有限差分法解矩形波导内场值、截止频率 MATLAB

利用有限差分法，解矩形波导内场解和截止频率：

这里以解TM11模为例，利用双重迭代法，每4次场值，更新一次Kc：

%%
%   求矩形波导中TM11模 截面内场分布、截止频率kc和特性阻抗Zc
%   2018/11/12
%
%% Init
clear; clc
w = 1.5;            %收敛因子
count = 20000;         %迭代次数
xa = 12; xb = 12;   %矩阵长宽度
h = pi / 7;
aa = h*(xa-1); bb = h*(xb-1);

%% 赋初值
% 内点初值
for i = 2 : (xa-1)
    for j = 2 : (xb - 1)
        u(i, j) = 1.15;
    end
end

% 边的初值
u(1,:) = 0; u(12,:) = 0;
u(:,1) = 0; u(:,12) = 0;

% kc初值
kc = 0.15;

%% 迭代更新
k = 0; k2 = 0;
sum1 = 0; sum2 = 0; sum = 0;

while k < count
    for i = 2 : xa-1
        for j = 2 : xb-1
               u(i,j) = u(i,j) + w*((u(i+1,j)+u(i,j+1)+u(i-1,j)+u(i,j-1))/(4-(kc*h)^2) -u(i,j));
        end
    end
    k = k + 1;
    k2 = k2 + 1;
    
    if(rem(k2,3) == 0)
        for i = 2 : xa-1
            for j = 2 : xb-1
                sum1 = sum1+ u(i,j)*(u(i+1,j)+u(i,j+1)+u(i-1,j)+u(i,j-1)-4*u(i,j));
                sum2 = sum2 + u(i,j)^2;
            end
        end
        kc = sqrt(-sum1/sum2)/h;
    end
end

%% 结果展示
surf(u);
title('TM_{11}模E_z值');
colorbar

c = 3 * 10^8;
fc = c * kc /(2*pi*10^6)

kc
kc0 = sqrt((pi/aa)^2+(pi/bb)^2)
max(max(u))

 最后效果如图所示：