clc

clear all

close all

dt $=0\mathrm{.}01$; $\%$timestep

dx $=0\mathrm{.}01$; $\%$ mesh size

n $=50$; $\%$number of node points

$\%$The size and the mesh of plate

Lx $=($n$-1)*$dx; $\%$ total length in X

Ly $=$ Lx; $\%$ total length in y

xs $=0$:dx:Lx; $\%$vector of positions in x

ys $=$ xs; $\%$vector of positions in y

$[$X $,$ Y$]=$ meshgrid$($xs $,$ ys$)$; $\%$mesh

Ta $=300$; $\%$k$,$ Temperture of the environment

dTmax $=200$; $\%$max temp above the temp of environment

alpha $=1$e$-4$; $\%$ thermal diffusivity of copper $($m$^2/$s$)$

$\%$ intial condition

T $=$ ones$($n$)*$Ta; $\%$uniform temperture

T$(30$:$45,30$:$45)=$ Ta $+$ dTmax; $\%$the hop part temp

c $=3$;

$\%$ c$0$ stands for Periodic boundaries

$\%$ c$1$ stands for Fixed temperature at the boundary

$\%$ c$2$ stands for Insulated BC

$\%$ c$3$ stands for Heat generation with fixed temp BC

q $=300000$; $\%$the power of the laser focusing on the center of the plate

cp $=1000$; $\%$heat capacity

rho $=1000$; $\%$density

for i$=1$:$30000\%$runnig thr simulation for $30,000$ cycles$=300$ s

Tx$1=$ circshift$($T$,[0,-1])$; $\%$ T shifted to the left $($aka the right neghbor$)$

Tx$\_1=$ circshift$($T$,[0,1])$; $\%$ T shifted to the right $($aka the left neghbor$)$

Ty$1=$ circshift$($T$,[-1,0])$; $\%$ T shifted upwards $($aka the below neghbor$)$

Ty$\_1=$ circshift$($T$,[1,0])$; $\%$ T shifted downwards $($aka the above neghbor$)$

$\%$the main heat equations

Told $=$ T;

T $=$ T $+$ alpha$*($Tx$1+$Tx$\_1+$ Ty$1+$Ty$\_1-4*$T$)/$dx$^2*$dt; $\%$ this is all there to it$!$

if c$==1\%$Fixed temperature at the boundary

T$(1,$:$)=$ Ta;

T$($n$,$:$)=$ Ta;

T$($:$,1)=$ Ta;

T$($:$,$n$)=$ Ta;

elseif c$==2\%$insulated BC

$\%$insulated edges

T$($:$,1)=$ Told$($:$,1)+$ alpha$*($Tx$1($:$,1)+$ Ty$1($:$,1)+$Ty$\_1($:$,1)-$

$3*$Told$($:$,1))/$dx$^2*$dt;

$\%$left edge,heat flows from right, top, bottom

T$($:$,$n$)=$ Told$($:$,$n$)+$ alpha$*($Tx$\_1($:$,$n$)+$ Ty$1($:$,$n$)+$Ty$\_1($:$,$n$)-3*$Told$($:$,$n$))/$

dx$^2*$dt;

$\%$right edge,heat flows from left, top, bottom

T$(1,$:$)=$ Told$(1,$:$)+$ alpha$*($Tx$1(1,$:$)+$ Ty$1(1,$:$)+$Tx$\_1(1,$:$)-$

$3*$Told$(1,$:$))/$dx$^2*$dt;

$\%$bottom,heat flows from right, top, left

T$($n$,$:$)=$ Told$($n$,$:$)+$ alpha$*($Tx$1($n$,$:$)+$ Tx$\_1($n$,$:$)+$Ty$\_1($n$,$:$)-3*$Told$($n$,$:$))/$

dx$^2*$dt;

$\%$top edge,heat flows from right, left, bottom

$\%$insulated coreners

T$(1,1)=$ Told$(1,1)+$ alpha$*($Tx$1(1,1)+$ Ty$1(1,1)-2*$Told$(1,1))/$dx$^2*$dt;

$\%$bottom left corner, heat flows from top and right

T$($n$,$n$)=$ Told$($n$,$n$)+$ alpha$*($Tx$\_1($n$,$n$)+$ Ty$\_1($n$,$n$)-2*$Told$($n$,$n$))/$dx$^2*$dt;

$\%$top right corner, heat flows from bottom and left

T$(1,$n$)=$ Told$(1,$n$)+$ alpha$*($Tx$\_1(1,$n$)+$ Ty$1(1,$n$)-2*$Told$(1,$n$))/$dx$^2*$dt;

$\%$bottom right corner, heat flows from top and left

T$($n$,1)=$ Told$($n$,1)+$ alpha$*($Tx$1($n$,1)+$ Ty$\_1($n$,1)-2*$Told$($n$,1))/$dx$^2*$dt;

$\%$top corner, heat flows from bottom and right

elseif c$==3$

T$(1,$:$)=$ Ta;

T$($n$,$:$)=$ Ta;

T$($:$,1)=$ Ta;

T$($:$,$n$)=$ Ta;

T$($n$/2,$ n$/2)=$ T$($n$/2,$ n$/2)+$ q$/($dx$*$dx$*$cp$*$rho$)*$dt;

end

if mod$($i$,100)==0\%$plotting every $100$th interation

pcolor$($X$,$ Y$,$ T$)$

$\%$caxis$([300,500])\%$this fixes the colorbar so that the colors slways

between $300-500$k

shading interp

colorbar$()$

$\%\%$alternative way to plot

$\%$ surf$($X$,$ Y $,$T$)$;

$\%$ axis$([0,$Lx$,0,$Ly$,$Ta$,$Ta$+$dTmax$])$;

$\%$ shading interp

title $([\text{'}$t$=\text{'},$ num$2$str$($dt$*$i$),\text{'}$ s$\text{'}])$

drawnow$()$

end

end

$!!$Can you edit this given matlab code to meet the requirements! $!$

The goal of this simulation is to figure out the intrinsic size of a point heat source and to practice mesh and

geometric convergence. Geometric convergence is defined as: "the smallest cell that you can use such that the

boundary conditions do not affect the thermal profile."

Start with the posted $2$D MATLAB simulation from the tutorial and apply a $5000$ W point source to the center

of the plate with $300$ K boundaries on all faces. From the results from this working $2$D simulation. Calculate

the width of the hot spot in the thermal profile. The definition of width you will use is also up to you, some

common choices are RMS width, standard deviation, and FWHM $($don$\text{'}$t forget to mention which one you

chose$).$

The steps I recommend are:

Pick how long to run your simulation and a converged time step $($up to you, but include in deliverable$).$

Pick a starting size of the plate $($also your choice$).$

Start with the thermal diffusivity for copper $(\{$: $1\mathrm{.}11$e$-4$m$^2//$s$)$ and converge the mesh, then the geometry.

As you change mesh, make sure you have defined your heat source in a way that scales with the mesh size

correctly!

Using the same mesh, use change the thermal diffusivity to that for acrylic $(1$e$-7$m$^2//$s $)$ and converge the

geometry.

For your deliverable, please include:

Proof of converged mesh for copper.

Proof of converged geometry for copper.

Proof of converged mesh for acrylic

Proof of converged geometry for acrylic.

Also include some short text comments on how you reasonably chose your timestep and total duration, as

well as why your plots make sense. $($hint: you can compare how mesh and geometry convergences differ

between the two materials$).$