1D Heat Conduction using explicit Finite Difference Method

Question

Derek Shaw 2016-12-15

1
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/316950-1d-heat-conduction-using-explicit-finite-difference-method

回答： Alugunuri 2023-2-8

Hello I am trying to write a program to plot the temperature distribution in a insulated rod using the explicit Finite Central Difference Method and 1D Heat equation. The rod is heated on one end at 400k and exposed to ambient temperature on the right end at 300k. I am using a time of 1s, 11 grid points and a .002s time step. When I plot it gives me a crazy curve which isn't right. I think I am messing up my initial and boundary conditions. Here is my code.

L=1;
t=1;
k=.001;
n=11;
nt=500;
dx=L/n;
dt=.002;
alpha=k*dt/dx^2;
T0(1)=400;
for j=1:nt
    for i=2:n
        T1(i)=T0(i)+alpha*(T0(i+1)-2*T0(i)+T0(i-1));
    end
    T0=T1;
end
plot(x,T1)

2 个评论
显示无隐藏无

KSSV 2016-12-15

First place, it is not giving any curve..there is a error in your code. Please recheck your code once.

Derek Shaw 2016-12-15

I am not sure I understand correctly. In the above I wrote this equation to be iterated

With Boundary conditions

and Initial Conditions

with T0=400k and TL=Ti=300k

I am not sure how to set these boundary conditions in the code. Or if there is a curve I need to derive before doing the iterations.

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

michio 2016-12-15

4
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/316950-1d-heat-conduction-using-explicit-finite-difference-method#answer_247287

编辑：michio 2016-12-15

在 MATLAB Online 中打开

It seems your initial condition and boundary conditon (x = L) are missing in the code. Try

 L=1;
 t=1;
 k=.001;
 n=11;
 nt=500;
 dx=L/n;
 dt=.002;
 alpha=k*dt/dx^2;
 T0=400*ones(1,n);
 T1=300*ones(1,n);
 T0(1) = 300;
 T0(end) = 300;
 for j=1:nt
    for i=2:n-1
        T1(i)=T0(i)+alpha*(T0(i+1)-2*T0(i)+T0(i-1));
    end
    T0=T1;
 end
 plot(T1)

http://jp.mathworks.com/matlabcentral/fileexchange/59916-simple-heat-equation-solver could make a good example for you.

9 个评论
显示 7更早的评论隐藏 7更早的评论

Torsten 2022-12-4

在 MATLAB Online 中打开

Yes, code should be

L=1;

t=1;

k=.001;

n=11;

nt=10000;

dx=L/n;

dt=.002;

alpha=k*dt/dx^2;

T0=300*ones(1,n+1);

T0(1) = 400;

T1 = T0;

for j=1:nt

for i=2:n

T1(i)=T0(i)+alpha*(T0(i+1)-2*T0(i)+T0(i-1));

end

T0=T1;

end

plot((0:n)*L/n,T1)

SYML2nd 2022-12-4

Thank you @Torsten

请先登录，再进行评论。

Answer 2

youssef aider 2019-2-12

4
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/316950-1d-heat-conduction-using-explicit-finite-difference-method#answer_360732

在 MATLAB Online 中打开

here is one, you can just change the boundaries

clear
clc
clf
% domain descritization
alpha = 0.05;
xmin = 0;
xmax = 0.2;
N = 100;
dx = (xmax-xmin)/(N-1);
x = xmin:dx:xmax;
dt = 4.0812E-5;
tmax = 1;
t = 0:dt:tmax;
% problem initialization
phi0 = ones(1,N)*300;
phiL = 230;
phiR = phiL;
% solving the problem
r = alpha*dt/(dx^2) % for stability, must be 0.5 or less
for j = 2:length(t) % for time steps
    phi = phi0;        
    for i = 1:N % for space steps
        if i == 1 || i == N
            phi(i) = phiL;
        else
            phi(i) = phi(i)+r*(phi(i+1)-2*phi(i)+phi(i-1));
        end
    end
    phi0 = phi;
    plot(x,phi0)
    shg
    pause(0.05)
end