SOLVING THE 2D HEAT EQUATION with a non-square domain

Hi,

Im trying to solve the THE 2D HEAT EQUATION.

My code currently solves it for a square domain id like it to solve it for a different non symetric domain

%Jorge Clares
%Numerical Methods Course Project
%Spring 2022

clear;
clc;
%% Creating the Geometry of the Cockpit
width  = 0.850;  %width in meters
height = 0.450; %Height in meters
NGX    = 100;       %Number of grid points in X direction
NGY    = 100;       %Number of grid points in Y direction 
DX     = width/(NGX-1);
DY     = height/(NGY-1);
%% Initial Conditions
T       = 25*ones(NGX,NGY);
Tback   = 100;    %Back Wall temp in Celcius
Tfront  = 20;    %Front Wall temp in Celcius
Ttop    = 20;      %Top Wall temp in Celcius
Tbottom = 80;   %Bottom Wall temp in Celcius

T(1,2:NGY-1)   = Tback;
T(2:NGX-1,NGY) = Ttop;
T(NGX,2:NGY-1) = Tfront;
T(2:NGX-1,1)   = Tbottom;

T(1,1)     = (Tback+Tbottom)/2;
T(NGX,1)   = (Tfront+Tbottom)/2;
T(1,NGY)   = (Tback+Ttop)/2;
T(NGX,NGY) = (Tfront+Ttop)/2;

%% Solving 2d Heat Equation
E = 1e-3;
Error = 5;

while (Error>E)
    Told           = T;
    for j          = 2:NGY-1
        for i      = 2:NGX -1
            T(i,j) = (T(i+1,j) + T(i-1,j)+ T(i,j+1) + T(i,j-1))/4;
        end
    end
    Error = sqrt(sumsqr(T-Told));
end

%% Ploting Results
x = 0:DX:width;
y = 0:DY:height;

colormap("jet")
contourf(x,y,T');
colorbar;
title('plot')
xlabel('Width')
ylabel('height')

[Torsten]

Can you find a mapping function that maps your domain onto a square or rectangular domain ?

[Jorge Arturo Clares Pastrana]

what do you mean?

[Torsten]

You can map a circle to a rectangle via the mapping

f(r,phi) = (r*cos(phi),r*sin(phi)) (polar coordinates)

Is something similar true for the 2d-domain on which you intend to solve the heat equation ?

[Jorge Arturo Clares Pastrana]

Yes, although currently my code already prints a square domain.

[Torsten]

Now I must ask: What do you mean ?

[Jorge Arturo Clares Pastrana]

I want a similar output, but instead of having a square geometry I want somethig like the following geometry

[Torsten]

I'd divide the geometry in two rectangles, apply your code to each rectangle separately and use the conditions of continuity of temperature and heat flux at the interface.