Generalized B-splines and local refinements Carla Manni Department - - PowerPoint PPT Presentation
Generalized B-splines and local refinements Carla Manni Department of Mathematics, University of Roma Tor Vergata collaboration with P. Costantini, F. Pelosi, H. Speleers 11-th MAIA Conference September 2530, 2013 Ettore
Carla Manni
Department of Mathematics, University of Roma “Tor Vergata” collaboration with
11-th MAIA Conference
September 25–30, 2013 “Ettore Majorana” Foundation and Centre, Erice
Generalized B-splines and local refinements – p. 1/50
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations Generalized B-splines
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations Generalized B-splines Local refinements
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations Generalized B-splines Local refinements Hierarchical bases for Generalized B-splines
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations Generalized B-splines Local refinements Hierarchical bases for Generalized B-splines Generalized B-splines over T-meshes
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations Generalized B-splines Local refinements Hierarchical bases for Generalized B-splines Generalized B-splines over T-meshes Generalized B-splines over triangles
Generalized B-splines and local refinements – p. 2/50
Bernstein-like representations
Ariadne’s thread
Generalized B-splines Local refinements Hierarchical bases for Generalized B-splines Generalized B-splines over T-meshes Generalized B-splines over triangles
Generalized B-splines and local refinements – p. 2/50
p
pi
p
i
t ∈ [0, 1], pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
Generalized B-splines and local refinements – p. 3/50
p
pi
p
i
t ∈ [0, 1], pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
B ´ EZIER CURVE
Generalized B-splines and local refinements – p. 3/50
p
pi
p
i
t ∈ [0, 1], pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
space: I
Pp
B ´ EZIER CURVE
Generalized B-splines and local refinements – p. 3/50
p
pi
p
i
t ∈ [0, 1], pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
space: I
Pp
basis: Bernstein pol.
B ´ EZIER CURVE
Generalized B-splines and local refinements – p. 3/50
p
piBi(t), t ∈ [a, b] pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
Generalized B-splines and local refinements – p. 4/50
p
piBi(t), t ∈ [a, b] pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
Bernstein-like representation
Generalized B-splines and local refinements – p. 4/50
p
piBi(t), t ∈ [a, b] pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
space: < B0, · · · , Bn >
Bernstein-like representation
Generalized B-splines and local refinements – p. 4/50
p
piBi, t ∈ [a, b] pi ∈ Rd
1 2 3 4 1 2 3 0.2 0.4 0.6 0.8 1
p0 p1 p2 p3 p4 p5
space: < B0, · · · , Bn > basis: ONTP basis (B-basis) Optimal Normalized Totally Positive
Bernstein-like representation
Generalized B-splines and local refinements – p. 4/50
p=3
Bernstein/B-splines bases are the ONTP bases for polynomials/piecewise polynomials
⇓
Generalized B-splines and local refinements – p. 5/50
p=3
Bernstein/B-splines bases are the ONTP bases for polynomials/piecewise polynomials
⇓
Generalized B-splines and local refinements – p. 5/50
in CAGD curves/surfaces are often subjected to constraints
Generalized B-splines and local refinements – p. 6/50
in CAGD curves/surfaces are often subjected to constraints reproduction constraints
exact reproduction of main curves/surfaces (conic sections, ...)
shape constraints
curvature orientation, torsion signs,...
tolerance constraints
...
Generalized B-splines and local refinements – p. 6/50
in CAGD curves/surfaces are often subjected to constraints reproduction constraints
exact reproduction of main curves/surfaces (conic sections, ...)
shape constraints
curvature orientation, torsion signs,...
tolerance constraints
... polynomials/ piecewise polynomials (B-splines) are not sufficient
Generalized B-splines and local refinements – p. 6/50
exponentials < 1, t, eωt, e−ωt >
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
trigonometrics < 1, t, cos(ωt), sin(ωt) >
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
trigonometrics < 1, t, cos(ωt), sin(ωt) >
ω : shape parameter cubic if ω → 0
Generalized B-splines and local refinements – p. 7/50
exponentials < 1, t, eωt, e−ωt >
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
variable degree < 1, t, tω, (1 − t)ω >
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
variable degree < 1, t, tω, (1 − t)ω >
ω : shape parameter cubic if ω = 3 linear as ω → +∞
exponentials < 1, t, eωt, e−ωt >
ω : shape parameter cubic as ω → 0 linear as ω → +∞
variable degree < 1, t, tω, (1 − t)ω >
ω : shape parameter cubic if ω = 3 linear as ω → +∞
...
Generalized B-splines and local refinements – p. 8/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1]
Generalized B-splines and local refinements – p. 9/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1] ONTP/Bernstein-like basis {B0, B1, B2, B3}:
Generalized B-splines and local refinements – p. 9/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1] ONTP/Bernstein-like basis {B0, B1, B2, B3}:
C2 ⇒easy to characterize/construct
B0(1) = B0′(1) = B0′′(1) = 0 B1(0) = B1(1) = B1′(1) = 0 B2(0) = B2′(0) = B2(1) = 0 B3(0) = B3′(0) = B3′′(0) = 0
0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B0 B1 B2 B3
Generalized B-splines and local refinements – p. 9/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1] ONTP/Bernstein-like basis {B0, B1, B2, B3}:
C2 ⇒easy to characterize/construct
B0(1) = B0′(1) = B0′′(1) = 0 B1(0) = B1(1) = B1′(1) = 0 B2(0) = B2′(0) = B2(1) = 0 B3(0) = B3′(0) = B3′′(0) = 0
control points: (0, b0), (ξ, b1), (1 − η, b2), (1, b3), 0 < ξ < 1 − η < 1,
Generalized B-splines and local refinements – p. 9/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1] ONTP/Bernstein-like basis {B0, B1, B2, B3}:
C2 ⇒easy to characterize/construct
B0(1) = B0′(1) = B0′′(1) = 0 B1(0) = B1(1) = B1′(1) = 0 B2(0) = B2′(0) = B2(1) = 0 B3(0) = B3′(0) = B3′′(0) = 0
control points: (0, b0), (ξ, b1), (1 − η, b2), (1, b3), 0 < ξ < 1 − η < 1, control polygon describes s(t) = 3
j=0 bjBj(t)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Generalized B-splines and local refinements – p. 9/50
Ex: < 1, t, u(t), v(t) > (≃ cubics) u, v ∈ C2, t ∈ [0, 1] ONTP/Bernstein-like basis {B0, B1, B2, B3}:
C2 ⇒easy to characterize/construct
B0(1) = B0′(1) = B0′′(1) = 0 B1(0) = B1(1) = B1′(1) = 0 B2(0) = B2′(0) = B2(1) = 0 B3(0) = B3′(0) = B3′′(0) = 0
control points: (0, b0), (ξ, b1), (1 − η, b2), (1, b3), 0 < ξ < 1 − η < 1, control polygon describes s(t) = 3
j=0 bjBj(t)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
properties of s by its control polygon
Generalized B-splines and local refinements – p. 9/50
I Pp =< 1, t, . . . , tp−2, tp−1, tp >
Generalized B-splines and local refinements – p. 10/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1]
Generalized B-splines and local refinements – p. 10/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1] < Dp−1u, Dp−1v > Chebyshev in [0, 1] and Extended Chebyshev in (0, 1)
Generalized B-splines and local refinements – p. 10/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1] < Dp−1u, Dp−1v > Chebyshev in [0, 1] and Extended Chebyshev in (0, 1) ⇓ I Pu,v
p
possesses a ONTP-basis
Generalized B-splines and local refinements – p. 10/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1] < Dp−1u, Dp−1v > Chebyshev in [0, 1] and Extended Chebyshev in (0, 1) ⇓ I Pu,v
p
possesses a ONTP-basis Ex:
u, v: trigonometric functions u, v: exponential functions u, v: variable degree ....
Generalized B-splines and local refinements – p. 10/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1] < Dp−1u, Dp−1v > Chebyshev in [0, 1] and Extended Chebyshev in (0, 1) ⇓ I Pu,v
p
possesses a ONTP-basis 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
smoothness between adjacent segments: easily described by control points
Generalized B-splines and local refinements – p. 11/50
smoothness between adjacent segments: easily described by control points
0.5 1 1.5 2 −4 −3 −2 −1 1 2 3 4 5 0.5 1 1.5 2 −4 −3 −2 −1 1 2 3
C1 cubics C1 Trig/Exp
Generalized B-splines and local refinements – p. 11/50
smoothness between adjacent segments: easily described by control points
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 cubics
Generalized B-splines and local refinements – p. 11/50
smoothness between adjacent segments: easily described by control points
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 cubics
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 exponential (cubics)
Generalized B-splines and local refinements – p. 11/50
I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >, p ≥ 2 t ∈ [0, 1]
I E ⊂ Cn: n + 1 dimensional EC space containing constants
I E is Extended Chebyshev (EC) in I if any non trivial element has at most n zeros in I
I E ⊂ Cn: n + 1 dimensional EC space containing constants B0, · · · Bn is a Bernstein-like basis of I E in [a, b] ⊂ I if B0, · · · Bn is NTP Bk vanishes exactly k times in a and n − k times in b
I E ⊂ Cn: n + 1 dimensional EC space containing constants B0, · · · Bn is a Bernstein-like basis of I E in [a, b] ⊂ I if B0, · · · Bn is NTP Bk 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 ⊂ Cn: n + 1 dimensional EC space containing constants B0, · · · Bn is a Bernstein-like basis of I E in [a, b] ⊂ I if B0, · · · Bn is NTP Bk 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
I E ⊂ Cn: n + 1 dimensional EC space containing constants B0, · · · Bn is a Bernstein-like basis of I E in [a, b] ⊂ I if B0, · · · Bn is NTP Bk 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
rational model:
I Pp → B-splines → NURBS
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
Generalized B-splines and local refinements – p. 13/50
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
select proper I
Pu,v
p :
good approximation properties
Generalized B-splines and local refinements – p. 13/50
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
select proper I
Pu,v
p :
good approximation properties exactly represent salient profiles
I Pu,v
p
:= < 1, t, . . . , tp−2, cos ωt, sin ωt > = TRIG I Pu,v
p
:= < 1, t, . . . , tp−2, cosh ωt, sinh ωt > = TRIG
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
select proper I
Pu,v
p :
good approximation properties exactly represent salient profiles
I Pu,v
p
:= < 1, t, . . . , tp−2, cos ωt, sin ωt >= TRIG I Pu,v
p
:= < 1, t, . . . , tp−2, cosh ωt, sinh ωt >= HYP
conic sections, helix, cycloid, ...
Generalized B-splines and local refinements – p. 13/50
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
select proper I
Pu,v
p :
good approximation properties describe sharp variations
I Pu,v
p
:= < 1, t, . . . , tp−2, eωt, e−ωt > = HYP(HYP) I Pu,v
p
:= < 1, t, . . . , tp−2, (1 − t)ω, tω > = VDP
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
select proper I
Pu,v
p :
good approximation properties describe sharp variations
I Pu,v
p
:= < 1, t, . . . , tp−2, eωt, e−ωt >= EXP = (HYP) I Pu,v
p
:= < 1, t, . . . , tp−2, (1 − t)ω, tω >= VDP
Generalized B-splines and local refinements – p. 13/50
rational model:
I Pp → B-splines → NURBS
alternative:
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
construct/analyse spline spaces with sections in I
Pu,v
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
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1}, {..., ui, vi, ...}, < 1, t, . . . , tp−2, ui(t), vi(t) >, < Dp−1ui, Dp−1vi > Chebyshev Dp−1vi(ξi) = 0, Dp−1vi(ξi+1) > 0, Dp−1ui(ξi) > 0, Dp−1ui(ξi+1) = 0,
Generalized B-splines and local refinements – p. 14/50
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1}, {..., ui, vi, ...}, < 1, t, . . . , tp−2, ui(t), vi(t) >, < Dp−1ui, Dp−1vi > Chebyshev Dp−1vi(ξi) = 0, Dp−1vi(ξi+1) > 0, Dp−1ui(ξi) > 0, Dp−1ui(ξi+1) = 0,
i,Ξ(t) :=
Dp−1vi(t) Dp−1vi(ξi+1)
t ∈ [ξi, ξi+1)
Dp−1ui+1(t) Dp−1ui+1(ξi+1)
t ∈ [ξi+1, ξi+2) elsewhere
Generalized B-splines and local refinements – p. 14/50
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1}, {..., ui, vi, ...}, < 1, t, . . . , tp−2, ui(t), vi(t) >, < Dp−1ui, Dp−1vi > Chebyshev Dp−1vi(ξi) = 0, Dp−1vi(ξi+1) > 0, Dp−1ui(ξi) > 0, Dp−1ui(ξi+1) = 0,
i,Ξ(t) :=
Dp−1vi(t) Dp−1vi(ξi+1)
t ∈ [ξi, ξi+1)
Dp−1ui+1(t) Dp−1ui+1(ξi+1)
t ∈ [ξi+1, ξi+2) elsewhere
i,Ξ(t) = t −∞
δ(p−1)
i,Ξ
i,Ξ
(s)ds − t
−∞
δ(p−1)
i+1,Ξ
B(p−1)
i+1,Ξ(s)ds
i,Ξ :=
1 +∞
−∞
B(p)
i,W,Ξ(s)ds
Generalized B-splines and local refinements – p. 14/50
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1}, {..., ui, vi, ...}, < 1, t, . . . , tp−2, ui(t), vi(t) >, < Dp−1ui, Dp−1vi > Chebyshev Dp−1vi(ξi) = 0, Dp−1vi(ξi+1) > 0, Dp−1ui(ξi) > 0, Dp−1ui(ξi+1) = 0,
i,Ξ(t) :=
Dp−1vi(t) Dp−1vi(ξi+1)
t ∈ [ξi, ξi+1)
Dp−1ui+1(t) Dp−1ui+1(ξi+1)
t ∈ [ξi+1, ξi+2) elsewhere B(1)
i,Ξ(t) :=
t−ξi ξi+1−ξi
t ∈ [ξi, ξi+1)
ξi+2−t ξi+2−ξi+1
t ∈ [ξi+1, ξi+2) elsewhere
i,Ξ(t) = t −∞
δ(p−1)
i,Ξ
i,Ξ
(s)ds − t
−∞
δ(p−1)
i+1,Ξ
B(p−1)
i+1,Ξ(s)ds
i,Ξ :=
1 +∞
−∞
B(p)
i,W,Ξ(s)ds
B-splines
B(p)
i,Ξ(t) = t −∞ δ(p−1) i,Ξ
B(p−1)
i,Ξ
(s)ds − t
−∞ δ(p−1) i+1,ΞB(p−1) i+1,Ξ(s)ds δ(p)
i,Ξ :=
1 +∞
−∞ B(p) i,Ξ(s)ds
Generalized B-splines and local refinements – p. 14/50
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1}, {..., ui, vi, ...}, < 1, t, . . . , tp−2, ui(t), vi(t) >, < Dp−1ui, Dp−1vi > Chebyshev Dp−1vi(ξi) = 0, Dp−1vi(ξi+1) > 0, Dp−1ui(ξi) > 0, Dp−1ui(ξi+1) = 0,
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1i,Ξ
B(1)
i,Ξ All Chebyshevian spline spaces good for design can be built by means of integral recurrence relations, [Mazure M.L., NM 2011]
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1} : knots W := {..., ωi, ...} : shape parameters
I Pui,vi
p
:=< 1, t, . . . , tp−2, cosh ωit, sinh ωit >
Exponential case: p = 3 EXP3 = I
Pu,v
3
:=< 1, t, eωt, e−ωt >
isomorphic to I
P3
Bernstein-like basis
1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25 0.5 0.75 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ω → 0: C2 cubic B-splines
Generalized B-splines and local refinements – p. 15/50
Ξ := {ξ1 ≤ ξ2 ≤ · · · ≤ ξn+p+1} : knots W := {..., ωi, ...} : shape parameters
I Pui,vi
p
:=< 1, t, . . . , tp−2, cosh ωit, sinh ωit >
Exponential case: p = 3 EXP3 = I
Pu,v
3
:=< 1, t, eωt, e−ωt >
isomorphic to I
P3
Bernstein-like basis
1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25 0.5 0.75 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ω = 3h
Generalized B-splines and local refinements – p. 15/50
{ 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
. . .
{ 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 {. . . , ui, vi, . . .}
Generalized B-splines and local refinements – p. 16/50
{ 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 {. . . , ui, vi, . . .}
−0.5 0.5 1 1.5 2 2.5 −1.5 −1 −0.5 0.5 1
Generalized B-splines and local refinements – p. 16/50
{ 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 {. . . , ui, vi, . . .}
straightforward multivariate extension via tensor product
Generalized B-splines and local refinements – p. 16/50
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
Bernstein like bases/control polygon
Generalized B-splines and local refinements – p. 17/50
I Pp =< 1, t, . . . , tp−2, tp−1, tp > ↓ I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
Bernstein like bases/control polygon Generalized B-splines: spline spaces with sections in I
Pu,v
p
with suitable bases for them (analogous to B-splines)
Generalized B-splines and local refinements – p. 17/50
local refinements are crucial in applications (geometric modelling, simulation,...)
Generalized B-splines and local refinements – p. 18/50
local refinements are crucial in applications (geometric modelling, simulation,...)
Generalized B-splines and local refinements – p. 18/50
local refinements are crucial in applications (geometric modelling, simulation,...)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
Generalized B-splines and local refinements – p. 18/50
local refinements are crucial in applications (geometric modelling, simulation,...)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
local refinements are crucial in applications (geometric modelling, simulation,...)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
Generalized B-splines and local refinements – p. 18/50
the tensor product structure prevents local refinements Alternatives (polynomial B-splines):
Generalized B-splines and local refinements – p. 19/50
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
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
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
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
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 B-splines and local refinements – p. 20/50
Generalized splines have global tensor-product structure
Generalized B-splines and local refinements – p. 20/50
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
[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 V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Generalized B-splines and local refinements – p. 21/50
[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 V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ is spanned by a tensor-product B-spline basis Bℓ:
Bℓ = {. . . , Bi,ℓ, . . .}
Generalized B-splines and local refinements – p. 21/50
[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 V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ is spanned by a tensor-product B-spline basis Bℓ:
Bℓ = {. . . , Bi,ℓ, . . .}
sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
Generalized B-splines and local refinements – p. 21/50
degree 1
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
Recursive definition (I) Initialization: H0 := B0
degree 1
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
Recursive definition (I) Initialization: H0 := B0 (II) construction of Hℓ+1 from Hℓ,
Hℓ+1 := Hℓ+1
C
∪ Hℓ+1
F
ℓ = 0, 1, . . . , N − 1
degree 1
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
Recursive definition (I) Initialization: H0 := B0 (II) construction of Hℓ+1 from Hℓ,
Hℓ+1 := Hℓ+1
C
∪ Hℓ+1
F
ℓ = 0, 1, . . . , N − 1
Hℓ+1
C
:= {Bi,ℓ ∈ Hℓ : supp(Bi,ℓ) ⊂ Ωℓ+1}
degree 1
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
Recursive definition (I) Initialization: H0 := B0 (II) construction of Hℓ+1 from Hℓ,
Hℓ+1 := Hℓ+1
C
∪ Hℓ+1
F
ℓ = 0, 1, . . . , N − 1
Hℓ+1
C
:= {Bi,ℓ ∈ Hℓ : supp(Bi,ℓ) ⊂ Ωℓ+1} Hℓ+1
F
:= {Bi,ℓ+1 ∈ Bℓ+1 : supp(Bi,ℓ+1) ⊂ Ωℓ+1}
Generalized B-splines and local refinements – p. 22/50
sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ is spanned by a tensor-product B-spline basis Bℓ:
Bℓ = {. . . , Bi,ℓ, . . .}
sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
Generalized B-splines and local refinements – p. 23/50
Generalized B-splines support a hierarchical refinement sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ spanned by a tensor-product Generalized B-spline basis
Bℓ:
Bi,ℓ , . . .}
sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
Generalized B-splines and local refinements – p. 23/50
Generalized B-splines support a hierarchical refinement sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ spanned by a tensor-product Generalized B-spline basis
Bℓ:
Bi,ℓ , . . .}
sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
⇒ similar recursive definition
Generalized B-splines and local refinements – p. 23/50
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
1D Example: Cubic B-spline basis
Generalized B-splines and local refinements – p. 24/50
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
1D Example: Cubic B-spline basis
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H0 = B0
Generalized B-splines and local refinements – p. 24/50
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
1D Example: Cubic B-spline basis
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H0 = B0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
B1
Generalized B-splines and local refinements – p. 24/50
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
1D Example: Cubic B-spline basis
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H1
C
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H1
F
∪
Generalized B-splines and local refinements – p. 24/50
[Vuong A.-V., Giannelli C., J¨ uttler B., Simeon B.; CMAME 2011]
1D Example: Cubic B-spline basis
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H1
C
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H1
F
∪
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H1 ↓
Generalized B-splines and local refinements – p. 24/50
Generalized B-splines support a hierarchical refinement 1D Example: EXP3 B-splines basis ωi = 50
Generalized B-splines and local refinements – p. 25/50
Generalized B-splines support a hierarchical refinement 1D Example: EXP3 B-splines basis ωi = 50
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H0 H1 ↓
Generalized B-splines and local refinements – p. 25/50
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines the functions in Hℓ obtained by the iterative procedure are linearly independent
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines the functions in Hℓ obtained by the iterative procedure are linearly independent the hierarchical bases Hℓ, for each ℓ, span nested spaces:
spanHℓ ⊆ spanHℓ+1
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines the functions in Hℓ obtained by the iterative procedure are linearly independent the hierarchical bases Hℓ, for each ℓ, span nested spaces:
spanHℓ ⊆ spanHℓ+1
positivity
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines the functions in Hℓ obtained by the iterative procedure are linearly independent the hierarchical bases Hℓ, for each ℓ, span nested spaces:
spanHℓ ⊆ spanHℓ+1
positivity partition of unity
The main properties of Hierarchical B-splines are inherited by Hierarchical GB-splines the functions in Hℓ obtained by the iterative procedure are linearly independent the hierarchical bases Hℓ, for each ℓ, span nested spaces:
spanHℓ ⊆ spanHℓ+1
positivity partition of unity by using truncated bases
[Giannelli, Jüttler, Speleers; AiCM 2013 ]
Generalized B-splines and local refinements – p. 26/50
Generalized B-splines: truncated hierarchical basis 1D Example: EXP3 B-splines basis ωi = 50
Generalized B-splines and local refinements – p. 27/50
Generalized B-splines: truncated hierarchical basis 1D Example: EXP3 B-splines basis ωi = 50
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
H0 H1 ↓
Generalized B-splines and local refinements – p. 27/50
Generalized B-splines: truncated hierarchical basis 1D Example: EXP3 B-splines basis ωi = 50
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1
T 0 T 1 ↓
Generalized B-splines and local refinements – p. 27/50
sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Generalized B-splines and local refinements – p. 28/50
sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ tensor-product (Generalized) B-splines sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
Generalized B-splines and local refinements – p. 28/50
sequence of N nested tensor-product spline spaces V0 ⊂ V1 ⊂ · · · ⊂ VN−1
V0 V1 V2
Vℓ tensor-product (Generalized) B-splines sequence of N nested domains
ΩN−1 ⊂ ΩN−2 ⊂ · · · ⊂ Ω0, ΩN = ∅
Ω2 Ω1 Ω0
hierarchical (Generalized) B-splines span the full space
{f : f|Ω0\Ωℓ+1 ∈ Vℓ|Ω0\Ωℓ+1, ℓ = 0, · · · , N − 1}
[Giannelli, Jüttler; JCAM 2013], [Speleers, Manni, 2013 preprint]
Generalized B-splines and local refinements – p. 28/50
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1
Generalized B-splines and local refinements – p. 29/50
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1 great flexibility
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1 great flexibility different section spaces at different levels
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1 great flexibility different section spaces at different levels the functions in Hℓ obtained by the iterative procedure remain linearly independent
the construction can be applied to a hierarchy of not nested spaces V0, V1, · · · , VN−1 great flexibility different section spaces at different levels the functions in Hℓ obtained by the iterative procedure remain linearly independent not nested spaces spanHℓ
[Manni, Pelosi, Speleers; 2013, to appear ]
Generalized B-splines and local refinements – p. 29/50
Hierarchical B-splines are particular bases of particular spline spaces on special rectangular partitions
Generalized B-splines and local refinements – p. 30/50
Generalized B-splines and local refinements – p. 30/50
T-mesh T
partition of a (rectangular) domain by rectangles: T-junctions (hanging vertices) are allowed
Generalized B-splines and local refinements – p. 30/50
T-mesh T
partition of a (rectangular) domain by rectangles: T-junctions (hanging vertices) are allowed
Sr
d(T ) :=
s(x, y) ∈ Cr, s(x, y)|τi ∈ I Pd1 × I Pd2, τi ∈ T ,
I Pd := q(z) =
d
zj , r = (r1, r2), d = (d1, d2)
[Deng, J.-S., Chen, F-L., Feng, Y.-Y., JCAM 2006] [Schumaker, L. L. and Wang, L., CAGD 2012] [Schumaker, L. L. and Wang, L., NM 2011]
Generalized B-splines and local refinements – p. 30/50
T-mesh T
partition of a (rectangular) domain by rectangles: T-junctions (hanging vertices) are allowed
Sr
d(T ) :=
s(x, y) ∈ Cr, s(x, y)|τi ∈ I Pd1 × I Pd2, τi ∈ T ,
I Pd := q(z) =
d
zj , r = (r1, r2), d = (d1, d2)
Generalized B-splines and local refinements – p. 30/50
[Mourrain, B., Math. Comp. 2013] dim(Sr
d(T )) =
F(d1 + 1)(d2 + 1) − Eh(d2 + 1)(r2 + 1) − Ev(d1 + 1)(r1 + 1) + V (r1 + 1)(r2 + 1) + homology term F : #faces, Eh : #hor.edges, Ev : #vert.edges, V : #int.vertices
Generalized B-splines and local refinements – p. 31/50
[Mourrain, B., Math. Comp. 2013] dim(Sr
d(T )) =
F(d1 + 1)(d2 + 1) − Eh(d2 + 1)(r2 + 1) − Ev(d1 + 1)(r1 + 1) + V (r1 + 1)(r2 + 1) + homology term F : #faces, Eh : #hor.edges, Ev : #vert.edges, V : #int.vertices
d ≥ 2r + 1,
dim(Sr
d(T )) =
F(d1 +1)(d2 +1)−Eh(d2 +1)(r2 +1)−Ev(d1 +1)(r1 +1)+V (r1 +1)(r2 +1)
[Deng, J.-S., Chen, F-L., Feng, Y.-Y. , JCAM 2006] [Schumaker, L. L. and Wang, L., 2011, CAGD 2012] [Schumaker, L. L. and Wang, L., NM 2011]
[Mourrain, B., Math. Comp. 2013] dim(Sr
d(T )) =
F(d1 + 1)(d2 + 1) − Eh(d2 + 1)(r2 + 1) − Ev(d1 + 1)(r1 + 1) + V (r1 + 1)(r2 + 1) + homology term F : #faces, Eh : #hor.edges, Ev : #vert.edges, V : #int.vertices
d ≥ 2r + 1,
dim(Sr
d(T )) =
F(d1 + 1)(d2 + 1) − Eh(d2 + 1)(r2 + 1) − Ev(d1 + 1)(r1 + 1) + V (r1 + 1)(r2 + 1)
C1 cubics: dim(S1
3(T )) = 4(Vb + V+) Vb : #b. vertices, V+ : #cross. vertices Ex: dim(S1
3(T )) = 4(9 + 1)
Generalized B-splines and local refinements – p. 31/50
d ≥ 2r + 1, rectangular domains: results based on
Bernstein representation minimal determining sets
[Alfeld, P ., Schumaker, L.L., CA 1987] [Alfeld P ., JCAM 2000] [Deng, J.-S., Chen, F-L., Feng, Y.-Y., JCAM 2006] [Schumaker, L. L. and Wang, L., 2011, preprint] [Schumaker, L. L. and Wang, L., NM 2011]
d ≥ 2r + 1, rectangular domains: results based on
Bernstein representation minimal determining sets
[Alfeld, P ., Schumaker, L.L., CA 1987] [Alfeld P ., JCAM 2000] [Deng, J.-S., Chen, F-L., Feng, Y.-Y., JCAM 2006] [Schumaker, L. L. and Wang, L., 2011, preprint] [Schumaker, L. L. and Wang, L., NM 2011]
smoothing cofactors
[Wang, R.-H., 2001] [Huang, Z.-J., Deng J.-S. Feng, Y.-Y., Chen, F .-L., JCM 2006]
Generalized B-splines and local refinements – p. 32/50
T-mesh: T
partition of a (rectangular) domain by rectangles so that T-junctions (hanging vertices) are allowed
Generalized B-splines and local refinements – p. 33/50
T-mesh: T
partition of a (rectangular) domain by rectangles so that T-junctions (hanging vertices) are allowed
d(T ) := {s(x, y) ∈ Cr, s(x, y)|τi ∈ I
Pu1,v1
d1
⊗ I Pu2,v2
d2
, τi ∈ T }, I Pu,v
p
:=< 1, t, . . . , tp−2, u(t), v(t) >
Generalized B-splines and local refinements – p. 33/50
suitable spaces : exponential, trigonometric
Generalized B-splines and local refinements – p. 34/50
suitable spaces : exponential, trigonometric smoothness cond.: Bernstein like representation
Generalized B-splines and local refinements – p. 34/50
suitable spaces : exponential, trigonometric smoothness cond.: Bernstein like representation
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 cubics
Generalized B-splines and local refinements – p. 34/50
suitable spaces : exponential, trigonometric smoothness cond.: Bernstein like representation
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 cubics
−1 1 2 0.5 1 1.5 2 5 10 15 20
C1 exponential (cubics)
Generalized B-splines and local refinements – p. 34/50
Generalized B-splines and local refinements – p. 35/50
1 2 3 1 2 3 −1 −0.5 0.5 1
C1 cubics
Generalized B-splines and local refinements – p. 35/50
1 2 3 1 2 3 −1 −0.5 0.5 1
C1 cubics
1 2 3 1 2 3 −1 −0.5 0.5 1
C1 trigonometric (cubics), ω = 2
5 π
Generalized B-splines and local refinements – p. 35/50
trigonometric/exponential C1 cubics:
dim(
S1
3(T )) = 4(Vb + V+) Vb : #b. vertices, V+ : #cross. vertices dim(S1
3(T )) = 4(9 + 1)
Generalized B-splines and local refinements – p. 36/50
Hierarchical bases, T-meshes: similar behavior of B-splines/GB-splines
Generalized B-splines and local refinements – p. 37/50
Hierarchical bases, T-meshes: similar behavior of B-splines/GB-splines Triangulations?
Generalized B-splines and local refinements – p. 37/50
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
ONTP basis {B0, B1, B2} B0(0) = 1, B0(1) = B′
0(1) = 0, · · ·
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
ONTP basis {B0, B1, B2} B0(0) = 1, B0(1) = B′
0(1) = 0, · · ·
Bernstein like representation control polygon for functions?
t / ∈< 1, u(t), v(t) >
No Greville abscissae
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
ONTP basis {B0, B1, B2} B0(0) = 1, B0(1) = B′
0(1) = 0, · · ·
control points f = b0B0 + b1B1 + b2B2 ∈ Pu,v
2 ⇓ (0, b0), (ξ, b1), (1 − ξ, b1), (1, b2) B0(t) = B2(1 − t) ξ = −1/B′
0(0) = 1/B′ 2(1)
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
ONTP basis {B0, B1, B2} B0(0) = 1, B0(1) = B′
0(1) = 0, · · ·
control points f = b0B0 + b1B1 + b2B2 ∈ Pu,v
2 ⇓ (0, b0), (ξ, b1), (1 − ξ, b1), (1, b2) B0(t) = B2(1 − t) ξ = −1/B′
0(0) = 1/B′ 2(1)
Generalized B-splines and local refinements – p. 38/50
Pu,v
2
:=< 1, u(t), v(t) >
ONTP basis {B0, B1, B2} B0(0) = 1, B0(1) = B′
0(1) = 0, · · ·
control points f = b0B0 + b1B1 + b2B2 ∈ Pu,v
2 ⇓ (0, b0), (ξ, b1), (1 − ξ, b1), (1, b2) B0(t) = B2(1 − t) ξ = −1/B′
0(0) = 1/B′ 2(1)
geometric properties of the usual control polygon
Generalized B-splines and local refinements – p. 38/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
Generalized B-splines and local refinements – p. 39/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω = 0.1 ω = 1.5 ω = 10
Generalized B-splines and local refinements – p. 39/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω = 0.1 ω = 1.5 ω = 10
ONTP basis B0,ω, B1,ω, B2,ω, ω → 0 quadratic Bernstein pol.
Generalized B-splines and local refinements – p. 39/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω = 0.1 ω = 1.5 ω = 10 ω = 1.5
Generalized B-splines and local refinements – p. 39/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω = 0.1 ω = 1.5 ω = 10 ω = 10
Generalized B-splines and local refinements – p. 39/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
Generalized B-splines and local refinements – p. 40/50
Hω :=< 1, cosh ωt, sinh ωt >, t ∈ [0, 1]
Generalized B-splines and local refinements – p. 40/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3
Generalized B-splines and local refinements – p. 41/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3 < 1, τ1, τ2, τ3 = 1 − τ1 − τ2, τ2
1 , τ2 2 , τ2 3 >,
Generalized B-splines and local refinements – p. 41/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3
Hω :=< 1, cosh ωτ1, sinh ωτ1, cosh ωτ2, sinh ωτ2, cosh ωτ3, sinh ωτ3 >,
Generalized B-splines and local refinements – p. 41/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3
Hω :=< 1, cosh ωτ1, sinh ωτ1, cosh ωτ2, sinh ωτ2, cosh ωτ3, sinh ωτ3 >,
dim(Hω) = 7
Generalized B-splines and local refinements – p. 41/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3
Hω :=< 1, cosh ωτ1, sinh ωτ1, cosh ωτ2, sinh ωτ2, cosh ωτ3, sinh ωτ3 >, Hω|τ 3=0 :=< 1, cosh ωτ1, sinh ωτ1 >,
Generalized B-splines and local refinements – p. 41/50
V1 V2 V3 X X = τ1V1 + τ2V2 + τ3V3 B200,ω(X) = B2,ω(τ1), B020,ω(X) = B2,ω(τ2), B002,ω(X) = B2,ω(τ3)
Generalized B-splines and local refinements – p. 42/50
B200,ω B020,ω B002,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = 0.1
Generalized B-splines and local refinements – p. 42/50
B200,ω B020,ω B002,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = 10
Generalized B-splines and local refinements – p. 42/50
V1 V2 V3 X B110,ω ???
Generalized B-splines and local refinements – p. 43/50
B110,ω ???
Generalized B-splines and local refinements – p. 43/50
B110,ω ??? 7 suitable interp. conditions to recover edge behavior
Generalized B-splines and local refinements – p. 43/50
B110,ω ??? 7 suitable interp. conditions to recover edge behavior
easy: 6 function values at ∗
Generalized B-splines and local refinements – p. 43/50
B110,ω ??? 7 suitable interp. conditions to recover edge behavior
easy: 6 function values at ∗ exotic: second derivative at one vertex to mimic the polynomial case
Generalized B-splines and local refinements – p. 43/50
B110,ω B101,ω B011,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = 0.1
Generalized B-splines and local refinements – p. 44/50
B110,ω B101,ω B011,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = 10
Generalized B-splines and local refinements – p. 44/50
B111,ω ???
Generalized B-splines and local refinements – p. 45/50
B111,ω = 1 −
Bijk,ω
Generalized B-splines and local refinements – p. 45/50
B111,ω = 1 −
Bijk,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = .1
Generalized B-splines and local refinements – p. 45/50
B111,ω = 1 −
Bijk,ω
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
ω = 10
Generalized B-splines and local refinements – p. 45/50
Bijk,ω ≥ 0
partition of unity
Generalized B-splines and local refinements – p. 46/50
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
Generalized B-splines and local refinements – p. 46/50
0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 u v 0.8 0.8 0.6 1 1 0.8 1
Generalized B-splines and local refinements – p. 46/50
NO Greville abscissae
Generalized B-splines and local refinements – p. 47/50
ω = 0.01 ω = 1.5
Generalized B-splines and local refinements – p. 47/50
ω = 1.5 ω = 2.57
Generalized B-splines and local refinements – p. 47/50
ω = 2.57 ω = 3
Generalized B-splines and local refinements – p. 47/50
ω = 3 ω = 10
Generalized B-splines and local refinements – p. 47/50
USUAL geometric interpretation
Generalized B-splines and local refinements – p. 48/50
USUAL geometric interpretation
ω = 0.1
Generalized B-splines and local refinements – p. 48/50
USUAL geometric interpretation
ω = 1.5
Generalized B-splines and local refinements – p. 48/50
USUAL geometric interpretation
ω = 10
Generalized B-splines and local refinements – p. 48/50
Bernstein-like representations
not confined to (piecewise) polynomial spaces
Generalized B-splines and local refinements – p. 49/50
Bernstein-like representations
not confined to (piecewise) polynomial spaces
Generalized (trigonometric/exponential/...) B-splines possible alternative to the rational model
Bernstein-like representations CAGD applications IgA applications
Generalized B-splines and local refinements – p. 49/50
Bernstein-like representations
not confined to (piecewise) polynomial spaces
Generalized (trigonometric/exponential/...) B-splines possible alternative to the rational model
Bernstein-like representations CAGD applications IgA applications
Local refinements B-splines/Generalized B-splines
Hierarchical bases T-meshes
Generalized B-splines and local refinements – p. 49/50
Bernstein-like representations
not confined to (piecewise) polynomial spaces
Generalized (trigonometric/exponential/...) B-splines possible alternative to the rational model
Bernstein-like representations CAGD applications IgA applications
Local refinements B-splines/Generalized B-splines
Hierarchical bases T-meshes
B-splines and GB-splines similar structure/properties thanks to 1D Bernstein-like representation.
Bernstein-like representations
not confined to (piecewise) polynomial spaces
Generalized (trigonometric/exponential/...) B-splines possible alternative to the rational model
Bernstein-like representations CAGD applications IgA applications
Local refinements B-splines/Generalized B-splines
Hierarchical bases T-meshes
B-splines and GB-splines similar structure/properties thanks to 1D Bernstein-like representation. Extending Bernstein representations/Generalized B-splines to triangles is not trivial
Generalized B-splines and local refinements – p. 49/50
Generalized B-splines and local refinements – p. 50/50