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\section{Steadystate Heat Refraction} 
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\label{STEADYSTATE HEAT REFRACTION} 
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In this chapter we demonstrate how to handle more complex geometries. 
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Steadystate heat refraction will give us an opportunity to investigate some of 
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the richer features that the \esc package has to offer. One of these is \pycad . 
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The advantage of using \pycad is that it offers an easy method for developing 
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and manipulating complex domains. In conjunction with \gmsh we can generate 
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finite element meshes that conform to our domain's shape providing accurate 
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modelling of interfaces and boundaries. Another useful function of \pycad is 
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that we can tag specific areas of our domain with labels as we construct them. 
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These labels can then be used in \esc to define properties like material 
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constants and source locations. 
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We proceed in this chapter by first looking at a very simple geometry. Whilst a 
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simple rectangular domain is not very interesting the example is elaborated upon 
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later by introducing an internal curved interface. 
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\section{Example 4: Creating the Domain with \pycad} 
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\label{example4} 
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\sslist{example04a.py} 
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We modify the example in Chapter~\ref{CHAP HEAT 2a} in two ways: we look at the 
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steady state case with slightly modified boundary conditions and use a more 
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flexible tool to generate the geometry. Let us look at the geometry first. 
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We want to define a rectangular domain of width $5 km$ and depth $6 km$ below 
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the surface of the Earth. The domain is subject to a few conditions. The 
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temperature is known at the surface and the basement has a known heat flux. Each 
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side of the domain is insulated and the aim is to calculate the final 
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temperature distribution. 
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In \pycad there are a few primary constructors that build upon each other to 
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define domains and boundaries. The ones we use are: 
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\begin{python} 
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from esys.pycad import * 
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Point() #Create a point in space. 
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Line() #Creates a line from a number of points. 
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CurveLoop() #Creates a closed loop from a number of lines. 
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PlaneSurface() #Creates a surface based on a CurveLoop 
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\end{python} 
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So to construct our domain as shown in \reffig{fig:pycad rec}, we first need to 
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create the corner points. From the corner points we build the four edges of the 
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rectangle. The four edges then form a closed loop which defines our domain as a 
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surface. We start by inputting the variables we need to construct the model. 
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\begin{python} 
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width=5000.0*m #width of model 
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depth=6000.0*m #depth of model 
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\end{python} 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/pycadrec}} 
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\caption{Example 4: Rectangular Domain for \pycad} 
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\label{fig:pycad rec} 
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\end{figure} 
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The variables are then used to construct the four corners of our domain, which 
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from the origin has the dimensions of $5000$ meters width and $6000$ meters 
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depth. This is done with the \verbPoint() function which accepts x, y and z 
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coordinates. Our domain is in two dimensions so z should always be zero. 
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\begin{python} 
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# Overall Domain 
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p0=Point(0.0, 0.0, 0.0) 
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p1=Point(0.0, depth, 0.0) 
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p2=Point(width, depth, 0.0) 
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p3=Point(width, 0.0, 0.0) 
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\end{python} 
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Now lines are defined using our points. This forms a rectangle around our 
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domain: 
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\begin{python} 
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l01=Line(p0, p1) 
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l12=Line(p1, p2) 
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l23=Line(p2, p3) 
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l30=Line(p3, p0) 
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\end{python} 
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Note that lines have a direction. These lines form the basis for our domain 
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boundary, which is a closed loop. 
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\begin{python} 
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c=CurveLoop(l01, l12, l23, l30) 
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\end{python} 
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Be careful to define the curved loop in an \textbf{anticlockwise} manner 
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otherwise the meshing algorithm may fail. 
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Finally we can define the domain as 
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\begin{python} 
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rec = PlaneSurface(c) 
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\end{python} 
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At this point the introduction of the curved loop seems to be unnecessary but 
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this concept plays an important role if holes are introduced. 
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Now we are ready to hand over the domain \verbrec to a mesher which 
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subdivides the domain into triangles (or tetrahedra in 3D). In our case we use 
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\gmsh. We create an instance of the \verbDesign class which will handle the 
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interface to \gmsh: 
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\begin{python} 
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from esys.pycad.gmsh import Design 
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d=Design(dim=2, element_size=200*m) 
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\end{python} 
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The argument \verbdim defines the spatial dimension of the domain\footnote{If 
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\texttt{dim}=3 the rectangle would be interpreted as a surface in the three 
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dimensional space.}. The second argument \verbelement_size defines the element 
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size which is the maximum length of a triangle edge in the mesh. The element 
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size needs to be chosen with care in order to avoid very dense meshes. If the 
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mesh is too dense, the computational time will be long but if the mesh is too 
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sparse, the modelled result will be poor. In our case with an element size of 
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$200$m and a domain length of $6000$m we will end up with about $\frac{6000m}{200m}=30$ 
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triangles in each spatial direction. So we end up with about $30 \times 30 = 
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900$ triangles which is a size that can be handled easily. 
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The domain \verbrec can simply be added to the \verbDesign; 
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\begin{python} 
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d.addItem(rec) 
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\end{python} 
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We have the plan to set a heat flux on the bottom of the domain. One can use 
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the masking technique to do this but \pycad offers a more convenient technique 
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called tagging. With this technique items in the domain are named using the 
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\verbPropertySet class. We can then later use this name to set values 
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specifically for those sample points located on the named items. Here we name 
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the bottom face of the domain where we will set the heat influx 
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\footnote{In some applications, eg. 
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when dealing with influxes, 
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it is required to have the surface being meshed without the need of explicitly 
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name the surface. In this case the line forming the surface of the domain need to be 
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added to the \texttt{Design} 
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using \texttt{d.addItem(rec, l01, l12, l23, l30)}}: 
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\begin{python} 
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ps=PropertySet("linebottom",l12)) 
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d.addItem(ps) 
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\end{python} 
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Now we are ready to hand over the \verbDesign to \FINLEY: 
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\begin{python} 
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from esys.finley import MakeDomain 
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domain=MakeDomain(d) 
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\end{python} 
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The \verbdomain object can now be used in the same way like the return object 
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of the \verbRectangle object we have used previously to generate a mesh. It 
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is common practice to separate the mesh generation from the PDE solution. 
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The main reason for this is that mesh generation can be computationally very 
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expensive in particular in 3D. So it is more efficient to generate the mesh 
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once and write it to a file. The mesh can then be read in every time a new 
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simulation is run. \FINLEY supports this in the following 
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way\footnote{An alternative is using the \texttt{dump} and \texttt{load} 
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functions. They work with a binary format and tend to be much smaller.}: 
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\begin{python} 
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# write domain to a text file 
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domain.write("example04.fly") 
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\end{python} 
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and then for reading in another script: 
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\begin{python} 
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# read domain from text file 
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from esys.finley import ReadMesh 
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domain =ReadMesh("example04.fly") 
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\end{python} 
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Before we discuss how to solve the PDE for this problem, it is useful to 
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present two additional options of the \verbDesign class. 
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These allow the user to access the script which is used by \gmsh to generate 
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the mesh as well as the generated mesh itself. This is done by setting specific 
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names for these files: 
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\begin{python} 
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d.setScriptFileName("example04.geo") 
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d.setMeshFileName("example04.msh") 
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\end{python} 
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Conventionally the extension \texttt{geo} is used for the script file of the 
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\gmsh geometry and the extension \texttt{msh} for the mesh file. Normally these 
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files are deleted after usage. 
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Accessing these files can be helpful to debug the generation of more complex 
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geometries. The geometry and the mesh can be visualised from the command line 
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using 
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\begin{verbatim} 
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gmsh example04.geo # show geometry 
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gmsh example04.msh # show mesh 
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\end{verbatim} 
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The mesh is shown in \reffig{fig:pycad rec mesh}. 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/simplemesh}} 
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\caption{Example 4a: Mesh over rectangular domain, see \reffig{fig:pycad rec}} 
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\label{fig:pycad rec mesh} 
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\end{figure} 
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\clearpage 
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\section{The Steadystate Heat Equation} 
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\sslist{example04b.py, cblib} 
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A temperature equilibrium or steady state is reached when the temperature 
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distribution in the model does not change with time. To calculate the steady 
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state solution the time derivative term in \refEq{eqn:Tform nabla} needs to be 
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set to zero; 
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\begin{equation}\label{eqn:Tform nabla steady} 
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\nabla \cdot \kappa \nabla T = q_H 
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\end{equation} 
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This PDE is easier to solve than the PDE in \refEq{eqn:hdgenf2}, as no time 
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steps (iterations) are required. The \verbD term from \refEq{eqn:hdgenf2} is 
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simply dropped in this case. 
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\begin{python} 
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mypde=LinearPDE(domain) 
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mypde.setValue(A=kappa*kronecker(model), Y=qH) 
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\end{python} 
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The temperature at the top face of the domain is known as \verbTtop~($=20 C$). 
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In \refSec{Sec:1DHDv0} we have already discussed how this constraint is added 
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to the PDE: 
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\begin{python} 
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x=Solution(domain).getX() 
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mypde.setValue(q=whereZero(x[1]sup(x[1])),r=Ttop) 
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\end{python} 
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Notice that we use the \verbsup function to calculate the maximum of $y$ 
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coordinates of the relevant sample points. 
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In all cases so far we have assumed that the domain is insulated which 
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translates into a zero normal flux $n \cdot \kappa \nabla T$, see 
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\refEq{eq:hom flux}. In the modelling setup of this chapter we want to set 
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the normal heat flux at the bottom to \verbqin while still maintaining 
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insulation at the left and right face. Mathematically we can express this as 
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\begin{equation} 
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n \cdot \kappa \nabla T = q_{S} 
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\label{eq:inhom flux} 
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\end{equation} 
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where $q_{S}$ is a function of its location on the boundary. Its value 
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becomes zero for locations on the left or right face of the domain while it has 
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the value \verbqin at the bottom face. 
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Notice that the value of $q_{S}$ at the top face is not relevant as we 
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prescribe the temperature here. 
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We could define $q_{S}$ by using the masking techniques demonstrated 
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earlier. The tagging mechanism provides an alternative and in many cases more 
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convenient way of defining piecewise constant functions such as 
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$q_{S}$. Recall now that the bottom face was denoted with the name 
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\verblinebottom when we defined the domain. 
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We can use this now to create $q_{S}$; 
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\begin{python} 
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qS=Scalar(0,FunctionOnBoundary(domain)) 
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qS.setTaggedValue("linebottom",qin) 
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\end{python} 
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In the first line \verbqS is defined as a scalar value over the sample points 
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on the boundary of the domain. It is initialised to zero for all sample points. 
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In the second statement the values for those sample points which are located on 
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the line marked by \verblinebottom are set to \verbqin. 
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The Neumann boundary condition assumed by \esc has the form 
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\begin{equation}\label{NEUMAN 2b} 
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n\cdot A \cdot\nabla u = y 
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\end{equation} 
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In comparison to the version in \refEq{NEUMAN 2} we have used so far the right 
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hand side is now the new PDE coefficient $y$. As we have not specified $y$ in 
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our previous examples, \esc has assumed the value zero for $y$. A comparison of 
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\refEq{NEUMAN 2b} and \refEq{eq:inhom flux} reveals that one needs to choose 
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$y=q_{S}$; 
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\begin{python} 
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qS=Scalar(0,FunctionOnBoundary(domain)) 
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qS.setTaggedValue("linebottom",qin) 
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mypde.setValue(y=qS) 
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\end{python} 
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To plot the results we use the \modmpl library as shown in 
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\refSec{Sec:2DHD plot}. For convenience the interpolation of the temperature to 
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a rectangular grid for contour plotting is made available via the 
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\verbtoRegGrid function in the \verbcblib module. Your result should look 
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similar to \reffig{fig:steady state heat}. 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/simpleheat}} 
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\caption{Example 4b: Result of simple steady state heat problem} 
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\label{fig:steady state heat} 
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\end{figure} 
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\clearpage 
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