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% Copyright (c) 20032010 by University of Queensland 
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\begin{figure}[h!] 
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\centerline{\includegraphics[width=4.in]{figures/onedheatdiff001}} 
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\caption{Example 1: Temperature differential along a single interface between two granite blocks.} 
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\label{fig:onedgbmodel} 
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\end{figure} 
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\section{Example 1: One Dimensional Heat Diffusion in Granite} 
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\label{Sec:1DHDv00} 
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The first model consists of two blocks of isotropic material, for instance granite, sitting next to each other. 
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Initial temperature in \textit{Block 1} is \verbT1 and in \textit{Block 2} is \verbT2. 
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We assume that the system is insulated. 
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What would happen to the temperature distribution in each block over time? 
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Intuition tells us that heat will be transported from the hotter block to the cooler one until both 
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blocks have the same temperature. 
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\subsection{1D Heat Diffusion Equation} 
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We can model the heat distribution of this problem over time using one dimensional heat diffusion equation\footnote{A detailed discussion on how the heat diffusion equation is derived can be found at \url{http://online.redwoods.edu/instruct/darnold/DEProj/sp02/AbeRichards/paper.pdf}}; 
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which is defined as: 
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\begin{equation} 
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\rho c\hackscore p \frac{\partial T}{\partial t}  \kappa \frac{\partial^{2} T}{\partial x^{2}} = q\hackscore H 
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\label{eqn:hd} 
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\end{equation} 
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where $\rho$ is the material density, $c\hackscore p$ is the specific heat and $\kappa$ is the thermal 
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conductivity\footnote{A list of some common thermal conductivities is available from Wikipedia \url{http://en.wikipedia.org/wiki/List_of_thermal_conductivities}}. Here we assume that these material 
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parameters are \textbf{constant}. 
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The heat source is defined by the right hand side of \refEq{eqn:hd} as $q\hackscore{H}$; this can take the form of a constant or a function of time and space. For example $q\hackscore{H} = q\hackscore{0}e^{\gamma t}$ where we have the output of our heat source decaying with time. There are also two partial derivatives in \refEq{eqn:hd}; $\frac{\partial T}{\partial t}$ describes the change in temperature with time while $\frac{\partial ^2 T}{\partial x^2}$ is the spatial change of temperature. As there is only a single spatial dimension to our problem, our temperature solution $T$ is only dependent on the time $t$ and our position along the iron bar $x$. 
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\subsection{PDEs and the General Form} 
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Potentially, it is possible to solve PDE \refEq{eqn:hd} analytically and obtain an exact solution to our problem. However, it is not always possible or practical to solve a problem this way. Alternatively, computers can be used to solve these kinds of problems. To do this, a numerical approach is required to discretise 
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the PDE \refEq{eqn:hd} in time and space, then we left with a finite number of equations for a finite number of spatial points and time steps in the model. While discretisation introduces approximations and a degree of error, a sufficiently sampled model is generally accurate enough to satisfy the requirements of the modeller. 
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Firstly, we discretise the PDE \refEq{eqn:hd} in time. This leaves us with a steady linear PDE which involves spatial derivatives only and needs to be solved in each time step to progress in time. \esc can help us here. 
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For time discretization we use the Backwards Euler approximation scheme\footnote{see \url{http://en.wikipedia.org/wiki/Euler_method}}. It is based on the 
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approximation 
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\begin{equation} 
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\frac{\partial T(t)}{\partial t} \approx \frac{T(t)T(th)}{h} 
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\label{eqn:beuler} 
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\end{equation} 
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for $\frac{\partial T}{\partial t}$ at time $t$ 
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where $h$ is the time step size. This can also be written as; 
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\begin{equation} 
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\frac{\partial T}{\partial t}(t^{(n)}) \approx \frac{T^{(n)}  T^{(n1)}}{h} 
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\label{eqn:Tbeuler} 
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\end{equation} 
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where the upper index $n$ denotes the n\textsuperscript{th} time step. So one has 
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\begin{equation} 
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\begin{array}{rcl} 
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t^{(n)} & = & t^{(n1)}+h \\ 
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T^{(n)} & = & T(t^{(n1)}) \\ 
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\end{array} 
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\label{eqn:Neuler} 
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\end{equation} 
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Substituting \refEq{eqn:Tbeuler} into \refEq{eqn:hd} we get; 
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\begin{equation} 
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\frac{\rho c\hackscore p}{h} (T^{(n)}  T^{(n1)})  \kappa \frac{\partial^{2} T^{(n)}}{\partial x^{2}} = q\hackscore H 
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\label{eqn:hddisc} 
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\end{equation} 
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Notice that we evaluate the spatial derivative term at current time $t^{(n)}$  therefore the name \textbf{backward Euler} scheme. Alternatively, one can evaluate the spatial derivative term at the previous time $t^{(n1)}$. This 
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approach is called the \textbf{forward Euler} scheme. This scheme can provide some computational advantages, which 
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are not discussed here. However, this scheme has a major disadvantage, namely depending on the 
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material parameter as well as the discretization of the spatial derivative term, the time step size $h$ needs to be chosen sufficiently small to achieve a stable temperature when progressing in time. The term \textit{stable} means 
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that the approximation of the temperature will not grow beyond its initial bounds and become nonphysical. 
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The backward Euler scheme, which we use here, is unconditionally stable meaning that under the assumption of 
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physically correct problem setup the temperature approximation remains physical for all time steps. 
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The user needs to keep in mind that the discretization error introduced by \refEq{eqn:beuler} 
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is sufficiently small, thus a good approximation of the true temperature is computed. It is 
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therefore crucial that the user remains critical about his/her results and for instance compares 
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the results for different time and spatial step sizes. 
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To get the temperature $T^{(n)}$ at time $t^{(n)}$ we need to solve the linear 
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differential equation \refEq{eqn:hddisc} which only includes spatial derivatives. To solve this problem 
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we want to use \esc. 
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In \esc any given PDE described by general form. For the purpose of this introduction we illustrate a simpler version of the general form for full linear PDEs which is available in the \esc user's guide. A simplified form that suits our heat diffusion problem\footnote{The form in the \esc users guide which uses the Einstein convention is written as 
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$(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}+D u =Y$} 
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is described by; 
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\begin{equation}\label{eqn:commonform nabla} 
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\nabla\cdot(A\cdot\nabla u) + Du = f 
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\end{equation} 
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where $A$, $D$ and $f$ are known values and $u$ is the unknown solution. The symbol $\nabla$ which is called the \textit{Nabla operator} or \textit{del operator} represents 
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the spatial derivative of its subject  in this case $u$. Lets assume for a moment that we deal with a onedimensional problem then ; 
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\begin{equation} 
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\nabla = \frac{\partial}{\partial x} 
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\end{equation} 
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and we can write \refEq{eqn:commonform nabla} as; 
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\begin{equation}\label{eqn:commonform} 
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A\frac{\partial^{2}u}{\partial x^{2}} + Du = f 
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\end{equation} 
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if $A$ is constant. To match this simplified general form to our problem \refEq{eqn:hddisc} 
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we rearrange \refEq{eqn:hddisc}; 
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\begin{equation} 
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\frac{\rho c\hackscore p}{h} T^{(n)}  \kappa \frac{\partial^2 T^{(n)}}{\partial x^2} = q\hackscore H + \frac{\rho c\hackscore p}{h} T^{(n1)} 
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\label{eqn:hdgenf} 
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\end{equation} 
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The PDE is now in a form that satisfies \refEq{eqn:commonform nabla} which is required for \esc to solve our PDE. This can be done by generating a solution for successive increments in the time nodes $t^{(n)}$ where 
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$t^{(0)}=0$ and $t^{(n)}=t^{(n1)}+h$ where $h>0$ is the step size and assumed to be constant. 
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In the following the upper index ${(n)}$ refers to a value at time $t^{(n)}$. Finally, by comparing \refEq{eqn:hdgenf} with \refEq{eqn:commonform} one can see that; 
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\begin{equation}\label{ESCRIPT SET} 
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u=T^{(n)}; 
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A = \kappa; D = \frac{\rho c \hackscore{p}}{h}; f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n1)} 
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\end{equation} 
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\subsection{Boundary Conditions} 
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\label{SEC BOUNDARY COND} 
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With the PDE sufficiently modified, consideration must now be given to the boundary conditions of our model. Typically there are two main types of boundary conditions known as \textbf{Neumann} and \textbf{Dirichlet} boundary conditions\footnote{More information on Boundary Conditions is available at Wikipedia \url{http://en.wikipedia.org/wiki/Boundary_conditions}}, respectively. 
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A \textbf{Dirichlet boundary condition} is conceptually simpler and is used to prescribe a known value to the unknown solution (in our example the temperature) on parts of the boundary or on the entire boundary of the region of interest. 
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We discuss Dirichlet boundary condition in our second example presented in Section~\ref{Sec:1DHDv0}. 
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We make the model assumption that the system is insulated so we need 
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to add an appropriate boundary condition to prevent 
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any loss or inflow of energy at boundary of our domain. Mathematically this is expressed by prescribing 
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the heat flux $\kappa \frac{\partial T}{\partial x}$ to zero. In our simplified one dimensional model this is expressed 
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in the form; 
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\begin{equation} 
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\kappa \frac{\partial T}{\partial x} = 0 
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\end{equation} 
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or in a more general case as 
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\begin{equation}\label{NEUMAN 1} 
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\kappa \nabla T \cdot n = 0 
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\end{equation} 
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where $n$ is the outer normal field \index{outer normal field} at the surface of the domain. 
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The $\cdot$ (dot) refers to the dot product of the vectors $\nabla T$ and $n$. In fact, the term $\nabla T \cdot n$ is the normal derivative of 
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the temperature $T$. Other notations used here are\footnote{The \esc notation for the normal 
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derivative is $T\hackscore{,i} n\hackscore i$.}; 
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\begin{equation} 
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\nabla T \cdot n = \frac{\partial T}{\partial n} \; . 
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\end{equation} 
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A condition of the type \refEq{NEUMAN 1} defines a \textbf{Neuman boundary condition} for the PDE. 
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The PDE \refEq{eqn:hdgenf} 
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and the Neuman boundary condition~\ref{eqn:hdgenf} (potentially together with the Dirichlet boundary conditions) define a \textbf{boundary value problem}. 
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It is the nature of a boundary value problem to allow making statements about the solution in the 
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interior of the domain from information known on the boundary only. In most cases 
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we use the term partial differential equation but in fact it is a boundary value problem. 
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It is important to keep in mind that boundary conditions need to be complete and consistent in the sense that 
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at any point on the boundary either a Dirichlet or a Neuman boundary condition must be set. 
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Conveniently, \esc makes default assumption on the boundary conditions which the user may modify where appropriate. 
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For a problem of the form in~\refEq{eqn:commonform nabla} the default condition\footnote{In the form of the \esc users guide which is using the Einstein convention is written as 
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$n\hackscore{j}A\hackscore{jl} u\hackscore{,l}=0$.} is; 
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\begin{equation}\label{NEUMAN 2} 
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n\cdot A \cdot\nabla u = 0 
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\end{equation} 
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which is used everywhere on the boundary. Again $n$ denotes the outer normal field. 
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Notice that the coefficient $A$ is the same as in the \esc PDE~\ref{eqn:commonform nabla}. 
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With the settings for the coefficients we have already identified in \refEq{ESCRIPT SET} this 
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condition translates into 
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\begin{equation}\label{NEUMAN 2b} 
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\kappa \frac{\partial T}{\partial x} = 0 
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\end{equation} 
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for the boundary of the domain. This is identical to the Neuman boundary condition we want to set. \esc will take care of this condition for us. We discuss the Dirichlet boundary condition later. 
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\subsection{Outline of the Implementation} 
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\label{sec:outline} 
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To solve the heat diffusion equation (equation \refEq{eqn:hd}) we write a simple \pyt script. At this point we assume that you have some basic understanding of the \pyt programming language. If not, there are some pointers and links available in Section \ref{sec:escpybas}. The script we discuss later in details have four major steps. Firstly, we need to define the domain where we want to 
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calculate the temperature. For our problem this is the joint blocks of granite which has a rectangular shape. Secondly, we need to define the PDE to solve in each time step to get the updated temperature. Thirdly, we need to define the coefficients of the PDE and finally we need to solve the PDE. The last two steps need to be repeated until the final time marker has been reached. The work flow is as follows: 
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\begin{enumerate} 
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\item create domain 
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\item create PDE 
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\item while end time not reached: 
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\begin{enumerate} 
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\item set PDE coefficients 
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\item solve PDE 
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\item update time marker 
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\end{enumerate} 
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\item end of calculation 
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\end{enumerate} 
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In the terminology of \pyt, the domain and PDE are represented by \textbf{objects}. The nice feature of an object is that it is defined by its usage and features 
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rather than its actual representation. So we will create a domain object to describe the geometry of the two 
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granite blocks. Then we define PDEs and spatially distributed values such as the temperature 
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on this domain. Similarly, to define a PDE object we use the fact that one needs only to define the coefficients of the PDE and solve the PDE. The PDE object has advanced features, but these are not required in simple cases. 
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\begin{figure}[t] 
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\centering 
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\includegraphics[width=6in]{figures/functionspace.pdf} 
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\label{fig:fs} 
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\caption{\esc domain construction overview} 
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\end{figure} 
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\subsection{The Domain Constructor in \esc} 
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\label{ss:domcon} 
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It is helpful to have a better understanding how spatially distributed value such as the temperature or PDE coefficients are interpreted in \esc. Again 
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from the user's point of view the representation of these spatially distributed values is not relevant. 
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There are various ways to construct domain objects. The simplest form is as rectangular shaped region with a length and height. There is 
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a ready to use function call for this. Besides the spatial dimensions the function call will require you to specify the number 
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elements or cells to be used along the length and height, see \reffig{fig:fs}. Any spatially distributed value 
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and the PDE is represented in discrete form using this element representation\footnote{We will use the finite element method (FEM), see \url{http://en.wikipedia.org/wiki/Finite_element_method} for details.}. Therefore we will have access to an approximation of the true PDE solution only. 
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The quality of the approximation depends  besides other factors mainly on the number of elements being used. In fact, the 
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approximation becomes better the more elements are used. However, computational costs and compute time grow with the number of 
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elements being used. It therefore important that you find the right balance between the demand in accuracy and acceptable resource usage. 
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In general, one can thinks about a domain object as a composition of nodes and elements. 
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As shown in \reffig{fig:fs}, an element is defined by the nodes used to describe its vertices. 
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To represent spatial distributed values the user can use 
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the values at the nodes, at the elements in the interior of the domain or at elements located at the surface of the domain. 
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The different approach used to represent values is called \textbf{function space} and is attached to all objects 
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in \esc representing a spatial distributed value such as the solution of a PDE. The three 
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function spaces we will use at the moment are; 
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\begin{enumerate} 
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\item the nodes, called by \verbContinuousFunction(domain) ; 
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\item the elements/cells, called by \verbFunction(domain) ; and 
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\item the boundary, called by \verbFunctionOnBoundary(domain) . 
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\end{enumerate} 
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A function space object such as \verbContinuousFunction(domain) has the method \verbgetX attached to it. This method returns the 
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location of the socalled \textbf{sample points} used to represent values with the particular function space attached to it. So the 
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call \verbContinuousFunction(domain).getX() will return the coordinates of the nodes used to describe the domain while 
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the \verbFunction(domain).getX() returns the coordinates of numerical integration points within elements, see 
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\reffig{fig:fs}. 
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This distinction between different representations of spatial distributed values 
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is important in order to be able to vary the degrees of smoothness in a PDE problem. 
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The coefficients of a PDE need not be continuous thus this qualifies as a \verbFunction() type. 
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On the other hand a temperature distribution must be continuous and needs to be represented with a \verbContinuousFunction() function space. 
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An influx may only be defined at the boundary and is therefore a \verb FunctionOnBoundary() object. 
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\esc allows certain transformations of the function spaces. A \verb ContinuousFunction() can be transformed into a \verbFunctionOnBoundary() 
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or \verbFunction(). On the other hand there is not enough information in a \verb FunctionOnBoundary() to transform it to a \verb ContinuousFunction() . 
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These transformations, which are called \textbf{interpolation} are invoked automatically by \esc if needed. 
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Later in this introduction we will discuss how 
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to define specific areas of geometry with different materials which are represented by different material coefficients such the 
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thermal conductivities $kappa$. A very powerful technique to define these types of PDE 
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coefficients is tagging. Blocks of materials and boundaries can be named and values can be defined on subregions based on their names. 
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This is simplifying PDE coefficient and flux definitions. It makes for much easier scripting. We will discuss this technique in Section~\ref{STEADYSTATE HEAT REFRACTION}. 
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\subsection{A Clarification for the 1D Case} 
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\label{SEC: 1D CLARIFICATION} 
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It is necessary for clarification that we revisit the general PDE from \refeq{eqn:commonform nabla} under the light of a two dimensional domain. \esc is inherently designed to solve problems that are greater than one dimension and so \refEq{eqn:commonform nabla} needs to be read as a higher dimensional problem. In the case of two spatial dimensions the \textit{Nabla operator} has in fact two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. In full, \refEq{eqn:commonform nabla} assuming a constant coefficient $A$, takes the form; 
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\begin{equation}\label{eqn:commonform2D} 
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A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}} 
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A\hackscore{01}\frac{\partial^{2}u}{\partial x\partial y} 
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A\hackscore{10}\frac{\partial^{2}u}{\partial y\partial x} 
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A\hackscore{11}\frac{\partial^{2}u}{\partial y^{2}} 
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+ Du = f 
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\end{equation} 
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Notice that for the higher dimensional case $A$ becomes a matrix. It is also 
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important to notice that the usage of the Nabla operator creates 
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a compact formulation which is also independent from the spatial dimension. 
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So to make the general PDE \refEq{eqn:commonform2D} one dimensional as 
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shown in \refEq{eqn:commonform} we need to set 
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\begin{equation} 
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A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0 
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\end{equation} 
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\subsection{Developing a PDE Solution Script} 
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\label{sec:key} 
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\sslist{example01a.py} 
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We will write a simple \pyt script which uses the \modescript, \modfinley and \modmpl modules. 
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By developing a script for \esc, the heat diffusion equation can be solved at successive time steps for a predefined period using our general form \refEq{eqn:hdgenf}. Firstly it is necessary to import all the libraries\footnote{The libraries contain predefined scripts that are required to solve certain problems, these can be simple like sine and cosine functions or more complicated like those from our \esc library.} 
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that we will require. 
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\begin{python} 
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from esys.escript import * 
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# This defines the LinearPDE module as LinearPDE 
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from esys.escript.linearPDEs import LinearPDE 
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# This imports the rectangle domain function from finley. 
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from esys.finley import Rectangle 
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# A useful unit handling package which will make sure all our units 
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# match up in the equations under SI. 
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from esys.escript.unitsSI import * 
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\end{python} 
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It is generally a good idea to import all of the \modescript library, although if the functions and classes required are known they can be specified individually. The function \verbLinearPDE has been imported explicitly for ease of use later in the script. \verbRectangle is going to be our type of model. The module \verb unitsSI provides support for SI unit definitions with our variables. 
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Once our library dependencies have been established, defining the problem specific variables is the next step. In general the number of variables needed will vary between problems. These variables belong to two categories. They are either directly related to the PDE and can be used as inputs into the \esc solver, or they are script variables used to control internal functions and iterations in our problem. For this PDE there are a number of constants which will need values. Firstly, the model upon which we wish to solve our problem needs to be defined. There are many different types of models in \modescript which we will demonstrate in later tutorials but for our granite blocks, we will simply use a rectangular model. 
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Using a rectangular model simplifies our granite blocks which would in reality be a \textit{3D} object, into a single dimension. The granite blocks will have a lengthways cross section that looks like a rectangle. As a result we do not need to model the volume of the block. There are four arguments we must consider when we decide to create a rectangular model, the model \textit{length}, \textit{width} and \textit{step size} in each direction. When defining the size of our problem it will help us determine appropriate values for our model arguments. If we make our dimensions large but our step sizes very small we will to a point, increase the accuracy of our solution. Unfortunately we also increase the number of calculations that must be solved per time step. This means more computational time is required to produce a solution. In this \textit{1D} problem, the bar is defined as being 1 metre long. An appropriate step size \verbndx would be 1 to 10\% of the length. Our \verbndy need only be 1, this is because our problem stipulates no partial derivatives in the $y$ direction. Thus the temperature does not vary with $y$. Hence, the model parameters can be defined as follows; note we have used the \verb unitsSI convention to make sure all our input units are converted to SI. 
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\begin{python} 
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mx = 500.*m #meters  model length 
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my = 100.*m #meters  model width 
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ndx = 50 # mesh steps in x direction 
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ndy = 1 # mesh steps in y direction 
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boundloc = mx/2 # location of boundary between the two blocks 
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\end{python} 
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The material constants and the temperature variables must also be defined. For the granite in the model they are defined as: 
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\begin{python} 
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#PDE related 
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rho = 2750. *kg/m**3 #kg/m^{3} density of iron 
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cp = 790.*J/(kg*K) # J/Kg.K thermal capacity 
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rhocp = rho*cp 
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kappa = 2.2*W/m/K # watts/m.Kthermal conductivity 
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qH=0 * J/(sec*m**3) # J/(sec.m^{3}) no heat source 
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T1=20 * Celsius # initial temperature at Block 1 
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T2=2273. * Celsius # base temperature at Block 2 
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\end{python} 
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Finally, to control our script we will have to specify our timing controls and where we would like to save the output from the solver. This is simple enough: 
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\begin{python} 
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t=0 * day #our start time, usually zero 
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tend=1. * day #  time to end simulation 
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outputs = 200 # number of time steps required. 
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h=(tendt)/outputs #size of time step 
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#user warning statement 
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print "Expected Number of time outputs is: ", (tendt)/h 
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i=0 #loop counter 
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\end{python} 
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Now that we know our inputs we will build a domain using the \verb Rectangle() function from \verb finley . The four arguments allow us to define our domain \verb model as: 
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\begin{python} 
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#generate domain using rectangle 
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blocks = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy) 
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\end{python} 
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\verb blocks now describes a domain in the manner of Section \ref{ss:domcon}. T 
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With a domain and all our required variables established, it is now possible to set up our PDE so that it can be solved by \esc. The first step is to define the type of PDE that we are trying to solve in each time step. In this example it is a single linear PDE\footnote{in comparison to a system of PDEs which will be discussed later.}. We also need to state the values of our general form variables. 
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\begin{python} 
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mypde=LinearPDE(blocks) 
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A=zeros((2,2))) 
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A[0,0]=kappa 
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mypde.setValue(A=A, D=rhocp/h) 
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\end{python} 
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In a many cases it may be possible to decrease the computational time of the solver if the PDE is symmetric. 
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Symmetry of a PDE is defined by; 
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\begin{equation}\label{eqn:symm} 
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A\hackscore{jl}=A\hackscore{lj} 
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\end{equation} 
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Symmetry is only dependent on the $A$ coefficient in the general form and the other coefficients $D$ as well as the right hand side $Y$ may take any value. From the above definition we can see that our PDE is symmetric. The \verb LinearPDE class provides the method \method{checkSymmetry} to check if the given PDE is symmetric. As our PDE is symmetrical we will enable symmetry via; 
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\begin{python} 
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myPDE.setSymmetryOn() 
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\end{python} 
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Next we need to establish the initial temperature distribution \verbT. We need to 
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assign the value \verbT1 to all sample points left to the contact interface at $x\hackscore{0}=\frac{mx}{2}$ 
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and the value \verbT2 right to the contact interface. \esc 
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provides the \verbwhereNegative function to construct this. In fact, 
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\verbwhereNegative returns the value $1$ at those sample points where the argument 
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has a negative value. Otherwise zero is returned. If \verbx are the $x\hackscore{0}$ 
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coordinates of the sample points used to represent the temperature distribution 
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then \verbx[0]boundloc gives us a negative value for 
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all sample points left to the interface and nonnegative value to 
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the right of the interface. So with; 
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\begin{python} 
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# ... set initial temperature .... 
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T= T1*whereNegative(x[0]boundloc)+T2*(1whereNegative(x[0]boundloc)) 
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\end{python} 
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we get the desired temperature distribution. To get the actual sample points \verbx we use 
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the \verbgetX() method of the function space \verbSolution(blocks) 
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which is used to represent the solution of a PDE; 
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\begin{python} 
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x=Solution(blocks).getX() 
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\end{python} 
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As \verbx are the sample points for the function space \verbSolution(blocks) 
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the initial temperature \verbT is using these sample points for representation. 
357 
Although \esc is trying to be forgiving with the choice of sample points and to convert 
358 
where necessary the adjustment of the function space is not always possible. So it is 
359 
advisable to make a careful choice on the function space used. 
360 

361 
Finally we will initialise an iteration loop to solve our PDE for all the time steps we specified in the variable section. As the right hand side of the general form is dependent on the previous values for temperature \verb T across the bar this must be updated in the loop. Our output at each time step is \verb T the heat distribution and \verb totT the total heat in the system. 
362 
\begin{python} 
363 
while t < tend: 
364 
i+=1 #increment the counter 
365 
t+=h #increment the current time 
366 
mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients 
367 
T=mypde.getSolution() #get the PDE solution 
368 
totE = integrate(rhocp*T) #get the total heat (energy) in the system 
369 
\end{python} 
370 
The last statement in this script calculates the total energy in the system as volume integral 
371 
of $\rho c\hackscore{p} T$ over the block. As the blocks are insulated no energy should be get lost or added. 
372 
The total energy should stay constant for the example discussed here. 
373 

374 
\subsection{Running the Script} 
375 
The script presented so for is available under 
376 
\verbexample01a.py. You can edit this file with your favourite text editor. 
377 
On most operating systems\footnote{The you can use \texttt{runescript} launcher is not supported under {\it MS Windows} yet.} you can use the \program{runescript} command 
378 
to launch {\it escript} scripts. For the example script use; 
379 
\begin{verbatim} 
380 
runescript example01a.py 
381 
\end{verbatim} 
382 
The program will print a progress report. Alternatively, you can use 
383 
the python interpreter directly; 
384 
\begin{verbatim} 
385 
python example01a.py 
386 
\end{verbatim} 
387 
if the system is configured correctly (Please talk to your system administrator). 
388 

389 
\begin{figure} 
390 
\begin{center} 
391 
\includegraphics[width=4in]{figures/ttblockspyplot150} 
392 
\caption{Example 1b: Total Energy in the Blocks over Time (in seconds).} 
393 
\label{fig:onedheatout1} 
394 
\end{center} 
395 
\end{figure} 
396 

397 
\subsection{Plotting the Total Energy} 
398 
\sslist{example01b.py} 
399 

400 
\esc does not include its own plotting capabilities. However, it is possible to use a variety of free \pyt packages for visualisation. 
401 
Two types will be demonstrated in this cookbook; \mpl\footnote{\url{http://matplotlib.sourceforge.net/}} and \verb VTK \footnote{\url{http://www.vtk.org/}} visualisation. 
402 
The \mpl package is a component of SciPy\footnote{\url{http://www.scipy.org}} and is good for basic graphs and plots. 
403 
For more complex visualisation tasks in particular when it comes to two and three dimensional problems it is recommended to us more advanced tools for instance \mayavi \footnote{\url{http://code.enthought.com/projects/mayavi/}} 
404 
which bases on the \verbVTK toolkit. We will discuss the usage of \verbVTK based 
405 
visualization in Chapter~\ref{Sec:2DHD} where will discuss a two dimensional PDE. 
406 

407 
For our simple problem we have two plotting tasks: Firstly we are interested in showing the 
408 
behaviour of the total energy over time and secondly in how the temperature distribution within the block is 
409 
developing over time. Lets start with the first task. 
410 

411 
The trick is to create a record of the time marks and the corresponding total energies observed. 
412 
\pyt provides the concept of lists for this. Before 
413 
the time loop is opened we create empty lists for the time marks \verbt_list and the total energies \verbE_list. 
414 
After the new temperature as been calculated by solving the PDE we append the new time marker and total energy 
415 
to the corresponding list using the \verbappend method. With these modifications the script looks as follows: 
416 
\begin{python} 
417 
t_list=[] 
418 
E_list=[] 
419 
# ... start iteration: 
420 
while t<tend: 
421 
t+=h 
422 
mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients 
423 
T=mypde.getSolution() #get the PDE solution 
424 
totE=integrate(rhocp*T) 
425 
t_list.append(t) # add current time mark to record 
426 
E_list.append(totE) # add current total energy to record 
427 
\end{python} 
428 
To plot $t$ over $totE$ we use the \mpl a module contained within \pylab which needs to be loaded before used; 
429 
\begin{python} 
430 
import pylab as pl # plotting package. 
431 
\end{python} 
432 
Here we are not using the \verbfrom pylab import * in order to avoid name clashes for function names 
433 
within \esc. 
434 

435 
The following statements are added to the script after the time loop has been completed; 
436 
\begin{python} 
437 
pl.plot(t_list,E_list) 
438 
pl.title("Total Energy") 
439 
pl.axis([0,max(t_list),0,max(E_list)*1.1]) 
440 
pl.savefig("totE.png") 
441 
\end{python} 
442 
The first statement hands over the time marks and corresponding total energies to the plotter. 
443 
The second statment is setting the title for the plot. The third statement 
444 
sets the axis ranges. In most cases these are set appropriately by the plotter. 
445 
The last statement renders the plot and writes the 
446 
result into the file \verbtotE.png which can be displayed by (almost) any image viewer. 
447 
As expected the total energy is constant over time, see \reffig{fig:onedheatout1}. 
448 

449 
\subsection{Plotting the Temperature Distribution} 
450 
\label{sec: plot T} 
451 
\sslist{example01c.py} 
452 
For plotting the spatial distribution of the temperature we need to modify the strategy we have used 
453 
for the total energy. Instead of producing a final plot at the end we will generate a 
454 
picture at each time step which can be browsed as slide show or composed to a movie. 
455 
The first problem we encounter is that if we produce an image in each time step we need 
456 
to make sure that the images previously generated are not overwritten. 
457 

458 
To develop an incrementing file name we can use the following convention. It is convenient to 
459 
put all image file showing the same variable  in our case the temperature distribution  
460 
into a separate directory. As part of the \verbos module\footnote{The \texttt{os} module provides 
461 
a powerful interface to interact with the operating system, see \url{http://docs.python.org/library/os.html}.} \pyt 
462 
provides the \verbos.path.join command to build file and 
463 
directory names in a platform independent way. Assuming that 
464 
\verbsave_path is name of directory we want to put the results the command is; 
465 
\begin{python} 
466 
import os 
467 
os.path.join(save_path, "tempT%03d.png"%i ) 
468 
\end{python} 
469 
where \verbi is the time step counter. 
470 
There are two arguments to the \verb join command. The \verb save_path variable is a predefined string pointing to the directory we want to save our data in, for example a single subfolder called \verb data would be defined by; 
471 
\begin{verbatim} 
472 
save_path = "data" 
473 
\end{verbatim} 
474 
while a subfolder of \verb data called \verb example01 would be defined by; 
475 
\begin{verbatim} 
476 
save_path = os.path.join("data","example01") 
477 
\end{verbatim} 
478 
The second argument of \verb join \xspace contains a string which is the file name or subdirectory name. We can use the operator \verb% to use the value of \verbi as part of our filename. The substring \verb %03d indicates that we want to substitude a value into the name; 
479 
\begin{itemize} 
480 
\item \verb 0 means that small numbers should have leading zeroes; 
481 
\item \verb 3 means that numbers should be written using at least 3 digits; and 
482 
\item \verb d means that the value to substitute will be an integer. 
483 
\end{itemize} 
484 
To actually substitute the value of \verbi into the name write \verb %i after the string. 
485 
When done correctly, the output files from this command would be place in the directory defined by \verb save_path as; 
486 
\begin{verbatim} 
487 
blockspyplot001.png 
488 
blockspyplot002.png 
489 
blockspyplot003.png 
490 
... 
491 
\end{verbatim} 
492 
and so on. 
493 

494 
A subfolder check/constructor is available in \esc. The command; 
495 
\begin{verbatim} 
496 
mkDir(save_path) 
497 
\end{verbatim} 
498 
will check for the existence of \verb save_path and if missing, make the required directories. 
499 

500 
We start by modifying our solution script from before. 
501 
Prior to the \verbwhile loop we will need to extract our finite solution points to a data object that is compatible with \mpl. First we create the node coordinates of the sample points used to represent 
502 
the temperature as a \pyt list of tuples or a \numpy array as requested by the plotting function. 
503 
We need to convert thearray \verbx previously set as \verbSolution(blocks).getX() into a \pyt list 
504 
and then to a \numpy array. The $x\hackscore{0}$ component is then extracted via an array slice to the variable \verbplx; 
505 
\begin{python} 
506 
import numpy as np # array package. 
507 
#convert solution points for plotting 
508 
plx = x.toListOfTuples() 
509 
plx = np.array(plx) # convert to tuple to numpy array 
510 
plx = plx[:,0] # extract x locations 
511 
\end{python} 
512 

513 
\begin{figure} 
514 
\begin{center} 
515 
\includegraphics[width=4in]{figures/blockspyplot001} 
516 
\includegraphics[width=4in]{figures/blockspyplot050} 
517 
\includegraphics[width=4in]{figures/blockspyplot200} 
518 
\caption{Example 1c: Temperature ($T$) distribution in the blocks at time steps $1$, $50$ and $200$.} 
519 
\label{fig:onedheatout} 
520 
\end{center} 
521 
\end{figure} 
522 

523 
For each time step we will generate a plot of the temperature distribution and save each to a file. We use the same 
524 
techniques provided by \mpl as we have used to plot the total energy over time. 
525 
The following is appended to the end of the \verb while loop and creates one figure of the temperature distribution. We start by converting the solution to a tuple and then plotting this against our \textit{x coordinates} \verb plx we have generated before. We add a title to the diagram before it is rendered into a file. 
526 
Finally, the figure is saved to a \verb*.png file and cleared for the following iteration. 
527 
\begin{python} 
528 
# ... start iteration: 
529 
while t<tend: 
530 
.... 
531 
T=mypde.getSolution() #get the PDE solution 
532 
tempT = T.toListOfTuples() # convert to a tuple 
533 
pl.plot(plx,tempT) # plot solution 
534 
# set scale (Temperature should be between Tref and T0) 
535 
pl.axis([0,mx,Tref*.9,T0*1.1]) 
536 
# add title 
537 
pl.title("Temperature across the blocks at time %e minutes"%(t/day)) 
538 
#save figure to file 
539 
pl.savefig(os.path.join(save_path,"tempT","blockspyplot%03d.png") %i) 
540 
\end{python} 
541 
Some results are shown in \reffig{fig:onedheatout}. 
542 

543 
\subsection{Make a video} 
544 
Our saved plots from the previous section can be cast into a video using the following command appended to the end of the script. \verb mencoder is Linux only however, and other platform users will need to use an alternative video encoder. 
545 
\begin{python} 
546 
# compile the *.png files to create a *.avi videos that show T change 
547 
# with time. This operation uses Linux mencoder. For other operating 
548 
# systems it is possible to use your favourite video compiler to 
549 
# convert image files to videos. 
550 

551 
os.system("mencoder mf://"+save_path+"/tempT"+"/*.png mf type=png:\ 
552 
w=800:h=600:fps=25 ovc lavc lavcopts vcodec=mpeg4 oac copy o \ 
553 
example01tempT.avi") 
554 
\end{python} 
555 
