Cookbook modifications Review -> Not quite finished yet.

 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 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