Biomedical Engineering Reference
In-Depth Information
6.2.2
C
S
S
S
OMPUTER
IMULATION
OF
YMMETRIC
WELLING
C
P
G
AND
ONTRACTION
OF
OLYELECTROLYTE
ELS
The numerical problem is solved in a fully Lagrangian sense: All field variables are
expressed as functions of time,
t
, and of the original particle location,
X
. The vector
g
quantity
effectively labels the particles of gel. This approach is natural for
problems of large deformation, for which the finite element mesh will undergo
significant deflection.
The governing differential equations are transformed into a system of algebraic
equations through a standard Galerkin finite element formalism (Segalman et al.,
1993, and Segalman and Witkowski, 1994, for instance). However, because of the
Lagrangian formulation, all interpolation is over a material manifold rather than over
space. For instance, the configuration field of the gel is interpolated:
X
g
Nodes
∧
(
)
=
(
)
∑
( )
i
I
btX
,
x t
φ
X
(6.12)
g
gi
g
i
=
1
where
. The other
fields are also represented as linear combinations of appropriate basis functions:
x
(
t
) is the location of particle
X
(attached to node
i
) at time
t
gi
gi
∧
∧
(
)
=
(
)
Nodes
)
=
∑
( )
s
ξ
tX
,
ξ
t
φ
X
g
s i
,
g
s
i
=
1
i
(
(
)
Nodes
∑
( )
H
ξ
tX
,
ξ
t
φ
X
(6.13)
g
Hi
,
g
H
i
=
1
i
Nodes
∧
(
)
=
(
)
∑
( )
p
ptX
,
p
t
φ
X
g
g
i
i
=
1
The preceding approximations are substituted into the governing equations and
the residuals are integrated with respect to the appropriate basis function over the
current volume of the gel. This process is reasonably standard. However, care should
be taken to use the chain rule in evaluating spatial derivatives:
−
1
∂
∂
x
X
∇=
∂
∂
(.)
X
g
(.)
(6.14)
g
g
The basis functions for pressure must be one order lower than the basis functions
for displacement such that the LBB condition is satisfied or else some additional
constraints must be imposed on the pressure field. (Both methods have been
employed successfully in the computer code.) Integration by parts must be done in
those equations involving second derivatives in the basis fields to accommodate low-
order basis functions.
A sequential solution strategy was chosen to solve the system of equations. First,
the mass transport equations were solved and then these results were piped into the
