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 1 ahallam 2597 2 jfenwick 3989 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 jfenwick 4154 % Copyright (c) 2003-2013 by University of Queensland 4 jfenwick 3989 5 ahallam 2597 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 jfenwick 3989 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 12 % 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 ahallam 2597 15 gross 2870 \section{Steady-state Heat Refraction} 16 \label{STEADY-STATE HEAT REFRACTION} 17 gross 2931 18 ahallam 2975 In this chapter we demonstrate how to handle more complex geometries. 19 gross 2931 20 ahallam 2979 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. 29 gross 2931 30 ahallam 2979 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. 33 ahallam 2597 34 caltinay 2982 \section{Example 4: Creating the Domain with \pycad} 35 ahallam 2977 \label{example4} 36 gross 2951 \sslist{example04a.py} 37 caltinay 2982 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. 40 ahallam 2597 41 ahallam 2979 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. 46 ahallam 2597 47 ahallam 2979 In \pycad there are a few primary constructors that build upon each other to 48 caltinay 2982 define domains and boundaries. The ones we use are: 49 ahallam 2801 \begin{python} 50 gross 2948 from esys.pycad import * 51 ahallam 2597 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 gross 2948 PlaneSurface() #Creates a surface based on a CurveLoop 55 ahallam 2801 \end{python} 56 ahallam 2979 So to construct our domain as shown in \reffig{fig:pycad rec}, we first need to 57 caltinay 2982 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 ahallam 2801 \begin{python} 61 gross 2948 width=5000.0*m #width of model 62 depth=-6000.0*m #depth of model 63 ahallam 2801 \end{python} 64 ahallam 3153 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} 70 71 ahallam 2979 The variables are then used to construct the four corners of our domain, which 72 caltinay 2982 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 ahallam 2801 \begin{python} 76 ahallam 2597 # Overall Domain 77 ahallam 2634 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 ahallam 2801 \end{python} 82 ahallam 2979 Now lines are defined using our points. This forms a rectangle around our 83 caltinay 2982 domain: 84 ahallam 2801 \begin{python} 85 ahallam 2597 l01=Line(p0, p1) 86 l12=Line(p1, p2) 87 l23=Line(p2, p3) 88 l30=Line(p3, p0) 89 ahallam 2801 \end{python} 90 caltinay 2982 Note that lines have a direction. These lines form the basis for our domain 91 ahallam 2979 boundary, which is a closed loop. 92 ahallam 2801 \begin{python} 93 ahallam 2597 c=CurveLoop(l01, l12, l23, l30) 94 ahallam 2801 \end{python} 95 ahallam 2979 Be careful to define the curved loop in an \textbf{anti-clockwise} manner 96 otherwise the meshing algorithm may fail. 97 gross 2948 Finally we can define the domain as 98 ahallam 2801 \begin{python} 99 gross 2948 rec = PlaneSurface(c) 100 \end{python} 101 ahallam 2979 At this point the introduction of the curved loop seems to be unnecessary but 102 caltinay 2982 this concept plays an important role if holes are introduced. 103 gross 2948 104 caltinay 2982 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 gross 2948 \begin{python} 109 from esys.pycad.gmsh import Design 110 d=Design(dim=2, element_size=200*m) 111 \end{python} 112 ahallam 2979 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 caltinay 2982 dimensional space.}. The second argument \verb|element_size| defines the element 115 ahallam 2979 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 caltinay 2982 $200$m and a domain length of $6000$m we will end up with about $\frac{6000m}{200m}=30$ 120 ahallam 2979 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 caltinay 2982 The domain \verb|rec| can simply be added to the \verb|Design|; 123 gross 2948 \begin{python} 124 d.addItem(rec) 125 \end{python} 126 caltinay 2982 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 gross 3644 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 jfenwick 4345 it is required to have the surface being meshed without the need of explicitly 135 gross 3644 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 gross 2948 \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 ahallam 2979 The \verb|domain| object can now be used in the same way like the return object 148 caltinay 2982 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 gross 2948 \begin{python} 157 caltinay 2982 # write domain to a text file 158 gross 2951 domain.write("example04.fly") 159 gross 2948 \end{python} 160 caltinay 2982 and then for reading in another script: 161 gross 2948 \begin{python} 162 # read domain from text file 163 from esys.finley import ReadMesh 164 gross 2951 domain =ReadMesh("example04.fly") 165 gross 2948 \end{python} 166 167 caltinay 2982 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 ahallam 2979 names for these files: 172 gross 2948 \begin{python} 173 gross 2951 d.setScriptFileName("example04.geo") 174 d.setMeshFileName("example04.msh") 175 gross 2948 \end{python} 176 caltinay 2982 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 ahallam 2979 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 gross 2948 \begin{verbatim} 183 gross 2951 gmsh example04.geo # show geometry 184 gmsh example04.msh # show mesh 185 gross 2948 \end{verbatim} 186 The mesh is shown in \reffig{fig:pycad rec mesh}. 187 ahallam 3153 188 gross 2948 \begin{figure}[ht] 189 ahallam 3153 \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 gross 2948 \end{figure} 193 ahallam 3153 \clearpage 194 gross 2948 195 \section{The Steady-state Heat Equation} 196 gross 2951 \sslist{example04b.py, cblib} 197 ahallam 2979 A temperature equilibrium or steady state is reached when the temperature 198 caltinay 2982 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 gross 2948 \begin{equation}\label{eqn:Tform nabla steady} 202 jfenwick 3308 -\nabla \cdot \kappa \nabla T = q_H 203 gross 2948 \end{equation} 204 ahallam 2979 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 gross 2948 \begin{python} 208 mypde=LinearPDE(domain) 209 mypde.setValue(A=kappa*kronecker(model), Y=qH) 210 \end{python} 211 caltinay 2982 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 gross 2948 \begin{python} 215 x=Solution(domain).getX() 216 mypde.setValue(q=whereZero(x[1]-sup(x[1])),r=Ttop) 217 \end{python} 218 ahallam 2979 Notice that we use the \verb|sup| function to calculate the maximum of $y$ 219 coordinates of the relevant sample points. 220 gross 2948 221 ahallam 2979 In all cases so far we have assumed that the domain is insulated which 222 caltinay 2982 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 gross 2948 \begin{equation} 227 jfenwick 3308 -n \cdot \kappa \nabla T = q_{S} 228 gross 2948 \label{eq:inhom flux} 229 \end{equation} 230 jfenwick 3308 where $q_{S}$ is a function of its location on the boundary. Its value 231 caltinay 2982 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 jfenwick 3308 Notice that the value of $q_{S}$ at the top face is not relevant as we 234 ahallam 2979 prescribe the temperature here. 235 jfenwick 3308 We could define $q_{S}$ by using the masking techniques demonstrated 236 ahallam 2979 earlier. The tagging mechanism provides an alternative and in many cases more 237 caltinay 2982 convenient way of defining piecewise constant functions such as 238 jfenwick 3308 $q_{S}$. Recall now that the bottom face was denoted with the name 239 caltinay 2982 \verb|linebottom| when we defined the domain. 240 jfenwick 3308 We can use this now to create $q_{S}$; 241 gross 2948 \begin{python} 242 qS=Scalar(0,FunctionOnBoundary(domain)) 243 qS.setTaggedValue("linebottom",qin) 244 \end{python} 245 ahallam 2979 In the first line \verb|qS| is defined as a scalar value over the sample points 246 caltinay 2982 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|. 249 gross 2948 250 caltinay 2982 The Neumann boundary condition assumed by \esc has the form 251 gross 2948 \begin{equation}\label{NEUMAN 2b} 252 n\cdot A \cdot\nabla u = y 253 \end{equation} 254 ahallam 2979 In comparison to the version in \refEq{NEUMAN 2} we have used so far the right 255 caltinay 2982 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 jfenwick 3308 $y=-q_{S}$; 259 gross 2948 \begin{python} 260 qS=Scalar(0,FunctionOnBoundary(domain)) 261 qS.setTaggedValue("linebottom",qin) 262 mypde.setValue(y=-qS) 263 \end{python} 264 caltinay 2982 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}. 269 gross 2948 270 \begin{figure}[ht] 271 ahallam 3153 \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 gross 2948 \end{figure} 275 ahallam 3153 \clearpage 276 gross 2948