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1、拉普拉斯方程有限差分解,均匀介质情况下,均匀介质情况下,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end end U(5,5)=100.0,for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j-1,k)+U(j,k+1)+U(j,k-1);U(j,k)=Unew(j,k);U(5,5)=100.0;endend end contour(U) axis equal,clear all %清除变量 Clc %清除command window m=10 %网格数目为10 for k=
2、1:mfor j=1:mU(j,k)=0;%赋初值end end,U =0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0,U(5,5)=100.0,U =0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0
3、0 0 0 0 0 0 0 0 0 00 0 0 0 100 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0,for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j- 1,k)+U(j,k+1)+U(j,k-1);%拉普拉斯方程有限差分解U(j,k)=Unew(j,k);U(m/2,m/2)=100.0;endend end,for i=1for k=2:m-1for
4、j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j-1,k)+U(j,k+1)+U(j,k-1);U(j,k)=Unew(j,k);U(m/2,m/2)=100.0;endend end,i=1,i=2,i=3,(for i=1:200) i=200,(for i=1:201) i=201,均匀介质情况下,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end end U(5,5)=100.0,for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j-1,k)+U(j,
5、k+1)+U(j,k-1);U(j,k)=Unew(j,k);U(5,5)=100.0;endend end contour(U) axis equal,均匀介质情况下,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end end U(5,5)=100.0 d=1,While d0.01 for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j-1,k)+U(j,k+1)+U(j,k-1);U(j,k)=Unew(j,k);U(5,5)=100.0;d=abs(U(j,k)- Unew(j,k);)ende
6、nd end contour(U) axis equal,均匀介质情况下,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end end U(5,5)=100.0,for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(U(j+1,k)+U(j-1,k)+U(j,k+1)+U(j,k-1);U(j,k)=Unew(j,k);U(5,5)=100.0;endend end contour(U) axis equal,2018/10/18,非均匀介质情况下,ep： 1 1 1 1 1 1 1 1 1 1 1 1 1
7、 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 10 1 1 1 1 1 10 10 10 1 1 1 1 1 10 10 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1,2018/10/18,非均匀介质情况下,2018/10/18,非均匀介质情况下,ep： for k=1:mfor j=1:mep(j,k)=1;end endfor k=4:6for j=4:6ep(j,k)=10;end end,2018/10/18,非均匀介质情况下,Unew(j,k)=1/4*(U(j+1
8、,k)+U(j-1,k)+U(j,k+1)+U(j,k-1);修改为：Unew(j,k)=1/4*(ep(j+1,k)*U(j+1,k)+ep(j-1,k)*U(j-1,k)+ep(j,k+1)*U(j,k+1)+ep(j,k-1)*U(j,k-1);,2018/10/18,非均匀介质情况下,U(5,5)=100.0for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(ep(j+1,k)*U(j+1,k)+ep(j-1,k)*U(j-1,k)+ep(j,k+1)*U(j,k+1)+ep(j,k-1)*U(j,k-1);U(j,k)=Unew(j,k);
9、U(5,5)=100.0;endend end contour(U) axis equal,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end endfor k=1:mfor j=1:mep(j,k)=1;end endfor k=4:6for j=4:6ep(j,k)=10;end end,2018/10/18,非均匀介质情况下,2018/10/18,非均匀介质情况下,ep：1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10
10、 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100,2018/10/18,第一类边界条件,clear all clc m=10 for k=1:mfor j=1:mU(j,k)=0;end endfor j=1for k=1
11、:mU(j,k)=10;end end,2018/10/18,第二类边界条件,for i=1:200for k=2:m-1for j=2:m-1Unew(j,k)=1/4*(ep(j+1,k)*U(j+1,k)+ep(j-1,k)*U(j-1,k)+ep(j,k+1)*U(j,k+1)+ep(j,k-1)*U(j,k-1);U(j,k)=Unew(j,k);U(5,5)=100.0;endend j=100for k=2:m-1Unew(j,k)=1/4*(ep(j+1,k)*U(j+1,k)+2*ep(j-1,k)*U(j-1,k)+ep(j,k+1)*U(j,k+1)+ep(j,k-1)*U(j,k-1);U(j,k)=Unew(j,k);endend,2018/10/18,本门课讲解的内容,梯度、散度和旋度 介绍拉普拉斯方程和泊松方程 均匀介质情况下 非均匀介质情况下 边界条件 第一类边界条件 第二类边界条件 解拉普拉斯方程 数值方法（有限差分法、有限元法） 解析法（电像法、分离变量法、保角变换、格林函数等）,
