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1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
11 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14 \begin{figure}[t]
15 \centerline{\includegraphics[width=4.in]{figures/twodheatdiff}}
16 \caption{2D model: granitic intrusion of sandstone country rock.}
17 \label{fig:twodhdmodel}
18 \end{figure}
19
20 \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 cross-section 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}
38 mx = 600*m #meters - model length
39 my = 600*m #meters - model width
40 ndx = 150 #mesh steps in x direction
41 ndy = 150 #mesh steps in y direction
42 r = 200*m #meters - radius of intrusion
43 ic = [300*m, 0] #coordinates of the centre of intrusion (meters)
44 qH=0.*J/(sec*m**3) #our heat source temperature is zero
45 \end{python}
46 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}
60 \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}
63 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^{(n-1)}
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 $f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n-1)}$ 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 \verb|kronecker| returns this matrix;
99 \begin{equation}
100 \verb|kronecker(model)| = \left[
101 \begin{array}{cc}
102 1 & 0 \\
103 0 & 1 \\
104 \end{array}
105 \right]
106 \end{equation}
107 As the argument \verb|model| represents a two dimensional domain the matrix is returned as $2 \times 2$ matrix
108 (In case of a three-dimensional 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 sub-domain.
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 (x-x\hackscore{0})^{2}+(y-y\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 semi-circle 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 $s-r < 0$ for all intrusion points and $s-r > 0$
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}
144 bound = length(x-ic)-r #where the boundary will be located
145 \end{python}
146 This definition of the boundary can now be used with \verb|whereNegative| command again to help define the material constants and temperatures in our domain.
147 If \verb|kappai| and \verb|kappac| are the
148 thermal conductivities for the intrusion material granite and for the surrounding sandstone we set;
149 \begin{python}
150 x=Function(model).getX()
151 bound = length(x-ic)-r
152 kappa = kappai * whereNegative(bound) + kappac * (1-whereNegative(bound))
153 mypde.setValue(A=kappa*kronecker(model))
154 \end{python}
155 Notice that we are using the sample points of the \verb|Function| function space as expected for the
156 PDE coefficient \verb|A|\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:
160 \begin{python}
161 #defining the initial temperatures.
162 x=Solution(model).getX()
163 bound = length(x-ic)-r
164 T= Ti*whereNegative(bound)+Tc*(1-whereNegative(bound))
165 \end{python}
166 where \verb|Ti| and \verb|Tc| 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 \verb|x| and \verb|bound| 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 \begin{figure}[h]
172 \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 \verb|numpy| 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}
190 import numpy as np
191 def toXYTuple(coords):
192 coords = np.array(coords.toListOfTuples()) #convert to Tuple
193 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 \verb|linspace| .
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
208 xi = np.linspace(0.0,mx,75)
209 yi = np.linspace(0.0,my,75)
210 \end{python}
211 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}
216 #grid the data.
217 zi = pl.matplotlib.mlab.griddata(coordX,coordY,tempT,xi,yi)
218 # contour the gridded data, plotting dots at the randomly spaced data points.
219 pl.matplotlib.pyplot.autumn()
220 pl.contourf(xi,yi,zi,10)
221 CS = pl.contour(xi,yi,zi,5,linewidths=0.5,colors='k')
222 pl.clabel(CS, inline=1, fontsize=8)
223 pl.axis([0,600,0,600])
224 pl.title("Heat diffusion from an intrusion.")
225 pl.xlabel("Horizontal Displacement (m)")
226 pl.ylabel("Depth (m)")
227 pl.savefig(os.path.join(save_path,"heatrefraction%03d.png") %i)
228 pl.clf()
229 \end{python}
230 The function \verb|pl.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
233
234
235 \section{Advanced Visualization using VTK}
236
237 \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 \verb|saveVTK| 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 \verb|saveVTK|.
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}
284 >> mayavi2 -d data.001.vtu -m Surface &
285 \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 \verb|saveVTK| call. In this menu item you can select other results
293 you may have written to the output file for visualization.
294
295 \textbf{For the advanced user}: Using the \modmpl to visualize spatially distributed data
296 is not MPI compatible. However, the \verb|saveVTK| function can be used with MPI. In fact,
297 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}.

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