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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
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12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14 \section{Steady-state Heat Refraction}
15 \label{STEADY-STATE HEAT REFRACTION}
16
17 In this chapter we demonstrate how to handle more complex geometries.
18
19 Steady-state heat refraction will give us an opportunity to investigate some of the richer features that the \esc package has to offer. One of these is \pycad . The advantage of using \pycad is that it offers an easy method for developing and manipulating complex domains. In conjunction with \gmsh we can generate finite element meshes that conform to our domain's shape providing accurate modelling of interfaces and boundaries. Another useful function of \pycad is that we can tag specific areas of our domain with labels as we construct them. These labels can then be used in \esc to define properties like material constants and source locations.
20
21 We proceed in this chapter by first looking at a very simple geometry. In fact, we solve
22 the steady-state heat equation over a rectangular domain. This is not a very interesting problem but then we extend our script by introducing an internal, curved interface.
23
24 \begin{figure}[ht]
25 \centerline{\includegraphics[width=4.in]{figures/pycadrec}}
26 \caption{Example 4: Rectangular Domain for \pycad.}
27 \label{fig:pycad rec}
28 \end{figure}
29
30 \section{Example 4: Creating the domain with \pycad}
31 \sslist{example04a.py}
32
33 We modify the example in Chapter~\ref{CHAP HEAT 2a} in two ways. Firstly, we look at the steady state
34 case with slightly modified boundary conditions and then we use a more flexible tool
35 to generate the geometry. Lets look at the geometry first.
36
37 We want to define a rectangular domain of width $5 km$ and depth $6 km$ below the surface of the Earth. The domain is subject to a few conditions. The temperature is known at the surface and the basement has a known heat flux. Each side of the domain is insulated and the aim is to calculate the final temperature distribution.
38
39 In \pycad there are a few primary constructors that build upon each other to define domains and boundaries;
40 the ones we use are:
41 \begin{python}
42 from esys.pycad import *
43 Point() #Create a point in space.
44 Line() #Creates a line from a number of points.
45 CurveLoop() #Creates a closed loop from a number of lines.
46 PlaneSurface() #Creates a surface based on a CurveLoop
47 \end{python}
48 So to construct our domain as shown in \reffig{fig:pycad rec}, we first need to create
49 the corner points. From the corner points we build the four edges of the rectangle. The four edges
50 then form a closed loop which defines our domain as a surface.
51 We start by inputting the variables we need to construct the model.
52 \begin{python}
53 width=5000.0*m #width of model
54 depth=-6000.0*m #depth of model
55 \end{python}
56 The variables are then used to construct the four corners of our domain, which from the origin has the dimensions of 5000 meters width and -6000 meters depth. This is done with the \verb Point() function which accepts x, y and z coordinates. Our domain is in two dimensions so z should always be zero.
57 \begin{python}
58 # Overall Domain
59 p0=Point(0.0, 0.0, 0.0)
60 p1=Point(0.0, depth, 0.0)
61 p2=Point(width, depth, 0.0)
62 p3=Point(width, 0.0, 0.0)
63 \end{python}
64 Now lines are defined using our points. This forms a rectangle around our domain;
65 \begin{python}
66 l01=Line(p0, p1)
67 l12=Line(p1, p2)
68 l23=Line(p2, p3)
69 l30=Line(p3, p0)
70 \end{python}
71 Notice that lines have a direction. These lines form the basis for our domain boundary, which is a closed loop.
72 \begin{python}
73 c=CurveLoop(l01, l12, l23, l30)
74 \end{python}
75 Be careful to define the curved loop in an \textbf{anti-clockwise} manner otherwise the meshing algorithm may fail.
76 Finally we can define the domain as
77 \begin{python}
78 rec = PlaneSurface(c)
79 \end{python}
80 At this point the introduction of the curved loop seems to be unnecessary but this concept plays an important role
81 if holes are introduced.
82
83 Now we are ready to handover the domain \verb|rec| to a mesher which subdivides the domain into triangles (or tetrahedron in 3D). In our case we use \gmsh. We create
84 an instance of the \verb|Design| class which will handle the interface to \gmsh:
85 \begin{python}
86 from esys.pycad.gmsh import Design
87 d=Design(dim=2, element_size=200*m)
88 \end{python}
89 The argument \verb|dim| defines the spatial dimension of the domain\footnote{If \texttt{dim}=3 the rectangle would be interpreted as a surface in the three dimensional space.}. The second argument \verb|element_size| defines element size which is the maximum length of a triangle edge in the mesh. The element size needs to be chosen with care in order to avoid very large meshes if you don't want to. In our case with an element size of $200$m
90 and a domain length of $6000$m we will end up with about $\frac{6000m}{200m}=30$ triangles in each spatial direction. So we end up with about $30 \times 30 = 900$ triangles which is a size that can be handled easily.
91 We can easily add our domain \verb|rec| to the \verb|Design|;
92 \begin{python}
93 d.addItem(rec)
94 \end{python}
95 We have the plan to set a heat flux on the bottom of the domain. One can use the masking technique to do this
96 but \pycad offers a more convenient technique called tagging. With this technique items in the domain are
97 named using the \verb|PropertySet| class. We can then later use this name to set values secifically for
98 those sample points located on the named items. Here we name the bottom face of the
99 domain where we will set the heat influx:
100 \begin{python}
101 ps=PropertySet("linebottom",l12))
102 d.addItem(ps)
103 \end{python}
104 Now we are ready to hand over the \verb|Design| to \FINLEY:
105 \begin{python}
106 from esys.finley import MakeDomain
107 domain=MakeDomain(d)
108 \end{python}
109 The \verb|domain| object can now be used in the same way like the return object of the \verb|Rectangle|
110 object we have used previously to generate a mesh. It is common practice to separate the
111 mesh generation from the PDE solution. The main reason for this is that mesh generation can be computationally very expensive in particular in 3D. So it is more efficient to generate the mesh once and write it to a file. The mesh
112 can then be read in every time a new simulation is run. \FINLEY supports this in the following
113 way~\footnote{An alternative are the \texttt{dump} and \texttt{load} functions. They using a binary format and tend to be much smaller.}
114 \begin{python}
115 # write domain to an text file
116 domain.write("example04.fly")
117 \end{python}
118 and then for reading in another script;
119 \begin{python}
120 # read domain from text file
121 from esys.finley import ReadMesh
122 domain =ReadMesh("example04.fly")
123 \end{python}
124
125 \begin{figure}[ht]
126 \centerline{\includegraphics[width=4.in]{figures/simplemesh}}
127 \caption{Example 4a: Mesh over rectangular domain, see \reffig{fig:pycad rec}.}
128 \label{fig:pycad rec mesh}
129 \end{figure}
130
131 Before we discuss how we solve the PDE for the
132 problem it is useful to present two additional options of the \verb|Design|.
133 They allow the user accessing the script which is used by \gmsh to generate the mesh as well as
134 the mesh as it has been generated by \gmsh. This is done by setting specific names for these files:
135 \begin{python}
136 d.setScriptFileName("example04.geo")
137 d.setMeshFileName("example04.msh")
138 \end{python}
139 Usually the extension \texttt{geo} is used for the script file of the \gmsh geometry and
140 the extension \texttt{msh} for the mesh file. Normally these files are deleted after usage.
141 Accessing these file can be helpful to debug the generation of more complex geometries. The geometry and the mesh can be visualised from the command line using
142 \begin{verbatim}
143 gmsh example04.geo # show geometry
144 gmsh example04.msh # show mesh
145 \end{verbatim}
146 The mesh is shown in \reffig{fig:pycad rec mesh}.
147 \begin{figure}[ht]
148 \centerline{\includegraphics[width=4.in]{figures/simpleheat}}
149 \caption{Example 4b: Result of simple steady state heat problem.}
150 \label{fig:steady state heat}
151 \end{figure}
152
153
154 \section{The Steady-state Heat Equation}
155 \sslist{example04b.py, cblib}
156
157 Steady state is reached in the temperature when it is not changing in time. So to calculate the
158 the steady state we set the time derivative term in \refEq{eqn:Tform nabla} to zero;
159 \begin{equation}\label{eqn:Tform nabla steady}
160 -\nabla \cdot \kappa \nabla T = q\hackscore H
161 \end{equation}
162 This PDE is easier to solve, than the PDE in \refEq{eqn:hdgenf2} in each time step. We can just drop
163 the \verb|D| term;
164 \begin{python}
165 mypde=LinearPDE(domain)
166 mypde.setValue(A=kappa*kronecker(model), Y=qH)
167 \end{python}
168 The temperature at the top face of the domain is known as \verb|Ttop| (=$20 C$). In \refSec{Sec:1DHDv0} we have
169 already discussed how this constraint is added to the PDE:
170 \begin{python}
171 x=Solution(domain).getX()
172 mypde.setValue(q=whereZero(x[1]-sup(x[1])),r=Ttop)
173 \end{python}
174 Notice that we use the \verb|sup| function to calculate the maximum of $y$ coordinates of the relevant sample points.
175
176 In all cases so far we have assumed that the domain is insulated which translates
177 into a zero normal flux $-n \cdot \kappa \nabla T$, see \refEq{eq:hom flux}. In the modeling
178 set-up of this chapter we want to set the normal heat flux at the bottom to \verb|qin| while we still
179 maintain insulation at the left and right face. Mathematically we can express this as the format
180 \begin{equation}
181 -n \cdot \kappa \nabla T = q\hackscore{S}
182 \label{eq:inhom flux}
183 \end{equation}
184 where $q\hackscore{S}$ is a function of its location on the boundary. Its value becomes zero
185 for locations on the left or right face of the domain while it has the value \verb|qin| at the bottom face.
186 Notice that the value of $q\hackscore{S}$ at the top face is not relevant as we prescribe the temperature here.
187 We could define $q\hackscore{S}$ by using the masking techniques demonstrated earlier. The tagging mechanism provides an alternative and in many cases more convenient way of defining piecewise
188 constant functions such as $q\hackscore{S}$. You need to remember now that we
189 marked the bottom face with the name \verb|linebottom| when we defined the domain.
190 We can use this now to create $q\hackscore{S}$;
191 \begin{python}
192 qS=Scalar(0,FunctionOnBoundary(domain))
193 qS.setTaggedValue("linebottom",qin)
194 \end{python}
195 In the first line \verb|qS| is defined as a scalar value over the sample points on the boundary of the domain. It is
196 initialized to zero for all sample points. In the second statement the values for those sample points which
197 on the line marked by \verb|linebottom| are set to \verb|qin|.
198
199 The Neuman boundary condition assumed by \esc has in fact the form
200 \begin{equation}\label{NEUMAN 2b}
201 n\cdot A \cdot\nabla u = y
202 \end{equation}
203 In comparison to the version in \refEq{NEUMAN 2} we have used so far the right hand side is now
204 the new PDE coefficient $y$. As we have not specific $y$ in our previous examples \esc has assumed
205 the value zero for $y$. A comparison of \refEq{NEUMAN 2b} and \refEq{eq:inhom flux} reveals that one need to
206 choose $y=-q\hackscore{S}$;
207 \begin{python}
208 qS=Scalar(0,FunctionOnBoundary(domain))
209 qS.setTaggedValue("linebottom",qin)
210 mypde.setValue(y=-qS)
211 \end{python}
212 To plot the results we are using the \modmpl library as shown \refSec{Sec:2DHD plot}. For convenience
213 the interpolation of the temperature to a rectangular grid for contour plotting is made available
214 via the \verb|toRegGrid| function in the \verb|cblib| module. Your result should look similar to
215 \reffig{fig:steady state heat}.
216
217 \begin{figure}[ht]
218 \centerline{\includegraphics[width=4.in]{figures/anticlineheatrefraction}}
219 \caption{Example 5a: Heat refraction model with point and line labels.}
220 \label{fig:anticlinehrmodel}
221 \end{figure}
222
223 \begin{figure}[ht]
224 \centerline{\includegraphics[width=4.in]{figures/heatrefraction}}
225 \caption{Example 5a: Temperature Distribution in the Heat Refraction Model.}
226 \label{fig:anticlinetemp}
227 \end{figure}
228
229 \section{Example 5: A Heat Refraction model}
230 \sslist{example05a.py and cblib.py}
231
232 Our heat refraction model will be a large anticlinal structure that is experiencing a constant temperature at the surface and a steady heat flux at it's base. Our aim is to show that the temperature flux across the surface is not linear from bottom to top but is in fact warped by the structure of the model and is heavily dependent upon the material properties and shape.
233
234
235 We modify the example of the previous section by subdividing the block into two parts. The curve
236 separating the two blocks is given as a spline, see \reffig{fig:anticlinehrmodel}. The data points
237 used to define the curve may be measurement but for simplicity we assume here that there coordinates are
238 known in analytic form.
239
240 There are two modes available in our example. When \verb modal=1 this indicates to the script that we wish to model an anticline. Otherwise when \verb modal=-1 this will model a syncline. The modal operator simply changes the orientation of the boundary function so that it is either upwards or downwards curving. A \verb save_path has also been defined so that we can easily separate our data from other examples and our scripts.
241
242 It is now possible to start defining our domain and boundaries.
243
244 The curve defining our clinal structure is located approximately in the middle of the domain and has a sinusoidal shape. We define the curve by generating points at discrete intervals; $51$ in this case, and then create a smooth curve through the points using the \verb Spline() function.
245 \begin{python}
246 # Material Boundary
247 x=[ Point(i*dsp\
248 ,-dep_sp+modal*orit*h_sp*cos(pi*i*dsp/dep_sp+pi))\
249 for i in range(0,sspl)\
250 ]
251 mysp = Spline(*tuple(x)) #*tuple() forces x to become a tuple
252 \end{python}
253 The start and end points of the spline can be returned to help define the material boundaries.
254 \begin{python}
255 x1=mysp.getStartPoint()
256 x2=mysp.getEndPoint()
257 \end{python}
258 The top block or material above the clinal/spline boundary is defined in a \textbf{anti-clockwise} manner by creating lines and then a closed loop. As we will be meshing the sub-domain we also need to assign it a planar surface.
259 \begin{python}
260 # TOP BLOCK
261 tbl1=Line(p0,x1)
262 tbl2=mysp
263 tbl3=Line(x2,p3)
264 l30=Line(p3, p0)
265 tblockloop = CurveLoop(tbl1,tbl2,tbl3,l30)
266 tblock = PlaneSurface(tblockloop)
267 \end{python}
268 We must repeat the above for every other sub-domain. In this example there is only one other, the bottom block. The process is fairly similar to the top block but with a few differences. The spline points must be reversed by setting the spline as negative.
269 \begin{python}
270 bbl4=-mysp
271 \end{python}
272 This reverse spline option unfortunately does not work for the getLoopCoords command, however, the \modmpl polygon tool will accept clock-wise oriented points so we can define a new curve.
273 \begin{python}
274 #clockwise check
275 bblockloop=CurveLoop(mysp,Line(x2,p2),Line(p2,p1),Line(p1,x1))
276 \end{python}
277 The last few steps in creating the model are: take the previously defined domain and sub-domain points and generate a mesh that can be imported into \esc.
278 To initialise the mesh it first needs some design parameters. In this case we have 2 dimensions \verb dim and a specified number of finite elements that need to be applied to the domain \verb element_size . It then becomes a simple task of adding the sub-domains and flux boundaries to the design. Each element of our model can be given an identifier which makes it easier to define the sub-domain properties in the solution script. This is done using the \verb PropertySet() function. The geometry and mesh are then saved so the \esc domain can be created.
279 \begin{python}
280 # Create a Design which can make the mesh
281 d=Design(dim=2, element_size=200)
282 # Add the sub-domains and flux boundaries.
283 d.addItems(PropertySet("top",tblock),PropertySet("bottom",bblock),\
284 PropertySet("linebottom",l12))
285 # Create the geometry, mesh and \esc domain
286 d.setScriptFileName(os.path.join(save_path,"example05.geo"))
287 d.setMeshFileName(os.path.join(save_path,"example05.msh"))
288 domain=MakeDomain(d,optimizeLabeling=True)
289 \end{python}
290 The creation of our domain and its mesh is complete.
291
292 With the mesh imported it is now possible to use our tagging property to set up our PDE coefficients. In this case $\kappa$ is set via the \verb setTaggedValue() function which takes two arguments, the name of the tagged points and the value to assign to them.
293 \begin{python}
294 # set up kappa (thermal conductivity across domain) using tags
295 kappa=Scalar(0,Function(domain))
296 kappa.setTaggedValue("top",2.0*W/m/K)
297 kappa.setTaggedValue("bottom",4.0*W/m/K)
298 \end{python}
299 No further changes are required to the PDE solution step, see \reffig{fig:anticlinetemp} for the result.
300
301 \begin{figure}
302 \centering
303 \subfigure[Temperature Depth Profile]{\includegraphics[width=3in]{figures/heatrefractiontdp.png}\label{fig:tdp}}
304 \subfigure[Temperature Gradient Depth Profile]{\includegraphics[width=3in]{figures/heatrefractiontgdp.png}\label{fig:tgdp}}
305 \subfigure[Thermal Conductivity Profile]{\includegraphics[width=3in]{figures/heatrefractiontcdp.png}\label{fig:tcdp}}
306 \subfigure[Heat Flow Depth Profile]{\includegraphics[width=3in]{figures/heatrefractionhf.png}\label{fig:hf}}
307 \caption{Example 5b: Depth profiles down centre of model.}
308 \label{figs:dps}
309 \end{figure}
310
311 \section{Line profiles of 2D data}
312 \sslist{example05b.py and cblib.py}
313
314 We want to investigate the profile of the data of the last example.
315 We are particularly interested in the depth profile of the heat flux which is
316 the second component of $-\kappa \nabla T$. We extend the script developed in the
317 previous section to show how for instance vertical profile can be plotted.
318
319 The first important information is that \esc assumes that $-\kappa \nabla T$ is not smooth and
320 will in fact represent the values at numerical interpolation points. This assumption is reasonable as
321 the flux is the product of the piecewise constant function $\kappa$ and
322 the gradient of the temperature $T$ which has a kink across the rock interface.
323 Before plotting this function we need to smooth it using the
324 \verb|Projector()| class;
325 \begin{python}
326 from esys.escript.pdetools import Projector
327 proj=Projector(domain)
328 qu=proj(-kappa*grad(T))
329 \end{python}
330 The \verb|proj| object provides a mechanism to distribute values given at the numerical interpolation points
331 - in this case the heat flux - to the nodes of the FEM mesh. \verb|qu| has the same attached function space
332 like the temperature \verb|T|. The smoothed flux is interpolated
333 to a regular $200\times 200$ grid;
334 \begin{python}
335 xiq,yiq,ziq = toRegGrid(qu[1],200,200)
336 \end{python}
337 using the \verb|toRegGrid| function from the cookbook library which we are using for the contour plot.
338 At return \verb|ziq[j,i]| is the value of vertical heat flux at point
339 (\verb|xiq[i]|,\verb|yiq[j]|). We can easily create deep profiles now by
340 plotting slices \verb|ziq[:,i]| over \verb|yiq|. The following script
341 creates a deep profile at $x_{0}=\frac{width}{2}$;
342 \begin{python}
343 cut=int(len(xiq)/2)
344 pl.plot(ziq[:,cut]*1000.,yiq)
345 pl.title("Vertical Heat Flow Depth Profile")
346 pl.xlabel("Heat Flow (mW/m^2)")
347 pl.ylabel("Depth (m)")
348 pl.savefig(os.path.join(save_path,"hf.png"))
349 \end{python}
350 This process can be repeated for various variations of the solution. In this case we have temperature, temperature gradient, thermal conductivity and heat flow \reffig{figs:dps}.
351
352 \begin{figure}[ht]
353 \centerline{\includegraphics[width=5.in]{figures/heatrefractionflux}}
354 \caption{Example 5c: Heat refraction model with gradient indicated by vectors.}
355 \label{fig:hr001qumodel}
356 \end{figure}
357
358 \section{Arrow plots in \mpl}
359 \sslist{example05c.py and cblib.py}
360 We would like to visualise the distribution of the flux $-\kappa \nabla T$ over the domain
361 and produce a plot like shown in \reffig{fig:hr001qumodel}.
362 The plot puts together three components. A contour plot of the temperature,
363 a colored representation of the two sub-domains where colour represents the thermal conductivity
364 in the particular region and finally the arrows representing the direction of flux.
365
366 Contours have already been discussed in \refSec{Sec:2DHD plot}. To show sub-domains,
367 we need to go back to \pycad data to get the points used to describe the boundary of the
368 sub-domains. We have created the \verb|CurveLoop| class object
369 \verb|tblockloop| to define the boundary of the upper sub-domain.
370 We use the \verb|getPolygon()| method of \verb|CurveLoop| to get
371 access to the \verb|Point|s used top define the boundary. The statement
372 \begin{python}
373 [ p.getCoordinates() for p in tblockloop.getPolygon() ])
374 \end{python}
375 creates a list of the node coordinates of all the points in question. In order
376 to simplify the selection of the $x$ and $y$ coordinates the list is converted
377 into \modnumpy array. To add the area colored in brown to the plot we use;
378 \begin{python}
379 import pylab as pl
380 import numarray as np
381 tpg=np.array([p.getCoordinates() for p in tblockloop.getPolygon() ])
382 pl.fill(tpg[:,0],tpg[:,1],'brown',label='2 W/m/k',zorder=-1000)
383 \end{python}
384 We can apply the same code to \verb|bblockloop| to a red area for this sub-domain to the block.
385
386 To plot vectors representing the flux orientation we use the
387 \verb|quiver| function in \pylab. The function places vectors at locations in the domain.
388 For instance one can plot vectors at the locations of the sample points used by \esc
389 to represent the flux \verb|-kappa*grad(T)|. As a vector is plotted at each sample point one typically ends
390 up with two many vectors. So one needs to select a subset of points:
391 First we create a coarse grid of points on a rectangular mesh, e.g. $20 \times 20$ points. Here we choose a grid of points which are located at the center of a \verb|nx| $\times$ \verb|ny| grid;
392 \begin{python}
393 dx = width/nx # x spacing
394 dy = depth/ny # y spacing
395 grid = [ ] # the grid points
396 for j in xrange(0,ny-1):
397 for i in xrange(0,nx-1):
398 grid.append([dx/2+dx*i,dy/2+dy*j])
399 \end{python}
400 With the \verb|Locator| \esc provides a mechanism to identify sample points that are closest
401 to the the grid points we have selected and to get the data at these data points;
402 \begin{python}
403 from esys.escript.pdetools import Locator
404 flux=-kappa*grad(T)
405 fluxLoc = Locator(flux.getFunctionSpace(),grid)
406 subflux= fluxLoc(flux)
407 \end{python}
408 \verb|subflux| gives now a list of flux component at certain sample points. To get the
409 list of the sample point coordinates one can use the \verb|getX()| method of the
410 \verb|Locator|;
411 \begin{python}
412 subfluxloc = fluxLoc.getX()
413 \end{python}
414 To simplify the selection of $x$ and $y$ components it is convenient
415 to transform \verb|subflux| and \verb|subfluxloc| to \numpy arrays
416 \verb|xflux|, \verb|flux|.
417 This function is implemented in the \verb|subsample|
418 in the \file{clib.py} file so we can use it in other examples. One can easily use this function
419 to create a vector plot of the flux;
420 \begin{python}
421 from cblib import subsample
422 xflux, flux=subsample(-kappa*grad(T), nx=20, ny=20)
423 pl.quiver(xflux[:,0],xflux[:,1],flux[:,0],flux[:,1], angles='xy',color="white")
424 \end{python}
425 We add title and labels;
426 \begin{python}
427 pl.title("Heat Refraction across a clinal structure.")
428 pl.xlabel("Horizontal Displacement (m)")
429 pl.ylabel("Depth (m)")
430 pl.title("Heat Refraction across a clinal structure \n with gradient quivers.")
431 pl.savefig(os.path.join(saved_path,"flux.png"))
432 \end{python}
433 to get the desired result, see \reffig{fig:hr001qumodel}.
434
435 \begin{figure}[ht]
436 \centerline{\includegraphics[width=4.in]{figures/heatrefraction2flux}}
437 \caption{Example 6: Heat refraction model with three blocks and heat flux.}
438 \label{fig:hr002qumodel}
439 \end{figure}
440
441 \section{Example 6:Fault and Overburden Model}
442 \sslist{example06.py and cblib.py}
443 A slightly more complicated model can be found in the examples \textit{heatrefraction2_solver.py} where three blocks are used within the model, see~\reffig{fig:hr002qumodel}. It is left to the reader to work through this example.
444
445

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