<s>
Isogeometric	B-Algorithm
analysis	I-Algorithm
is	O
a	O
computational	O
approach	O
that	O
offers	O
the	O
possibility	O
of	O
integrating	O
finite	B-Application
element	I-Application
analysis	I-Application
(	O
FEA	O
)	O
into	O
conventional	O
NURBS-based	O
CAD	B-Application
design	I-Application
tools	O
.	O
</s>
<s>
Currently	O
,	O
it	O
is	O
necessary	O
to	O
convert	O
data	O
between	O
CAD	B-Application
and	O
FEA	O
packages	O
to	O
analyse	O
new	O
designs	O
during	O
development	O
,	O
a	O
difficult	O
task	O
since	O
the	O
two	O
computational	O
geometric	O
approaches	O
are	O
different	O
.	O
</s>
<s>
Isogeometric	B-Algorithm
analysis	I-Algorithm
employs	O
complex	O
NURBS	B-Algorithm
geometry	O
(	O
the	O
basis	O
of	O
most	O
CAD	B-Application
packages	O
)	O
in	O
the	O
FEA	O
application	O
directly	O
.	O
</s>
<s>
A	O
reference	O
free	B-Application
software	I-Application
implementation	O
of	O
some	O
isogeometric	B-Algorithm
analysis	I-Algorithm
methods	O
is	O
GeoPDEs	O
.	O
</s>
<s>
For	O
instance	O
,	O
PetIGA	O
is	O
an	O
open	O
framework	O
for	O
high	O
performance	O
isogeometric	B-Algorithm
analysis	I-Algorithm
heavily	O
based	O
on	O
PETSc	B-Language
.	O
</s>
<s>
Furthermore	O
,	O
G+Smo	O
is	O
an	O
open	O
C++	O
library	O
for	O
isogeometric	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
FEAP	O
is	O
a	O
finite	B-Application
element	I-Application
analysis	I-Application
program	O
which	O
includes	O
an	O
Isogeometric	B-Algorithm
analysis	I-Algorithm
library	O
FEAP	O
IsoGeometric	B-Algorithm
(	O
Version	O
FEAP84	O
&	O
Version	O
FEAP85	O
)	O
.	O
</s>
<s>
Isogeometric	B-Algorithm
analysis	I-Algorithm
presents	O
two	O
main	O
advantages	O
with	O
respect	O
to	O
the	O
finite	B-Application
element	I-Application
method	I-Application
:	O
</s>
<s>
Wave	O
propagation	O
problems	O
,	O
arising	O
for	O
instance	O
in	O
cardiac	B-Algorithm
electrophysiology	I-Algorithm
,	O
acoustics	O
and	O
elastodynamics	O
,	O
are	O
better	O
described	O
,	O
thanks	O
to	O
the	O
reduction	O
of	O
numerical	B-Algorithm
dispersion	I-Algorithm
and	O
dissipation	O
errors	O
.	O
</s>
<s>
A	O
patch	O
is	O
mapped	O
from	O
a	O
reference	O
rectangle	O
in	O
two	O
dimensions	O
and	O
from	O
a	O
reference	O
cuboid	B-General_Concept
in	O
three	O
dimensions	O
:	O
it	O
can	O
be	O
seen	O
as	O
the	O
entire	O
computational	O
domain	O
or	O
a	O
smaller	O
portion	O
of	O
it	O
.	O
</s>
<s>
Once	O
a	O
definition	O
of	O
knot	O
vector	O
is	O
provided	O
,	O
several	O
types	O
of	O
basis	O
functions	O
can	O
be	O
introduced	O
in	O
this	O
context	O
,	O
such	O
as	O
B-splines	B-Algorithm
,	O
NURBS	B-Algorithm
and	O
T-splines	B-Algorithm
.	O
</s>
<s>
B-splines	B-Algorithm
can	O
be	O
derived	O
recursively	O
from	O
a	O
piecewise	O
constant	O
function	O
with	O
:	O
</s>
<s>
Using	O
De	B-Algorithm
Boor	I-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
generate	O
B-splines	B-Algorithm
of	O
arbitrary	O
order	O
:	O
</s>
<s>
B-splines	B-Algorithm
that	O
are	O
generated	O
in	O
this	O
way	O
own	O
both	O
the	O
partition	O
of	O
unity	O
and	O
positivity	O
properties	O
,	O
i.e	O
:	O
</s>
<s>
So	O
as	O
to	O
calculate	O
derivatives	B-Algorithm
or	O
order	O
of	O
the	O
B-splines	B-Algorithm
of	O
degree	O
,	O
another	O
recursive	O
formula	O
can	O
be	O
employed	O
:	O
</s>
<s>
A	O
B-spline	B-Algorithm
curve	O
can	O
be	O
written	O
in	O
the	O
following	O
way	O
:	O
</s>
<s>
An	O
extension	O
to	O
the	O
two	O
dimensional	O
case	O
can	O
be	O
easily	O
obtained	O
from	O
B-splines	B-Algorithm
curves	O
.	O
</s>
<s>
In	O
particular	O
B-spline	B-Algorithm
surfaces	I-Algorithm
are	O
introduced	O
as	O
:	O
</s>
<s>
Finally	O
,	O
B-splines	B-Algorithm
solids	O
,	O
that	O
need	O
three	O
sets	O
of	O
B-splines	B-Algorithm
basis	O
functions	O
and	O
a	O
tensor	O
of	O
control	O
points	O
,	O
can	O
be	O
defined	O
as	O
:	O
</s>
<s>
B-splines	B-Algorithm
,	O
due	O
to	O
their	O
intrinsic	O
structure	O
,	O
are	O
not	O
able	O
to	O
generate	O
properly	O
circular	O
shapes	O
for	O
instance	O
.	O
</s>
<s>
In	O
order	O
to	O
circumvent	O
this	O
issue	O
,	O
non-uniform	B-Algorithm
rational	I-Algorithm
B-splines	I-Algorithm
,	O
also	O
known	O
as	O
NURBS	B-Algorithm
,	O
are	O
introduced	O
in	O
the	O
following	O
way	O
:	O
</s>
<s>
where	O
is	O
a	O
one	O
dimensional	O
B-spline	B-Algorithm
,	O
is	O
referred	O
to	O
as	O
weighting	O
function	O
,	O
and	O
finally	O
is	O
the	O
weight	O
.	O
</s>
<s>
Following	O
the	O
idea	O
developed	O
in	O
the	O
subsection	O
about	O
B-splines	B-Algorithm
,	O
NURBS	B-Algorithm
curve	I-Algorithm
are	O
generated	O
as	O
follows	O
:	O
</s>
<s>
The	O
extension	O
of	O
NURBS	B-Algorithm
basis	O
functions	O
to	O
manifolds	O
of	O
higher	O
dimensions	O
(	O
for	O
instance	O
2	O
and	O
3	O
)	O
is	O
given	O
by	O
:	O
</s>
