Question

in math lab how to simulated heat conduction and how to show it in a video sinulation

bottom code is a simulation of just the hear conduction without the video

clear,pack,clc

% material properties

k=200

cp=921

rho=2500

alpha=k/(cp*rho);

% introduce discretization constants

% time

delta_t=0.1;

%space

delta_L=0.01; % this is in meters

% how many points

N=100;

% constants a1, and a2

a1=1-2*(alpha*delta_t)/(delta_L^2);

a2=(alpha*delta_t)/(delta_L^2);

% introduce initial and boundary conditions

% initial condition

T0=zeros(N,1);

T0(1:floor(N/3))=30;

% boundary condition

% this is the left point next to the bar

T0_m1=500;

% this is the right point next to the bar

T0_Np1=10;

b=zeros(N,1);

b(1)=a2*T0_m1;

b(N)=a2*T0_Np1;

% define the A matrix

A=zeros(N,N);

for i=1:N

if i==1

A(1,1)=a1; A(1,2)=a2;

end

if i==N

A(N,N)=a1; A(N,N-1)=a2;

end

if (i>=2)&&(i<=N-1)

A(i,i)=a1;

A(i,i-1)=a2;

A(i,i+1)=a2;

end

end

% number of simulation steps

sim_steps=10000;

RESULT=zeros(N,sim_steps);

for j=1:sim_steps

if j==1

RESULT(:,1)=T0; % initial temperature

else

RESULT(:,j)=A*RESULT(:,j-1)+b;

end

end

imagesc(RESULT)

% RESULT is number of points X time

% middle point plotting

plot(RESULT(50,:))

hold on

plot(RESULT(80,:),'r')

Answer 1

I have modified parts of the code to store animation in the form of video file.

clear;
pack;clc;

% material properties
k=200;
cp=921;
rho=2500;
alpha=k/(cp*rho);

% introduce discretization constants
% time
delta_t=0.1;
%space
delta_L=0.01; % this is in meters

% how many points
N=100;

% constants a1, and a2

a1=1-2*(alpha*delta_t)/(delta_L^2);
a2=(alpha*delta_t)/(delta_L^2);

% introduce initial and boundary conditions

% initial condition
T0=zeros(N,1);
T0(1:floor(N/3))=30;

% boundary condition

% this is the left point next to the bar
T0_m1=500;

% this is the right point next to the bar
T0_Np1=10;
b=zeros(N,1);
b(1)=a2*T0_m1;
b(N)=a2*T0_Np1;

% define the A matrix
A=zeros(N,N);

for i=1:N
if i==1
A(1,1)=a1; A(1,2)=a2;
end
if i==N
A(N,N)=a1; A(N,N-1)=a2;
end
if (i>=2)&&(i<=N-1)
A(i,i)=a1;
A(i,i-1)=a2;
A(i,i+1)=a2;
end

end

% number of simulation steps
sim_steps=10000;

RESULT=zeros(N,sim_steps);

figure()
reruns=1; % number of times movie is to play
fps=2; % frames per second

writerObj = VideoWriter('heat_conduction.avi');
open(writerObj);

for j=1:sim_steps

if j==1
RESULT(:,1)=T0; % initial temperature
else

RESULT(:,j)=A*RESULT(:,j-1)+b;
end

imagesc(RESULT(:,j))
drawnow;
Frames(:,i)=getframe;
writeVideo(writerObj,Frames(:,i))
end
close(writerObj);

%imagesc(RESULT)

% RESULT is number of points X time

% middle point plotting

% plot(RESULT(50,:))
%
% hold on
%
% plot(RESULT(80,:),'r')