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% Copyright (c) 20032010 by University of Queensland 
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% Licensed under the Open Software License version 3.0 
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\begin{figure}[t] 
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\centerline{\includegraphics[width=4.in]{figures/twodheatdiff}} 
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\caption{Example 3: 2D model: granitic intrusion of sandstone country rock.} 
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\label{fig:twodhdmodel} 
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\end{figure} 
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\sslist{example03a.py and cblib.py} 
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Building upon our success from the 1D models, it is now prudent to expand our domain by another dimension. 
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For this example we use a very simple magmatic intrusion as the basis for our model. The simulation will be a single event where some molten granite has formed a cylindrical dome at the base of some cold sandstone country rock. Assuming that the cylinder is very long 
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we model a crosssection as shown in \reffig{fig:twodhdmodel}. We will implement the same 
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diffusion model as we have use for the granite blocks in \refSec{Sec:1DHDv00} 
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but will add the second spatial dimension and show how to define 
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variables depending on the location of the domain. 
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We use \file{onedheatdiff001b.py} as the starting point for develop this model. 
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\section{Example 3: Two Dimensional Heat Diffusion for a basic Magmatic Intrusion} 
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\label{Sec:2DHD} 
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To expand upon our 1D problem, the domain must first be expanded. In fact, we have solved a two dimensional problem already but essentially ignored the second dimension. In our definition phase we create a square domain in $x$ and $y$\footnote{In \esc the notation 
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$x\hackscore{0}$ and $x\hackscore{1}$ is used for $x$ and $y$, respectively.} that is $600$ meters along each side \reffig{fig:twodhdmodel}. Now we set the number of discrete spatial cells to 150 in both direction and the radius of the intrusion to $200$ meters with the centre located at the $300$ meter mark on the $x$axis. Thus, the domain variables are; 
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\begin{python} 
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mx = 600*m #meters  model length 
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my = 600*m #meters  model width 
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ndx = 150 #mesh steps in x direction 
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ndy = 150 #mesh steps in y direction 
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r = 200*m #meters  radius of intrusion 
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ic = [300*m, 0] #coordinates of the centre of intrusion (meters) 
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qH=0.*J/(sec*m**3) #our heat source temperature is zero 
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\end{python} 
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As before we use 
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\begin{python} 
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model = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy) 
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\end{python} 
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to generate the domain. 
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There are two fundamental changes to the PDE that we have discussed in \refSec{Sec:1DHDv00}. Firstly, 
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because the material coefficients for granite and sandstone are different, we need to deal with 
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PDE coefficients which vary with their location in the domain. Secondly, we need 
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to deal with the second spatial dimension. We can investigate these two modification at the same time. 
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In fact, the temperature model \refEq{eqn:hd} we utilised in \refSec{Sec:1DHDv00} applied for the 
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1D case with a constant material parameter only. For the more general case examined in this chapter, the correct model equation is 
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\begin{equation} 
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\rho c\hackscore p \frac{\partial T}{\partial t}  \frac{\partial }{\partial x} \kappa \frac{\partial T}{\partial x}  \frac{\partial }{\partial y} \kappa \frac{\partial T}{\partial y} = q\hackscore H 
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\label{eqn:hd2} 
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\end{equation} 
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Notice, that for the 1D case we have $\frac{\partial T}{\partial y}=0$ and 
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for the case of constant material parameters $\frac{\partial }{\partial x} \kappa = \kappa \frac{\partial }{\partial x}$ thus this new equation coincides with a simplified model equation for this case. It is more convenient 
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to write this equation using the $\nabla$ notation as we have already seen in \refEq{eqn:commonform nabla}; 
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\begin{equation}\label{eqn:Tform nabla} 
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\rho c\hackscore p \frac{\partial T}{\partial t} 
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\nabla \cdot \kappa \nabla T = q\hackscore H 
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\end{equation} 
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We can easily apply the backward Euler scheme as before to end up with 
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\begin{equation} 
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\frac{\rho c\hackscore p}{h} T^{(n)} \nabla \cdot \kappa \nabla T^{(n)} = q\hackscore H + \frac{\rho c\hackscore p}{h} T^{(n1)} 
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\label{eqn:hdgenf2} 
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\end{equation} 
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which is very similar to \refEq{eqn:hdgenf} used to define the temperature solution in the simple 1D case. 
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The difference is in the second order derivate term $\nabla \cdot \kappa \nabla T^{(n)}$. Under 
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the light of the more general case we need to revisit the \esc PDE form as 
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shown in \ref{eqn:commonform2D}. For the 2D case with variable PDE coefficients 
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the form needs to be read as 
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\begin{equation}\label{eqn:commonform2D 2} 
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\frac{\partial }{\partial x} A\hackscore{00}\frac{\partial u}{\partial x} 
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\frac{\partial }{\partial x} A\hackscore{01}\frac{\partial u}{\partial y} 
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\frac{\partial }{\partial y} A\hackscore{10}\frac{\partial u}{\partial x} 
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\frac{\partial }{\partial x} A\hackscore{11}\frac{\partial u}{\partial y} 
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+ Du = f 
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\end{equation} 
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So besides the settings $u=T^{(n)}$, $D = \frac{\rho c \hackscore{p}}{h}$ and 
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$f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n1)}$ as we have used before (see \refEq{ESCRIPT SET}) we need to set 
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\begin{equation}\label{eqn: kappa general} 
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A\hackscore{00}=A\hackscore{11}=\kappa; A\hackscore{01}=A\hackscore{10}=0 
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\end{equation} 
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The fundamental difference to the 1D case is that $A\hackscore{11}$ is not set to zero but $\kappa$, 
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which brings in the second dimension. Important to notice that the coefficients 
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of the PDE may depend on their location in the domain and does not influence the usage of the PDE form. This is very convenient as we can introduce spatial dependence to the PDE coefficients without modification to the way we call the PDE solver. 
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A very convenient way to define the matrix $A$ in \refEq{eqn: kappa general} can be carried out using the 
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Kronecker $\delta$ symbol\footnote{see \url{http://en.wikipedia.org/wiki/Kronecker_delta}}. The 
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\esc function \verbkronecker returns this matrix; 
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\begin{equation} 
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\verbkronecker(model) = \left[ 
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\begin{array}{cc} 
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1 & 0 \\ 
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0 & 1 \\ 
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\end{array} 
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\right] 
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\end{equation} 
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As the argument \verbmodel represents a two dimensional domain the matrix is returned as $2 \times 2$ matrix 
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(In case of a threedimensional domain a $3 \times 3$ matrix is returned). The call 
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\begin{python} 
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mypde.setValue(A=kappa*kronecker(model),D=rhocp/h) 
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\end{python} 
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sets the PDE coefficients according to \refEq{eqn: kappa general}. 
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We need to check the boundary conditions before we turn to the question: how we set $\kappa$. As 
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pointed out in \refEq{NEUMAN 2} makes certain assumptions on the boundary conditions. In our case 
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this assumptions translates to; 
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\begin{equation} 
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n \cdot \kappa \nabla T^{(n)} = 
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n\hackscore{0} \cdot \kappa \frac{\partial T^{(n)}}{\partial x}  n\hackscore{1} \cdot \kappa \frac{\partial T^{(n)}}{\partial y} = 0 
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\label{eq:hom flux} 
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\end{equation} 
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which sets the normal component of the heat flux $ \kappa \cdot (\frac{\partial T^{(n)}}{\partial x}, \frac{\partial T^{(n)}}{\partial y})$ to zero. This means that the regions is insulated which is what we want. 
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On the left and right face of the domain where we have $(n\hackscore{0},n\hackscore{1} ) = (\mp 1,0)$ 
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this means $\frac{\partial T^{(n)}}{\partial x}=0$ and on the top and bottom on the domain 
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where we have $(n\hackscore{0},n\hackscore{1} ) = (\pm 1,0)$ this is $\frac{\partial T^{(n)}}{\partial y}=0$. 
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\section{Setting Variable PDE Coefficients} 
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Now we need to look into the problem of how we define the material coefficients 
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$\kappa$ and $\rho c\hackscore p$ depending on their location in the domain. 
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We can make use of the technique used in the granite block example in \refSec{Sec:1DHDv00} 
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to set up the initial temperature. However, 
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the situation is more complicated here as we have to deal with a 
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curved interface between the two subdomain. 
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Prior to setting up the PDE, the interface between the two materials must be established. 
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The distance $s\ge 0$ between two points $[x,y]$ and $[x\hackscore{0},y\hackscore{0}]$ in Cartesian coordinates is defined as: 
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\begin{equation} 
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(xx\hackscore{0})^{2}+(yy\hackscore{0})^{2} = s^{2} 
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\end{equation} 
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If we define the point $[x\hackscore{0},y\hackscore{0}]$ as $ic$ which denotes the centre of the semicircle of our intrusion, then for all the points $[x,y]$ in our model we can calculate a distance to $ic$. 
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All the points that fall within the given radius $r$ of our intrusion will have a corresponding 
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value $s < r$ and all those belonging to the country rock will have a value $s > r$. By subtracting $r$ from both of these conditions we find $sr < 0$ for all intrusion points and $sr > 0$ 
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for all country rock points. 
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Defining these conditions within the script is quite simple and is done using the following command: 
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\begin{python} 
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bound = length(xic)r #where the boundary will be located 
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\end{python} 
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This definition of the boundary can now be used with \verbwhereNegative command again to help define the material constants and temperatures in our domain. 
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If \verbkappai and \verbkappac are the 
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thermal conductivities for the intrusion material granite and for the surrounding sandstone, then we set; 
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\begin{python} 
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x=Function(model).getX() 
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bound = length(xic)r 
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kappa = kappai * whereNegative(bound) + kappac * (1whereNegative(bound)) 
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mypde.setValue(A=kappa*kronecker(model)) 
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\end{python} 
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Notice that we are using the sample points of the \verbFunction function space as expected for the 
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PDE coefficient \verbA\footnote{For the experienced user: use \texttt{x=mypde.getFunctionSpace("A").getX()}.} 
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The corresponding statements are used to set $\rho c\hackscore p$. 
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Our PDE has now been properly established. The initial conditions for temperature are set out in a similar manner: 
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\begin{python} 
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#defining the initial temperatures. 
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x=Solution(model).getX() 
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bound = length(xic)r 
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T= Ti*whereNegative(bound)+Tc*(1whereNegative(bound)) 
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\end{python} 
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where \verbTi and \verbTc are the initial temperature 
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in the regions of the granite and surrounding sandstone, respectively. It is important to 
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notice that we reset \verbx and \verbbound to refer to the appropriate 
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sample points of a PDE solution\footnote{For the experienced user: use \texttt{x=mypde.getFunctionSpace("r").getX()}.}. 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/heatrefraction001.png}} 
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\centerline{\includegraphics[width=4.in]{figures/heatrefraction030.png}} 
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\centerline{\includegraphics[width=4.in]{figures/heatrefraction200.png}} 
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\caption{Example 3a: 2D model: Total temperature distribution ($T$) at time step $1$, $20$ and $200$. Contour lines show temperature.} 
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\label{fig:twodhdans} 
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\end{figure} 
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\section{Contouring \esc data using \modmpl.} 
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\label{Sec:2DHD plot} 
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To complete our transition from a 1D to a 2D model we also need to modify the 
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plotting procedure. As before we use the \modmpl to do the plotting 
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in this case a contour plot. For 2D contour plots \modmpl expects that the 
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data are regularly gridded. We have no control over the location and ordering of the sample points 
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used to represent the solution. Therefore it is necessary to interpolate our solution onto a regular grid. 
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In \refSec{sec: plot T} we have already learned how to extract the $x$ coordinates of sample points as 
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\verbnumpy array to hand the values to \modmpl. This can easily be extended to extract both the 
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$x$ and the $y$ coordinates; 
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\begin{python} 
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import numpy as np 
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def toXYTuple(coords): 
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coords = np.array(coords.toListOfTuples()) #convert to Tuple 
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coordX = coords[:,0] #X components. 
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coordY = coords[:,1] #Y components. 
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return coordX,coordY 
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\end{python} 
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For convenience we have put this function into \file{clib.py} file so we can use this 
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function in other scripts more easily. 
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We now generate a regular $100 \times 100$ grid over the domain ($mx$ and $my$ 
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are the dimensions in the $x$ and $y$ directions) which is done using the \modnumpy function \verblinspace . 
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\begin{python} 
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from clib import toXYTuple 
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# get sample points for temperature as for contouring 
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coordX, coordY = toXYTuple(T.getFunctionSpace().getX()) 
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# create regular grid 
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xi = np.linspace(0.0,mx,75) 
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yi = np.linspace(0.0,my,75) 
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\end{python} 
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The values \verb[xi[k], yi[l]] are the grid points. 
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The remainder of our contouring commands reside within a \verb while loop so that a new contour is generated for each time step. For each time step the solution must be regridded for \modmpl using the \verb griddata function. This function interprets irregularly located values \verb tempT at locations defined by \verb coordX and \verb coordY as values at the new coordinates of a rectangular grid defined by 
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\verb xi and \verb yi . The output is \verb zi . It is now possible to use the \verb contourf function which generates colour filled contours. The colour gradient of our plots is set via the command \verb pl.matplotlib.pyplot.autumn() , other colours are listed on the \modmpl web page\footnote{see \url{http://matplotlib.sourceforge.net/api/}}. Our results are then contoured, visually adjusted using the \modmpl functions and then saved to file. \verb pl.clf() clears the figure in readiness for the next time iteration. 
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\begin{python} 
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#grid the data. 
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zi = pl.matplotlib.mlab.griddata(coordX,coordY,tempT,xi,yi) 
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# contour the gridded data, plotting dots at the randomly spaced data points. 
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pl.matplotlib.pyplot.autumn() 
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pl.contourf(xi,yi,zi,10) 
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CS = pl.contour(xi,yi,zi,5,linewidths=0.5,colors='k') 
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pl.clabel(CS, inline=1, fontsize=8) 
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pl.axis([0,600,0,600]) 
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pl.title("Heat diffusion from an intrusion.") 
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pl.xlabel("Horizontal Displacement (m)") 
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pl.ylabel("Depth (m)") 
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pl.savefig(os.path.join(save_path,"Tcontour%03d.png") %i) 
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pl.clf() 
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\end{python} 
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The function \verbpl.contour is used to add labeled contour lines to the plot. 
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The results for selected time steps are shown in \reffig{fig:twodhdans}. 
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\section{Advanced Visualization using VTK} 
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\sslist{example03b.py} 
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An alternative approach to \modmpl for visualization is the usage of a package which base on 
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visualization tool kit (VTK) library\footnote{see \url{http://www.vtk.org/}}. There are variety 
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of packages available. Here we use the package \mayavi\footnote{see \url{http://code.enthought.com/projects/mayavi/}} as an example. 
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\mayavi is an interactive, GUI driven tool which is 
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really designed to visualize large three dimensional data sets where \modmpl 
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is not suitable. But it is very useful when it comes to two dimensional problems. 
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The decision of which tool is the best can be subjective and the user should determine which package they require and are most comfortable with. The main difference between using \mayavi (and other VTK based tools) 
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or \modmpl is that the actual visualization is detached from the 
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calculation by writing the results to external files 
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and importing them into \mayavi. In 3D the best camera position for rendering a scene is not obvious 
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before the results are available. Therefore the user may need to try 
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different position before the best is found. Moreover, in many cases a 3D interactive 
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visualization is the only way to really understand the results (e.g. using stereographic projection). 
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To write the temperatures at each time step to data files in the VTK file format one 
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needs to insert a \verbsaveVTK call into the code; 
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\begin{python} 
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while t<=tend: 
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i+=1 #counter 
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t+=h #current time 
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mypde.setValue(Y=qH+T*rhocp/h) 
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T=mypde.getSolution() 
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saveVTK(os.path.join(save_path,"data.%03d.vtu"%i, T=T) 
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\end{python} 
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The data files, eg. \file{data.001.vtu}, contains all necessary information to 
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visualize the temperature and can directly processed by \mayavi. Notice that there is no 
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regridding required. It is recommended to use the file extension \file{.vtu} for files 
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created by \verbsaveVTK. 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/ScreeshotMayavi2n1}} 
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\caption{Example 3b: \mayavi start up Window.} 
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\label{fig:mayavi window} 
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\end{figure} 
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\begin{figure}[ht] 
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\centerline{\includegraphics[width=4.in]{figures/ScreeshotMayavi2n2}} 
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\caption{Example 3b: \mayavi data control window.} 
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\label{fig:mayavi window2} 
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\end{figure} 
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After you run the script you will find the 
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result files \file{data.*.vtu} in the result directory \file{data/example03}. Run the 
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command 
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\begin{python} 
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>> mayavi2 d data.001.vtu m Surface & 
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\end{python} 
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from the result directory. \mayavi will start up a window similar to \reffig{fig:mayavi window}. 
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The right hand side shows the temperature at the first time step. To show 
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the results at other time steps you can click at the item \texttt{VTK XML file (data.001.vtu) (timeseries)} 
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at the top left hand side. This will bring up a new window similar to the window shown in \reffig{fig:mayavi window2}. By clicking at the arrows in the top right corner you can move forwards and backwards in time. 
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You will also notice the text \textbf{T} next to the item \texttt{Point scalars name}. The 
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name \textbf{T} corresponds to the keyword argument name \texttt{T} we have used 
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in the \verbsaveVTK call. In this menu item you can select other results 
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you may have written to the output file for visualization. 
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\textbf{For the advanced user}: Using the \modmpl to visualize spatially distributed data 
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is not MPI compatible. However, the \verbsaveVTK function can be used with MPI. In fact, 
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the function collects the values of the sample points spread across processor ranks into a single. 
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For more details on writing scripts for parallel computing please consult the \emph{user's guide}. 