Biomedical Engineering Reference
In-Depth Information
19.3 Numerical Application
In order to treat the strongly coupled multiphasic and multiphysical problem numer-
ically, the FE solver P
ANDAS
1
is used. The primary variables of the present initial-
boundary-value problem (IVBP) are the solid displacement
u
S
with corresponding
test function
δ
u
S
associated with the momentum balance (
19.7
) of the overall aggre-
gate, the effective pore pressures
p
ξR
with test functions
δp
ξR
corresponding to the
volume balances (
19.5
) and (
19.6
) of the interstitial fluid and the blood plasma, and
the concentration
c
m
with test function
δc
m
belonging to the concentration balance
(
19.4
) of the therapeutic agent. After a transformation of the local balance equations
into weak formulations, the momentum balance of the overall aggregate yields
G
u
S
≡
T
·
grad
δ
u
S
d
v
−
ρ
g
·
δ
u
S
d
v
−
t
·
δ
u
S
d
a
=
0
,
(19.17)
Ω
Ω
Γ
t
where
t
Tn
is the external stress vector acting on the boundary of the overall
aggregate and
n
is the outward-oriented unit surface normal. The weak form of the
liquid constituents reads
=
n
ξ
S
+
n
ξ
div
(
u
S
)
S
δp
ξR
d
v
G
p
ξ
≡
Ω
n
ξ
w
ξ
·
grad
δp
ξR
d
v
v
ξ
δp
ξR
d
a
−
+
Γ
v
ξ
¯
=
0
,
(19.18)
Ω
v
ξ
n
ξ
w
ξ
·
where
n
is the efflux of liquid volume. Finally, the weak formulation of
the concentration balance is
¯
=
n
I
c
m
S
+
n
I
c
m
div
(
u
S
)
S
δc
m
d
v
G
c
m
≡
Ω
n
I
c
m
w
D
·
grad
δc
m
d
v
+
¯
ı
D
¯
ı
D
δc
m
d
a
=
−
0
,
(19.19)
Ω
Γ
¯
ı
D
=
n
I
c
m
w
D
·
where
n
is the molar efflux of the therapeutic agent.
The spatial discretization of the coupled solid-fluid-transport problem within a
u
S
-
p
BR
-
p
IR
-
c
m
-formulation requires mixed finite elements (see, e.g., Ellsiepen,
1999
) with a simultaneous approximation of all primary variables. A standard
Galerkin method is applied using extended Taylor-Hood elements with quadratic
shape functions for
u
S
and linear shape functions for
p
IR
,
p
BR
and
c
m
in order
to obtain a stable numerical solution. This leads to a differential-algebraic system
of equations, which is solved in a monolithic manner with an implicit Euler time-
integration scheme.
1
P
orous media
A
daptive
N
onlinear finite element solver based on
D
ifferential
A
lgebraic
S
ystems
