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Question 4 (20 marks)
Laplace’s equation: Determine the distribution of temperature in a rectangular plane section, subject to a temperature distribution around its edges as follows:
The section shape, the boundary temperature distribution section and two chosen nodes are shown in the figure.
The temperature distribution is described by Laplace’s equation. Solving this equation by the finite difference method to nodes 1 and 2 of the mesh shown in the figure. This gives
(233.33 +T2 + 0 100 - 4T1)/h^2 = 2
(333.33 + 250 + 0 + T1 - 4T2)/h^2 = 2
where and are the unknown temperatures at nodes 1 and 2, respectively, and
Rearranging these equations gives
[■(-4&1@1&-4)][■(T_1@T_2 )]=[■(-333.33@-583.33)][■(-4&1@1&-4)][■(T_1@T_2 )]=[■(-333.33@-583.33)]
Solving this equation, we have and .
If we require a more accurate solution of Laplace’s equation, then we must use more nodes and the computation burden increases rapidly. Write a MATLAB script to solve the Laplace’s equation for any number of nodes in a square domain only. Run your codes for number of nodes
4 Comments
Accepted Answer
Rik
on 30 Jun 2020
You should learn about the linspace function. I would also suggest you check if you have x and y correct. I kept your convention.
You were also limiting the axis so changing the number of nodes doesn't behave as expected.
Lx = 3; Ly = 2;
nx = 120; ny = 12; hx = Lx/nx; hy = Ly/ny;
by0 = linspace(0,Lx,nx+1); %by0 = 0*[0:hx:Lx];
byn = 200+(100/3)*linspace(0,Lx,nx+1).^2; %byn = 200+(100/3)*[0:hx:Lx].^2;
bx0 = 100*linspace(0,Ly,ny+1); %bx0 = 100*[0:hy:Ly];
bxn = 250*linspace(0,Ly,ny+1); %bxn = 250*[0:hy:Ly];
F = zeros(nx+1,ny+1); G = F;
a = ellipgen(nx,hx,ny,hy,G,F,bx0,bxn,by0,byn);
aa = flipud(a); colormap(gray)
surfl(aa)
xlabel('x direction')
ylabel('y direction')
zlabel('Temperature')
axis([0 nx 0 ny 0 500]) %axis([0 12 0 12 0 500])
2 Comments
Rik
on 30 Jun 2020
I already edited that line for you. The only thing that is left for you is to figure out how you can run the same code with different inputs. Hint: use a for-loop, wrap your code in a function, or both.
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