Please help as I am struggling to solve this homework problem.

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Sam
Sam 2012-8-20
编辑： Walter Roberson 2024-8-6
An H-section copper conductor carries an overload current of 54000 A. Under steady state conditions, the surface temperature is 60°C.
The dimensions of the bar are given in Figure 1 overleaf, and you are asked to use a numerical method to produce a mesh plot of the temperature distribution and a graph of the temperature distribution along the line y = 0.
The electrical resistivity, ρ, of copper is 2 × 10-8 Ω m, and the thermal conductivity, κ, is 0.311 kW m-1 K-1.
The governing differential equation is ∂2T + ∂2T + g = 0 (1)
∂x2 ∂y2 κ where g is the (constant) generation rate of heat per unit volume:
g = i2ρ (2) and i is the current density.
Using an appropriate Numerical Technique:
Determine the temperature distribution to an accuracy of 4 significant figures
You are expected to hand in the following by the submission date:
Hints
The Gauss-Seidel technique used in Tutorial 9 can be adapted to the present problem by including the additional term g/κ in the finite difference equation. You can generate a mask array as in Tutorial 9 to specify the region of the array covered by the H cross- section.
Try calculating the solution over a square mesh for a range of step lengths h = 5 mm, 2.5 mm, 1.25 mm...
You can if you wish make use of the symmetry of the problem and model just one quarter of the conductor. However, the boundary conditions then become more difficult to apply.
The H section is of width 40mm, depth 30mm and cut outs on top and bottom of depth 10x width 20 mm
Any help would be most appreciated
Cheers
Sam
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Steven Lord
Steven Lord 2024-8-6
Since this is a homework assignment, show us the code you've written to try to solve the problem and ask a specific question about where you're having difficulty and we may be able to provide some guidance.
If you aren't sure where to start because you're not familiar with how to write MATLAB code, I suggest you start with the free MATLAB Onramp tutorial to quickly learn the essentials of MATLAB.
If you aren't sure where to start because you're not familiar with the mathematics you'll need to solve the problem, I recommend asking your professor and/or teaching assistant for help.

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回答（1 个）

Elizabeth
Elizabeth 2012-8-21
Don't know if this is any assistance to you but--Here is my code for the Gauss-Seidel iterative method:
function [ X, Ntot ] = gs( A, b, tol, nmax )
%GS is a function that utilizes the Gauss-Seidel iterative method in order to
% to formulate an approximation to the linear algebra problem Ax=b
%Input
% A is an NxN nonsingular matrix
% b is an Nx1 matrix
% tol is the tolerance for convergence
% Nmax is the maximum number of iterations
%Output
% X is the Gauss-Seidel approximation to the soln of Ax=b
% Ntot is the number of iterations it took to reach convergence
N=length(b) ;
p=zeros(N,1);
tol=eps;
for k=1:nmax %for each iteration
for j=1:N %to solve for the unknown variable Xj
if j==1
X(1)=(b(1)-A(1,2:N)*p(2:N))/A(1,1); %x1 in terms other variables which are unknowns
elseif j==N
X(N)=(b(N)-A(N,1:N-1)*(X(1:N-1))')/A(N,N); %xN written in terms other variables
else
X(j)=(b(j)-A(j,1:j-1)*X(1:j-1)'-A(j,j+1:N)*p(j+1:N))/A(j,j); %xj written in terms of other variables
end
end
p=X';
r=max(abs(A*p-b));
if r< tol
X=p;
Ntot=k;
disp('The problem has converged to a soln');
end
end
Ntot=k;
if k>nmax
disp('Warning: the solution did not converge.');
end
end
  2 个评论
Jan
Jan 2012-8-21
Thge convergence criterion is useless in your code, because the iteration is not stopped. A missing convergence is not detected, because for k=1:nmax will never set the value of k to nmax+1, such that the test if k>nmax will be FALSE in every case. Better:
converged = false;
while k <= nmax
...
if r < tol
converged = true;
break;
end
end

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