Biomedical Engineering Reference
In-Depth Information
Fig. 7.13
Nodal point
P
, and its surrounding neighbour nodes in a structured
a
2D grid and
b
3D
grid
Table 7.1
The coefficients at each node of a control volume in 1D, 2D, and 3D
a
E
a
W
a
S
a
N
a
B
a
T
e
A
e
x
PE
w
A
w
x
WP
1D
-
-
-
-
e
A
e
x
PE
w
A
w
x
WP
s
A
s
y
SP
n
A
n
y
PN
2D
-
-
e
A
e
x
PE
w
A
w
x
WP
s
A
s
y
SP
n
A
n
y
PN
b
A
b
z
BP
t
A
t
z
PT
3D
Extension to 2D and 3D
The discretisation method for 2D and 3D domains follows
the same steps to determine a nodal point, P with its neighbouring nodes being E,
W, (east, west) and S, N (south and north for 2D) and E, W, S, N and T, B (top and
bottom for 3D). The corresponding cell volume faces have the notation of
e, w, s, t
,
and
b
and is illustrated in Fig.
7.13
.
The diffusion equation in 2D and 3D are:
∂φ
∂x
∂φ
∂y
∂
∂x
∂
∂y
0
=
+
∂φ
∂x
∂φ
∂y
∂φ
∂z
∂
∂x
∂
∂y
∂
∂z
and
0
=
+
+
The discretised equations can be summarised in the general form as:
a
nb
φ
nb
+
a
P
φ
P
=
b
where the subscript
nb
represents the neighbouring nodes. The coefficients for each
neighbouring node is summarised in Table
7.1
for 1D, 2D, and 3D.
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