Generalized B-splines and local refinements Carla Manni Department - PowerPoint PPT Presentation

Unifying approach: Bernstein-like basis Ex: < 1 , t, u ( t ) , v ( t ) > ( ≃ cubics ) u, v ∈ C 2 , t ∈ [0 , 1] ONTP/Bernstein-like basis { B 0 , B 1 , B 2 , B 3 } : B 0 (1) = B 0 ′ (1) = B 0 ′′ (1) = 0 C 2 ⇒ easy to characterize/construct B 1 (0) = B 1 (1) = B 1 ′ (1) = 0 B 2 (0) = B 2 ′ (0) = B 2 (1) = 0 B 3 (0) = B 3 ′ (0) = B 3 ′′ (0) = 0 control points: (0 , b 0 ) , ( ξ, b 1 ) , (1 − η, b 2 ) , (1 , b 3 ) , 0 < ξ < 1 − η < 1 , control polygon describes s ( t ) = � 3 j =0 b j B j ( t ) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Generalized B-splines and local refinements – p. 9/50

Unifying approach: Bernstein-like basis Ex: < 1 , t, u ( t ) , v ( t ) > ( ≃ cubics ) u, v ∈ C 2 , t ∈ [0 , 1] ONTP/Bernstein-like basis { B 0 , B 1 , B 2 , B 3 } : B 0 (1) = B 0 ′ (1) = B 0 ′′ (1) = 0 C 2 ⇒ easy to characterize/construct B 1 (0) = B 1 (1) = B 1 ′ (1) = 0 B 2 (0) = B 2 ′ (0) = B 2 (1) = 0 B 3 (0) = B 3 ′ (0) = B 3 ′′ (0) = 0 control points: (0 , b 0 ) , ( ξ, b 1 ) , (1 − η, b 2 ) , (1 , b 3 ) , 0 < ξ < 1 − η < 1 , control polygon describes s ( t ) = � 3 j =0 b j B j ( t ) 2 1.8 1.6 1.4 1.2 1 0.8 properties of s by its control polygon 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Generalized B-splines and local refinements – p. 9/50

Unifying approach: P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > I Generalized B-splines and local refinements – p. 10/50

Unifying approach: P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p Generalized B-splines and local refinements – p. 10/50

Unifying approach: P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p < D p − 1 u, D p − 1 v > Chebyshev in [0 , 1] and Extended Chebyshev in (0 , 1) Generalized B-splines and local refinements – p. 10/50

Unifying approach: ONTP-basis P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p < D p − 1 u, D p − 1 v > Chebyshev in [0 , 1] and Extended Chebyshev in (0 , 1) ⇓ P u,v possesses a ONTP-basis I p Generalized B-splines and local refinements – p. 10/50

Unifying approach: ONTP-basis P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p < D p − 1 u, D p − 1 v > Chebyshev in [0 , 1] and Extended Chebyshev in (0 , 1) ⇓ P u,v possesses a ONTP-basis I p Ex: u, v : trigonometric functions u, v : exponential functions u, v : variable degree .... Generalized B-splines and local refinements – p. 10/50

Unifying approach: ONTP-basis P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p < D p − 1 u, D p − 1 v > Chebyshev in [0 , 1] and Extended Chebyshev in (0 , 1) ⇓ P u,v possesses a ONTP-basis I p Bernstein-like representations [Goodman, T.N.T., Mazure, M.-L., JAT, 2001] [Mainar, E., Pe˜ na, J.M., S´ anchez-Reyes, J, CAGD 2001] [Carnicer, Mainar, Pe˜ na; CA 2004] [Mazure, M.-L., CA, 2005] [Costantini, P ., Lyche, T., Manni, C., NM, 2005] .... Generalized B-splines and local refinements – p. 10/50

Unifying approach: Bernstein-like basis smoothness between adjacent segments: easily described by control points Generalized B-splines and local refinements – p. 11/50

Unifying approach: Bernstein-like basis smoothness between adjacent segments: easily described by control points 5 3 4 2 3 1 2 0 1 0 −1 −1 −2 −2 −3 −3 −4 −4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 C 1 cubics C 1 Trig/Exp Generalized B-splines and local refinements – p. 11/50

Unifying approach: Bernstein-like basis smoothness between adjacent segments: easily described by control points 20 15 10 5 0 2 1.5 2 1 1 0.5 0 0 −1 C 1 cubics Generalized B-splines and local refinements – p. 11/50

Unifying approach: Bernstein-like basis smoothness between adjacent segments: easily described by control points 20 15 10 5 0 2 1.5 2 1 1 0.5 0 0 −1 C 1 cubics 20 15 10 5 0 2 1.5 2 1 1 0.5 0 0 −1 C 1 exponential (cubics) Generalized B-splines and local refinements – p. 11/50

Spaces good for design P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) >, p ≥ 2 t ∈ [0 , 1] I p

Spaces good for design E ⊂ C n : n + 1 dimensional EC space containing constants I I E is Extended Chebyshev (EC) in I if any non trivial element has at most n zeros in I

Spaces good for design E ⊂ C n : n + 1 dimensional EC space containing constants I B 0 , · · · B n is a Bernstein-like basis of I E in [ a, b ] ⊂ I if B 0 , · · · B n is NTP B k vanishes exactly k times in a and n − k times in b

Spaces good for design E ⊂ C n : n + 1 dimensional EC space containing constants I B 0 , · · · B n is a Bernstein-like basis of I E in [ a, b ] ⊂ I if B 0 , · · · B n is NTP B k vanishes exactly k times in a and n − k times in b A Bernstein-like basis of I E is the ONTP basis of I E

Spaces good for design E ⊂ C n : n + 1 dimensional EC space containing constants I B 0 , · · · B n is a Bernstein-like basis of I E in [ a, b ] ⊂ I if B 0 , · · · B n is NTP B k vanishes exactly k times in a and n − k times in b A Bernstein-like basis of I E is the ONTP basis of I E I E possesses a Bernstein-like basis in any [ a, b ] ⊂ I iff { f ′ : f ∈ I E } is an Extended Chebyshev space in I

Spaces good for design E ⊂ C n : n + 1 dimensional EC space containing constants I B 0 , · · · B n is a Bernstein-like basis of I E in [ a, b ] ⊂ I if B 0 , · · · B n is NTP B k vanishes exactly k times in a and n − k times in b A Bernstein-like basis of I E is the ONTP basis of I E I E possesses a Bernstein-like basis in any [ a, b ] ⊂ I iff { f ′ : f ∈ I E } is an Extended Chebyshev space in I in I E all classical geometric design algorithms can be developed for the Bernstein-like basis (blossoms) ⇒ I E is good for design true under less restrictive hypoteses [Goodman, T.N.T., Mazure, M.-L., JAT, 2001], [Carnicer, Mainar, Pe˜ na; CA 2004], [Mazure, M.-L., AiCM, 2004], [Mazure, M.-L., CA, 2005], [Costantini, P ., Lyche, T., Manni, C., NM, 2005], [Mazure, M.-L., NM, 2011] ... Generalized B-splines and local refinements – p. 12/50

Alternatives to the rational model

Alternatives to the rational model rational model: I P p → B-splines → NURBS

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p Generalized B-splines and local refinements – p. 13/50

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v select proper I p : good approximation properties Generalized B-splines and local refinements – p. 13/50

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v select proper I p : good approximation properties exactly represent salient profiles P u,v < 1 , t, . . . , t p − 2 , cos ωt, sin ωt > = TRIG I := p P u,v < 1 , t, . . . , t p − 2 , cosh ωt, sinh ωt > = TRIG I := p

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v select proper I p : good approximation properties exactly represent salient profiles P u,v < 1 , t, . . . , t p − 2 , cos ωt, sin ωt > = TRIG I := p P u,v < 1 , t, . . . , t p − 2 , cosh ωt, sinh ωt > = HYP I := p conic sections, helix, cycloid, ... Generalized B-splines and local refinements – p. 13/50

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v select proper I p : good approximation properties describe sharp variations < 1 , t, . . . , t p − 2 , e ωt , e − ωt > = HYP ( HYP ) P u,v I := p < 1 , t, . . . , t p − 2 , (1 − t ) ω , t ω > = VDP P u,v I := p

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v select proper I p : good approximation properties describe sharp variations < 1 , t, . . . , t p − 2 , e ωt , e − ωt > = EXP = ( HYP ) P u,v I := p < 1 , t, . . . , t p − 2 , (1 − t ) ω , t ω > = VDP P u,v I := p Generalized B-splines and local refinements – p. 13/50

Alternatives to the rational model rational model: I P p → B-splines → NURBS P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > alternative: I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p P u,v construct/analyse spline spaces with sections in I p with suitable bases for them (analogous to B-splines) [Lyche, CA 1985] [Schumaker, L.L.; 1993], [Koch, P .E, Lyche, T.; Computing 1993], [Maruˇ sic, M., Rogina, M.; JCAM 1995], [Kvasov, B.I., Sattayatham, P .; JCAM 1999], [Costantini, P .; CAGD 2000], [Costantini, P ., Manni, C.; RM 2006] [Wang Fang; JCAM 2008], [Kavcic, Rogina, Bosner, Math. Comput. in Simulation, 2010], . . . Generalized B-splines and local refinements – p. 13/50

Generalized B-splines Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } , { ..., u i , v i , ... } , < 1 , t, . . . , t p − 2 , u i ( t ) , v i ( t ) >, < D p − 1 u i , D p − 1 v i > Chebyshev D p − 1 v i ( ξ i ) = 0 , D p − 1 v i ( ξ i +1 ) > 0 , D p − 1 u i ( ξ i ) > 0 , D p − 1 u i ( ξ i +1 ) = 0 , Generalized B-splines and local refinements – p. 14/50

Generalized B-splines Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } , { ..., u i , v i , ... } , < 1 , t, . . . , t p − 2 , u i ( t ) , v i ( t ) >, < D p − 1 u i , D p − 1 v i > Chebyshev D p − 1 v i ( ξ i ) = 0 , D p − 1 v i ( ξ i +1 ) > 0 , D p − 1 u i ( ξ i ) > 0 , D p − 1 u i ( ξ i +1 ) = 0 ,  D p − 1 v i ( t )  t ∈ [ ξ i , ξ i +1 )   D p − 1 v i ( ξ i +1 )  B (1) D p − 1 u i +1 ( t ) � i, Ξ ( t ) := t ∈ [ ξ i +1 , ξ i +2 )  D p − 1 u i +1 ( ξ i +1 )    0 elsewhere Generalized B-splines and local refinements – p. 14/50

Generalized B-splines Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } , { ..., u i , v i , ... } , < 1 , t, . . . , t p − 2 , u i ( t ) , v i ( t ) >, < D p − 1 u i , D p − 1 v i > Chebyshev D p − 1 v i ( ξ i ) = 0 , D p − 1 v i ( ξ i +1 ) > 0 , D p − 1 u i ( ξ i ) > 0 , D p − 1 u i ( ξ i +1 ) = 0 ,  D p − 1 v i ( t )  t ∈ [ ξ i , ξ i +1 )   D p − 1 v i ( ξ i +1 )  B (1) D p − 1 u i +1 ( t ) � i, Ξ ( t ) := t ∈ [ ξ i +1 , ξ i +2 )  D p − 1 u i +1 ( ξ i +1 )    0 elsewhere i, Ξ ( t ) = � t ( s )d s − � t B ( p ) δ ( p − 1) B ( p − 1) δ ( p − 1) B ( p − 1) � −∞ � � −∞ � i +1 , Ξ � i +1 , Ξ ( s )d s i, Ξ i, Ξ 1 δ ( p ) � i, Ξ := � + ∞ B ( p ) −∞ � i,W, Ξ ( s )d s Generalized B-splines and local refinements – p. 14/50

Generalized B-splines Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } , { ..., u i , v i , ... } , < 1 , t, . . . , t p − 2 , u i ( t ) , v i ( t ) >, < D p − 1 u i , D p − 1 v i > Chebyshev D p − 1 v i ( ξ i ) = 0 , D p − 1 v i ( ξ i +1 ) > 0 , D p − 1 u i ( ξ i ) > 0 , D p − 1 u i ( ξ i +1 ) = 0 ,   D p − 1 v i ( t )   t − ξ i t ∈ [ ξ i , ξ i +1 ) t ∈ [ ξ i , ξ i +1 )     D p − 1 v i ( ξ i +1 ) ξ i +1 − ξ i   B (1) D p − 1 u i +1 ( t ) B (1) ξ i +2 − t � i, Ξ ( t ) := i, Ξ ( t ) := t ∈ [ ξ i +1 , ξ i +2 ) t ∈ [ ξ i +1 , ξ i +2 )  D p − 1 u i +1 ( ξ i +1 )  ξ i +2 − ξ i +1       0 elsewhere 0 elsewhere i, Ξ ( t ) = � t ( s )d s − � t B ( p ) δ ( p − 1) B ( p − 1) δ ( p − 1) B ( p − 1) � −∞ � � −∞ � i +1 , Ξ � i +1 , Ξ ( s )d s i, Ξ i, Ξ 1 δ ( p ) � i, Ξ := � + ∞ B ( p ) −∞ � i,W, Ξ ( s )d s B-splines i, Ξ ( t ) = � t ( s )d s − � t B ( p ) −∞ δ ( p − 1) B ( p − 1) −∞ δ ( p − 1) i +1 , Ξ B ( p − 1) i +1 , Ξ ( s )d s i, Ξ i, Ξ 1 δ ( p ) i, Ξ := � + ∞ −∞ B ( p ) i, Ξ ( s )d s Generalized B-splines and local refinements – p. 14/50

Generalized B-splines Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } , { ..., u i , v i , ... } , < 1 , t, . . . , t p − 2 , u i ( t ) , v i ( t ) >, < D p − 1 u i , D p − 1 v i > Chebyshev D p − 1 v i ( ξ i ) = 0 , D p − 1 v i ( ξ i +1 ) > 0 , D p − 1 u i ( ξ i ) > 0 , D p − 1 u i ( ξ i +1 ) = 0 , 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 B (1) B (1) � i, Ξ i, Ξ All Chebyshevian spline spaces good for design can be built by means of integral recurrence relations, [Mazure M.L., NM 2011]

Generalized B-splines: exponential (hyperbolic) Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } : knots W := { ..., ω i , ... } : shape parameters P u i ,v i := < 1 , t, . . . , t p − 2 , cosh ω i t, sinh ω i t > I p Exponential case: p = 3 := < 1 , t, e ωt , e − ωt > P u,v EXP 3 = I isomorphic to I P 3 3 Bernstein-like basis 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 0 1 0 0.5 1 0 0.25 0.5 0.75 1 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 C 2 cubic B-splines ω → 0 : Generalized B-splines and local refinements – p. 15/50

Generalized B-splines: exponential (hyperbolic) Ξ := { ξ 1 ≤ ξ 2 ≤ · · · ≤ ξ n + p +1 } : knots W := { ..., ω i , ... } : shape parameters P u i ,v i := < 1 , t, . . . , t p − 2 , cosh ω i t, sinh ω i t > I p Exponential case: p = 3 := < 1 , t, e ωt , e − ωt > P u,v EXP 3 = I isomorphic to I P 3 3 Bernstein-like basis 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 0 0.25 0.5 0.75 1 0 1 0 0.5 1 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 ω = 3 h Generalized B-splines and local refinements – p. 15/50

Generalized B-splines: properties B ( p ) { � i, Ξ ( t ) , i = 1 , . . . } , Properties analogous to classical B-splines positivity partition of unity: p ≥ 2 compact support smoothness derivatives local linear independence . . .

Generalized B-splines: properties B ( p ) { � i, Ξ ( t ) , i = 1 , . . . } , Properties analogous to classical B-splines positivity partition of unity: p ≥ 2 compact support smoothness derivatives local linear independence . . . shape properties { . . . , u i , v i , . . . } Generalized B-splines and local refinements – p. 16/50

Generalized B-splines: properties B ( p ) { � i, Ξ ( t ) , i = 1 , . . . } , Properties analogous to classical B-splines positivity partition of unity: p ≥ 2 1 compact support 0.5 smoothness 0 derivatives −0.5 local linear independence −1 . . . −1.5 shape properties { . . . , u i , v i , . . . } −0.5 0 0.5 1 1.5 2 2.5 trig. and exp. parts can be mixed Generalized B-splines and local refinements – p. 16/50

Generalized B-splines: properties B ( p ) { � i, Ξ ( t ) , i = 1 , . . . } , Properties analogous to classical B-splines positivity partition of unity: p ≥ 2 compact support smoothness derivatives local linear independence . . . shape properties { . . . , u i , v i , . . . } trig. and exp. parts can be mixed straightforward multivariate extension via tensor product Generalized B-splines and local refinements – p. 16/50

Summary

Summary P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p

Summary P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p Bernstein like bases/control polygon Generalized B-splines and local refinements – p. 17/50

Summary P p = < 1 , t, . . . , t p − 2 , t p − 1 , t p > I ↓ P u,v := < 1 , t, . . . , t p − 2 , u ( t ) , v ( t ) > I p Bernstein like bases/control polygon P u,v Generalized B-splines: spline spaces with sections in I p with suitable bases for them (analogous to B-splines) Generalized B-splines and local refinements – p. 17/50

Local Refinements local refinements are crucial in applications (geometric modelling, simulation,...) Generalized B-splines and local refinements – p. 18/50

Local Refinements local refinements are crucial in applications (geometric modelling, simulation,...) Generalized B-splines and local refinements – p. 18/50

Local Refinements local refinements are crucial in applications (geometric modelling, simulation,...) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Generalized B-splines and local refinements – p. 18/50

Local Refinements local refinements are crucial in applications (geometric modelling, simulation,...) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Local Refinements local refinements are crucial in applications (geometric modelling, simulation,...) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Generalized B-splines and local refinements – p. 18/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): Generalized B-splines and local refinements – p. 19/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): T-splines/Analysis-Suitable T-splines [Bazilevs, Y., et al. CMAME 2010], [Beirão da Veiga, et al. CMAME, 2012]... Generalized B-splines and local refinements – p. 19/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): T-splines/Analysis-Suitable T-splines [Bazilevs, Y., et al. CMAME 2010], [Beirão da Veiga, et al. CMAME, 2012]... LR splines [Dokken T., Lyche T., Pettersen K.F ., CAGD 2013], Generalized B-splines and local refinements – p. 19/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): T-splines/Analysis-Suitable T-splines [Bazilevs, Y., et al. CMAME 2010], [Beirão da Veiga, et al. CMAME, 2012]... LR splines [Dokken T., Lyche T., Pettersen K.F ., CAGD 2013], Hierarchical bases Generalized B-splines and local refinements – p. 19/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): T-splines/Analysis-Suitable T-splines [Bazilevs, Y., et al. CMAME 2010], [Beirão da Veiga, et al. CMAME, 2012]... LR splines [Dokken T., Lyche T., Pettersen K.F ., CAGD 2013], Hierarchical bases Splines over T-meshes Generalized B-splines and local refinements – p. 19/50

DRAWBACKS of tensor product structures the tensor product structure prevents local refinements Alternatives (polynomial B-splines): T-splines/Analysis-Suitable T-splines [Bazilevs, Y., et al. CMAME 2010], [Beirão da Veiga, et al. CMAME, 2012]... LR splines [Dokken T., Lyche T., Pettersen K.F ., CAGD 2013], Hierarchical bases Splines over T-meshes B-splines on triangulations Generalized B-splines and local refinements – p. 19/50

Generalized Splines: local refinements? Generalized B-splines and local refinements – p. 20/50

Generalized Splines: local refinements? Generalized splines have global tensor-product structure Generalized B-splines and local refinements – p. 20/50

Generalized Splines: local refinements? Generalized splines have global tensor-product structure some localization techniques can be applied to (some) generalized spline spaces. Hierarchical generalized splines Generalized splines over T-meshes Quadratic Generalized splines over triangulations Generalized B-splines and local refinements – p. 20/50

Hierarchical model [Forsey, D.R., Bartels R.H., CG 1988], [Kraft R., Bartels R.H., Surf. Fitt. Mult. Meth. 1997], [Rabut C., 2005] [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011], [ Giannelli C., J¨ uttler B., Speleers, H.; CAGD 2012], [Bracco C., et al., JCAM 2014] sequence of N nested tensor-product spline spaces V 0 ⊂ V 1 ⊂ · · · ⊂ V N − 1 V 2 V 1 V 0 Generalized B-splines and local refinements – p. 21/50

Hierarchical B-spline model [Forsey, D.R., Bartels R.H., CG 1988], [Kraft R., Bartels R.H., Surf. Fitt. Mult. Meth. 1997], [Rabut C., 2005] [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011], [ Giannelli C., J¨ uttler B., Speleers, H.; CAGD 2012], [Bracco C., et al., JCAM 2014] sequence of N nested tensor-product spline spaces V 0 ⊂ V 1 ⊂ · · · ⊂ V N − 1 V 2 V 1 V 0 V ℓ is spanned by a tensor-product B-spline basis B ℓ : B ℓ = { . . . , B i,ℓ , . . . } Generalized B-splines and local refinements – p. 21/50

Hierarchical B-spline model [Forsey, D.R., Bartels R.H., CG 1988], [Kraft R., Bartels R.H., Surf. Fitt. Mult. Meth. 1997], [Rabut C., 2005] [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011], [ Giannelli C., J¨ uttler B., Speleers, H.; CAGD 2012], [Bracco C., et al., JCAM 2014] sequence of N nested tensor-product spline spaces V 0 ⊂ V 1 ⊂ · · · ⊂ V N − 1 V 2 V 1 V 0 V ℓ is spanned by a tensor-product B-spline basis B ℓ : B ℓ = { . . . , B i,ℓ , . . . } sequence of N nested domains Ω N − 1 ⊂ Ω N − 2 ⊂ · · · ⊂ Ω 0 , Ω N = ∅ Ω 2 Ω 1 Ω 0 Generalized B-splines and local refinements – p. 21/50

Hierarchical B-spline model degree 1 Recursive definition (I) Initialization: H 0 := B 0 [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]

Hierarchical B-spline model degree 1 Recursive definition (I) Initialization: H 0 := B 0 (II) construction of H ℓ +1 from H ℓ , H ℓ +1 := H ℓ +1 ∪ H ℓ +1 C F ℓ = 0 , 1 , . . . , N − 1 [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]

Hierarchical B-spline model degree 1 Recursive definition (I) Initialization: H 0 := B 0 (II) construction of H ℓ +1 from H ℓ , H ℓ +1 := H ℓ +1 ∪ H ℓ +1 C F ℓ = 0 , 1 , . . . , N − 1 := { B i,ℓ ∈ H ℓ : supp ( B i,ℓ ) �⊂ Ω ℓ +1 } H ℓ +1 C [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]

Hierarchical B-spline model degree 1 Recursive definition (I) Initialization: H 0 := B 0 (II) construction of H ℓ +1 from H ℓ , H ℓ +1 := H ℓ +1 ∪ H ℓ +1 C F ℓ = 0 , 1 , . . . , N − 1 := { B i,ℓ ∈ H ℓ : supp ( B i,ℓ ) �⊂ Ω ℓ +1 } H ℓ +1 C := { B i,ℓ +1 ∈ B ℓ +1 : supp ( B i,ℓ +1 ) ⊂ Ω ℓ +1 } H ℓ +1 F [Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011] Generalized B-splines and local refinements – p. 22/50

Hierarchical B-spline model sequence of N nested tensor-product spline spaces V 0 ⊂ V 1 ⊂ · · · ⊂ V N − 1 V 2 V 1 V 0 V ℓ is spanned by a tensor-product B-spline basis B ℓ : B ℓ = { . . . , B i,ℓ , . . . } sequence of N nested domains Ω N − 1 ⊂ Ω N − 2 ⊂ · · · ⊂ Ω 0 , Ω N = ∅ Ω 2 Ω 1 Ω 0 Generalized B-splines and local refinements – p. 23/50

Hierarchical Generalized B-spline model Generalized B-splines support a hierarchical refinement sequence of N nested tensor-product spline spaces V 0 ⊂ V 1 ⊂ · · · ⊂ V N − 1 V 2 V 1 V 0 V ℓ spanned by a tensor-product Generalized B-spline basis � B ℓ : B ℓ = { . . . , � � B i,ℓ , . . . } sequence of N nested domains Ω N − 1 ⊂ Ω N − 2 ⊂ · · · ⊂ Ω 0 , Ω N = ∅ Ω 2 Ω 1 Ω 0 Generalized B-splines and local refinements – p. 23/50