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15 \section{Steady-state Heat Refraction}
18 In this chapter we demonstrate how to handle more complex geometries.
20 Steady-state heat refraction will give us an opportunity to investigate some of
21 the richer features that the \esc package has to offer. One of these is \pycad .
22 The advantage of using \pycad is that it offers an easy method for developing
23 and manipulating complex domains. In conjunction with \gmsh we can generate
24 finite element meshes that conform to our domain's shape providing accurate
25 modelling of interfaces and boundaries. Another useful function of \pycad is
26 that we can tag specific areas of our domain with labels as we construct them.
27 These labels can then be used in \esc to define properties like material
28 constants and source locations.
30 We proceed in this chapter by first looking at a very simple geometry. Whilst a
31 simple rectangular domain is not very interesting the example is elaborated upon
32 later by introducing an internal curved interface.
34 \section{Example 4: Creating the Domain with \pycad}
35 \label{example4}
36 \sslist{example04a.py}
37 We modify the example in Chapter~\ref{CHAP HEAT 2a} in two ways: we look at the
38 steady state case with slightly modified boundary conditions and use a more
39 flexible tool to generate the geometry. Let us look at the geometry first.
41 We want to define a rectangular domain of width $5 km$ and depth $6 km$ below
42 the surface of the Earth. The domain is subject to a few conditions. The
43 temperature is known at the surface and the basement has a known heat flux. Each
44 side of the domain is insulated and the aim is to calculate the final
45 temperature distribution.
47 In \pycad there are a few primary constructors that build upon each other to
48 define domains and boundaries. The ones we use are:
49 \begin{python}
50 from esys.pycad import *
51 Point() #Create a point in space.
52 Line() #Creates a line from a number of points.
53 CurveLoop() #Creates a closed loop from a number of lines.
54 PlaneSurface() #Creates a surface based on a CurveLoop
55 \end{python}
56 So to construct our domain as shown in \reffig{fig:pycad rec}, we first need to
57 create the corner points. From the corner points we build the four edges of the
58 rectangle. The four edges then form a closed loop which defines our domain as a
59 surface. We start by inputting the variables we need to construct the model.
60 \begin{python}
61 width=5000.0*m #width of model
62 depth=-6000.0*m #depth of model
63 \end{python}
65 \begin{figure}[ht]
66 \centerline{\includegraphics[width=4.in]{figures/pycadrec}}
67 \caption{Example 4: Rectangular Domain for \pycad}
68 \label{fig:pycad rec}
69 \end{figure}
71 The variables are then used to construct the four corners of our domain, which
72 from the origin has the dimensions of $5000$ meters width and $-6000$ meters
73 depth. This is done with the \verb|Point()| function which accepts x, y and z
74 coordinates. Our domain is in two dimensions so z should always be zero.
75 \begin{python}
76 # Overall Domain
77 p0=Point(0.0, 0.0, 0.0)
78 p1=Point(0.0, depth, 0.0)
79 p2=Point(width, depth, 0.0)
80 p3=Point(width, 0.0, 0.0)
81 \end{python}
82 Now lines are defined using our points. This forms a rectangle around our
83 domain:
84 \begin{python}
85 l01=Line(p0, p1)
86 l12=Line(p1, p2)
87 l23=Line(p2, p3)
88 l30=Line(p3, p0)
89 \end{python}
90 Note that lines have a direction. These lines form the basis for our domain
91 boundary, which is a closed loop.
92 \begin{python}
93 c=CurveLoop(l01, l12, l23, l30)
94 \end{python}
95 Be careful to define the curved loop in an \textbf{anti-clockwise} manner
96 otherwise the meshing algorithm may fail.
97 Finally we can define the domain as
98 \begin{python}
99 rec = PlaneSurface(c)
100 \end{python}
101 At this point the introduction of the curved loop seems to be unnecessary but
102 this concept plays an important role if holes are introduced.
104 Now we are ready to hand over the domain \verb|rec| to a mesher which
105 subdivides the domain into triangles (or tetrahedra in 3D). In our case we use
106 \gmsh. We create an instance of the \verb|Design| class which will handle the
107 interface to \gmsh:
108 \begin{python}
109 from esys.pycad.gmsh import Design
110 d=Design(dim=2, element_size=200*m)
111 \end{python}
112 The argument \verb|dim| defines the spatial dimension of the domain\footnote{If
113 \texttt{dim}=3 the rectangle would be interpreted as a surface in the three
114 dimensional space.}. The second argument \verb|element_size| defines the element
115 size which is the maximum length of a triangle edge in the mesh. The element
116 size needs to be chosen with care in order to avoid very dense meshes. If the
117 mesh is too dense, the computational time will be long but if the mesh is too
118 sparse, the modelled result will be poor. In our case with an element size of
119 $200$m and a domain length of $6000$m we will end up with about $\frac{6000m}{200m}=30$
120 triangles in each spatial direction. So we end up with about $30 \times 30 =
121 900$ triangles which is a size that can be handled easily.
122 The domain \verb|rec| can simply be added to the \verb|Design|;
123 \begin{python}
124 d.addItem(rec)
125 \end{python}
126 We have the plan to set a heat flux on the bottom of the domain. One can use
127 the masking technique to do this but \pycad offers a more convenient technique
128 called tagging. With this technique items in the domain are named using the
129 \verb|PropertySet| class. We can then later use this name to set values
130 specifically for those sample points located on the named items. Here we name
131 the bottom face of the domain where we will set the heat influx
132 \footnote{In some applications, eg.
133 when dealing with influxes,
134 it is required to have the surface being meshed without the need of explicitly
135 name the surface. In this case the line forming the surface of the domain need to be
136 added to the \texttt{Design}
137 using \texttt{d.addItem(rec, l01, l12, l23, l30)}}:
138 \begin{python}
139 ps=PropertySet("linebottom",l12))
140 d.addItem(ps)
141 \end{python}
142 Now we are ready to hand over the \verb|Design| to \FINLEY:
143 \begin{python}
144 from esys.finley import MakeDomain
145 domain=MakeDomain(d)
146 \end{python}
147 The \verb|domain| object can now be used in the same way like the return object
148 of the \verb|Rectangle| object we have used previously to generate a mesh. It
149 is common practice to separate the mesh generation from the PDE solution.
150 The main reason for this is that mesh generation can be computationally very
151 expensive in particular in 3D. So it is more efficient to generate the mesh
152 once and write it to a file. The mesh can then be read in every time a new
153 simulation is run. \FINLEY supports this in the following
154 way\footnote{An alternative is using the \texttt{dump} and \texttt{load}
155 functions. They work with a binary format and tend to be much smaller.}:
156 \begin{python}
157 # write domain to a text file
158 domain.write("example04.fly")
159 \end{python}
160 and then for reading in another script:
161 \begin{python}
162 # read domain from text file
163 from esys.finley import ReadMesh
164 domain =ReadMesh("example04.fly")
165 \end{python}
167 Before we discuss how to solve the PDE for this problem, it is useful to
168 present two additional options of the \verb|Design| class.
169 These allow the user to access the script which is used by \gmsh to generate
170 the mesh as well as the generated mesh itself. This is done by setting specific
171 names for these files:
172 \begin{python}
173 d.setScriptFileName("example04.geo")
174 d.setMeshFileName("example04.msh")
175 \end{python}
176 Conventionally the extension \texttt{geo} is used for the script file of the
177 \gmsh geometry and the extension \texttt{msh} for the mesh file. Normally these
178 files are deleted after usage.
179 Accessing these files can be helpful to debug the generation of more complex
180 geometries. The geometry and the mesh can be visualised from the command line
181 using
182 \begin{verbatim}
183 gmsh example04.geo # show geometry
184 gmsh example04.msh # show mesh
185 \end{verbatim}
186 The mesh is shown in \reffig{fig:pycad rec mesh}.
188 \begin{figure}[ht]
189 \centerline{\includegraphics[width=4.in]{figures/simplemesh}}
190 \caption{Example 4a: Mesh over rectangular domain, see \reffig{fig:pycad rec}}
191 \label{fig:pycad rec mesh}
192 \end{figure}
193 \clearpage
195 \section{The Steady-state Heat Equation}
196 \sslist{example04b.py, cblib}
197 A temperature equilibrium or steady state is reached when the temperature
198 distribution in the model does not change with time. To calculate the steady
199 state solution the time derivative term in \refEq{eqn:Tform nabla} needs to be
200 set to zero;
201 \begin{equation}\label{eqn:Tform nabla steady}
202 -\nabla \cdot \kappa \nabla T = q_H
203 \end{equation}
204 This PDE is easier to solve than the PDE in \refEq{eqn:hdgenf2}, as no time
205 steps (iterations) are required. The \verb|D| term from \refEq{eqn:hdgenf2} is
206 simply dropped in this case.
207 \begin{python}
208 mypde=LinearPDE(domain)
209 mypde.setValue(A=kappa*kronecker(model), Y=qH)
210 \end{python}
211 The temperature at the top face of the domain is known as \verb|Ttop|~($=20 C$).
212 In \refSec{Sec:1DHDv0} we have already discussed how this constraint is added
213 to the PDE:
214 \begin{python}
215 x=Solution(domain).getX()
216 mypde.setValue(q=whereZero(x[1]-sup(x[1])),r=Ttop)
217 \end{python}
218 Notice that we use the \verb|sup| function to calculate the maximum of $y$
219 coordinates of the relevant sample points.
221 In all cases so far we have assumed that the domain is insulated which
222 translates into a zero normal flux $-n \cdot \kappa \nabla T$, see
223 \refEq{eq:hom flux}. In the modelling set-up of this chapter we want to set
224 the normal heat flux at the bottom to \verb|qin| while still maintaining
225 insulation at the left and right face. Mathematically we can express this as
226 \begin{equation}
227 -n \cdot \kappa \nabla T = q_{S}
228 \label{eq:inhom flux}
229 \end{equation}
230 where $q_{S}$ is a function of its location on the boundary. Its value
231 becomes zero for locations on the left or right face of the domain while it has
232 the value \verb|qin| at the bottom face.
233 Notice that the value of $q_{S}$ at the top face is not relevant as we
234 prescribe the temperature here.
235 We could define $q_{S}$ by using the masking techniques demonstrated
236 earlier. The tagging mechanism provides an alternative and in many cases more
237 convenient way of defining piecewise constant functions such as
238 $q_{S}$. Recall now that the bottom face was denoted with the name
239 \verb|linebottom| when we defined the domain.
240 We can use this now to create $q_{S}$;
241 \begin{python}
242 qS=Scalar(0,FunctionOnBoundary(domain))
243 qS.setTaggedValue("linebottom",qin)
244 \end{python}
245 In the first line \verb|qS| is defined as a scalar value over the sample points
246 on the boundary of the domain. It is initialised to zero for all sample points.
247 In the second statement the values for those sample points which are located on
248 the line marked by \verb|linebottom| are set to \verb|qin|.
250 The Neumann boundary condition assumed by \esc has the form
251 \begin{equation}\label{NEUMAN 2b}
252 n\cdot A \cdot\nabla u = y
253 \end{equation}
254 In comparison to the version in \refEq{NEUMAN 2} we have used so far the right
255 hand side is now the new PDE coefficient $y$. As we have not specified $y$ in
256 our previous examples, \esc has assumed the value zero for $y$. A comparison of
257 \refEq{NEUMAN 2b} and \refEq{eq:inhom flux} reveals that one needs to choose
258 $y=-q_{S}$;
259 \begin{python}
260 qS=Scalar(0,FunctionOnBoundary(domain))
261 qS.setTaggedValue("linebottom",qin)
262 mypde.setValue(y=-qS)
263 \end{python}
264 To plot the results we use the \modmpl library as shown in
265 \refSec{Sec:2DHD plot}. For convenience the interpolation of the temperature to
266 a rectangular grid for contour plotting is made available via the
267 \verb|toRegGrid| function in the \verb|cblib| module. Your result should look
268 similar to \reffig{fig:steady state heat}.
270 \begin{figure}[ht]
271 \centerline{\includegraphics[width=4.in]{figures/simpleheat}}
272 \caption{Example 4b: Result of simple steady state heat problem}
273 \label{fig:steady state heat}
274 \end{figure}
275 \clearpage

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