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 1 ahallam 2401 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 jfenwick 2881 % Copyright (c) 2003-2010 by University of Queensland 5 ahallam 2401 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 gross 2905 \begin{figure}[h!] 15 \centerline{\includegraphics[width=4.in]{figures/onedheatdiff001}} 16 \caption{Temperature differential along a single interface between two granite blocks.} 17 \label{fig:onedgbmodel} 18 \end{figure} 19 ahallam 2801 20 gross 2905 \section{One Dimensional Heat Diffusion in Granite} 21 \label{Sec:1DHDv00} 22 gross 2878 23 gross 2905 The first model consists of two blocks of isotropic material, for instance granite, sitting next to each other. 24 Initially, \textit{Block 1} is of a temperature 25 \verb|T1| and \textit{Block 2} is at a temperature \verb|T2|. 26 We assume that the system is insulated. 27 What would happen to the temperature distribution in each block over time? 28 Intuition tells us that heat will transported from the hotter block to the cooler until both 29 blocks have the same temperature. 30 31 ahallam 2495 \subsection{1D Heat Diffusion Equation} 32 We can model the heat distribution of this problem over time using the 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}}; 33 ahallam 2494 which is defined as: 34 ahallam 2401 \begin{equation} 35 \rho c\hackscore p \frac{\partial T}{\partial t} - \kappa \frac{\partial^{2} T}{\partial x^{2}} = q\hackscore H 36 \label{eqn:hd} 37 \end{equation} 38 gross 2861 where $\rho$ is the material density, $c\hackscore p$ is the specific heat and $\kappa$ is the thermal 39 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 40 parameters are \textbf{constant}. 41 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 $x$. 42 ahallam 2401 43 gross 2861 \subsection{PDEs and the General Form} 44 Potentially, it is now possible to solve PDE \refEq{eqn:hd} analytically and this would produce 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 discretised 45 gross 2905 the PDE \refEq{eqn:hd} in time and space so finally we are left with a finite number of equations for a finite number of spatial and time steps in the model. While discretization introduces approximations and a degree of error, we find that a sufficiently sampled model is generally accurate enough for the requirements of the modeller. 46 gross 2477 47 gross 2861 Firstly, we will discretise the PDE \refEq{eqn:hd} in the time direction which will 48 leave as with a steady linear PDE which is involving spatial derivatives only and needs to be solved in each time 49 step to progress in time - \esc can help us here. 50 51 For the discretization in time we will use is the Backwards Euler approximation scheme\footnote{see \url{http://en.wikipedia.org/wiki/Euler_method}}. It bases on the 52 approximation 53 \begin{equation} 54 \frac{\partial T(t)}{\partial t} \approx \frac{T(t)-T(t-h)}{h} 55 \label{eqn:beuler} 56 \end{equation} 57 for $\frac{\partial T}{\partial t}$ at time $t$ 58 where $h$ is the time step size. This can also be written as; 59 \begin{equation} 60 \frac{\partial T}{\partial t}(t^{(n)}) \approx \frac{T^{(n)} - T^{(n-1)}}{h} 61 \label{eqn:Tbeuler} 62 \end{equation} 63 where the upper index $n$ denotes the n\textsuperscript{th} time step. So one has 64 \begin{equation} 65 \begin{array}{rcl} 66 t^{(n)} & = & t^{(n-1)}+h \\ 67 T^{(n)} & = & T(t^{(n-1)}) \\ 68 \end{array} 69 \label{eqn:Neuler} 70 \end{equation} 71 Substituting \refEq{eqn:Tbeuler} into \refEq{eqn:hd} we get; 72 \begin{equation} 73 \frac{\rho c\hackscore p}{h} (T^{(n)} - T^{(n-1)}) - \kappa \frac{\partial^{2} T^{(n)}}{\partial x^{2}} = q\hackscore H 74 \label{eqn:hddisc} 75 \end{equation} 76 Notice that we evaluate the spatial derivative term at current time $t^{(n)}$ - therefore the name \textbf{backward Euler} scheme. Alternatively, one can use evaluate the spatial derivative term at the previous time $t^{(n-1)}$. This 77 approach is called the \textbf{forward Euler} scheme. This scheme can provide some computational advantages which 78 we are not discussed here but has the major disadvantage that depending on the 79 gross 2905 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 80 that the approximation of the temperature will not grow beyond its initial bounds and becomes non-physical. 81 gross 2861 The backward Euler which we use here is unconditionally stable meaning that under the assumption of 82 physically correct problem set-up the temperature approximation remains physical for all times. 83 The user needs to keep in mind that the discretization error introduced by \refEq{eqn:beuler} 84 is sufficiently small so a good approximation of the true temperature is calculated. It is 85 therefore crucial that the user remains critical about his/her results and for instance compares 86 the results for different time and spatial step sizes. 87 88 To get the temperature $T^{(n)}$ at time $t^{(n)}$ we need to solve the linear 89 differential equation \refEq{eqn:hddisc} which is only including spatial derivatives. To solve this problem 90 we want to to use \esc. 91 92 \esc interfaces with any given PDE via a general form. For the purpose of this introduction we will illustrate a simpler version of the full linear PDE general form which is available in the \esc user's guide. A simplified form that suits our heat diffusion problem\footnote{In the form of the \esc users guide which using the Einstein convention is written as 93 gross 2477 $-(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}+D u =Y$} 94 ahallam 2606 is described by; 95 gross 2477 \begin{equation}\label{eqn:commonform nabla} 96 jfenwick 2657 -\nabla\cdot(A\cdot\nabla u) + Du = f 97 ahallam 2411 \end{equation} 98 gross 2861 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 99 ahallam 2495 the spatial derivative of its subject - in this case $u$. Lets assume for a moment that we deal with a one-dimensional problem then ; 100 gross 2477 \begin{equation} 101 \nabla = \frac{\partial}{\partial x} 102 \end{equation} 103 gross 2861 and we can write \refEq{eqn:commonform nabla} as; 104 ahallam 2411 \begin{equation}\label{eqn:commonform} 105 gross 2477 -A\frac{\partial^{2}u}{\partial x^{2}} + Du = f 106 ahallam 2411 \end{equation} 107 gross 2861 if $A$ is constant. To match this simplified general form to our problem \refEq{eqn:hddisc} 108 we rearrange \refEq{eqn:hddisc}; 109 ahallam 2411 \begin{equation} 110 ahallam 2645 \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^{(n-1)} 111 ahallam 2494 \label{eqn:hdgenf} 112 \end{equation} 113 ahallam 2775 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 114 ahallam 2495 $t^{(0)}=0$ and $t^{(n)}=t^{(n-1)}+h$ where $h>0$ is the step size and assumed to be constant. 115 gross 2861 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} it can be seen that; 116 gross 2862 \begin{equation}\label{ESCRIPT SET} 117 gross 2861 u=T^{(n)}; 118 ahallam 2494 A = \kappa; D = \frac{\rho c \hackscore{p}}{h}; f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n-1)} 119 \end{equation} 120 121 ahallam 2495 \subsection{Boundary Conditions} 122 gross 2878 \label{SEC BOUNDARY COND} 123 gross 2862 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. 124 gross 2905 A \textbf{Dirichlet boundary condition} is conceptually simpler and is used to prescribe a known value to the unknown - in our example the temperature - on parts of the boundary or on the entire boundary of the region of interest. 125 We discuss Dirichlet boundary condition in our second example presented in Section~\ref{Sec:1DHDv0}. 126 ahallam 2495 127 gross 2905 We make the model assumption that the system is insulated so we need 128 to add an appropriate boundary condition to prevent 129 any loss or inflow of energy at boundary of our domain. Mathematically this is expressed by prescribing 130 the heat flux $\kappa \frac{\partial T}{\partial x}$ to zero. In our simplified one dimensional model this is expressed 131 gross 2862 in the form; 132 ahallam 2494 \begin{equation} 133 gross 2862 \kappa \frac{\partial T}{\partial x} = 0 134 ahallam 2494 \end{equation} 135 gross 2862 or in a more general case as 136 \begin{equation}\label{NEUMAN 1} 137 \kappa \nabla T \cdot n = 0 138 \end{equation} 139 where $n$ is the outer normal field \index{outer normal field} at the surface of the domain. 140 gross 2905 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 141 gross 2862 the temperature $T$. Other notations which are used are\footnote{The \esc notation for the normal 142 derivative is $T\hackscore{,i} n\hackscore i$.}; 143 ahallam 2645 \begin{equation} 144 gross 2862 \nabla T \cdot n = \frac{\partial T}{\partial n} \; . 145 ahallam 2645 \end{equation} 146 gross 2862 A condition of the type \refEq{NEUMAN 1} defines a \textbf{Neuman boundary condition} for the PDE. 147 ahallam 2494 148 gross 2905 The PDE \refEq{eqn:hdgenf} 149 and the Neuman boundary condition~\ref{eqn:hdgenf} (potentially together with the Dirichlet boundary condition set) define a \textbf{boundary value problem}. 150 gross 2862 It is a nature of a boundary value problem that it allows to make statements on the solution in the 151 interior of the domain from information known on the boundary only. In most cases 152 we use the term partial differential equation but in fact mean a boundary value problem. 153 It is important to keep in mind that boundary conditions need to be complete and consistent in the sense that 154 at any point on the boundary either a Dirichlet or a Neuman boundary condition must be set. 155 156 gross 2878 Conveniently, \esc makes default assumption on the boundary conditions which the user may modify where appropriate. 157 gross 2862 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 158 $n\hackscore{j}A\hackscore{jl} u\hackscore{,l}=0$.} is; 159 \begin{equation}\label{NEUMAN 2} 160 n\cdot A \cdot\nabla u = 0 161 \end{equation} 162 which is used everywhere on the boundary. Again $n$ denotes the outer normal field. 163 Notice that the coefficient $A$ is the same as in the \esc PDE~\ref{eqn:commonform nabla}. 164 With the settings for the coefficients we have already identified in \refEq{ESCRIPT SET} this 165 condition translates into 166 gross 2867 \begin{equation}\label{NEUMAN 2b} 167 gross 2862 \kappa \frac{\partial T}{\partial x} = 0 168 \end{equation} 169 gross 2905 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 will discuss the Dirichlet boundary condition later. 170 gross 2862 171 gross 2870 \subsection{Outline of the Implementation} 172 \label{sec:outline} 173 To solve the heat diffusion equation (equation \refEq{eqn:hd}) we will 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 will discuss later in details will have four major steps. Firstly we need to define the domain where we want to 174 gross 2905 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 175 gross 2878 we need to solve in each time step to get the updated temperature. Thirdly we need to define the 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. As a work flow this takes the form; 176 \begin{enumerate} 177 \item create domain 178 \item create PDE 179 \item while end time not reached: 180 \begin{enumerate} 181 \item set PDE coefficients 182 \item solve PDE 183 \item update time marker 184 \end{enumerate} 185 \item end of calculation 186 \end{enumerate} 187 gross 2870 In the terminology of \pyt the domain and PDE are represented by \textbf{objects}. The nice feature of an object is that it defined by it usage and features 188 gross 2905 rather than its actual representation. So we will create a domain object to describe the geometry of the two 189 granite blocks. The main feature 190 gross 2870 of the object we will use is the fact that we can define PDEs and spatially distributed values such as the temperature 191 on a domain. In fact the domain object has many more features - most of them you will 192 never use and do not need to understand. Similar a PDE object is defined by the fact that we can define the coefficients of the PDE and solve the PDE. At a 193 gross 2878 later stage you may use more advanced features of the PDE class but you need to worry about them only at the point when you use them. 194 gross 2870 195 196 \begin{figure}[t] 197 \centering 198 \includegraphics[width=6in]{figures/functionspace.pdf} 199 \label{fig:fs} 200 \caption{\esc domain construction overview} 201 \end{figure} 202 203 \subsection{The Domain Constructor in \esc} 204 \label{ss:domcon} 205 It is helpful to have a better understanding how spatially distributed value such as the temperature or PDE coefficients are interpreted in \esc. Again 206 from the user's point of view the representation of these spatially distributed values is not relevant. 207 208 There are various ways to construct domain objects. The simplest form is as rectangular shaped region with a length and height. There is 209 a ready to use function call for this. Besides the spatial dimensions the function call will require you to specify the number 210 gross 2905 elements or cells to be used along the length and height, see \reffig{fig:fs}. Any spatially distributed value 211 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. 212 gross 2870 The quality of the approximation depends - besides other factors- mainly on the number of elements being used. In fact, the 213 approximation becomes better the more elements are used. However, computational costs and compute time grow with the number of 214 gross 2878 elements being used. It therefore important that you find the right balance between the demand in accuracy and acceptable resource usage. 215 gross 2870 216 In general, one can thinks about a domain object as a composition of nodes and elements. 217 gross 2905 As shown in \reffig{fig:fs}, an element is defined by the nodes used to describe its vertices. 218 gross 2870 To represent spatial distributed values the user can use 219 the values at the nodes, at the elements in the interior of the domain or at elements located at the surface of the domain. 220 The different approach used to represent values is called \textbf{function space} and is attached to all objects 221 in \esc representing a spatial distributed value such as the solution of a PDE. The three 222 function spaces we will use at the moment are; 223 \begin{enumerate} 224 \item the nodes, called by \verb|ContinuousFunction(domain)| ; 225 \item the elements/cells, called by \verb|Function(domain)| ; and 226 \item the boundary, called by \verb|FunctionOnBoundary(domain)| . 227 \end{enumerate} 228 A function space object such as \verb|ContinuousFunction(domain)| has the method \verb|getX| attached to it. This method returns the 229 location of the so-called \textbf{sample points} used to represent values with the particular function space attached to it. So the 230 call \verb|ContinuousFunction(domain).getX()| will return the coordinates of the nodes used to describe the domain while 231 the \verb|Function(domain).getX()| returns the coordinates of numerical integration points within elements, see 232 gross 2905 \reffig{fig:fs}. 233 gross 2870 234 gross 2878 This distinction between different representations of spatial distributed values 235 gross 2870 is important in order to be able to vary the degrees of smoothness in a PDE problem. 236 The coefficients of a PDE need not be continuous thus this qualifies as a \verb|Function()| type. 237 On the other hand a temperature distribution must be continuous and needs to be represented with a \verb|ContinuousFunction()| function space. 238 An influx may only be defined at the boundary and is therefore a \verb FunctionOnBoundary() object. 239 \esc allows certain transformations of the function spaces. A \verb ContinuousFunction() can be transformed into a \verb|FunctionOnBoundary()| 240 or \verb|Function()|. On the other hand there is not enough information in a \verb FunctionOnBoundary() to transform it to a \verb ContinuousFunction() . 241 gross 2878 These transformations, which are called \textbf{interpolation} are invoked automatically by \esc if needed. 242 gross 2870 243 Later in this introduction we will discuss how 244 to define specific areas of geometry with different materials which are represented by different material coefficients such the 245 gross 2878 thermal conductivities $kappa$. A very powerful technique to define these types of PDE 246 gross 2870 coefficients is tagging. Blocks of materials and boundaries can be named and values can be defined on subregions based on their names. 247 gross 2878 This is simplifying PDE coefficient and flux definitions. It makes for much easier scripting. We will discuss this technique in Section~\ref{STEADY-STATE HEAT REFRACTION}. 248 gross 2870 249 250 \subsection{A Clarification for the 1D Case} 251 gross 2861 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; 252 gross 2477 \begin{equation}\label{eqn:commonform2D} 253 -A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}} 254 -A\hackscore{01}\frac{\partial^{2}u}{\partial x\partial y} 255 -A\hackscore{10}\frac{\partial^{2}u}{\partial y\partial x} 256 -A\hackscore{11}\frac{\partial^{2}u}{\partial y^{2}} 257 + Du = f 258 \end{equation} 259 ahallam 2606 Notice that for the higher dimensional case $A$ becomes a matrix. It is also 260 ahallam 2495 important to notice that the usage of the Nabla operator creates 261 a compact formulation which is also independent from the spatial dimension. 262 gross 2861 So to make the general PDE \refEq{eqn:commonform2D} one dimensional as 263 shown in \refEq{eqn:commonform} we need to set 264 ahallam 2606 \begin{equation} 265 ahallam 2494 A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0 266 gross 2477 \end{equation} 267 268 gross 2867 269 ahallam 2495 \subsection{Developing a PDE Solution Script} 270 ahallam 2801 \label{sec:key} 271 gross 2878 \sslist{onedheatdiffbase.py} 272 gross 2870 We will write a simple \pyt script which uses the \modescript, \modfinley and \modmpl modules. 273 gross 2905 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.} 274 ahallam 2495 that we will require. 275 ahallam 2775 \begin{python} 276 ahallam 2495 from esys.escript import * 277 ahallam 2606 # This defines the LinearPDE module as LinearPDE 278 from esys.escript.linearPDEs import LinearPDE 279 # This imports the rectangle domain function from finley. 280 from esys.finley import Rectangle 281 # A useful unit handling package which will make sure all our units 282 # match up in the equations under SI. 283 from esys.escript.unitsSI import * 284 ahallam 2775 \end{python} 285 gross 2878 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 \verb|LinearPDE| has been imported explicitly for ease of use later in the script. \verb|Rectangle| is going to be our type of model. The module \verb unitsSI provides support for SI unit definitions with our variables. 286 gross 2477 287 gross 2905 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. 288 ahallam 2401 289 gross 2905 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 \verb|ndx| would be 1 to 10\% of the length. Our \verb|ndy| 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. 290 ahallam 2775 \begin{python} 291 gross 2905 mx = 500.*m #meters - model length 292 my = 100.*m #meters - model width 293 ndx = 50 # mesh steps in x direction 294 ndy = 1 # mesh steps in y direction 295 boundloc = mx/2 # location of boundary between the two blocks 296 ahallam 2775 \end{python} 297 gross 2905 The material constants and the temperature variables must also be defined. For the granite in the model they are defined as: 298 ahallam 2775 \begin{python} 299 ahallam 2495 #PDE related 300 gross 2905 rho = 2750. *kg/m**3 #kg/m^{3} density of iron 301 cp = 790.*J/(kg*K) # J/Kg.K thermal capacity 302 gross 2878 rhocp = rho*cp 303 gross 2905 kappa = 2.2*W/m/K # watts/m.Kthermal conductivity 304 gross 2878 qH=0 * J/(sec*m**3) # J/(sec.m^{3}) no heat source 305 gross 2905 T1=20 * Celsius # initial temperature at Block 1 306 T2=2273. * Celsius # base temperature at Block 2 307 ahallam 2775 \end{python} 308 ahallam 2495 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: 309 ahallam 2775 \begin{python} 310 gross 2878 t=0 * day #our start time, usually zero 311 tend=1. * day # - time to end simulation 312 ahallam 2495 outputs = 200 # number of time steps required. 313 h=(tend-t)/outputs #size of time step 314 ahallam 2606 #user warning statement 315 print "Expected Number of time outputs is: ", (tend-t)/h 316 i=0 #loop counter 317 ahallam 2775 \end{python} 318 gross 2905 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: 319 ahallam 2775 \begin{python} 320 ahallam 2606 #generate domain using rectangle 321 gross 2905 blocks = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy) 322 ahallam 2775 \end{python} 323 gross 2905 \verb blocks now describes a domain in the manner of Section \ref{ss:domcon}. T 324 ahallam 2658 325 ahallam 2775 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. 326 \begin{python} 327 gross 2905 mypde=LinearPDE(blocks) 328 gross 2878 A=zeros((2,2))) 329 A[0,0]=kappa 330 gross 2905 mypde.setValue(A=A, D=rhocp/h) 331 ahallam 2775 \end{python} 332 gross 2878 In a many cases it may be possible to decrease the computational time of the solver if the PDE is symmetric. 333 Symmetry of a PDE is defined by; 334 ahallam 2495 \begin{equation}\label{eqn:symm} 335 A\hackscore{jl}=A\hackscore{lj} 336 \end{equation} 337 gross 2878 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; 338 ahallam 2775 \begin{python} 339 ahallam 2495 myPDE.setSymmetryOn() 340 ahallam 2775 \end{python} 341 gross 2905 Next we need to establish the initial temperature distribution \verb|T|. We need to 342 assign the value \verb|T1| to all sample points left to the contact interface at $x\hackscore{0}=\frac{mx}{2}$ 343 and the value \verb|T2| right to the contact interface. \esc 344 provides the \verb|whereNegative| function to construct this. In fact, 345 \verb|whereNegative| returns the value $1$ at those sample points where the argument 346 has a negative value. Otherwise zero is returned. If \verb|x| are the $x\hackscore{0}$ 347 coordinates of the sample points used to represent the temperature distribution 348 then \verb|x[0]-boundloc| gives us a negative value for 349 all sample points left to the interface and non-negative value to 350 the right of the interface. So with; 351 gross 2878 \begin{python} 352 # ... set initial temperature .... 353 gross 2905 T= T1*whereNegative(x[0]-boundloc)+T2*(1-whereNegative(x[0]-boundloc)) 354 gross 2878 \end{python} 355 gross 2905 we get the desired temperature distribution. To get the actual sample points \verb|x| we use 356 the \verb|getX()| method of the function space \verb|Solution(blocks)| 357 which is used to represent the solution of a PDE; 358 gross 2878 \begin{python} 359 gross 2905 x=Solution(blocks).getX() 360 \end{python} 361 As \verb|x| are the sample points for the function space \verb|Solution(blocks)| 362 the initial temperature \verb|T| is using these sample points for representation. 363 Although \esc is trying to be forgiving with the choice of sample points and to convert 364 where necessary the adjustment of the function space is not always possible. So it is 365 advisable to make a careful choice on the function space used. 366 367 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. 368 \begin{python} 369 gross 2878 while t < tend: 370 i+=1 #increment the counter 371 t+=h #increment the current time 372 mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients 373 T=mypde.getSolution() #get the PDE solution 374 totE = integrate(rhocp*T) #get the total heat (energy) in the system 375 \end{python} 376 The last statement in this script calculates the total energy in the system as volume integral 377 gross 2905 of $\rho c\hackscore{p} T$ over the block. As the blocks are insulated no energy should be get lost or added. 378 The total energy should stay constant for the example discussed here. 379 ahallam 2401 380 gross 2905 \subsection{Running the Script} 381 The script presented so for is available under 382 \verb|onedheatdiffbase.py|. You can edit this file with your favourite text editor. 383 On most operating systems\footnote{The you can use \texttt{escript} launcher is not supported under {\it MS Windows} yet.} you can use the \program{escript} command 384 to launch {\it escript} scripts. For the example script use; 385 \begin{verbatim} 386 escript onedheatdiffbase.py 387 \end{verbatim} 388 The program will print a progress report. Alternatively, you can use 389 the python interpreter directly; 390 \begin{verbatim} 391 python onedheatdiffbase.py 392 \end{verbatim} 393 if the system is configured correctly (Please talk to your system administrator). 394 395 \begin{figure} 396 \begin{center} 397 \includegraphics[width=4in]{figures/ttblockspyplot150} 398 \caption{Total Energy in the Blocks over Time (in seconds).} 399 \label{fig:onedheatout1} 400 \end{center} 401 \end{figure} 402 403 gross 2878 \subsection{Plotting the Total Energy} 404 \sslist{onedheatdiff001.py} 405 406 gross 2905 \esc does not include its own plotting capabilities. However, it is possible to use a variety of free \pyt packages for visualisation. 407 gross 2878 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. 408 The \mpl package is a component of SciPy\footnote{\url{http://www.scipy.org}} and is good for basic graphs and plots. 409 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/}} 410 gross 2905 which bases on the \verb|VTK| toolkit. We will discuss the usage of \verb|VTK| based 411 gross 2878 visualization in Chapter~\ref{Sec:2DHD} where will discuss a two dimensional PDE. 412 413 For our simple problem we have two plotting tasks: Firstly we are interested in showing the 414 gross 2905 behaviour of the total energy over time and secondly in how the temperature distribution within the block is 415 gross 2878 developing over time. Lets start with the first task. 416 417 The trick is to create a record of the time marks and the corresponding total energies observed. 418 \pyt provides the concept of lists for this. Before 419 the time loop is opened we create empty lists for the time marks \verb|t_list| and the total energies \verb|E_list|. 420 After the new temperature as been calculated by solving the PDE we append the new time marker and total energy 421 to the corresponding list using the \verb|append| method. With these modifications the script looks as follows: 422 ahallam 2775 \begin{python} 423 gross 2878 t_list=[] 424 E_list=[] 425 # ... start iteration: 426 while t