Biomedical Engineering Reference
In-Depth Information
ˇ being ˇ a constant of order 10
7
for
double precision calculations. The second set of (
3
) is eliminated and the Navier-
Stokes equations become:
conditioning; generally is assigned to
D
C
C
K
u
:
u
M
C
K
C
D
F
(13)
where
K
is the so-called penalty matrix:
Z
M
ij
K
QM
p
Q
t
M
p
P
Q
t
u
D
and
D
D
L
i
L
j
de
(14)
e
Under such conditions the pressure is eliminated as a field variable since it can
be recovered by the approximation of (
12
). If the standard Galerkin formulation is
applied it is necessary to use compatible spaces for the velocity and the pressure in
order to satisfy the LBB stability condition. This often excludes the use of the equal
order interpolation functions for both fields. In order to avoid oscillatory results
the numerical problem is eliminated by proper evaluation of the integrals for the
stiffness matrix where penalty terms are calculated using a numerical integration
rule of an order less than that required to integrate them exactly. This technique
of under-integrating the penalty terms is known in the literature as the reduced
integration.
3.3
Streamline Upwind/Petrov Galerkin Method
In a Galerkin formulation there is no doubt that the most difficult problem arises
because of the nonlinear convective term in (
3
). In blood flow high Reynolds
numbers appear and loss of unicity of solution, hydrodynamical instabilities and
turbulence are caused by this apparently innocent term. The numerical scheme
requires a stabilization technique in order to avoid oscillations in the numerical
solution. The most appropriate technique to solve these problems is the Streamline
upwind/Petrov Galerkin method, SUPG-method [
21
,
22
]. The goal of this technique
is the elimination of the instability problems of the Galerkin formulation by
introducing an artificial dissipation. The method uses modified velocity shape
functions, W
i
, for the convective terms:
K
SUPG
u
r
N
i
W
i
D
N
i
C
(15)
k
u
k
where K
SUPG
denotes the upwind parameter that controls the factor of upwind
weighting. This parameter controls the amount of upwind weighting and is defined
on an element as:
1
2
i
.P e
e
/
u
i
h
i
;
k
SUPG
D
i
D
1; 2; 3
(16)
