numerical methods assignment 2

8/11/2019 Numerical Methods Assignment 2

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Steady state square plateStarting from a 3 x 3 square plate in the example, the code is written to allow for simple refinement.

The code is first tested without a hole in the centre.

Steady state square plate with 3x3 cells



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It was found that after several refinements to 81 x 81, the values in the original cells are within 0.05%of the previous refinement.


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Table showing percentage change in results from previous refinement for a steady state square

plate

Cell Number

(Based on 3x3

grid)

% Change in values from previous

refinement

3x3 9x9 27x27 81x811 0 1.1403 32.5492 0.0450

2 0 1.4195 0.1575 0.0169

3 0 2.9435 47.4714 0.0152

4 0 1.4863 28.6655 0.0168

5 0 1.6517 0.1925 0.0200

6 0 1.6562 39.7151 0.0180

7 0 1.6783 25.1864 0.0087

8 0 1.6330 0.1797 0.0216

9 0 1.0371 33.2635 0.0197

The results from several points were plotted out on a graph, to better visualise the effect of the grid

refinement.

Graph of temperature (C) against number of cells in grid

Steady state square plate with hole

For the case of the square plate with a hole, we have decided to set the boundaries at the hole to

30C for realism and simplicity of calculation since we can eliminate the number of unknowns in the

system of equations for those cells that have been replaced by the hole.

0

50

100

150

200

250

1 9 81 729 6561

Temperature(degreesCelcius)

Number of cells

Cell 1

Cell 5

Cell 9


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Steady state square plate with hole with 3x3 cells



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It was found that after refinements to 81 x 81, the values in the original 8 cells (centre cell omitted,

as it is set at 30C) are within 5% of the previous refinement.

Table showing percentage change in results from previous refinement for a steady state square

plate with hole

Cell Number

(Based on 3x3

grid)

% Change in values from previous

refinement

3x3 9x9 27x27 81x81

1 0 22.4944 8.4426 1.7417

2 0 20.4441 10.4796 4.1920

3 0 12.1164 20.7259 3.6241

4 0 27.4608 44.7039 2.3313

6 0 11.8615 42.7340 4.2270

7 0 30.0400 70.4730 1.3141

8 0 20.9673 9.3775 3.3623

9 0 8.9047 49.9211 3.7554

Since a 27x27 grid showed fairly accurate results, with less than 5% change after refinement, we

decided to use this grid for the transient simulation, as the 81x81 grid would be too time-consuming.

The results from several points were plotted out on a graph, to better visualise the effect of the grid

refinement.


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Graph of temperature (C) against number of cells in grid

Transient square plate with holeA fully implicit and a Crank-Nicolson scheme were carried out for the transient problem. The

simulation was initialized with the plate at 30C. The time steps run were 0.5s, 0.05s, 0.005s and

0.0005s. The simulation was run for a 30-second duration, and the steadiness of the result was

ascertained by comparing to the data produced at 27 seconds.

Table showing percentage difference between data at 30s and 27s for fully implicit scheme

Cell

number

% difference between result at 27s and 30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s

1 0.0220 0.0176 0.0176 0.0176

2 0.0940 0.0783 0.0783 0.0769

3 0.2102 0.1768 0.1741 0.1741

4 0.0071 0.0057 0.0057 0.0057

6 0.2346 0.2016 0.1992 0.1969

7 0.0172 0.0158 0.0158 0.0158

8 0.1098 0.0981 0.0958 0.0958

9 0.3153 0.2739 0.2713 0.2713

0

20

40

60

80

100

120

1 9 81 729 6561

Temperature(degre

esCelcius)

Number of cells

Cell 1

Cell 4

Cell 7

Cell 9


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Table showing percentage difference between data at 30s and 27s for Crank-Nicolson scheme

Cell

number

% difference between result at 27s and 30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s

1 0.0176 0.0176 0.0176 0.0176

2 0.0769 0.0769 0.0769 0.0769

3 0.1741 0.1741 0.1741 0.1741

4 0.0057 0.0057 0.0057 0.0057

6 0.1992 0.1969 0.1969 0.1969

7 0.0158 0.0158 0.0158 0.0158

8 0.0958 0.0958 0.0958 0.0958

9 0.2687 0.2713 0.2713 0.2713

We can assume the result has reached steady-state, since the percentage difference is minimal for

both schemes. At higher values of time step, the Crank-Nicolson scheme showed greaterconvergence, which is affected little by the size of the time step.

Next, the effect of time step refinement on the results of the simulation was investigated. First, the

results of each scheme were compared against its previous time step refinement.

Table showing percentage difference in results at 30s from previous time step refinement for fully

implicit scheme

Cell

number

% difference from previous time-step

refinement at 30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s

1 0 0.0055 0.0011 0.0000

2 0 0.0256 0.0028 0.0000

3 0 0.0584 0.0053 0.0013

4 0 0.0014 0.0000 0.0000

6 0 0.0632 0.0070 0.0000

7 0 0.0053 0.0000 0.0000

8 0 0.0303 0.0023 0.0000

9 0 0.0818 0.0077 0.0026


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Table showing percentage difference in results at 30s from previous time step refinement for

Crank-Nicolson scheme

Cell

number

% difference from previous time-step

refinement at 30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s1 0 0.0000 0.0000 0.0000

2 0 0.0000 0.0000 0.0000

3 0 0.0013 0.0000 0.0000

4 0 0.0000 0.0000 0.0000

6 0 0.0023 0.0000 0.0000

7 0 0.0000 0.0000 0.0000

8 0 0.0000 0.0000 0.0000

9 0 0.0000 0.0000 0.0000

We notice that the time step refinement has little effect on the results of the simulation for both

schemes, and especially so for the Crank-Nicolson scheme, which showed almost zero change in

results.

Next, we analysed the graphs of temperature against time for cell 3 for each scheme.

Graph of temperature (C) against time (s) for cell 3 for different time step sizes for fully implicit

scheme

30

35

40

45

50

55

60

65

70

75

80

0 5 10 15 20 25 30


Time (s)

Graph of Temperature (degrees

Celcius) against Time (s) for Cell 3

dt=0.5

dt=0.05

dt=0.005

dt=0.0005


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Graph of temperature (C) against time (s) for cell 3 for different time step sizes for Crank-Nicolson

scheme

From the graphs, we notice very slight variations between the time steps, from about 3 to 18

seconds of the simulation in the fully implicit scheme. For the Crank-Nicolson scheme, no such

variation can be seen. Nevertheless, the lack of variation for both schemes implies that a time step

of 0.5s was good enough for the simulation.

Lastly, the results at 30 seconds for both schemes were compared against the steady-state results

attained earlier. This comparison was done for all time step sizes.

Table showing percentage difference in results at 30s from steady-state results for fully implicit

scheme

Cell

number

% difference from steady-state results at

30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s

1 0.0308 0.0253 0.0242 0.02422 0.1322 0.1066 0.1037 0.1037

3 0.3005 0.2422 0.2369 0.2356

4 0.0085 0.0071 0.0071 0.0071

6 0.3476 0.2846 0.2776 0.2776

7 0.0290 0.0237 0.0237 0.0237

8 0.1771 0.1468 0.1445 0.1445

9 0.4856 0.4043 0.3966 0.3941

30

35

40

45

50

55

60

65

70

75

80

0 5 10 15 20 25 30


Time (s)

Graph of Temperature (degrees

Celcius) against Time (s) for Cell 3

dt=0.5

dt=0.05

dt=0.005

dt=0.0005


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Table showing percentage difference in results at 30s from steady-state results for Crank-Nicolson

scheme

Cell

number

% difference from steady-state results at

30s

dt=0.5s dt=0.05s dt=0.005s dt=0.0005s1 0.0242 0.0242 0.0242 0.0242

2 0.1037 0.1037 0.1037 0.1037

3 0.2343 0.2356 0.2356 0.2356

4 0.0071 0.0071 0.0071 0.0071

6 0.2753 0.2776 0.2776 0.2776

7 0.0237 0.0237 0.0237 0.0237

8 0.1445 0.1445 0.1445 0.1445

9 0.3941 0.3941 0.3941 0.3941

Both schemes showed a strong similarity with the steady-state results at 30s. At larger time steps,

the Crank-Nicolson scheme performed slightly better than the fully implicit scheme.

The variation of temperature against time at cell 3 was compared between the fully implicit and

Crank-Nicolson schemes for a time step size of 0.005s.

Graph of temperature (C) against time (s) for cell 3 for a time step size of 0.005s for fully implicit

and Crank-Nicolson schemes.

There was little difference between the two schemes for this time step size. This implies that the

results of the simulation are not strongly dependent on the scheme type.

For a better visualisation of how the temperature distribution changes with time, the contour plots

at 3 second intervals are attached in Appendix A for both the schemes at a time step of 0.005s. The

MATLAB code used to produce the results are attached in Appendix B.

Discussions

30

35

40

45

50

55

60

65

70

75

80

0 10 20 30

Temperature(degre

esCelcius)

Time (s)

Crank-Nicolson

Fully Implicit


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Appendix A

Fully implicit

3s

6s

9s


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12s

15s

18s


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21s

24s


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27s

30s


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Crank-Nicolson Scheme

3s

6s

9s


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12s

15s

18s


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21s

24s


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27s

30s


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Appendix B

Steady state, no hole

clear alln=81*81; %change to appropriate size

A=zeros(n, n);BT=zeros(n,1);

k=1000; %conductivitydelta_x=0.1*3/sqrt(n);q=500*10^3;

%setting the corner boundaries%bottom left cornerA(1,1)=2;

A(1,1+sqrt(n))=-1;A(1,2)=-1;

%top right cornerA(n,n)=4;A(n,n-sqrt(n))=-1;A(n,n-1)=-1;

%top left cornerA(sqrt(n),sqrt(n))=4;A(sqrt(n),sqrt(n)*2)=-1;A(sqrt(n),sqrt(n)-1)=-1;

%bottom right cornerA(n-sqrt(n)+1,n-sqrt(n)+1)=2;A(n-sqrt(n)+1,n-sqrt(n)+2)=-1;A(n-sqrt(n)+1,n-sqrt(n)*2+1)=-1;

%setting the left side boundariesfori=2:(sqrt(n)-1)

A(i,i)=3;A(i,i+sqrt(n))=-1;A(i,i+1)=-1;A(i,i-1)=-1;

end

%setting the right side boundariesfori=(n-sqrt(n)+2):(n-1)

A(i,i)=3;A(i,i+1)=-1;A(i,i-1)=-1;A(i,i-sqrt(n))=-1;

end

%setting the top boundariesfori=2:(sqrt(n)-1)

A(i*sqrt(n),i*sqrt(n))=5;A(i*sqrt(n),(i-1)*sqrt(n))=-1;

A(i*sqrt(n),(i+1)*sqrt(n))=-1;A(i*sqrt(n),i*sqrt(n)-1)=-1;


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end

%setting the bottom boundariesfori=1:(sqrt(n)-2)

A(i*sqrt(n)+1,i*sqrt(n)+1)=3;A(i*sqrt(n)+1,(i-1)*sqrt(n)+1)=-1;

A(i*sqrt(n)+1,(i+1)*sqrt(n)+1)=-1;A(i*sqrt(n)+1,i*sqrt(n)+2)=-1;

end

%setting internal cellsfori=1:(sqrt(n)-2)

forj=2:(sqrt(n)-1)A(i*sqrt(n)+j,i*sqrt(n)+j)=4;A(i*sqrt(n)+j,i*sqrt(n)+j+sqrt(n))=-1; A(i*sqrt(n)+j,i*sqrt(n)+j-sqrt(n))=-1; A(i*sqrt(n)+j,i*sqrt(n)+j-1)=-1;A(i*sqrt(n)+j,i*sqrt(n)+j+1)=-1;

end

end

%setting top and left boundary temperaturesfori=1:sqrt(n)

BT(i*sqrt(n))=BT(i*sqrt(n))+200; %top boundaryBT(i)=BT(i)+q*delta_x/k; %left boundary

end

T=A\BT;T1=zeros(sqrt(n),sqrt(n));

fori=1:n

ifrem(i,sqrt(n))==0T1(sqrt(n),i/sqrt(n))=T(i);

elseT1(rem(i,sqrt(n)), ceil(i/sqrt(n)))=T(i);

endend

%out put array with original 9-cell temperaturesifn==9

Tout=T;elseifn==81

Tout=[T(11);T(14);T(17);T(38);T(41);T(44);T(65);T(68);T(71);]; elseifn==27*27

Tout=[T(135-4);T(135-4-9);T(135-4-9-9);T(378-4);T(378-4-9);T(378-4-9-9);T(621-4);T(621-4-9);T(621-4-9-9);];

elseifn==81*81Tout=[T(1134-13);T(1134-13-27);T(1134-13-54);T(3321-

13);T(3321-13-27);T(3321-13-54);T(5508-13);T(5508-13-27);T(5508-13-54);];end

endend

endclfcontourf(T1)


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dlmwrite('T_9.txt',Tout,'delimiter','\t')

Steady-state, with hole

clear all

n1=9;n=81*81; % set to appropriate size

A=zeros(n, n);BT=zeros(n,1);

k=1000; %conductivitydelta_x=0.1*3/sqrt(n);q=500*10^3;

k1=500; %conductivity of insulated centre

%setting the corner boundaries%bottom left cornerA(1,1)=2;A(1,1+sqrt(n))=-1;A(1,2)=-1;

%top right cornerA(n,n)=4;A(n,n-sqrt(n))=-1;A(n,n-1)=-1;

%top left corner

A(sqrt(n),sqrt(n))=4;A(sqrt(n),sqrt(n)*2)=-1;A(sqrt(n),sqrt(n)-1)=-1;

%bottom right cornerA(n-sqrt(n)+1,n-sqrt(n)+1)=2;A(n-sqrt(n)+1,n-sqrt(n)+2)=-1;A(n-sqrt(n)+1,n-sqrt(n)*2+1)=-1;


A(i,i)=3;A(i,i+sqrt(n))=-1;A(i,i+1)=-1;A(i,i-1)=-1;

end


A(i,i)=3;A(i,i+1)=-1;A(i,i-1)=-1;A(i,i-sqrt(n))=-1;

end


A(i*sqrt(n),i*sqrt(n))=5;


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A(i*sqrt(n),(i-1)*sqrt(n))=-1;A(i*sqrt(n),(i+1)*sqrt(n))=-1;A(i*sqrt(n),i*sqrt(n)-1)=-1;

end

%setting the bottom boundaries

fori=1:(sqrt(n)-2)A(i*sqrt(n)+1,i*sqrt(n)+1)=3;A(i*sqrt(n)+1,(i-1)*sqrt(n)+1)=-1;A(i*sqrt(n)+1,(i+1)*sqrt(n)+1)=-1;A(i*sqrt(n)+1,i*sqrt(n)+2)=-1;

end


forj=1:(sqrt(n)-2)A(i+j*sqrt(n),i+j*sqrt(n))=4;A(i+j*sqrt(n),i+(j+1)*sqrt(n))=-1;A(i+j*sqrt(n),i+(j-1)*sqrt(n))=-1;

A(i+j*sqrt(n),i+j*sqrt(n)-1)=-1;A(i+j*sqrt(n),i+j*sqrt(n)+1)=-1;

endend

%setting top and left boundary temperaturesfori=1:sqrt(n)

BT(i*sqrt(n))=BT(i*sqrt(n))+200; %top boundaryBT(i)=BT(i)+q*delta_x/k; %left boundary

end

%setting the cells in the hole to constant 30 degreesfori=1:(sqrt(n)/3)

forj=1:(sqrt(n)/3)A(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n),:)=0; A(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n), j+(sqrt(n)/3)+(i-

1+sqrt(n)/3)*sqrt(n))=1;BT(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n))=30;

endend

T=A\BT;

T1=zeros(sqrt(n),sqrt(n));fori=1:n



endend

%output array wiith original 9-cell temperaturesifn==9

Tout=T;elseifn==81


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Tout=[T(11);T(14);T(17);T(38);T(41);T(44);T(65);T(68);T(71);];elseifn==27*27

Tout=[T(135-4);T(135-4-9);T(135-4-9-9);T(378-4);T(378-4-9);T(378-4-9-9);T(621-4);T(621-4-9);T(621-4-9-9);];


13);T(3321-13-27);T(3321-13-54);T(5508-13);T(5508-13-27);T(5508-13-54);];endend

endendclfcontourf(T1)%dlmwrite('T_9.txt',Tout,'delimiter','\t')

Transient, with hole

clear all

n=27*27;rho=8000;dt=0.005;theta=0.5; %theta=0.5 for Crank-Nicolson, 1 for fully implicit

A=zeros(n, n);BT=zeros(n,1);T=zeros(n,1); %initial condition arrayT(:,1)=30;dist=3;

k=1000; %conductivitydelta_x=0.1*3/sqrt(n);

delta_v=delta_x^2;q=500*10^3;

total_t=30;clf

fort=dt:dt:total_t

%setting the corner boundaries%bottom left cornerA(1,1)=rho*delta_v/dt+theta*delta_x*(2*k); A(1,1+sqrt(n))=-theta*k*delta_x;A(1,2)=-theta*k*delta_x;BT(1)=k*delta_x*q*delta_x/k+rho*delta_v/dt*T(1)-(1-theta)*(k*delta_x)*(2*T(1)-T(2)-T(1+sqrt(n)));

%top right cornerA(n,n)=rho*delta_v/dt+theta*delta_x*(4*k); A(n,n-sqrt(n))=-theta*k*delta_x;A(n,n-1)=-theta*k*delta_x;BT(n)=k*delta_x*200+rho*delta_v/dt*T(n)-(1-theta)*(k*delta_x)*(4*T(n)-T(n-sqrt(n))-T(n-1));

%top left cornerA(sqrt(n),sqrt(n))=rho*delta_v/dt+theta*delta_x*(4*k);

A(sqrt(n),sqrt(n)*2)=-theta*k*delta_x; A(sqrt(n),sqrt(n)-1)=-theta*k*delta_x;


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BT(sqrt(n))=k*delta_x*(200+q*delta_x/k)+rho*delta_v/dt*T(sqrt(n))-(1-theta)*(k*delta_x)*(4*T(sqrt(n))-T(sqrt(n)-1)-T(sqrt(n)*2));

%bottom right cornerA(n-sqrt(n)+1,n-sqrt(n)+1)=rho*delta_v/dt+theta*delta_x*(2*k); A(n-sqrt(n)+1,n-sqrt(n)+2)=-theta*k*delta_x;

A(n-sqrt(n)+1,n-sqrt(n)*2+1)=-theta*k*delta_x; BT(n-sqrt(n)+1)=rho*delta_v/dt*T(n-sqrt(n)+1)-(1-theta)*(k*delta_x)*(2*T(n-sqrt(n)+1)-T(n-sqrt(n)+2)-T(n-sqrt(n)*2+1));


A(i,i)=rho*delta_v/dt+theta*delta_x*(3*k); A(i,i+1)=-theta*k*delta_x;A(i,i-1)=-theta*k*delta_x;A(i,i+sqrt(n))=-theta*k*delta_x;BT(i)=k*delta_x*q*delta_x/k+rho*delta_v/dt*T(i)-(1-

theta)*(k*delta_x)*(3*T(i)-T(i+1)-T(i-1)-T(i+sqrt(n))); end


A(i,i)=rho*delta_v/dt+theta*delta_x*(3*k); A(i,i+1)=-theta*k*delta_x;A(i,i-1)=-theta*k*delta_x;A(i,i-sqrt(n))=-theta*k*delta_x;BT(i)=rho*delta_v/dt*T(i)-(1-theta)*(k*delta_x)*(3*T(i)-T(i+1)-T(i-1)-

T(i-sqrt(n)));end


A(i*sqrt(n),i*sqrt(n))=rho*delta_v/dt+theta*delta_x*(5*k); A(i*sqrt(n),(i-1)*sqrt(n))=-theta*k*delta_x; A(i*sqrt(n),(i+1)*sqrt(n))=-theta*k*delta_x; A(i*sqrt(n),i*sqrt(n)-1)=-theta*k*delta_x; BT(i*sqrt(n))=k*delta_x*200+rho*delta_v/dt*T(i*sqrt(n))-(1-

theta)*(k*delta_x)*(5*T(i*sqrt(n))-T((i-1)*sqrt(n))-T((i+1)*sqrt(n))-T(i*sqrt(n)-1));end

%setting the bottom boundariesfori=1:(sqrt(n)-2)

A(i*sqrt(n)+1,i*sqrt(n)+1)=rho*delta_v/dt+theta*delta_x*(3*k); A(i*sqrt(n)+1,(i-1)*sqrt(n)+1)=-theta*k*delta_x; A(i*sqrt(n)+1,(i+1)*sqrt(n)+1)=-theta*k*delta_x; A(i*sqrt(n)+1,i*sqrt(n)+2)=-theta*k*delta_x; BT(i*sqrt(n)+1)=rho*delta_v/dt*T(i*sqrt(n)+1)-(1-

theta)*k*delta_x*(3*T(i*sqrt(n)+1)-T((i-1)*sqrt(n)+1)-T((i+1)*sqrt(n)+1)-T(i*sqrt(n)+2));end


forj=1:(sqrt(n)-2)A(i+j*sqrt(n),i+j*sqrt(n))=theta*delta_x*(4*k)+(rho*delta_v/dt); %aPA(i+j*sqrt(n),i+(j+1)*sqrt(n))=-theta*k*delta_x; %aSA(i+j*sqrt(n),i+(j-1)*sqrt(n))=-theta*k*delta_x; %aNA(i+j*sqrt(n),i+j*sqrt(n)-1)=-theta*k*delta_x; %aWA(i+j*sqrt(n),i+j*sqrt(n)+1)=-theta*k*delta_x; %aE


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BT(i+j*sqrt(n))=(rho*delta_v/dt)*T(i+j*sqrt(n))-(1-theta)*(delta_x*k)*(4*T(i+j*sqrt(n))-T(i+(j+1)*sqrt(n))-T(i+(j-1)*sqrt(n))-T(i+j*sqrt(n)-1)-T(i+j*sqrt(n)+1));

endend

%setting the cells in the hole to constant 30 degreesfori=1:(sqrt(n)/3)

forj=1:(sqrt(n)/3)A(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n),:)=0; A(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n), j+(sqrt(n)/3)+(i-

1+sqrt(n)/3)*sqrt(n))=1;BT(j+(sqrt(n)/3)+(i-1+sqrt(n)/3)*sqrt(n))=30;

endend

T=A\BT;T1=zeros(sqrt(n),sqrt(n));

fori=1:n



endendifn==9 %sets output cells for 3x3, 9x9, 27x27 and 81x81

Tout=T;elseifn==81

Tout=[T(11);T(14);T(17);T(38);T(41);T(44);T(65);T(68);T(71);];elseifn==27*27

Tout=[T(135-4);T(135-4-9);T(135-4-9-9);T(378-4);T(378-4-9);T(378-4-9-9);T(621-4);T(621-4-9);T(621-4-9-9);];


13);T(3321-13-27);T(3321-13-54);T(5508-13);T(5508-13-27);T(5508-13-54);];end

endend

endifrem(t,3)==0%set when to produce an output file and plot. just change thenumber in the rem function to how many seconds you want a plotfigurecontourf(T1)dlmwrite(sprintf('T_9_%f.txt',t),Tout,'delimiter','\t') %output file nameis T_9_(number of seconds simulation has run). Just copy and paste outputinto excel it will be in a columnendend

numerical methods assignment 2

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