Question

Consider a rectangular plate of width L and height W. Three of the sides (left, bottom, and right side) are maintained at a constant temperature of T1 whereas the top side is maintained at T2. Write a general computer program to solve for the steady state temperature solution of the plate using finite difference techniques and the Gauss-Seidel iterative method for any arbitrary grid mesh m x n. Obtain a temperature solution using a grid mesh of 10x10, 20x20, and 30x30 for the following conditions: L = 40 cm, W = 60 cm, T1 = 20℃, and T2 = 95℃. Compare in terms of percent error the three numerical solutions to the closed form solution at the following five points (L and W represent the width and height of the plate.

Answer 1

Solution:

This is matlab program to solve the steady state solution of the plate, try this code:

clear all

close all

%Specify grid size

Nx = 10;

Ny = 10;

%Specify boundary conditions

Tbottom = 100

Ttop = 150

Tleft = 250

Tright = 300

% initialize coefficient matrix and constant vector with zeros

A = zeros(Nx*Ny);

C = zeros(Nx*Ny,1);

% initial 'guess' for temperature distribution

T(1:Nx*Ny,1) = 100;

% Build coefficient matrix and constant vector

% inner nodes

for n = 2:(Ny-1)

for m = 2:(Nx-1)

i = (n-1)*Nx + m;

A(i,i+Nx) = 1;

A(i,i-Nx) = 1;

A(i,i+1) = 1;

A(i,i-1) = 1;

A(i,i) = -4;

end

end

% Edge nodes

% bottom

for m = 2:(Nx-1)

%n = 1

i = m;

A(i,i+Nx) = 1;

A(i,i+1) = 1;

A(i,i-1) = 1;

A(i,i) = -4;

C(i) = -Tbottom;

end

%top:

for m = 2:(Nx-1)

% n = Ny

i = (Ny-1)*Nx + m;

A(i,i-Nx) = 1;

A(i,i+1) = .5;

A(i,i-1) = .5;

A(i,i) = -5;

C(i) = -Ttop;

end

%left:

for n=2:(Ny-1)

%m = 1

i = (n-1)*Nx + 1;

A(i,i+Nx) = .5;

A(i,i+1) = 1;

A(i,i-Nx) = .5;

A(i,i) = -2;

end

%right:

for n=2:(Ny-1)

%m = Nx

i = (n-1)*Nx + Nx;

A(i,i+Nx) = 1;

A(i,i-1) = 1;

A(i,i-Nx) = 1;

A(i,i) = -4;

C(i) = -Tright;

DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE

end

% Corners

%bottom left (i=1):

i=1

A(i,Nx+i) = 1;

A(i,2) = 1;

A(i,1) = -4;

C(i) = -(Tbottom + Tleft);

%bottom right:

i = Nx

A(i,Nx+i) = 1;

A(i,2) = 1;

A(i,1) = -4;

C(Nx) = -(Tbottom + Tright);

%top left:

i = (Ny-1)*Nx + 1;

A(i,i+1) = .5

A(i,i) =

%top right:

i = Nx*Ny;

DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE

%Solve using Gauss-Seidel

residual = 100;

iterations = 0;

while (residual > 0.0001) % The residual criterion is 0.0001 in this example

7% You can test different values

iterations = iterations+1

%Transfer the previously computed temperatures to an array Told

Told = T;

%Update estimate of the temperature distribution

INSERT GAUSS-SEIDEL ITERATION HERE

%compute residual

deltaT = abs(T - Told);

residual = max(deltaT);

end

iterations % report the number of iterations that were executed

%Now transform T into 2-D network so it can be plotted.

delta_x = 0.03/(Nx+1)

delta_y = 0.03/(Ny+1)

for n=1:Ny

for m=1:Nx

i = (n-1)*Nx + m;

T2d(m,n) = T(i);

x(m) = m*delta_x;

y(n) = n*delta_y;

end

end

T2d

surf(x,y,T2d)

figure

contour(x,y,T2d)