Biomedical Engineering Reference
In-Depth Information
diffusion in steady state. We further simplify the problem by working through a
one-dimensional steady state diffusion equation to demonstrate the application of
discretisation techniques to attain the algebraic form of the governing equation. It
will be seen that this discretised equation can be extended to accommodate for two-
and three-dimensional diffusion problems.
7.2.3.1
Finite Difference Discretisation
Mesh creation
The first step is to create a geometric domain with discrete nodal
points. Let us consider a general nodal point
P
and its surrounding neighbouring
nodal points to the west and east,
W
and
E
, for the one-dimensional geometry as
shown in Fig.
7.11
. The uniform grid has spacing of
x
.
Fig. 7.11
Representation of
the
uniform grid
spacing
along the
x
direction for the
one-dimensional geometry
Discretisation
The next step is to discretise the governing equation around the nodal
point
P
. We focus only on one point at a time to establish a general set of equations
which then can be applied to other nodal points. To derive a suitable expression for
the FD method, Eq. (7.23) needs to be expanded into its non-conservative form,
which is given by
d
2
φ
dx
2
d
dx
dφ
dx
+
0
=
+
S
φ
(7.24)
By applying the central differencing of the first order derivative and second order
derivative equations (see Fig.
7.3
) the discretised terms of Eq. (7.24) are
d
2
φ
dx
2
d
dx
=
(
E
−
W
)
2
x
dφ
dx
=
(
φ
E
−
φ
W
)
2
x
P
(
φ
E
−
2
φ
P
+
φ
W
)
;
;
=
x
2
Putting it together we get:
(
E
−
W
)
2
x
(
φ
E
−
φ
W
)
2
x
(
φ
E
−
2
φ
P
+
φ
W
)
x
2
+
P
+
S
φ
=
0
(7.25)
It is convenient to re-arrange and group the terms together, which gives
(
E
−
φ
E
+
φ
W
+
2
P
x
2
φ
P
W
)
4
x
2
P
x
2
(
E
−
W
)
4
x
2
P
x
2
=
+
−
+
S
φ
(7.26)
The coefficients of
φ
E
and
φ
W
and
φ
P
can be defined as
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