Information Technology Reference
In-Depth Information
By considering homogeneous boundary conditions equal to zero and periodic
boundary conditions in all directions, the strong form of the problem can then
be stated as follows: over the spatial domain
ʩ
and the time interval ]0
,T
[, given
ˆ
0
:
ʩ
−ₒ
R
, find
ˆ
:
ʩ
×
[0
,T
]
−ₒ
R
such that
⊧
⊨
∂t
=
ʔ
(1 +
ʔ
)
2
ˆ
ˆ
+
ˆ
3
∂ˆ
−
on
ʩ
×
]0
,T
]
(11)
⊩
on
ʩ
ˆ
(
x
,
0) =
ˆ
0
(
x
)
3
corresponds to the Sobolev
space of square integrable functions with square integrable first, second deriva-
tives and third derivatives. A weak form is obtained by multiplying equation (11)
by a test functions
q
3
is defined, where
The functional space
V∈H
H
∈H
3
, and integrating by parts. One can state the problem
as: find
ˆ
∈V
and
˃
∈V
such that for all
q
∈V
and
w
∈V
q,
∂ˆ
∂t
)
ˆ
+
ˆ
3
))
ʩ
−
3
q,
3
ˆ
)
ʩ
= 0 (12)
+(
∇
q,
∇
((1
−
2(
ʔq, ʔˆ
)
ʩ
+(
∇
∇
ʩ
To derive a finite approximation to the problem, we pick the finite dimensional
space
V
h
ↂV
and derive a semi discrete formulation. The problem is then to
find
ˆ
h
h
such that for all
q
h
h
∈V
∈V
0=
q
h
,
∂ˆ
h
∂t
)
ˆ
h
+
ˆ
h
3
))
ʩ
q
h
,
+(
∇
∇
((1
−
ʩ
2(
ʔq
h
,ʔˆ
h
)
ʩ
+(
3
q
h
,
3
ˆ
h
)
ʩ
= 0
−
∇
∇
(13)
h
is spanned by the linear combi-
We again suppose that the discrete space
V
2
-continuous B-spline basis functions.
An example modeling crack propagation in a ductile material [18] is shown in
Figure 3. A circular notch of radius 20
ˀ/
3 is set at the center of a crystalline
lattice, defined as
ˆ
(
t
=0
,
x
)=0
.
49 + cos(q
x
x
)cos
q
y
y/
√
3
nation of basis functions
N
A
,whichare
C
0
.
5cos
2q
y
y/
√
3
.
(14)
where q
x
and q
y
are equal to 1
.
16
/
√
2and1
.
15
/
√
2, respectively. These assigned
values, different from the equilibrium values that q
x
and q
y
are supposed to take,
induce stretching in both the
x
and
y
directions of 16% and 15%, respectively,
and explain the propagating crack in the domain. With regards to the numerical
simulation, this problem is not as hard to solve as the one presented previously,
and requires no more than 4 nonlinear Newton iterations per time step.
−
6 Conclusions
In this paper, a scalable implementation of isogeometric analysis is presented.
The framework is built in such a way that so as to have the the user only
worry about coding the discrete variational formulation of the problem. Besides
