , ∀ q ∈ P , q � = p } Bisector of { p,q } is a conic: ( x − p ) t M p ( x − p ) = ( x − q ) t M q ( x − q ) The Du et Wang variant V ( p ) = { x ∈ R d : d x ( p , x ) ≤ d x ( q , x ) , ∀ q ∈ P , q � = p } MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 7 / 49

Anisotropic Voronoi diagrams Definition of Anisotropic Voronoi Diagrams [Labelle and Shewchuk 03] ◮ Each site is within its cells ◮ Cells may be not connected. ◮ The dual need not be a triangulation. MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 8 / 49

Anisotropic Voronoi diagrams Meshing Algorithm I Meshing Algorithm from AVD [ Labelle and Shewchuk 03] Summary of the algorithm ◮ Anisotropic Voronoi diagram computed as a lower enveloppe: { f i ( x ) = ( x − p i ) t M p i ( x − p i ) } ◮ refine the Voronoi diagram by adding new sites on bisectors ◮ until bisectors are straigth enough , so that Voronoi cells are connected and the dual is a triangulation. Remarks ◮ Terminates and works only in 2D ◮ No need to maintain the exact anisotropic Voronoi diagram : a loose version is enough, where only the main connected component of each new site is computed. MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 9 / 49

Anisotropic Voronoi diagrams Meshing Algorithm II Anisotropic Voronoi diagrams from power diagrams Boissonnat et al. Linearization of the distance function x t = ( x , y , x 2 , xy , y 2 ) ˆ x ( x , y ) − → x , ˆ ˆ i = ( M i p i , − M xx i , − 1 / 2 M xy i , − M yy p t site( p i , M i ) − → p i , ˆ i ) d p i ( x , p i ) 2 x t M i x − 2 p t i M i x + p t = i M i p i t ˆ x + p t = − 2 ˆ p i i M i p i Anisotropic Voronoi diagram from a power diagram ◮ Compute the power diagram, in dimension d + d ( d + 1) / 2 = 5, p i � 2 − p t � of Σ = { σ i ( ˆ p i , � ˆ i M i p i ) , i = 1 · · · n } ◮ Intersect it with the 2 manifold M = { x , y , x 2 , xy , y 2 } ◮ Project the result in R 2 MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 10 / 49

Anisotropic Voronoi diagrams Meshing Algorithm II Meshing Algorithm from AVD II Boissonnat et al. Algorithms ◮ Maintain the power diagram V (Σ) of Σ in dimension 5, ◮ Maintain the set of vertices of the intersection V (Σ) � M ◮ Refine inserting new sites at Voronoi vertices until the set of dual triangles form a triangulation and are well shaped (according to local metrics) Remarks ◮ Terminates and works in 2D ◮ Problems for termination in higher dimension MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 11 / 49

Anisotropic Delaunay meshes Anisotropic Metric and Space Transform Metric definition An anisotropic metric in R d is defined in some basis by a symmetric positive definite d × d matrix M M -distance between two points a and b � ( a − b ) t M ( a − b ) d M ( a , b ) = F M ↓ Associated space transform ∃ matrix F M such that det( F M ) > 0 and F t M F M = M . � ( a − b ) t F t d M ( a , b ) = M F M ( a − b ) = � F M ( a − b ) � MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 12 / 49

Anisotropic Delaunay meshes Delaunay triangulation in a uniform anisotropic metric metric M � ( a − b ) t M ( a − b ) M -distance, d M ( a , b )= = � F M ( a − b ) � M -Volume M -sphere: C M ( c , r ) = { x : d M ( c , x ) = r } M -ball: B M ( c , r ) = { x : d M ( c , x ) ≤ r } M -balls are ellipses F M − → with axes along eigenvectors of M M -circumball of a k -simplex τ : among the M -ball circumscribing τ the one with smallest radius. Delaunay triangulation Del M ( V ): the M -circumball of each d -simplex is empty F − 1 M Del ( F M ( V )) − → Del M ( V ) MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 13 / 49

Anisotropic Delaunay meshes Simplex Quality in Anisotropic Metric A k -simplex τ in a metric M C M ( τ )( c M ( τ ) , r M ( τ )) the M -circumsphere of τ e M ( τ ) the shortest edge ρ M ( τ ) = r M ( τ ) e M ( τ ) the radius-edge ratio � 1 k the sliverity ratio. � Vol M ( τ ) σ M ( τ ) = e k M ( τ ) Slivers Let ρ 0 and σ 0 be two constants. With respect to the metric M , the k -simplex τ is: • well shaped if ρ M ( τ ) ≤ ρ 0 and σ M ( τ ) ≥ σ 0 • a sliver if ρ M ( τ ) ≤ ρ 0 and σ M ( τ ) ≤ σ 0 • a k -sliver if it is a sliver and all its ( k − 1)-faces are well-shaped. MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 14 / 49

Anisotropic Delaunay meshes Locally Uniform Anisotropic Delaunay Meshes A mesh such that: the star of each vertex is Delaunay and well shaped wrt the metric at that vertex. V set of vertices, v ∈ V : M v metric at v Del v ( V ) Delaunay triangulation of V computed with metric M v S v : the star of v in Del v ( V ) MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 15 / 49

Anisotropic Delaunay meshes Inconsistencies x S w v ( vwx ) w S v v y w ( wxy ) Inconsistency : some simplex τ with vertices { v , w , . . . } appears in star S v but not in star S w . MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 16 / 49

Anisotropic Delaunay meshes Meshing Algorithm Overview of the meshing algorithm ◮ Maintain the set of stars S ( V ) = { S v : v ∈ V } ◮ Refine V until stars are well shaped and consistent. ◮ Consistent stars are stitched MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 17 / 49