Biomedical Engineering Reference
In-Depth Information
arithmetic operations required for the Gaussian elimination to perform on the zero
entries may not be as significant comparing to the Thomas algorithm. Nevertheless,
this is not true when a number of grid points is used to better predict the temperature
distribution across the plate. This is because of the additional and more cumbersome
numerical computations (multiplication and divisions) that have to be performed on
the matrix entries. The algorithm degenerates and becomes inefficient once the order
of the matrix becomes higher ( > 10).
The Jacobi method
To illustrate the
Jacobi method
, the resulting set of algebraic
equations as previously derived is rewritten
3000
T
1
−
1000
T
2
+
0
×
T
3
+
0
×
T
4
=
203000
−
1000
T
1
+
2000
T
2
−
1000
T
3
+
0
×
T
4
=
3000
0
×
T
1
−
1000
T
2
+
2000
T
3
−
1000
T
4
=
3000
0
×
T
1
+
0
×
T
2
−
1000
T
3
+
3000
T
4
=
803000
The above set of equations can be reorganized so that the required variable is on the
left hand side of the equation.
T
1
=
+
(
T
2
/
3)
(203
/
3)
T
2
=
(
T
1
/
2)
+
(
T
3
/
2)
+
3
/
2
T
3
=
(
T
2
/
2)
+
(
T
4
/
2)
+
3
/
2
T
4
=
(
T
3
/
3)
+
(803
/
30)
By employing initial guesses:
T
(0)
1
T
(0)
2
T
(0)
3
T
(0)
4
=
=
=
=
100, the nodal
temperatures for the first iteration are determined as:
T
1
(1)
=
(100
/
3)
+
(203
/
3)
=
101
.
000
T
2
(1)
=
+
+
=
(100
/
2)
(100
/
2)
3
/
2
101
.
500
T
3
(1)
=
(100
/
2)
+
(100
/
2)
+
3
/
2
=
101
.
500
T
4
(1)
=
(100
/
3)
+
(803
/
3)
=
301
.
000
The above first iteration values of
T
(1)
1
101,
T
(1)
2
101
.
5,
T
(1)
3
101
.
5 and
T
(1)
4
=
=
=
=
317
3
are substituted back into the system of equations; the second iteration yields
T
1
(2)
=
(101
.
5
/
3)
+
(203
/
3)
=
101
.
500
T
2
(2)
=
(101
/
2)
+
(101
.
5
/
2)
+
3
/
2
=
102
.
750
T
3
(2)
=
(101
.
5
/
2)
+
(301
/
2)
+
3
/
2
=
202
.
750
T
4
(2)
=
(101
.
5
/
3)
+
(803
/
3)
=
301
.
500
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