工程电磁场实验报告(仿真)
一．实验原理——有限差分法介绍
有限差分法（FiniteDifferential Method）是基于差分原理的一种数值计算法。其基本思想：将场域离散为许多小网格，应用差分原理，将求解连续函数 的泊松方程的问
题转换为求解网格节点上 的差分方程组的问题。
1．1二维泊松方程的差分格式
图1-1有限差分的网格分割
二维静电场边值问题：
∂2ϕ∂2ϕρ
+=−=F 22
ε∂x ∂y
(1-1)
ϕ
L
=f (s )
(1-2)
通常将场域分成足够小的正方形网格,网格线之间的距离为h , 节点0, 1, 2, 3, 4上的电位分别用ϕ0, ϕ1, ϕ2, ϕ3和ϕ4表示。
设函数ϕ在x 0处可微，则沿x 方向在x 0处的泰勒公式展开为
ϕχ=
将χ=
K =0
∑
n
ϕ(K )
K !
(χ−χ0)K +ο(χ−χ0)
(
n
)
（1-3）
χ1和χ3分别代入式（1－3）,得
∂ϕ12∂2ϕ13∂3ϕ
ϕ1=ϕ0+h () 0+h (20+h (3) 0+⋅⋅⋅⋅⋅⋅
∂x 2! 3! ∂x ∂x ∂ϕ12∂2ϕ13∂3ϕ
ϕ3=ϕ0−h () 0+h (20−h (3) 0+⋅⋅⋅⋅⋅⋅
∂x 2! 3! ∂x ∂x
（1-4）（1-5）
由（1-4）-（1-5）得
(
(1-4)+(1-5)得
∂ϕϕ−ϕ3
x =x 0≈1
2h ∂x
（1-6）
ϕ−2ϕ0+ϕ3∂2ϕ
(2) x =x 0≈1∂x h 2
同理
（1-7）
(
ϕ−ϕ3∂ϕ
) y =y 0≈1∂y 2h
（1-8）
ϕ−2ϕ0+ϕ3∂2ϕ
(2y =y 0≈1
h 2∂y
，得到泊松方程的五点差分格式将式(1-7)、（1-9）代入式（1-1）
（1-9）
ϕ1+ϕ2+ϕ3+ϕ4−4ϕ0=Fh 2 ⇒ ϕ0=(ϕ1+ϕ2+ϕ3+ϕ4−Fh 2)
当场域中ρ=0, 得到拉普拉斯方程的五点差分格式
14
ϕ1+ϕ2+ϕ3+ϕ4−4ϕ0=0 ⇒ ϕ0=(ϕ1+ϕ2+ϕ3+ϕ4)
1.2边界条件的离散化处理
1
14
•
2
若场域离散为矩形网格(如图1-2示)，差分格式为：图1-2边界条件的离散化处
(1-10)
1111(ϕ+ϕ) +(ϕ+ϕ) −(+2ϕ0=F 12242222h 1h 2h 1h 2
（1）第一类边界条件:给边界离散节点直接赋已知电位值
（2）对称边界条件:合理减小计算场域，差分格式为：
ϕ0=(2ϕ1+ϕ2+ϕ4−h 2F )
14
(1-11)
图1-3边界条件的离散化处理
（3）第二类边界条件:边界线与网格线相重合的差分格式：
(
ϕ−ϕ0∂ϕ
0≈1=f 2∂n h
, ϕ0=ϕ1−f 2h
(1-12)
(4)介质分界面衔接条件的差分格式
ϕ0=(
其中
122K
ϕ1+ϕ2+ϕ3+ϕ4)
41+K 1+K
(1-13)
K =εa εb
1.3差分方程组的求解方法
(1)高斯——赛德尔迭代法
k +1) (k +1) (k ) (k ) 2
ϕi (, k j +1) =[ϕi (−1, j +ϕi , j −1+ϕi +1, j +ϕi , j +1−Fh ]
1
4
图1-4高斯——赛德尔迭代法
式中：i , j =1, 2, ⋅⋅⋅⋅⋅⋅，k =0, 1, 2, ⋅⋅⋅⋅⋅⋅
•迭代顺序可按先行后列，或先列后行进行。
•迭代过程遇到边界节点时，代入边界值或边界差分格式，直到所有节点电位满足
i (, k j +l ) −ϕi (, k j )
（2）超松弛迭代法
α1) (k ) (k ) 2(k )
ϕi (, k j +1) =ϕi (, k j ) +[ϕi (−k 1+, 1j ) +ϕi (, k j +−1+ϕi +1, j +ϕi , j +1−Fh −4ϕi , j ]
4
式中：α是加速收敛因子(11000
1.7269
1.8174
1.83143
1.85122
1.87133
1.90171
2.0发散
(1-15)
最佳收敛因子的经验公式：
α0=
21+sin()
p
（正方形场域、正方形网格）
P指在正方形场区域中网格数。
α0=2−π2
11
（矩形场域、正方形网格）+22
p q
•迭代收敛的速度与电位初始值的给定及网格剖分精细有关
•迭代收敛的速度与工程精度要求有关
+l ) (N ) i (, N −ϕ
借助计算机进行计算时，其程序框图1－5
所示
二、实验内容
1、设计编程语言解决下面的问题
试用超松弛迭代法求解接地金属槽内电位的分布已知：a =4cm ，h =给定边值给定初值误差范围选取
ϕ=100V
a
=10mm 4
ϕ=0
如图1－6示
ϕi (. 0j ) =0
ϕ=0
ε=10−5α=?
计算：迭代次数N=?
ϕi , j 分布。
ϕ=0
图1-6接地金属槽的网格剖
2、设计编程语言解决下面的问题
按对称场差分格式求解电位的分布给定边值：如图1-7示给定初值ϕi . j =
ϕ2−ϕ1
p
(j −1) =
100
(j −1) 40
误差范围:
ε=10−5
计算：1）迭代次数N ，ϕ
i , j ，将计算结果保存到文件中；
图1-7接地金属槽内半场域的网
2）按电位差Δϕ=10画出槽中等位线分布图。
三．实验求解过程
1. 试用超松弛迭代法求解接地金属槽内电位的分布采用MATLAB 所编写的程序如下：
%用超松弛迭代法求解矩形槽内电位分布hx=5;hy=5;%设置网格节点数v1=ones(hx,hy);%设置行列二维数组m=4;n=4;%横纵向网格数v1(hy,:)=ones(1,hx)*0;%下行的边界条件值v1(1,:)=ones(1,hx)*100;%上行的边界条件值for i=1:hy%左右两边的边界条件值
v1(i,1)=0;v1(i,hx)=0;end
mid=v1;
%计算超松弛迭代因子w=2/(1+sin(pi/m));
ww=[w1.01.71.81.831.851.871.9];for i1=1:8
v1=mid;v2=v1;%构造一个矩阵V2，用来存放中间值maxt=1;%每次迭代完成，前后矩阵中的最大差值t=0;k=0;%迭代次数while (maxt>1e-5)%由V1开始迭代，算出V2，迭代精度为e-5
k=k+1;maxt=0;
for i=2:hy-1%从2行到hy-1行进行循环
for j=2:hx-1%从2行到hx-1行进行循环
v2(i,j)=v1(i,j)+(v1(i,j+1)+v1(i+1,j)+v2(i-1,j)+v2(i,j-1)-4*v1(i,j))*ww(i1)/4;
t=abs(v2(i,j)-v1(i,j));if(t>maxt)maxt=t;end end v1=v2;end end
disp('采用的收敛因子为:')disp(ww(i1))
disp('迭代次数为：')disp(k)
disp('所求的各点电位值为：')disp(v1)end
———————————————————————————————————————
输出的结果如下：
———————————————————————————————————————采用的收敛因子为:
1.1716迭代次数为：
12
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1
迭代次数为：
22
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000采用的收敛因子为:1.2000迭代次数为：
11
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.3000迭代次数为：
14
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
采用的收敛因子为:
1.4000迭代次数为：
18
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.5000迭代次数为：
24
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.6000迭代次数为：
32
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.7000迭代次数为：
45
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
采用的收敛因子为:
1.8000迭代次数为：
70
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.8300迭代次数为：
85
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.8500迭代次数为：
98
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————采用的收敛因子为:
1.8700迭代次数为：
114
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
采用的收敛因子为:
1.9000迭代次数为：
150
所求的各点电位值为：
0100.0000100.0000100.00000042.857152.678642.85710018.750025.000018.7500007.14299.82147.1429000000
———————————————————————————————————————在选取收敛因子为经验值即1.1716时，画出的电位分布图如下：（左边为三维，
右边为二维）
2. 按对称场差分格式求解电位的分布
采用MATLAB 所编写的程序如下：
%按对称场差分格式求解电位的分布hx=41;hy=41;%设置网格节点数v1=ones(hy,hx);%设置行列二维数组m=40;n=40;%横纵向网格数v1(hy,:)=ones(1,hx)*0;%下行的边界条件值v1(1,:)=ones(1,hx)*100;%上行的边界条件值for i=1:hy%左右两边的边界条件值
v1(i,1)=0;v1(i,hx)=0;end
%计算超松弛迭代因子w=2/(1+sin(pi/m));v2=v1;%构造一个矩阵V2，用来存放中间值maxt=1;%每次迭代完成，前后矩阵中的最大差值t=0;k=0;%迭代次数
while (maxt>1e-5)%由V1开始迭代，算出V2，迭代精度为e-5
k=k+1;
maxt=0;
for i=2:hy-1%从2行到hy-1行进行循环
for j=2:21%从2行到hx-1行进行循环
if(j==21)%第21列特殊情况特殊考虑，为超松弛迭代
v2(i,j)=v1(i,j)+(v1(i+1,j)+v2(i-1,j)+2*v2(i,j-1)-4*v1(i,j))*w/4;
else %其他情况下超松弛迭代
v2(i,j)=v1(i,j)+(v1(i,j+1)+v1(i+1,j)+v2(i-1,j)+v2(i,j-1)-4*v1(i,j))*w/4;
end
t=abs(v2(i,j)-v1(i,j));
if(t>maxt)maxt=t;
end
end
v1=v2;
end
end
%由对称性得到右半边的电位值
for j=2:20
v1(:,hx+1-j)=v1(:,j);
end
disp('采用的收敛因子为:')
disp(w)
disp('迭代次数为：')
disp(k)
disp('所求的各点电位值为：')
disp(v1)
subplot(1,1,1),v=0:10:100;[A,b]=contour(rot90(rot90(v1)),v);
clabel(A,b);
set(gca,'xtick',0:1:41,'ytick',0:1:41)
gtext('0V');gtext('0V');gtext('0V');gtext('100V');
gtext('按对称场差分格式求解电位的分布')
hold on
grid on
hold off
———————————————————————————————————————程序输出的结果如下：
所求的各点电位值为：（是一个49*49介次的方阵）
Columns 1through 8（第一列到第八列）
0100.000049.931630.098120.734615.5784100.0000100.000069.628178.854749.726261.916637.262149.384129.203340.2502100.0000100.000083.874086.939869.701674.908458.107364.438648.905155.6170100.0000100.000088.976990.414278.553481.206969.121672.659860.834664.9243
012.375623.722633.508241.645948.289753.675758.0403
010.201419.803328.414235.880842.220047.538251.9738
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03.34526.65019.876612.990415.962018.767721.3897
03.03586.03808.974711.816614.538017.117719.5388
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02.28874.55676.78398.951311.041513.039214.9311
02.08594.15396.18648.166810.079811.911613.6501
01.90113.78665.64107.44989.199410.877412.4727
01.73203.45025.14126.79218.39069.925811.3875
01.57663.14114.68156.18667.64539.047710.3847
01.43332.85604.25735.62736.95628.23519.4555
01.30082.59213.86455.10916.31737.48088.5922
01.17772.34703.49964.62755.72296.77857.7877
01.06302.11863.15944.17825.16826.12287.0360
00.95581.90512.84123.75794.64895.50866.3313
00.85521.70462.54253.36314.16104.93125.6687
00.76051.51582.26092.99093.70104.38655.0432
00.67081.33711.99452.63873.26543.87074.4507
00.58561.16741.74142.30392.85143.38023.8870
00.50431.00531.49971.98432.45592.91163.3484
00.42630.84991.26781.67762.07642.46182.8314
00.35110.70001.04421.38181.71032.02792.3324
00.27820.55460.82741.09491.35531.60701.8484
00.20710.41290.61590.81511.00891.19631.3761
00.13730.27380.40840.54050.66910.79330.9126
00.06840.13640.20360.26940.33350.39540.4548
0000000
Columns 9through 16
100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000
91.473092.277692.902993.396393.789594.104094.355094.5534
83.200284.734685.937686.892987.657688.271488.762689.1516
75.386477.523179.219980.580281.676782.561583.272583.8372
68.162470.751572.838774.531275.907477.025477.928578.6487
61.587464.481866.852368.798470.396371.704472.767273.6189
55.665158.735961.290363.413965.175166.628667.817268.7745
50.363553.506356.159358.391760.261761.817663.098464.1354
45.631248.766451.448853.731855.662557.281758.623359.7154
41.409244.479247.137749.424051.375053.023454.397955.5224
37.638540.603743.198845.451747.389949.039150.422351.5597
34.263737.098239.602141.794143.693845.320746.692747.8260
31.234933.923336.317438.428740.270641.857143.201944.3174
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15.284816.807418.210819.489720.639921.658522.543523.2934
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5.00185.52076.00446.45016.85537.21787.53577.8074
4.36884.82265.24575.63575.99036.30776.58626.8242
3.76384.15514.52014.85655.16265.43675.67715.8827
3.18283.51403.82294.10784.36704.59914.80284.9770
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1.02601.13301.23281.32491.40871.48381.54981.6062
0.51140.56470.61450.66040.70220.73960.77250.8006
0000000
Columns 17through 24
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89.453389.678289.833989.925589.955789.925589.833989.6782
84.276284.604184.831484.965185.009384.965184.831484.6041
79.210279.630579.922480.094380.151180.094379.922479.6305
74.285274.785575.133575.338775.406575.338775.133574.7855
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60.580361.235861.695061.966962.057061.966961.695061.2358
56.416957.096857.574257.857457.951257.857457.574257.0968
52.467853.160353.647753.937254.033253.937253.647753.1603
48.734449.429149.919050.210550.307250.210549.919049.4291
45.214545.902646.388846.678546.774746.678546.388845.9026
41.903842.577943.055343.339943.434543.339943.055342.5779
38.795739.450139.914340.191540.283640.191539.914339.4501
35.882336.512636.960337.228037.317037.228036.960336.5126
33.154833.757734.186534.443134.528534.443134.186533.7577
30.603631.176731.584931.829331.910731.829331.584931.1767
28.218728.760629.147129.378629.455829.378629.147128.7606
25.990026.500126.864127.082427.155127.082426.864126.5001
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6.05236.18516.28056.33796.35716.33796.28056.1851
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4.22054.31344.38024.42034.43384.42034.38024.3134
3.34593.41973.47263.50453.51523.50453.47263.4197
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Columns 25through 32
100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000
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89.453389.151688.762688.271487.657686.892985.937684.7346
84.276283.837283.272582.561581.676780.580279.219977.5231
79.210278.648777.928577.025475.907474.531272.838770.7515
74.285273.618972.767271.704470.396368.798466.852364.4818
69.526468.774567.817266.628665.175163.413961.290358.735900
64.953364.135463.098461.817660.261758.391756.159353.5063
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Columns 33through 40
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83.200281.206978.553474.908469.701661.916649.726230.0981
75.386472.659869.121664.438658.107349.384137.262120.7346
68.162464.924360.834655.617048.905140.250229.203315.5784
61.587458.040353.675748.289741.645933.508223.722612.3756
55.665151.973847.538242.220035.880828.414219.803310.2014
50.363546.651542.283237.171331.243024.464416.87528.6266
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41.409237.887933.879429.357724.315318.774012.79536.4869
37.638534.277430.500226.296921.673816.660411.31385.7222
34.263731.083127.547123.655719.422914.879710.07755.0881
31.234928.244224.949221.356117.482313.35819.02864.5525
28.508425.709722.649419.337315.792012.04198.12614.0933
26.046223.436820.601417.551814.306410.89147.34063.6948
23.815721.389718.767715.962012.99049.87666.65013.3452
21.788819.538817.117714.538011.81668.97476.03803.0358
19.941617.858915.626213.255810.76328.16775.49152.7599
18.253316.329014.272512.09569.81287.44125.00042.5123
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8.74427.78776.77855.72294.62753.49962.34701.1777
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6.36895.66874.93124.16103.36312.54251.70460.85520
5.6670
5.0018
4.3688
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Column 41
05.04324.45073.88703.34842.83142.33241.84841.37610.91260.454804.38653.87073.38022.91162.46182.02791.60701.19630.79330.395403.70103.26542.85142.45592.07641.71031.35531.00890.66910.333502.99092.63872.30391.98431.67761.38181.09490.81510.54050.269402.26091.99451.74141.49971.26781.04420.82740.61590.40840.203601.51581.33711.16741.00530.84990.70000.55460.41290.27380.136400.76050.67080.58560.50430.42630.35110.27820.20710.13730.06840
———————————————————————————————————————
采用的收敛因子为:
1.8545
迭代次数为：
102
按电位差Δϕ=10画出槽中等位线分布图：
四．实验总结
在编写程序前，我考虑到第二题，即按对称场差分格式求解电位的分布时，还要按电位差Δϕ=10画出槽中等位线分布图。此时相到MATLAB 在数据数据处理方面独特的优势，所以选择MATLAB 软件来编写程序。另外，MATLAB 程序的特点与C++原理相通，所以，在掌握C++的基础上，稍加学习，就会掌握基本的MATLAB 编程方法。
在编写这两个程序时，我参考了《电磁场数值计算法与MATLAB 实现》（华中科技大学出版社，何红雨编著）这本书。
分析上述求解的输出结果，发现，第一题采用超松弛迭代法时，在加速因子不同的情况下，迭代的次数明显不同的。迭代次数先减少再增加，其中当加速因子为1.2时迭代次数最少，只有11次；当加速因子为1.1716时迭代次数有12次，次加速因子使用经验公式算出的。当加速因子为2.0时，程序将无法输出结果。通过对比第一题中加速因子为1.2和1.1716可以发现，经验公式只基于前人经验所得，不是经过严格的推导，所以不一定是收敛最快的加速因子。
称场差分格式求解电位分布的方法，使求解问题规模直接减半，这对于节约求解时间，减少求解工作量有直接的作用。在今后的工作中，要灵活分析具体问题，合理简化问题。
通过实验，我学习了一些基础的MATLAB 编程方法，对于今后的应用有很大帮助。在具体的实验过程中，加深了我对书本上抽象知识的理解。总之，此次实验很锻炼我的能力。