## An Example Temperature Distribution in a Heated Metal Plate

A typical example of an elliptic partial differential equation involves the solution of a steady-state heat-flow problem. For example, if a thin steel plate, 10x10 cm, has one of the edges held at 100°C and the other three edges at 0°C, what are the steady-state temperatures within the plate? For simplicity, we assume that heat is not lost through the faces of the plate.

We subdivide the plate by means of a grid with h — k= 0.5 cm, thus creating a lattice of size 20 x 20. At equilibrium, heat flows in the x-axis direction into a lattice element at a rate proportional to the temperature of the adjoining element in the *-axis, and flows out of the element at a rate proportional to the temperature of the element. The same is true in the y-axis direction. This model gives rise to an elliptic partial differential equation of the form of equation 12-2. The time and the thermal conductivity k of the material do not enter into the equation.

We will use equation 12-16 to calculate the temperature at each lattice point; the temperature at a lattice point is the average of the temperatures of the four surrounding lattice points. Thus we have generated a system of 400 simultaneous linear equations in 400 unknowns. Although most of the terms in a given equation are zero, the problem is still unmanageable. However, we can solve the system by an iterative method, as described below.

Figure 12-2 shows part of the spreadsheet used to solve the system; each cell of the 20 x 20 array corresponds to a lattice point. The formula in cell B6 is

You can Fill Down the formula into 20 rows and then Fill Right into 20 columns to create the 20 x 20 array.

Since cell B6 refers to cell B7 and B7 similarly refers to B6, we have created a circular reference, a formula that refers to itself, either directly or indirectly. In fact, the spreadsheet contains a large number of circular references. A circular reference is usually an error; Excel displays the "Cannot resolve circular references" error message and puts a zero in the cell. In this case, however, the circular reference is intentional. We can make Excel recalculate the value in each cell, using the result of the previous iteration.

A |
I |
C |
D |
E |
F |
LâJ |
Jjj |
I |
J K |
L |
M |
LU |
0 |
PJ | ||

5 |
0 |
0 |
0 |
0 |
0 |
0 |
□ |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 | |

6 |
0 |
0.25 |
0,49 |
0.72 |
0.94 |
1,1 |
1.3 |
1.4 |
2 |
2 |
2 |
2 |
2 |
2 |
1.4 |
1.3 |

7 |
0 |
0.50 |
1.0 |
1.5 |
2 |
2 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |

8 |
0 |
0.77 |
2 |
2 |
3 |
3 |
4 |
4 |
5 |
S |
S |
S |
5 |
5 |
4 |
4 |

9 |
0 |
1.1 |
2 |
3 |
4 |
5 |
■ 5 |
6 |
7 |
7 |
7 |
7 |
7 |
7 |
8 |
5 |

Nt |
0 |
1.4 |
3 |
4 |
5 |
6 |
: 7 |
8 |
8 |
9 |
9 |
9 |
9 |
8 |
8 |
7 |

0 |
2 |
3 |
S |
6 |
8 |
9 |
10 |
10 |
11 |
11 |
11 |
11 |
10 |
1Ü |
9 | |

12 |
0 |
2 |
4 |
6 |
8 |
9 |
11 |
12 |
13 |
13 |
14 |
14 |
13 |
13 |
12 |
11 |

13 |
0 |
3 |
5 |
7 |
9 |
11 |
13 |
14 |
15 |
16 |
16 |
16 |
16 |
15 |
14 |
13 |

14" |
0 |
3 |
6 |
9 |
11 |
14 |
16 |
17 |
18 |
19 |
20 |
20 |
19 |
1S |
17 |
16 |

15 |
0 |
4 |
7 |
10 |
13 |
16 |
18 |
20 |
22 |
23 |
23 |
23 |
23 |
22 |
20 |
18 |

16 |
0 |
4 |
9 |
12 |
16 |
19 |
22 |
24 |
25 |
26 |
27 |
27 |
26 |
25 |
24 |
22 |

17 |
0 |
5 |
10 |
15 |
19 |
23 |
26 |
28 |
30 |
31 |
31 |
31 |
31 |
30 |
28 |
26 |

18 |
0 |
6 |
12 |
18 |
22 |
27 |
30 |
33 |
35 |
36 |
37 |
37 |
36 |
35 |
33 |
30 |

~W |
0 |
8 |
15 |
21 |
27 |
v -v. |
35 |
38 |
40 |
42 |
42 |
42 |
42 |
40 |
38 |
35 |

20 |
0 |
9 |
16 |
25 |
32 |
37 |
41 |
44 |
46 |
48 |
49 |
49 |
48 |
46 |
44 |
41 |

21 |
0 |
12 |
22 |
31 |
38 |
44 |
48 |
51 |
54 |
55 |
56 |
56 |
55 |
54 |
51 |
481 |

22 |
0 |
15 |
28 |
36 |
46 |
52 |
56 |
59 |
62 |
63 |
64 |
64 |
63 |
62 |
59 |
56 |

23 |
a |
20 |
36 |
48 |
56 |
62 |
66 |
68 |
70 |
71 |
72 |
72 |
71 |
70 |
68 |
66 |

24 |
0 |
30 |
43 |
61 |
68 |
73 |
76 |
78 |
80 |
81 |
81 |
81 |
31 |
80 |
78 |
76 |

25 |
0 |
50 |
69 |
78 |
83 |
86 |
68 |
89 |
90 |
90 |
90 |
90 |
90 |
90 |
89 |
88 |

26 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |
100 |

Figure 12-2. Solving an elliptic PDE using intentional circular references. The worksheet shows part of the 20 x 20 array of lattice points representing the temperature distribution in a metal plate; the gray cells represent the temperature at the edges of the plate, (folder 'Chapter 12 (PDE) Examples, workbook 'Elliptic PDE', sheet 'Temp in a Plate')

To "turn on" iteration, choose Tools->Options-*Calculation and check the iteration box. Unless you change the default settings for iteration, Microsoft Excel stops calculating after 100 iterations or after all values in the circular reference change by less than 0.001 between iterations, whichever comes first. When you press OK the iterative circular reference calculations will begin.

Temperature Distribution in a Metal Plate a 90-100

Figure 12-3. Temperature distribution in a metal plate, (folder 'Chapter 12 (PDE) Examples, workbook 'Elliptic PDE', sheet 'Temp in a Plate')

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