IGA main concepts
The computational domain ⌦ is parametrized by spline/NURBS:
F : b⌦! ⌦, where F(⇠, ⌘) =P
iCiBi(⇠, ⌘), and Ci are the control points
G. Sangalli (UNIPV) Splines and PDEs: Recent Advances July 3 - July 7, 2017 5 / 13
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IGA main concepts
The computational domain ⌦ is parametrized by spline/NURBS:
F : b⌦! ⌦, where F(⇠, ⌘) =P
iCiBi(⇠, ⌘), and Ci are the control points
F !
Bi(⇠, ⌘) Bi· F(x, y)
G. Sangalli (UNIPV) Splines and PDEs: Recent Advances July 3 - July 7, 2017 5 / 13
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IGA main concepts
The computational domain ⌦ is parametrized by spline/NURBS:
F : b⌦! ⌦, where F(⇠, ⌘) =P
iCiBi(⇠, ⌘), and Ci are the control points
F !
IGA has been proposed by structural engineers:
Isoparametric paradigm: push-forward of splines/NURBS basis functions on the computational domain ⌦ to approximate the solution of the PDE.
Finite Element kind method (splines etc. replace piecewise polynomials)
G. Sangalli (UNIPV) Splines and PDEs: Recent Advances July 3 - July 7, 2017 5 / 13
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SINITTA
Isoparametric concept I
ˆ
F
initial
def ormed
u
Initial geometry parametrization
Displacement vector
⌦deformed is parametrized by (F +u|{z}· F
bu
)
Isoparametric construction
The unknown written w.r.t. the parametric variables (bu : b⌦! ⌦deformed) is represented by the same fuctions that are used for the geometry parametrization F : b⌦! ⌦initial
G. Sangalli (UNIPV)
2 -
Splines and PDEs: Recent Advances July 3 - July 7, 2017 6 / 13Isoparametric concept I
ˆ
F
initial
def ormed
u
Initial geometry parametrization
Displacement vector
⌦deformed is parametrized by (F +u|{z}· F
bu
)
Isoparametric construction
The unknown written w.r.t. the parametric variables (bu : b⌦! ⌦deformed) is represented by the same fuctions that are used for the geometry parametrization F : b⌦! ⌦initial
G. Sangalli (UNIPV)
I T
Splines and PDEs: Recent Advances July 3 - July 7, 2017 6 / 13Isoparametric concept II
F
initial
ˆ
G. Sangalli (UNIPV)
I o
Splines and PDEs: Recent AdvancesT
July 3 - July 7, 2017 7 / 13Isoparametric concept II
ˆ
F
initial
c
def ormed
a constantcbelongs to the isoparametric space if the same constant bc =c· F is represented in the basis over b⌦...
ok for splines and NURBS (partition of the unity)
G. Sangalli (UNIPV) I TSplines and PDEs: Recent Advances July 3 - July 7, 2017 8 / 13
Isoparametric concept II
ˆ
F
initial
c
def ormed
a constantcbelongs to the isoparametric space if the same constant bc =c· F is represented in the basis over b⌦... ok for splines and NURBS (partition of the unity)
G. Sangalli (UNIPV)
Tia
Splines and PDEs: Recent Advances July 3 - July 7, 2017 8 / 13Isoparametric concept II
F
initial
ˆ
def ormed
u
a constantcbelongs to the isoparametric space if the same constant bc=c· F is represented in the basis over b⌦... ok for splines and NURBS (partition of the unity)
u(x, y) = A⇥x
y⇤
is inanyisoparametric space since u(⇠, ⌘) = A Fb ⇥⇠
⌘
⇤=AhF
1(⇠,⌘) F2(⇠,⌘)
iis a vector field whose components are linear combination of the parametrization components F1(⇠, ⌘) and F2(⇠, ⌘), and then is represented in the same basis
G. Sangalli (UNIPV) Splines and PDEs: Recent Advances July 3 - July 7, 2017 9 / 13
=D
- #
Isoparametric concept II
ˆ
F
initial
def ormed
u
The isoparametric space contains any rigid body motion u(x, y) = c + R⇥x
y⇤ that form the kernel of the internal elastic energy
G. Sangalli (UNIPV) tSplines and PDEs: Recent Advances July 3 - July 7, 2017 10 / 13
IGA main side-effect
The computational domain ⌦ is parametrized by spline/NURBS:
F : b⌦! ⌦, where F(⇠, ⌘) =P
iCiBi(⇠, ⌘), and Ci are the control points
F !
Isogeometric spaces are smooth
G. Sangalli (UNIPV)
