Biomedical Engineering Reference
In-Depth Information
3.1
Mixed Finite Element Model
The mixed model is a natural formulation in which the weak forms of (
3
)areused
to construct the finite element method. The resulting finite element model is termed
the velocity-pressure model or mixed model. Developing a Galerkin formulation the
weak forms of (
3
) results in the following finite element equations:
:
u
M
C
C
.
u
/
u
C
Ku
QP
D
F
Q
T
u
D
0
(6)
where the superpose dot represents a time derivative. Considering
N
and
L
the
element interpolation functions for the velocity and pressure, the elements of the
matrices at the finite element level are defined as:
Z
Z
N
i
u
x
@N
j
@x
de
u
y
@N
j
@y
u
z
@N
j
@
z
M
ij
D
N
i
N
j
de
I
C
ij
D
C
C
e
e
Z
@N
i
@x
de
Z
@N
j
@x
@N
i
@y
@N
j
@y
@N
i
@
z
@N
j
@
z
@N
i
@˛
Q
ik
D
K
ij
D
C
C
I
L
k
(7)
e
e
and the resulting equation system is:
C
u
P
F
0
C
K
Q
:
u
M
C
D
(8)
Q
T
0
The above partitioned system (
8
) with a null submatrix could in principle be
solved in several ways. However, it can be asked under which conditions it can
be safely solved. This problem results from the incompressibility condition. In
simple terms, we want to obtain, in the linear space U of all admissible solutions,
the velocity field
u
belonging to a subspace I
h
U , associated to the space of
incompressible deformations. This subspace is given as:
n
u
h
0
o
I
h
U
h
Qu
h
D
2
W
D
(9)
The solution I
h
should then lie on the null space of
Q
that must be zero. The
numerical problem described above is eliminated by proper choice for the finite
element spaces of the velocity and pressure fields; in other words the evaluation
of the integrals for the stiffness matrix where velocity and pressure interpolations
appear must satisfy the Babuska-Brezzi compatibility condition the so called LBB
condition [
15
-
17
] that states velocity and pressure spaces can not be chosen
arbitrarily and a link between them is necessary.
In this chapter the numerical procedure for the transient non-Newton inelastic
Navier-Stokes equations uses the Galerkin-finite element method with implicit
time discretization. Considering a 3D analysis hexahedral meshes often provide
the best quality solution as errors due to numerical diffusion are reduced whenever
