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13 


14 
\section{Two Dimensional Heat Diffusion for a basic Magmatic Intrusion} 
\begin{figure}[t] 

\sslist{twodheatdiff001.py and cblib.py} 


\label{Sec:2DHD} 


Building upon our success from the 1D models, it is now prudent to expand our domain by another dimension. For this example we will be using a very simple magmatic intrusion as the basis for our model. The simulation will be a single event where some molten granite has formed a hemispherical dome at the base of some cold sandstone country rock. A hemisphere is symmetric so taking a crosssection through its centre will effectively model a 3D problem in 2D. New concepts will include nonlinear boundaries and the ability to prescribe location specific variables. 





\begin{figure}[h!] 

15 
\centerline{\includegraphics[width=4.in]{figures/twodheatdiff}} 
\centerline{\includegraphics[width=4.in]{figures/twodheatdiff}} 
16 
\caption{2D model: granitic intrusion of sandstone country rock.} 
\caption{2D model: granitic intrusion of sandstone country rock.} 
17 
\label{fig:twodhdmodel} 
\label{fig:twodhdmodel} 
18 
\end{figure} 
\end{figure} 
19 


20 
To expand upon our 1D problem, the domain must first be expanded. This will be done in our definition phase by creating a square domain in $x$ and $y$ that is 600 meters along each side \reffig{fig:twodhdmodel}. The number of discrete spatial cells will be 100 in either direction. The radius of the intrusion will be 200 meters with the centre located at the 300 meter mark on the xaxis. The domain variables are; 
\sslist{twodheatdiff001.py and cblib.py} 
21 


22 

Building upon our success from the 1D models, it is now prudent to expand our domain by another dimension. 
23 

For this example we will be using a very simple magmatic intrusion as the basis for our model. The simulation will be a single event where some molten granite has formed a cylindrical dome at the base of some cold sandstone country rock. Assuming that the cylinder is very long 
24 

we model a crosssection as shown in \reffig{fig:twodhdmodel}. We will implement the same 
25 

diffusion model as we have use for the granite blocks in \refSec{Sec:1DHDv00} 
26 

but will add the second spatial dimension and show how to define 
27 

variables depending on the location of the domain. 
28 

We use \file{onedheatdiff001b.py} as the starting point for develop this model. 
29 


30 

\section{Two Dimensional Heat Diffusion for a basic Magmatic Intrusion} 
31 

\label{Sec:2DHD} 
32 


33 

To expand upon our 1D problem, the domain must first be expanded. In fact, we have solved a two dimensional problem already but didn't put much 
34 

attention to the second dimension. This will be changed now. 
35 

In our definition phase by creating a square domain in $x$ and $y$\footnote{In \esc the notation 
36 

$x\hackscore{0}$ and $x\hackscore{1}$ is used for $x$ and $y$, respectively.} that is $600$ meters along each side \reffig{fig:twodhdmodel}. The number of discrete spatial cells will be 100 in either direction. The radius of the intrusion will be $200$ meters with the centre located at the $300$ meter mark on the $x$axis. The domain variables are; 
37 
\begin{python} 
\begin{python} 
38 
mx = 600*m #meters  model length 
mx = 600*m #meters  model length 
39 
my = 600*m #meters  model width 
my = 600*m #meters  model width 
40 
ndx = 100 #mesh steps in x direction 
ndx = 150 #mesh steps in x direction 
41 
ndy = 100 #mesh steps in y direction 
ndy = 150 #mesh steps in y direction 
42 
r = 200*m #meters  radius of intrusion 
r = 200*m #meters  radius of intrusion 
43 
ic = [300, 0] #centre of intrusion (meters) 
ic = [300*m, 0] #coordinates of the centre of intrusion (meters) 
44 
q=0.*Celsius #our heat source temperature is now zero 
qH=0.*J/(sec*m**3) #our heat source temperature is zero 
45 
\end{python} 
\end{python} 
46 
The next step is to define our variables for each material in the model in a manner similar to the previous tutorial. Note that each material has its own unique set of values. The time steps and set up for the domain remain as in Section \ref{sec:key}. Prior to setting up the PDE the boundary between the two materials must be established. The distance $s$ between two points in Cartesian coordinates is defined as: 
As before we use 
47 

\begin{python} 
48 

model = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy) 
49 

\end{python} 
50 

to generate the domain. 
51 


52 

There are two fundamental changes to the PDE has we have discussed PDEs in \refSec{Sec:1DHDv00}. Firstly, 
53 

as the material coefficients for granite and sandstone is different, we need to deal with 
54 

PDE coefficients which vary with there location in the domain. Secondly, we need 
55 

to deal with the second spatial dimension. We will look at these two modification at the same time. 
56 

In fact our temperature model \refEq{eqn:hd} we have used in \refSec{Sec:1DHDv00} applies for the 
57 

1D case with constant material parameter only. For the more general case we are interested 
58 

in this chapter the correct model equation is 
59 
\begin{equation} 
\begin{equation} 
60 
(x_{1}x_{0})^{2}+(y_{1}y_{0})^{2} = s^{2} 
\rho c\hackscore p \frac{\partial T}{\partial t}  \frac{\partial }{\partial x} \kappa \frac{\partial T}{\partial x}  \frac{\partial }{\partial y} \kappa \frac{\partial T}{\partial y} = q\hackscore H 
61 

\label{eqn:hd2} 
62 
\end{equation} 
\end{equation} 
63 
If we define the point $[x_{0},y_{0}]$ as $c$ which denotes the centre of the semicircle of our intrusion, then for all the points $[x,y]$ in our model we can calculate a distance to $c$. All the points that fall within the radius $r$ of our intrusion will have a corresponding value $s < r$ and all those belonging to the country rock will have a value $s > r$. By subtracting $r$ from both of these conditions we find $sr < 0$ for all intrusion points and $sr > 0$ for all country rock points. Defining these conditions within the script is quite simple and is done using the following command: 
Notice, that for the 1D case we have $\frac{\partial T}{\partial y}=0$ and 
64 

for the case of constant material parameters $\frac{\partial }{\partial x} \kappa = \kappa \frac{\partial }{\partial x}$ so this new equation coincides with simplified model equation for this case. It is more convenient 
65 

to write this equation using the $\nabla$ notation as we have already seen in \refEq{eqn:commonform nabla}; 
66 

\begin{equation}\label{eqn:Tform nabla} 
67 

\rho c\hackscore p \frac{\partial T}{\partial t} 
68 

\nabla \cdot \kappa \nabla T = q\hackscore H 
69 

\end{equation} 
70 

We can easily apply the backward Euler scheme as before to end up with 
71 

\begin{equation} 
72 

\frac{\rho c\hackscore p}{h} T^{(n)} \nabla \cdot \kappa \nabla T^{(n)} = q\hackscore H + \frac{\rho c\hackscore p}{h} T^{(n1)} 
73 

\label{eqn:hdgenf2} 
74 

\end{equation} 
75 

which is very similar to \refEq{eqn:hdgenf} used to define the temperature update in the simple 1D case. 
76 

The difference is in the second order derivate term $\nabla \cdot \kappa \nabla T^{(n)}$. Under 
77 

the light of the more general case we need to revisit the \esc PDE form as 
78 

shown in \ref{eqn:commonform2D}. For the 2D case with variable PDE coefficients 
79 

the form needs to be read as 
80 

\begin{equation}\label{eqn:commonform2D 2} 
81 

\frac{\partial }{\partial x} A\hackscore{00}\frac{\partial u}{\partial x} 
82 

\frac{\partial }{\partial x} A\hackscore{01}\frac{\partial u}{\partial y} 
83 

\frac{\partial }{\partial y} A\hackscore{10}\frac{\partial u}{\partial x} 
84 

\frac{\partial }{\partial x} A\hackscore{11}\frac{\partial u}{\partial y} 
85 

+ Du = f 
86 

\end{equation} 
87 

So besides the settings $u=T^{(n)}$, $D = \frac{\rho c \hackscore{p}}{h}$ and 
88 

$f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n1)}$ as we have used before (see \refEq{ESCRIPT SET}) we need to set 
89 

\begin{equation}\label{eqn: kappa general} 
90 

A\hackscore{00}=A\hackscore{11}=\kappa; A\hackscore{01}=A\hackscore{10}=0 
91 

\end{equation} 
92 

The fundamental difference to the 1D case is that $A\hackscore{11}$ is not set to zero but $\kappa$ 
93 

which brings in the second dimension. Important to notice that the fact that the coefficients 
94 

of the PDE may depend on their location in the domain now does not influence the usage of the PDE form. This is very convenient as we can introduce spatial dependence to the PDE coefficients without modification to the way we call the PDE solver. 
95 


96 

A very convenient way to define the matrix $A$ is required in \refEq{eqn: kappa general} is using the 
97 

Kronecker $\delta$ symbol\footnote{see \url{http://en.wikipedia.org/wiki/Kronecker_delta}}. The 
98 

\esc function \verbkronecker returns this matrix; 
99 

\begin{equation} 
100 

\verbkronecker(model) = \left[ 
101 

\begin{array}{cc} 
102 

1 & 0 \\ 
103 

0 & 1 \\ 
104 

\end{array} 
105 

\right] 
106 

\end{equation} 
107 

As the argument \verbmodel represents a two dimensional domain the matrix is returned as $2 \times 2$ matrix 
108 

(In case of a threedimensional domain a $3 \times 3$ matrix is returned). The call 
109 

\begin{python} 
110 

mypde.setValue(A=kappa*kronecker(model),D=rhocp/h) 
111 

\end{python} 
112 

sets the PDE coefficients according to \refEq{eqn: kappa general}. 
113 


114 

Before we turn the question how we set $\kappa$ we need to check the boundary conditions. As 
115 

pointed out in \refEq{NEUMAN 2} makes certain assumptions on the boundary conditions. In our case 
116 

this assumptions translates to; 
117 

\begin{equation} 
118 

n\hackscore{0} \cdot \kappa \frac{\partial T^{(n)}}{\partial x}  n\hackscore{1} \cdot \kappa \frac{\partial T^{(n)}}{\partial y} = 0 
119 

\end{equation} 
120 

which sets the normal component of the heat flux $ \kappa \cdot (\frac{\partial T^{(n)}}{\partial x}, \frac{\partial T^{(n)}}{\partial y})$ to zero. This means that the regions is insulated which is what we want. 
121 

On the left and right face of the domain where we have $(n\hackscore{0},n\hackscore{1} ) = (\mp 1,0)$ 
122 

this means $\frac{\partial T^{(n)}}{\partial x}=0$ and on the top and bottom on the domain 
123 

where we have $(n\hackscore{0},n\hackscore{1} ) = (\pm 1,0)$ this is $\frac{\partial T^{(n)}}{\partial y}=0$. 
124 


125 

\section{Setting Variable PDE Coefficients} 
126 

Now we need to look into the problem how we define the material coefficients 
127 

$\kappa$ and $\rho c\hackscore p$ depending on there location in the domain. 
128 

We have used the technique we discuss here already when we set up the initial 
129 

temperature in the granite block example in \refSec{Sec:1DHDv00}. However, 
130 

the situation is more complicated here as we have to deal with a 
131 

curved interface between the two subdomain. 
132 


133 

Prior to setting up the PDE the interface between the two materials must be established. 
134 

The distance $s\ge 0$ between two points $[x,y]$ and $[x\hackscore{0},y\hackscore{0}]$ in Cartesian coordinates is defined as: 
135 

\begin{equation} 
136 

(xx\hackscore{0})^{2}+(yy\hackscore{0})^{2} = s^{2} 
137 

\end{equation} 
138 

If we define the point $[x\hackscore{0},y\hackscore{0}]$ as $ic$ which denotes the centre of the semicircle of our intrusion, then for all the points $[x,y]$ in our model we can calculate a distance to $ic$. 
139 

All the points that fall within the given radius $r$ of our intrusion will have a corresponding 
140 

value $s < r$ and all those belonging to the country rock will have a value $s > r$. By subtracting $r$ from both of these conditions we find $sr < 0$ for all intrusion points and $sr > 0$ 
141 

for all country rock points. 
142 

Defining these conditions within the script is quite simple and is done using the following command: 
143 
\begin{python} 
\begin{python} 
144 
bound = length(xic)r #where the boundary will be located 
bound = length(xic)r #where the boundary will be located 
145 
\end{python} 
\end{python} 
146 
This definition of the boundary can now be used with the \verb wherePositive() and \verb whereNegative() commands to help define the material constants and temperatures in our domain. By examining the general form we solved in the earlier tutorials, it is obvious that both \verb A and \verb D depend on the predefined variables. To set these variables accordingly and complete our PDE we use: 
This definition of the boundary can now be used with \verbwhereNegative command again to help define the material constants and temperatures in our domain. 
147 

If \verbkappai and \verbkappac are the 
148 

thermal conductivities for the intrusion material granite and for the surrounding sandstone we set; 
149 
\begin{python} 
\begin{python} 
150 
A = (kappai)*whereNegative(bound)+(kappac)*wherePositive(bound) 
x=Function(model).getX() 
151 
D = (rhocpi/h)*whereNegative(bound)+(rhocpc/h)*wherePositive(bound) 
bound = length(xic)r 
152 

kappa = kappai * whereNegative(bound) + kappac * (1whereNegative(bound)) 
153 
mypde.setValue(A=A*kronecker(model),D=D,d=eta,y=eta*Tc) 
mypde.setValue(A=kappa*kronecker(model)) 
154 
\end{python} 
\end{python} 
155 

Notice that we are using the sample points of the \verbFunction function space as expected for the 
156 

PDE coefficient \verbA\footnote{For the experience user: use \texttt{x=mypde.getFunctionSpace("A").getX()}.} 
157 

The corresponding statements are used to set $\rho c\hackscore p$. 
158 


159 
Our PDE has now been properly established. The initial conditions for temperature are set out in a similar matter: 
Our PDE has now been properly established. The initial conditions for temperature are set out in a similar matter: 
160 
\begin{python} 
\begin{python} 
161 
#defining the initial temperatures. 
#defining the initial temperatures. 
162 
T= Ti*whereNegative(bound)+Tc*wherePositive(bound) 
x=Solution(model).getX() 
163 

bound = length(xic)r 
164 

T= Ti*whereNegative(bound)+Tc*(1whereNegative(bound)) 
165 
\end{python} 
\end{python} 
166 
The iteration process now begins as before, but using our new conditions for \verb D as defined above. 
where \verbTi and \verbTc are the initial temperature 
167 

in the regions of the granite and surrounding sandstone, respectively. It is important to 
168 

notice that we have reset \verbx and \verbbound to refer to the appropriate 
169 

sample points of a PDE solution\footnote{For the experience user: use \texttt{x=mypde.getFunctionSpace("r").getX()}.}. 
170 


171 
\subsection{Contouring escript data} 
\begin{figure}[h] 
172 
It is possible to contour our solution using \modmpl . Unfortunately the \modmpl contouring function only accepts regularly gridded data. As our solution is not regularly gridded, it is necessary to interpolate our solution onto a regular grid. First we extract the model coordinates using \verb getX these are then transformed to a \verb numpy array known as a \verb tuple . Ths multidimensional array is then broken down into individual $x$ and $y$ arrays. This is a one step process using the function \verb toXYTuple . We also need to generate our regular grid which is done using the \modnumpy function \verb linspace . 
\centerline{\includegraphics[width=4.in]{figures/heatrefraction001.png}} 
173 

\centerline{\includegraphics[width=4.in]{figures/heatrefraction030.png}} 
174 

\centerline{\includegraphics[width=4.in]{figures/heatrefraction200.png}} 
175 

\caption{2D model: Total temperature distribution ($T$) at time step $1$, $20$ and $200$. Contour lines show temperature.} 
176 

\label{fig:twodhdans} 
177 

\end{figure} 
178 


179 

\section{Contouring \esc data using \modmpl.} 
180 

To complete our transition from a 1D to a 2D model we also need to modify the 
181 

plotting procedure. As before we use the \modmpl to do the plotting 
182 

in this case a contour plot. For 2D contour plots \modmpl expects that the 
183 

data are regularly gridded. We have no control on the location and ordering of the sample points 
184 

used to represent the solution. Therefore it is necessary to interpolate our solution onto a regular grid. 
185 


186 

In \refSec{sec: plot T} we have already learned how to extract the $x$ coordinates of sample points as 
187 

\verbnumpy array to hand the values to \modmpl. This can easily be extended to extract both the 
188 

$x$ and the $y$ coordinates; 
189 
\begin{python} 
\begin{python} 
190 
# rearrage mymesh to suit solution function space for contouring 
import numpy as np 
191 
oldspacecoords=model.getX() 
def toXYTuple(coords): 
192 
coords=Data(oldspacecoords, T.getFunctionSpace()) 
coords = np.array(coords.toListOfTuples()) #convert to Tuple 
193 
coordX, coordY = toXYTuple(coords) 
coordX = coords[:,0] #X components. 
194 

coordY = coords[:,1] #Y components. 
195 

return coordX,coordY 
196 

\end{python} 
197 

For convenience we have put this function into \file{clib.py} file so we can use this 
198 

function in other scripts more easily. 
199 


200 


201 

We now generate a regular $100 \times 100$ grid over the domain ($mx$ and $my$ 
202 

are the dimensions in $x$ and $y$ direction) which is done using the \modnumpy function \verblinspace . 
203 

\begin{python} 
204 

from clib import toXYTuple 
205 

# get sample points for temperature as for contouring 
206 

coordX, coordY = toXYTuple(T.getFunctionSpace().getX()) 
207 
# create regular grid 
# create regular grid 
208 
xi = np.linspace(0.0,mx,100) 
xi = np.linspace(0.0,mx,75) 
209 
yi = np.linspace(0.0,my,100) 
yi = np.linspace(0.0,my,75) 
210 
\end{python} 
\end{python} 
211 
The remainder of our contouring commands reside within a \verb while loop so that a new contour is generated for each time step. For each time step the solution must be regridded for \modmpl using the \verb griddata function. This function interprets an irregular grid and solution from \verb tempT , \verb xi and \verb yi . This is transformed to the new coordinates defined by \verb coordX and \verb coordY with an output \verb zi . It is now possible to use the \verb contourf function which generates colour filled contours. The colour gradient of our plots is set via the command \verb pl.matplotlib.pyplot.autumn() , other colours are listed on the \modmpl web page. Our results are then contoured, visually adjusted using the \modmpl functions and then saved to file. \verb pl.clf() clears the figure in readiness for the next time iteration. 
The values \verb[xi[k], yi[l]] are the grid points. 
212 


213 

The remainder of our contouring commands reside within a \verb while loop so that a new contour is generated for each time step. For each time step the solution must be regridded for \modmpl using the \verb griddata function. This function interprets a potentially irregularly located values \verb tempT at locations defined by \verb coordX and \verb coordY as values at the new coordinates of a rectangular grid defined by 
214 

\verb xi and \verb yi . The output is \verb zi . It is now possible to use the \verb contourf function which generates colour filled contours. The colour gradient of our plots is set via the command \verb pl.matplotlib.pyplot.autumn() , other colours are listed on the \modmpl web page\footnote{see \url{http://matplotlib.sourceforge.net/api/}}. Our results are then contoured, visually adjusted using the \modmpl functions and then saved to file. \verb pl.clf() clears the figure in readiness for the next time iteration. 
215 
\begin{python} 
\begin{python} 
216 
#grid the data. 
#grid the data. 
217 
zi = pl.matplotlib.mlab.griddata(coordX,coordY,tempT,xi,yi) 
zi = pl.matplotlib.mlab.griddata(coordX,coordY,tempT,xi,yi) 
227 
pl.savefig(os.path.join(save_path,"heatrefraction%03d.png") %i) 
pl.savefig(os.path.join(save_path,"heatrefraction%03d.png") %i) 
228 
pl.clf() 
pl.clf() 
229 
\end{python} 
\end{python} 
230 

The function \verbpl.contour is used to add labeled contour lines to the plot. 
231 

The results for selected time steps are shown in \reffig{fig:twodhdans}. 
232 



\begin{figure}[h!] 


\centerline{\includegraphics[width=4.in]{figures/heatrefraction050}} 


\caption{2D model: Total temperature distribution ($T$) at time $t=50$.} 


\label{fig:twodhdans} 


\end{figure} 

233 



\subsection{Advanced Visualization using VTK} 

234 


235 
\subsubsection{Parallel scripts (MPI)} 
\section{Advanced Visualization using VTK} 
236 
In some of the example files for this cookbook the plotting commands are a little different. 

237 
For example, 
\sslist{twodheatdiffvtk.py} 
238 

An alternative approach to \modmpl for visualization is the usage of a package which base on 
239 

visualization tool kit (VTK) library\footnote{see \url{http://www.vtk.org/}}. There is a variety 
240 

of package available. Here we will use the package \mayavi\footnote{see \url{http://code.enthought.com/projects/mayavi/}} as an example. 
241 


242 

\mayavi is an interactive, GUI driven tool which is 
243 

really designed to visualize large three dimensional data sets where \modmpl 
244 

is not suitable. But it is very useful when it comes to two dimensional problems. 
245 

The decision which tool is best is finally the user's decision. The main 
246 

difference between using \mayavi (and other VTK based tools) 
247 

or \modmpl is the fact that actually visualization is detached from the 
248 

calculation by writing the results to external files 
249 

and import them into \mayavi. In 3D where the best camera position for rendering a scene is not obvious 
250 

before the results are available. Therefore the user may need to try 
251 

different position before the best is found. Moreover, in many cases in 3D the interactive 
252 

visualization is the only way to really understand the results (e.g. using stereographic projection). 
253 


254 

To write the temperatures at each time step to data files in the VTK file format one 
255 

needs to insert a \verbsaveVTK call into the code; 
256 

\begin{python} 
257 

while t<=tend: 
258 

i+=1 #counter 
259 

t+=h #current time 
260 

mypde.setValue(Y=qH+T*rhocp/h) 
261 

T=mypde.getSolution() 
262 

saveVTK(os.path.join(save_path,"data.%03d.vtu"%i, T=T) 
263 

\end{python} 
264 

The data files, eg. \file{data.001.vtu}, contains all necessary information to 
265 

visualize the temperature and can directly processed by \mayavi. Notice that there is no 
266 

regridding required. It is recommended to use the file extension \file{.vtu} for files 
267 

created by \verbsaveVTK. 
268 


269 

\begin{figure}[h] 
270 

\centerline{\includegraphics[width=4.in]{figures/ScreeshotMayavi2n1}} 
271 

\caption{\mayavi start up Window.} 
272 

\label{fig:mayavi window} 
273 

\end{figure} 
274 


275 

\begin{figure}[h] 
276 

\centerline{\includegraphics[width=4.in]{figures/ScreeshotMayavi2n2}} 
277 

\caption{\mayavi data control window.} 
278 

\label{fig:mayavi window2} 
279 

\end{figure} 
280 

After you have run the script you will find the 
281 

result files \file{data.*.vtu} in the result directory \file{data/twodheatdiff}. Run the 
282 

command 
283 
\begin{python} 
\begin{python} 
284 
if getMPIRankWorld() == 0: 
>> mayavi2 d data.001.vtu m Surface & 

pl.savefig(os.path.join(save_path+"/totT","ttrodpyplot%03d.png")%i) 


pl.clf() 

285 
\end{python} 
\end{python} 
286 

from the result directory. \mayavi will start up a window similar to \reffig{fig:mayavi window}. 
287 

The right hand side shows the temperature at the first time step. To show 
288 

the results at other time steps you can click at the item \texttt{VTK XML file (data.001.vtu) (timeseries)} 
289 

at the top left hand side. This will bring up a new window similar to tye window shown in \reffig{fig:mayavi window2}. By clicking at the arrows in the top right corner you can move forwards and backwards in time. 
290 

You will also notice the text \textbf{T} next to the item \texttt{Point scalars name}. The 
291 

name \textbf{T} corresponds to the keyword argument name \texttt{T} we have used 
292 

in the \verbsaveVTK call. In this menu item you can select other results 
293 

you may have written to the output file for visualization. 
294 


295 
The additional \verb if statement is not necessary for normal desktop use. 
\textbf{For the advanced user}: Using the \modmpl to visualize spatially distributed data 
296 
It becomes important for scripts run on parallel computers. 
is not MPI compatible. However, the \verbsaveVTK function can be used with MPI. In fact, 
297 
Its purpose is to ensure that only one copy of the file is written. 
the function collects the values of the sample points spread across processor ranks into a single. 
298 
For more details on writing scripts for parallel computing please consult the \emph{user's guide}. 
For more details on writing scripts for parallel computing please consult the \emph{user's guide}. 