Chapter 6 Experimental Mapping Method

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180 ENGINEERING ELECTROMAGNETICS FIGURE 6.8 The problem of Figs. 6.6 and 6.7 is divided into smaller squares. Values obtained on the nine successive traverses are listed in order in the columns.
EXPERIMENTAL MAPPING METHODS 181 If each of the nine initial values were set equal to zero, it is interesting to note that ten traverses would be required. The cost of having a computer do these additional traverses is probably much less than the cost of the programming necessary to make decent initial estimates. Since there is a large difference in potential from square to square, we should not expect our answers to be accurate to the tenth of a volt shown (and perhaps not to the nearest volt). Increased accuracy comes from dividing each square into four smaller squares, not from finding the potential to a larger number of significant figures at each corner. In Fig. 6.8, which shows only one of the symmetrical halves plus an additional column, this subdivision is accomplished, and the potential at the newly created corners is estimated by applying (4) directly where possible and diagonally when necessary. The set of estimated values appears at the top of each column, and the values produced by the successive traverses appear in order below. Here nine sets of values are required, and it might be noted that no values change on the last traverse (a necessary condition for the last traverse), and only one value changes on each of the preceding three traverses. No value in the bottom four rows changes after the second traverse; this results in a great saving in time, forif none of the fourpotentials in (4) changes, the answeris of course unchanged. For this problem, it is possible to compare our final values with the exact potentials, obtained by evaluating some infinite series, as discussed at the end of the following chapter. At the point for which the original estimate was 53.2, the final value for the coarse grid was 52.6, the final value for the finer grid was 53.6, and the final value fora 16 16 grid is 53.93 V to two decimals, using data obtained with the following Fortran program: 1 DIMENSION A {17,17},B{17,17} 2 DO 6 I=2,17 3 DO 5 J=1,17 4 A{I,J}=0. 5 CONTINUE 6 CONTINUE 7 DO 9 J=2,16 8 A{I,J}=100. 9 CONTINUE 10 A{1,1}=50. 11 A{1,17}=50. 12 DO 16 I=2,16 13 DO 15 J=2,16 14 A{I,J}={A{I,J-1}+A{I-1,J}+A{I,J+1}+A{I+1,J}}/4. 15 CONTINUE 16 CONTINUE 17 DO 23 I=2,16 18 DO 22 J=2,16
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Chapter 6 Experimental Mapping Method

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