Topics in Computer Aided Geometric Design, Erice May 12-19, 1990 - PowerPoint PPT Presentation

Topics in Computer Aided Geometric Design, Erice May 12-19, 1990

Smooth Simplex Splines for the Powell-Sabin 12-split Tom Lyche Centre of Mathematics for Applications, Department of Mathematics, University of Oslo September 28, 2013

Outline Simplex splines A quadratic simplex spline basis for PS12 on one triangle Higher degree splines on the 12-split

PART I: simplex splines

Schoenberg’s View of the Bivariate B-Spline In a letter from Iso Schoenberg to Phillip Davis from 1965: “ A sketch of the spline function z = M ( x , y ; z 0 , z 1 , z 2 , z 3 , z 4 )”

Simplex spline definitions ◮ geometric ◮ variational ◮ recurrence relation

Recurrence Relation � � ∆ n f ( v 0 x 0 + v 1 x 1 + · · · + v n x n ) dv 1 · · · dv n f := [ x 0 ,..., x n ]

Bivariate simplex spline properties ◮ Let X be a collection of d + 3 points x 1 , . . . , x d +3 in R 2 ◮ A simplex spline S = S [ X ] : R 2 → R with knots X is a nonnegative piecewise polynomial ◮ the degree is d ◮ the support is the convex hull of X ◮ the knotlines are the edges in the complete graph of X ◮ a knot line has multiplicity m if it contains m + 1 of the points in X ◮ S ∈ C d − m across a knotline of multiplicity m

Some simplex spline spaces ◮ Triangulate a slab , de Boor, 1976. ◮ Complete Configurations , Hakopian[1981], Dahmen, Michelli[1983], ◮ Pull apart , Dahmen, Micchelli[1982], H¨ ollig[1982], Dahmen, Micchelli,Seidel[Erice 1990], ◮ Delaunay configurations , Neamtu[2000-2007]

”There is no clever way to implement the recurrence relation once the standard recipe for constructing spaces of simplex spline functions has been followed” Tom Grandine, 1987

What should be the space of Simplex splines on a triangulation?

PART II: A Simplex spline basis for PS12 on one triangle Cohen, E., T. Lyche, and R. F. Riesenfeld, A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines, Math. Comp., 82(2013), 1667-1707

The PS12-split (Powell,Sabin 1977)

The PS12-split

The PS12 spline space S 1 ) = { f ∈ C 1 ( 2 ( ) : f | ∆ i ∈ Π 2 , i = 1 , . . . , 12 } dim( S 1 2 (∆ PS 12 )) = 12

Computing with PS12 ◮ Bernstein-B´ ezier methods, ◮ FEM nodal basis, (Oswald) ◮ minimal determining set (Alfeld, Schumaker, Sorokina) ◮ subdivision (Dyn, Lyche, Davydov, Yeo) 3 A Hermite Sub division Scheme for PS-12 ◮ quadratic S(implex) - splines (Cohen, Lyche, Riesenfeld) 1. subdivision Initialization step Fig. 2. Sub dividing the PS-12 split elemen t. A circle around a v ertex means that b oth the function v alue and the gradien t are kno wn at that v ertex. 2.1. Initializati on. The �rst step in the computation of suc h an elemen t in v olv es the computation of its v alue and gradien t at the midp oin ts a; b; c of the triangle T , (see, Fig. 1). Here w e use the form ula f = ( f + f ) = 2 � ( r f � r f ) � ( A � C ) = 8 b A C A C for the function v alue at the midp oin t b of AC . F or the gradien t w e �rst compute the directional deriv ativ e in the direction AC at b ( A � C ) � r f = 2( f � f ) � ( r f + r f ) � ( A � C ) = 2 : b A C A C Com bining this v alue with the giv en v alue of the cross-deriv ativ e at b , w e can calculate r f . F or the other midp oin ts w e use similar form ulae. b These form ulae are obtained from the observ ation that along eac h side of 1 T the PS-12 split elemen t is a piecewise quadratic C -spline with a knot at the midp oin t. 2.2. The General Sub division Step. F or the �rst sub division step (see Fig. 1) w e use the follo wing form ulae: f = ( f + f ) = 2 ( r f r f ) ( b C ) = 8 � � � � e b C b C f = ( f + f ) = 2 � ( r f � r f ) � ( a � C ) = 8 g a C a C f = ( f + f ) = 2 � ( r f � r f ) � ( b � a ) = 8 d b a b a r f = ( r f + r f ) = 2 (1) e b C r f = ( r f + r f ) = 2 g a C ( a b ) r f = 2( f f ) ( r f + r f ) ( a b ) = 2 � � � � � � d a b a b ( C � d ) � r f = 2( f � f ) � r f � ( C � d ) : d C d C >F rom the last t w o v alues w e can solv e for r f . Similar form ulae are used d for the t w o other corner triangles Abc and B ca and w e obtain the v alues and gradien ts at lo cations sho wn to the righ t in Fig. 2. This pro cess can no w b e con tin ued for as man y lev els of re�nemen t as desired. Fig. 3 displa ys a PS-12 split elemen t obtained from random initial data. The implemen tation w as done using Mathematica.

3 corner S-splines for the quadratic case; support 1/4

6 half support S-splines for the quadratic case

3 trapezoidal support S-splines for the quadratic case

Properties ◮ Local linear independence, ◮ nonnegative partition of unity, ◮ stable recurrence relations, ◮ fast pyramidal evaluation algorithms, ◮ differentiation formula, ◮ L p stable basis, ◮ subdivision algorithms of Oslo- and Lane,Riesenfeld type, ◮ quadratic convergence of control mesh, ◮ well conditioned collocation matrices for Lagrange and Hermite interpolation, ◮ explicit dual functionals, ◮ dual polynomials and Marsden-like identity.

Marsden-like identity Univariate quadratic: (1 − yx ) 2 = � j B j , 2 ( x )(1 − yt j +1 )(1 − yt j +2 ) Quadratic S-splines: x ∈ ∆ , y ∈ R 2 12 (1 − y T x ) 2 = � S j , 2 ( x )(1 − y T p ∗ j , 1 )(1 − y T p ∗ j , 2 ) . j =1 3 � 6 � 5 6 9 5 � 12 � 11 � 7 � 10 10 � 1 7 8 � 4 � 8 � 9 � 2 � 3 1 4 2 [ p ∗ 1 , 1 , . . . , p ∗ 12 , 1 ] := [ p 1 , p 1 , p 4 , p 4 , p 2 , p 2 , p 5 , p 5 , p 3 , p 3 , p 6 , p 6 ] , [ p ∗ 1 , 2 , . . . , p ∗ 12 , 2 ] := [ p 1 , p 4 , p 10 , p 2 , p 2 , p 5 , p 10 , p 3 , p 3 , p 6 , p 10 , p 1 ] , ◮ 1 = � 12 j =1 S j , 2 ( x ) , x ∈ ∆ , ◮ x = � 12 m j := ( p ∗ j , 1 + p ∗ j =1 S j , 2 ( x ) m j , x ∈ ∆ , j , 2 ) / 2 .

domain- and control mesh The control points are at a distance O ( h 2 ) from the surface, where h is the longest side of the triangle..

Dual functionals Univariate quadratic: λ j f := 2 f ( t j +3 / 2 ) − 1 2 f ( t j +1 ) − 1 2 f ( t j +2 ) , λ i B j , 2 = δ ij Quadratic S-splines: λ j f := 2 f ( m j ) − 1 j , 1 ) − 1 2 f ( p ∗ 2 f ( p ∗ j , 2 ) λ i S j , 2 = δ ij

C 1 smoothness is controlled as in the polynomial B´ ezier case. 5 5 6 6 4 4 8 7 3 7 9 3 8 10 11 2 2 11 9 10 12 12 1 1

PART III: Higher degree splines on the 12-split Joint work with Georg Muntingh

Smooth splines on triangulation ◮ Consider a general triangulation in the plane ◮ consider a subdivided triangulation with the 12-split on each triangle and a piecewise polynomial of degree d on this triangulation ◮ d = 2 necessary and sufficient for C 1 ◮ d = 5 necessary and sufficient for C 2

Dimensions of S r d ( ) S r ) = { f ∈ C r ( d ( ) : f | ∆ i ∈ Π d , i = 1 , . . . , 12 } Theorem For any integers d , r with d ≥ 0 and d ≥ r ≥ − 1 ) = 1 2( r + 1)( r + 2) + 9 dim S r d ( 2( d − r )( d − r + 1) d − r + 3 � 2( d − 2 r − 1)( d − 2 r ) + + ( r − 2 j + 1) + , j =1 (1) where z + := max { 0 , z } for any real z.

Proof One cell and three flaps. Use cell dimension formula in Lai-Schumaker book.

Dimensions of S r d ( ) d/r − 1 0 1 2 3 4 5 6 7 8 9 10 11 0 12 1 1 36 10 3 2 72 31 12 6 3 120 64 30 16 10 4 180 109 60 34 21 15 5 252 166 102 61 39 27 21 6 336 235 156 100 66 46 34 28 7 432 316 222 151 102 73 54 42 36 8 540 409 300 214 150 109 81 63 51 45 9 660 514 390 289 210 154 117 91 73 61 55 10 792 631 492 376 282 211 162 127 102 84 72 66 11 936 760 606 475 366 280 216 172 138 114 96 84 78

An interesting family on ◮ For any positive integer n consider on the spline space S 2 n − 1 3 n − 1 ( ) of splines of smoothness 2 n − 1 and degree 3 n − 1 . ◮ n = 1: C 1 quadratics ◮ n = 2: C 3 quintics ◮ n = 3: C 5 octic 2 n 2 + 9 ◮ dim S 2 n − 1 ) = 15 3 n − 1 ( 2 n .

Hermite degrees of freedom S 3 5 ( ) ◮ dim S 3 5 ( ) = 39 ◮ 10 derivatives at 3 corners ◮ 3 first order cross boundary derivatives ◮ 6 second order cross boundary derivatives ◮ Connects to neighboring triangles with smoothness C 2 .

Spline space on triangulation of smoothness C n ◮ Consider a triangulation in the plane ◮ use S 2 n − 1 3 n − 1 ( ) on each triangle ◮ get a global spline space of smoothness C n ◮ for n = 2 we get a C 2 spline space of dimension 10 V + 3 E ◮ for n ≥ 1 we get a C n spline space of dimension n (2 n + 1) V + 1 2 n ( n + 1) E

Simplex spline basis for S 3 5 ( ) 2 1 1 1 2 3 1 6 1 5 2 4 2 1 3 6 6 6 1 2 1 2 1 1 1 1 2 2 4 2 4 2 2 2 1 1 6 3 6 3

Simplex spline basis for S 3 5 ( ) 1.0 1.0 0.5 0.5 0.0 0.0 1.0 0.4 0.3 0.5 0.2 0.1 0.0 0.0 0.0 0.0 0.5 0.5 1.0 1.0 1.0 0.5 1.0 0.5 0.0 0.4 0.3 0.0 0.2 0.4 0.1 0.2 0.0 0.0 0.0 0.0 0.5 0.5 1.0 1.0