Autonomous Underwater Vehicle

Project No. 03

Project Title:

Submitted By: Khalil Ahmad

Submitted To: Dr. Khalid Pervaiz

Dated: 13-07-2014

Institute of space Technology Islamabad, Pakistan

Laplace Equation

Then the Laplace equation for two dimensional rectangular steel block is shown below

0

2

2

2

2

=

+

y

u

x

u

and its fde becomes

uij = ui+1,j + ui-1,j + ui,j+1 + ui,j-1

4

Consider the problem below we use Laplace equation to calculate temperature distribution

T = 50

T = 100 T = 0

T = 25

Matlab Program:

function [x,y,T]= LaplaceExplicit(n,m,Dx,Dy)

clc

echo off;

n = 10;

m = 20;

nx = 30;

ny = 24;

Dx = 1;

Dy = 1;

%numgrid(n,m);

R = 5.0;

T = R*ones(n+1,m+1); % All T(i,j) = 1 includes all boundary conditions

x = [0:Dx:n*Dx];

y = [0:Dy:m*Dy]; % x and y vectors

for i = 1:n % Boundary conditions at j = m+1 and j = 1

T(i,m+1) = 50; % Top boundary temperature

T(i,1) = 25; % Bottom Boundary temperature

end

for j = 1:m

T(n+1,j) = 0; % Left Boundary temperature

T(1,j) = 100; % Bottom Boundary temperature

end;

TN = T; % TN = new iteration for solution

err = TN-T;

% Iterative procedure

epsilon = 1e-5; % tolerance for convergence

imax = 1000; % maximum number of iterations allowed

k = 1; % initial index value for iteration

% Calculation loop

while k