Biomedical Engineering Reference
In-Depth Information
a good alignment between mesh edges and flow exists [
4
]; in this work a spatial
discretization with isoparametric brick elements of low order with trilinear approx-
imation for the velocity components and element constant pressure is adopted:
X
8
D
D
u
.
x
;t/
N
i
.
x
/
u
i
.t /
and p.t/
Mp
c
.t /
(10)
i
D
1
where
u
i
and p
c
are the unknown element velocity node values and the pressure
element center value, respectively. At each time step Picard iteration is applied to
linearize the non-linear convection and diffusion terms; the method is based on a
pressure correction [
11
,
18
]. The essential steps of the algorithm at a time or iteration
are:
1. Calculation of an auxiliary velocity field from the equations of motion using
known pressure values from the previous time step or previous iteration step;
2. Calculation of the pressure correction using lumped mass matrix;
3. Pressure updating;
4. Calculation of the divergence free velocity field;
5. Calculation of the apparent viscosity.
This method developed for obtaining a divergence-free velocity field has been
based on Chorin's method [
19
] and validated by other authors.
3.2
Penalty Finite Element Model
The incompressibility constraint given by ©
ii
0 is difficult to implement due to
the zero divergence condition for the velocity field. The incompressible problem
may be stated as a constrained minimization of a functional. The penalty function
method, like the Lagrange multiplier method, allows us to reformulate a problem
with constraints as one without constraints [
15
,
17
]. Using the penalty function
method proposed by Courant [
20
], the problem is transformed into the minimization
of the unconstrained augmented functional:
D
Z
2
.
u
/
D
.
u
/
C
."
ii
.
u
//
dV
(11)
V
Considering the pseudo-constitutive relation for the incompressibility constraint
the second set of (
3
) is replaced by:
r
:
u
D
p=
(12)
where is the penalty parameter. If is too small compressibility and pressure
errors will occur and an excessively large value may result in numerical ill
