Information Technology Reference
In-Depth Information
In the second section we give a brief definition of what spline functions are,
followed by a section that describes the main ideas behind the IGA framework.
In the fourth section we describe some of the PetIGA data-structures and their
parallelism. In the last section, we show how the framework can be applied to
the phase-field models mentioned before.
2 Spline Spaces
To define a univariate B-spline basis one needs to specify the number
n
of basis
functions wanted, the polynomial degree
p
of the basis and a knot vector
ʞ
=
{
ʶ
1
,...,ʶ
1
,ʶ
2
,...,ʶ
2
,...,ʶ
m
,...,ʶ
m
}
,
(1)
r
1
times
r
2
times
r
m
times
i
=1
r
i
=
n
+
p
+1and
ʶ
i
<ʶ
i
+1
. The B-spline basis functions are piecewise
polynomials of degree
p
on the subdivision
ʶ
=
m
with
.
A stable way of generating them involves using the Cox-de Boor recursion
algorithm [5], which receives as inputs
p
and
ʞ
. Knot multiplicity is an essential
ingredient in spline theory, since it allows to control the smoothness of the basis.
Indeed, if a breakpoint
ʶ
j
has multiplicity
r
j
, then the basis functions have at
least
ʱ
j
:=
p
{
ʶ
1
,...,ʶ
m
}
−
r
j
continuous derivatives at
ʶ
j
. The vector
ʱ
:=
{
ʱ
1
,...,ʱ
m
}
collects the basis regularities.
The space of spline functions is denoted by
n
i
=1
. The multi-
variate cases are defined by tensor products of univariate spaces. As an example,
the bivariate spline space is defined by
B
i
}
p
α
S
:= span
{
p
1
,p
2
α
1
,
α
2
p
1
p
2
α
2
S
=
S
α
1
↗S
.
3 Isogeometric Analysis Concept
Spline spaces are one of the main theories that the Computer Assisted De-
sign (CAD) community uses to model geometries on a computer [5]. They are
used as basis functions to parameterize euclidean subsets. The main concept
behind Isogeometric analysis (IGA) is to use the same spline functions as basis
functions for the Galerkin approximation of partial differential equations. As is
done with isoparametric finite elements methods, the parameterization, here-
after denoted as
F
:[0
,
1]
d
n
, called
patch
by the CAD community and
geometric mapping
by the IGA community, can then be used to induce an
approximation space on the domain of the equation,
ʩ
=
F
([0
,
1]
d
).
It is important to note that the tensor product nature of a multivariate basis
induces a structured grid on the parametric space. That is, we end up with a set
of basis on a structured grid. An ecient implementation should take this into
account.
Besides the possibility of
h
-refinement and
p
-refinement, spline functions also
allow the possibility of a new type of refinement, called
k
-refinement in the
IGA literature. Within
k
-refinement, the basis has a higher regularity than only
continuous. This opened another pathway of application for the IGA framework,
namely, the discretization of higher-order differential operators [8].
ₒ
R
