Explanation of parallel plates capacitor implementation with finite element methods and with in-homogenous domain.

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function cap
close all;
hx = 0.002;
vx = 0:hx:0.1;
hy = 0.002;
h = hx;
vy = 0:hy:0.1;
nx = length(vx);
ny = length(vy);
n = nx*ny;
A = zeros(n);
b = zeros(n,1);
eps1 = 4
eps2 = 14
for ix = 1:nx
for jy = 1:ny
i = (jy-1) * nx + ix; % the global index of the ix,jy vertex
x = vx(ix); % the geometrical x coordinate
y = vy(jy); % the geometrical y coordinate
if ix == 1
% (1)
A(i,i) = 1;
b(i) = 0;
elseif ix == nx
% (2)
A(i,i) = 1;
b(i) = 10;
elseif jy == 1
if ix < (nx+1)/2
% (3)
A(i,i) = eps1*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps1*(1/hy^2);
elseif ix > (nx+1)/2
% (4)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
else
% (5)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = (eps1+eps2)*(1/hy^2);
end
elseif jy == ny
% (6)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-nx) = eps2*(2/hy^2);
elseif jy < (ny+1)/2 && ix < (nx+1)/2
% (7)
A(i,i) = eps1*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-nx) = eps1*(1/hy^2);
A(i,i+nx) = eps1*(1/hy^2);
elseif jy > (ny+1)/2 || ix > (nx+1)/2
% (8)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = eps2*(1/hy^2);
elseif jy < (ny+1)/2
% (9)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = ((eps1+eps2)/2)*(1/hy^2);
A(i,i-nx) = ((eps1+eps2)/2)*(1/hy^2);
elseif ix < (nx+1)/2
% (10)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = eps1*(1/hy^2);
else
% (11)
A(i,i) = (eps1+(eps2*3))*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = ((eps1+eps2)/2)*(1/hy^2);
end
end
end
% solve the problem
u = A\b;
s = reshape(u,nx,ny);
[X,Y] = meshgrid(vx,vy);
surf(X,Y,s)
end

Answers (1)

Shubham Khatri
Shubham Khatri on 16 Feb 2021
Hello,
As far as the code is concerned, Please remove the last 'end' from the code for it to work as in the code below.
close all;
hx = 0.002;
vx = 0:hx:0.1;
hy = 0.002;
h = hx;
vy = 0:hy:0.1;
nx = length(vx);
ny = length(vy);
n = nx*ny;
A = zeros(n);
b = zeros(n,1);
eps1 = 4
eps2 = 14
for ix = 1:nx
for jy = 1:ny
i = (jy-1) * nx + ix; % the global index of the ix,jy vertex
x = vx(ix); % the geometrical x coordinate
y = vy(jy); % the geometrical y coordinate
if ix == 1
% (1)
A(i,i) = 1;
b(i) = 0;
elseif ix == nx
% (2)
A(i,i) = 1;
b(i) = 10;
elseif jy == 1
if ix < (nx+1)/2
% (3)
A(i,i) = eps1*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps1*(1/hy^2);
elseif ix > (nx+1)/2
% (4)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
else
% (5)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = (eps1+eps2)*(1/hy^2);
end
elseif jy == ny
% (6)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-nx) = eps2*(2/hy^2);
elseif jy < (ny+1)/2 && ix < (nx+1)/2
% (7)
A(i,i) = eps1*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i-nx) = eps1*(1/hy^2);
A(i,i+nx) = eps1*(1/hy^2);
elseif jy > (ny+1)/2 || ix > (nx+1)/2
% (8)
A(i,i) = eps2*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = eps2*(1/hy^2);
elseif jy < (ny+1)/2
% (9)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = eps1*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = ((eps1+eps2)/2)*(1/hy^2);
A(i,i-nx) = ((eps1+eps2)/2)*(1/hy^2);
elseif ix < (nx+1)/2
% (10)
A(i,i) = (eps1+eps2)*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = eps1*(1/hy^2);
else
% (11)
A(i,i) = (eps1+(eps2*3))*(-(2/hx^2)-(2/hy^2));
A(i,i+1) = eps2*((1/hx^2)-(1/2*x*hx));
A(i,i-1) = ((eps1+eps2)/2)*((1/hx^2)-(1/2*x*hx));
A(i,i+nx) = eps2*(1/hy^2);
A(i,i-nx) = ((eps1+eps2)/2)*(1/hy^2);
end
end
end
% solve the problem
u = A\b;
s = reshape(u,nx,ny);
[X,Y] = meshgrid(vx,vy);
surf(X,Y,s)
Hope it helps

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