Biomedical Engineering Reference
In-Depth Information
~
Ę“
Wt
Q
Wt
Fig. 5.6
Stress tensor extrapolation from elements K adjacent to the boundary
where l denotes the distance of the masses m
1
and m
2
from the center mass m
3
(see
Fig.
5.4
b). The aerodynamic quantities are defined by
L.t/ D h
s
Z
M.t/ Dh
s
Z
2
2
X
X
2j
n
j
dS;
ij
n
j
r
ort
dS; (5.87)
i
Wt
Wt
j
D
1
i;j
D
1
where r
ort
1
D.x
2
x
C2
/, r
ort
2
D x
1
x
C1
, .x
C1
;x
C2
/ D .0;L=2/ and the
components
ij
are defined by (
5.2
).
The evaluation of the lift L and the moment M at time t D t
n
C
1
from the
approximate solution U
h
D .v
h
;p
h
/ can be carried out in two ways.
Stress Tensor Extrapolation
One possibility is to compute the components
ij
of the stress tensor at time t D
t
n
C
1
on the elements K 2
T
h
adjacent to the boundary
Q
Wt
, extrapolate
ij
to
Q
Wt
,
see Fig.
5.6
, and then to compute L and M by the integration along
Q
Wt
.
This approach can be improved by using the idea of superconvergence of gradient
in the center of triangles, cf. [
14
,
110
]. It means that for any vertex A 2
Q
Wt
the value
of the stress tensor
ij
can be reconstructed using least squares approximation from
the patches of neighboring elements. See Fig.
5.7
.
Weak Formulation of Lift Force
In this paragraph
Q
Wt
represents again the interface between the structure and the
domain
t
. The Navier-Stokes equations in the ALE form discretized with respect
to time at instant t WD t
n
C
1
can be expressed component-wise as
Search WWH ::
Custom Search