PowellSabin spline based multilevel preconditioners for 4th order - PowerPoint PPT Presentation

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin spline based multilevel preconditioners for 4th order elliptic equations on the sphere Jan Maes, Adhemar Bultheel Department of Computer Science Katholieke Universiteit Leuven Bremen, November 9, 2006

Introduction Powell–Sabin splines Numerical simulation with PS splines References Outline Introduction 1 Powell–Sabin splines 2 The space of Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Numerical simulation with PS splines 3 The biharmonic equation in the plane The hierarchical basis preconditioner Optimal multilevel preconditioners The biharmonic equation on the sphere References 4

Introduction Powell–Sabin splines Numerical simulation with PS splines References Introduction

Introduction Powell–Sabin splines Numerical simulation with PS splines References The biharmonic equation: plate bending problems � ∂ 2 � � ∂ 2 � ∂ x 2 + ∂ 2 ∂ x 2 + ∂ 2 ∇ 4 u ( x , y ) = u ( x , y ) = f ( x , y ) ∂ y 2 ∂ y 2 u ( x , y ) is the vertical displacement due to an external force. boundary conditions u = 0 ∂ u ∂ n = 0

Introduction Powell–Sabin splines Numerical simulation with PS splines References Design of car windscreens

Introduction Powell–Sabin splines Numerical simulation with PS splines References Design of aircrafts Courtesy of University of Bath

Introduction Powell–Sabin splines Numerical simulation with PS splines References Comparison with a classical method Error × 1000 � �� � Classical method Optimal PS spline based multilevel preconditioner

Introduction Powell–Sabin splines Numerical simulation with PS splines References Plate bending in a spherical geometry Physical geodesy Oceanography Meteorology Earth dynamics

Introduction Powell–Sabin splines Numerical simulation with PS splines References The pole problem Spherical coordinates give rise to the “pole problem” Therefore we will explore a different approach based on homogeneous polynomials in R 3 .

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin splines

Introduction Powell–Sabin splines Numerical simulation with PS splines References Splines as mathematical building blocks                                             

Introduction Powell–Sabin splines Numerical simulation with PS splines References Bernstein–Bézier representation = ⇒ Pierre Étienne Bézier (1910-1999)

Introduction Powell–Sabin splines Numerical simulation with PS splines References Stitching together Bézier triangles = ⇒ No C 1 continuity at the red curve

Introduction Powell–Sabin splines Numerical simulation with PS splines References C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Introduction Powell–Sabin splines Numerical simulation with PS splines References C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Introduction Powell–Sabin splines Numerical simulation with PS splines References C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Introduction Powell–Sabin splines Numerical simulation with PS splines References The dimension of S 1 2 (∆ PS ) ? There is exactly one solution s ∈ S 1 2 (∆ PS ) to the Hermite interpolation problem s ( V i ) = α i , ∀ V i ∈ ∆ , i = 1 , . . . , N . D x s ( V i ) = β i , D y s ( V i ) = γ i , The dimension of S 1 2 (∆ PS ) is 3 N . Therefore we need 3 N basis functions.

Introduction Powell–Sabin splines Numerical simulation with PS splines References The dimension of S 1 2 (∆ PS ) ? There is exactly one solution s ∈ S 1 2 (∆ PS ) to the Hermite interpolation problem s ( V i ) = α i , ∀ V i ∈ ∆ , i = 1 , . . . , N . D x s ( V i ) = β i , D y s ( V i ) = γ i , The dimension of S 1 2 (∆ PS ) is 3 N . Therefore we need 3 N basis functions.

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles Three locally supported basis functions per vertex

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles The control triangle is tangent to the PS spline surface.

Introduction Powell–Sabin splines Numerical simulation with PS splines References Powell–Sabin B-splines with control triangles It ‘controls’ the local shape of the spline surface.

Introduction Powell–Sabin splines Numerical simulation with PS splines References Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 \ { 0 } ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of the unit sphere S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Introduction Powell–Sabin splines Numerical simulation with PS splines References Spherical Powell–Sabin splines s ( v i ) = f i , D g i s ( v i ) = f gi , D h i s ( v i ) = f hi , ∀ v i ∈ ∆ has a unique solution in S 1 2 (∆ PS )

Introduction Powell–Sabin splines Numerical simulation with PS splines References 1 − 1 connection with bivariate PS splines ⇒ | v | 2 B ij ( v | v | ) ⇒ ← − Spherical PS B- piecewise trivari- Restriction to the spline B ij ( v ) ate polynomial of plane tangent to degree 2 that is S at v i ∈ ∆ homogeneous of degree 2

Introduction Powell–Sabin splines Numerical simulation with PS splines References 1 − 1 connection with bivariate PS splines Let T i be the plane tangent to S at vertex v i Radial projection: R i v := v := v | v | ∈ S , v ∈ T i ⊂ ∆ PS be its Define ∆ i as the star of v i in ∆ , and let ∆ PS i PS 6-split. Theorem Let s ∈ S 1 ) . Let s be the restriction of | v | 2 s ( v / | v | ) to T i . 2 (∆ PS i 2 ( R − 1 Then s is in S 1 ) and ∆ PS i i s ( v i ) = s ( v i ) , D g i s ( v i ) = D g i s ( v i ) , D h i s ( v i ) = D h i s ( v i ) .

Introduction Powell–Sabin splines Numerical simulation with PS splines References 1 − 1 connection with bivariate PS splines Let T i be the plane tangent to S at vertex v i Radial projection: R i v := v := v | v | ∈ S , v ∈ T i ⊂ ∆ PS be its Define ∆ i as the star of v i in ∆ , and let ∆ PS i PS 6-split. Theorem Let s ∈ S 1 ) . Let s be the restriction of | v | 2 s ( v / | v | ) to T i . 2 (∆ PS i 2 ( R − 1 Then s is in S 1 ) and ∆ PS i i s ( v i ) = s ( v i ) , D g i s ( v i ) = D g i s ( v i ) , D h i s ( v i ) = D h i s ( v i ) .

Introduction Powell–Sabin splines Numerical simulation with PS splines References Spherical B-splines with control triangles

Introduction Powell–Sabin splines Numerical simulation with PS splines References Applications on a spherical domain Approximation of a mesh: consider the triangles of the original triangular mesh as control triangles of a PS spline.