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a new almost completed version of the cookbook
1
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4 % Copyright (c) 2003-2010 by University of Queensland
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9 % Licensed under the Open Software License version 3.0
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13
14 \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
20 \section{One Dimensional Heat Diffusion in Granite}
21 \label{Sec:1DHDv00}
22
23 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 \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 which is defined as:
34 \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 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
43 \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 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
47 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 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 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 $-(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}+D u =Y$}
94 is described by;
95 \begin{equation}\label{eqn:commonform nabla}
96 -\nabla\cdot(A\cdot\nabla u) + Du = f
97 \end{equation}
98 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 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 \begin{equation}
101 \nabla = \frac{\partial}{\partial x}
102 \end{equation}
103 and we can write \refEq{eqn:commonform nabla} as;
104 \begin{equation}\label{eqn:commonform}
105 -A\frac{\partial^{2}u}{\partial x^{2}} + Du = f
106 \end{equation}
107 if $A$ is constant. To match this simplified general form to our problem \refEq{eqn:hddisc}
108 we rearrange \refEq{eqn:hddisc};
109 \begin{equation}
110 \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 \label{eqn:hdgenf}
112 \end{equation}
113 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 $t^{(0)}=0$ and $t^{(n)}=t^{(n-1)}+h$ where $h>0$ is the step size and assumed to be constant.
115 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 \begin{equation}\label{ESCRIPT SET}
117 u=T^{(n)};
118 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 \subsection{Boundary Conditions}
122 \label{SEC BOUNDARY COND}
123 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 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
127 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 in the form;
132 \begin{equation}
133 \kappa \frac{\partial T}{\partial x} = 0
134 \end{equation}
135 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 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 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 \begin{equation}
144 \nabla T \cdot n = \frac{\partial T}{\partial n} \; .
145 \end{equation}
146 A condition of the type \refEq{NEUMAN 1} defines a \textbf{Neuman boundary condition} for the PDE.
147
148 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 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 Conveniently, \esc makes default assumption on the boundary conditions which the user may modify where appropriate.
157 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 \begin{equation}\label{NEUMAN 2b}
167 \kappa \frac{\partial T}{\partial x} = 0
168 \end{equation}
169 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
171 \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 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 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 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 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 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 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
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 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 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 elements being used. It therefore important that you find the right balance between the demand in accuracy and acceptable resource usage.
215
216 In general, one can thinks about a domain object as a composition of nodes and elements.
217 As shown in \reffig{fig:fs}, an element is defined by the nodes used to describe its vertices.
218 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 \reffig{fig:fs}.
233
234 This distinction between different representations of spatial distributed values
235 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 These transformations, which are called \textbf{interpolation} are invoked automatically by \esc if needed.
242
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 thermal conductivities $kappa$. A very powerful technique to define these types of PDE
246 coefficients is tagging. Blocks of materials and boundaries can be named and values can be defined on subregions based on their names.
247 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
249
250 \subsection{A Clarification for the 1D Case}
251 \label{SEC: 1D CLARIFICATION}
252 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;
253 \begin{equation}\label{eqn:commonform2D}
254 -A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}}
255 -A\hackscore{01}\frac{\partial^{2}u}{\partial x\partial y}
256 -A\hackscore{10}\frac{\partial^{2}u}{\partial y\partial x}
257 -A\hackscore{11}\frac{\partial^{2}u}{\partial y^{2}}
258 + Du = f
259 \end{equation}
260 Notice that for the higher dimensional case $A$ becomes a matrix. It is also
261 important to notice that the usage of the Nabla operator creates
262 a compact formulation which is also independent from the spatial dimension.
263 So to make the general PDE \refEq{eqn:commonform2D} one dimensional as
264 shown in \refEq{eqn:commonform} we need to set
265 \begin{equation}
266 A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0
267 \end{equation}
268
269
270 \subsection{Developing a PDE Solution Script}
271 \label{sec:key}
272 \sslist{onedheatdiffbase.py}
273 We will write a simple \pyt script which uses the \modescript, \modfinley and \modmpl modules.
274 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.}
275 that we will require.
276 \begin{python}
277 from esys.escript import *
278 # This defines the LinearPDE module as LinearPDE
279 from esys.escript.linearPDEs import LinearPDE
280 # This imports the rectangle domain function from finley.
281 from esys.finley import Rectangle
282 # A useful unit handling package which will make sure all our units
283 # match up in the equations under SI.
284 from esys.escript.unitsSI import *
285 \end{python}
286 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.
287
288 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.
289
290 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.
291 \begin{python}
292 mx = 500.*m #meters - model length
293 my = 100.*m #meters - model width
294 ndx = 50 # mesh steps in x direction
295 ndy = 1 # mesh steps in y direction
296 boundloc = mx/2 # location of boundary between the two blocks
297 \end{python}
298 The material constants and the temperature variables must also be defined. For the granite in the model they are defined as:
299 \begin{python}
300 #PDE related
301 rho = 2750. *kg/m**3 #kg/m^{3} density of iron
302 cp = 790.*J/(kg*K) # J/Kg.K thermal capacity
303 rhocp = rho*cp
304 kappa = 2.2*W/m/K # watts/m.Kthermal conductivity
305 qH=0 * J/(sec*m**3) # J/(sec.m^{3}) no heat source
306 T1=20 * Celsius # initial temperature at Block 1
307 T2=2273. * Celsius # base temperature at Block 2
308 \end{python}
309 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:
310 \begin{python}
311 t=0 * day #our start time, usually zero
312 tend=1. * day # - time to end simulation
313 outputs = 200 # number of time steps required.
314 h=(tend-t)/outputs #size of time step
315 #user warning statement
316 print "Expected Number of time outputs is: ", (tend-t)/h
317 i=0 #loop counter
318 \end{python}
319 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:
320 \begin{python}
321 #generate domain using rectangle
322 blocks = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy)
323 \end{python}
324 \verb blocks now describes a domain in the manner of Section \ref{ss:domcon}. T
325
326 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.
327 \begin{python}
328 mypde=LinearPDE(blocks)
329 A=zeros((2,2)))
330 A[0,0]=kappa
331 mypde.setValue(A=A, D=rhocp/h)
332 \end{python}
333 In a many cases it may be possible to decrease the computational time of the solver if the PDE is symmetric.
334 Symmetry of a PDE is defined by;
335 \begin{equation}\label{eqn:symm}
336 A\hackscore{jl}=A\hackscore{lj}
337 \end{equation}
338 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;
339 \begin{python}
340 myPDE.setSymmetryOn()
341 \end{python}
342 Next we need to establish the initial temperature distribution \verb|T|. We need to
343 assign the value \verb|T1| to all sample points left to the contact interface at $x\hackscore{0}=\frac{mx}{2}$
344 and the value \verb|T2| right to the contact interface. \esc
345 provides the \verb|whereNegative| function to construct this. In fact,
346 \verb|whereNegative| returns the value $1$ at those sample points where the argument
347 has a negative value. Otherwise zero is returned. If \verb|x| are the $x\hackscore{0}$
348 coordinates of the sample points used to represent the temperature distribution
349 then \verb|x[0]-boundloc| gives us a negative value for
350 all sample points left to the interface and non-negative value to
351 the right of the interface. So with;
352 \begin{python}
353 # ... set initial temperature ....
354 T= T1*whereNegative(x[0]-boundloc)+T2*(1-whereNegative(x[0]-boundloc))
355 \end{python}
356 we get the desired temperature distribution. To get the actual sample points \verb|x| we use
357 the \verb|getX()| method of the function space \verb|Solution(blocks)|
358 which is used to represent the solution of a PDE;
359 \begin{python}
360 x=Solution(blocks).getX()
361 \end{python}
362 As \verb|x| are the sample points for the function space \verb|Solution(blocks)|
363 the initial temperature \verb|T| is using these sample points for representation.
364 Although \esc is trying to be forgiving with the choice of sample points and to convert
365 where necessary the adjustment of the function space is not always possible. So it is
366 advisable to make a careful choice on the function space used.
367
368 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.
369 \begin{python}
370 while t < tend:
371 i+=1 #increment the counter
372 t+=h #increment the current time
373 mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients
374 T=mypde.getSolution() #get the PDE solution
375 totE = integrate(rhocp*T) #get the total heat (energy) in the system
376 \end{python}
377 The last statement in this script calculates the total energy in the system as volume integral
378 of $\rho c\hackscore{p} T$ over the block. As the blocks are insulated no energy should be get lost or added.
379 The total energy should stay constant for the example discussed here.
380
381 \subsection{Running the Script}
382 The script presented so for is available under
383 \verb|onedheatdiffbase.py|. You can edit this file with your favourite text editor.
384 On most operating systems\footnote{The you can use \texttt{run-escript} launcher is not supported under {\it MS Windows} yet.} you can use the \program{run-escript} command
385 to launch {\it escript} scripts. For the example script use;
386 \begin{verbatim}
387 run-escript onedheatdiffbase.py
388 \end{verbatim}
389 The program will print a progress report. Alternatively, you can use
390 the python interpreter directly;
391 \begin{verbatim}
392 python onedheatdiffbase.py
393 \end{verbatim}
394 if the system is configured correctly (Please talk to your system administrator).
395
396 \begin{figure}
397 \begin{center}
398 \includegraphics[width=4in]{figures/ttblockspyplot150}
399 \caption{Total Energy in the Blocks over Time (in seconds).}
400 \label{fig:onedheatout1}
401 \end{center}
402 \end{figure}
403
404 \subsection{Plotting the Total Energy}
405 \sslist{onedheatdiff001.py}
406
407 \esc does not include its own plotting capabilities. However, it is possible to use a variety of free \pyt packages for visualisation.
408 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.
409 The \mpl package is a component of SciPy\footnote{\url{http://www.scipy.org}} and is good for basic graphs and plots.
410 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/}}
411 which bases on the \verb|VTK| toolkit. We will discuss the usage of \verb|VTK| based
412 visualization in Chapter~\ref{Sec:2DHD} where will discuss a two dimensional PDE.
413
414 For our simple problem we have two plotting tasks: Firstly we are interested in showing the
415 behaviour of the total energy over time and secondly in how the temperature distribution within the block is
416 developing over time. Lets start with the first task.
417
418 The trick is to create a record of the time marks and the corresponding total energies observed.
419 \pyt provides the concept of lists for this. Before
420 the time loop is opened we create empty lists for the time marks \verb|t_list| and the total energies \verb|E_list|.
421 After the new temperature as been calculated by solving the PDE we append the new time marker and total energy
422 to the corresponding list using the \verb|append| method. With these modifications the script looks as follows:
423 \begin{python}
424 t_list=[]
425 E_list=[]
426 # ... start iteration:
427 while t<tend:
428 t+=h
429 mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients
430 T=mypde.getSolution() #get the PDE solution
431 totE=integrate(rhocp*T)
432 t_list.append(t) # add current time mark to record
433 E_list.append(totE) # add current total energy to record
434 \end{python}
435 To plot $t$ over $totE$ we use the \mpl a module contained within \pylab which needs to be loaded before used;
436 \begin{python}
437 import pylab as pl # plotting package.
438 \end{python}
439 Here we are not using the \verb|from pylab import *| in order to avoid name clashes for function names
440 within \esc.
441
442 The following statements are added to the script after the time loop has been completed;
443 \begin{python}
444 pl.plot(t_list,E_list)
445 pl.title("Total Energy")
446 pl.axis([0,max(t_list),0,max(E_list)*1.1])
447 pl.savefig("totE.png")
448 \end{python}
449 The first statement hands over the time marks and corresponding total energies to the plotter.
450 The second statment is setting the title for the plot. The third statement
451 sets the axis ranges. In most cases these are set appropriately by the plotter.
452 The last statement renders the plot and writes the
453 result into the file \verb|totE.png| which can be displayed by (almost) any image viewer.
454 As expected the total energy is constant over time, see \reffig{fig:onedheatout1}.
455
456 \subsection{Plotting the Temperature Distribution}
457 \label{sec: plot T}
458 \sslist{onedheatdiff001b.py}
459 For plotting the spatial distribution of the temperature we need to modify the strategy we have used
460 for the total energy. Instead of producing a final plot at the end we will generate a
461 picture at each time step which can be browsed as slide show or composed to a movie.
462 The first problem we encounter is that if we produce an image in each time step we need
463 to make sure that the images previously generated are not overwritten.
464
465 To develop an incrementing file name we can use the following convention. It is convenient to
466 put all image file showing the same variable - in our case the temperature distribution -
467 into a separate directory. As part of the \verb|os| module\footnote{The \texttt{os} module provides
468 a powerful interface to interact with the operating system, see \url{http://docs.python.org/library/os.html}.} \pyt
469 provides the \verb|os.path.join| command to build file and
470 directory names in a platform independent way. Assuming that
471 \verb|save_path| is name of directory we want to put the results the command is;
472 \begin{python}
473 import os
474 os.path.join(save_path, "tempT%03d.png"%i )
475 \end{python}
476 where \verb|i| is the time step counter.
477 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 sub-folder called \verb data would be defined by;
478 \begin{verbatim}
479 save_path = "data"
480 \end{verbatim}
481 while a sub-folder of \verb data called \verb onedheatdiff001 would be defined by;
482 \begin{verbatim}
483 save_path = os.path.join("data","onedheatdiff001")
484 \end{verbatim}
485 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 increment our file names with the value \verb|i| denoting a incrementing counter. The sub-string \verb %03d does this by defining the following parameters;
486 \begin{itemize}
487 \item \verb 0 becomes the padding number;
488 \item \verb 3 tells us the amount of padding numbers that are required; and
489 \item \verb d indicates the end of the \verb % operator.
490 \end{itemize}
491 To increment the file name a \verb %i is required directly after the operation the string is involved in. When correctly implemented the output files from this command would be place in the directory defined by \verb save_path as;
492 \begin{verbatim}
493 blockspyplot.png
494 blockspyplot.png
495 blockspyplot.png
496 ...
497 \end{verbatim}
498 and so on.
499
500 A sub-folder check/constructor is available in \esc. The command;
501 \begin{verbatim}
502 mkDir(save_path)
503 \end{verbatim}
504 will check for the existence of \verb save_path and if missing, make the required directories.
505
506 We start by modifying our solution script from before.
507 Prior to the \verb|while| 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
508 the temperature as a \pyt list of tuples or a \numpy array as requested by the plotting function.
509 We need to convert thearray \verb|x| previously set as \verb|Solution(blocks).getX()| into a \pyt list
510 and then to a \numpy array. The $x\hackscore{0}$ component is then extracted via an array slice to the variable \verb|plx|;
511 \begin{python}
512 import numpy as np # array package.
513 #convert solution points for plotting
514 plx = x.toListOfTuples()
515 plx = np.array(plx) # convert to tuple to numpy array
516 plx = plx[:,0] # extract x locations
517 \end{python}
518
519 \begin{figure}
520 \begin{center}
521 \includegraphics[width=4in]{figures/blockspyplot001}
522 \includegraphics[width=4in]{figures/blockspyplot050}
523 \includegraphics[width=4in]{figures/blockspyplot200}
524 \caption{Temperature ($T$) distribution in the blocks at time steps $1$, $50$ and $200$.}
525 \label{fig:onedheatout}
526 \end{center}
527 \end{figure}
528
529 For each time step we will generate a plot of the temperature distribution and save each to a file. We use the same
530 techniques provided by \mpl as we have used to plot the total energy over time.
531 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.
532 Finally, the figure is saved to a \verb|*.png| file and cleared for the following iteration.
533 \begin{python}
534 # ... start iteration:
535 while t<tend:
536 ....
537 T=mypde.getSolution() #get the PDE solution
538 tempT = T.toListOfTuples() # convert to a tuple
539 pl.plot(plx,tempT) # plot solution
540 # set scale (Temperature should be between Tref and T0)
541 pl.axis([0,mx,Tref*.9,T0*1.1])
542 # add title
543 pl.title("Temperature across the blocks at time %e minutes"%(t/day))
544 #save figure to file
545 pl.savefig(os.path.join(save_path,"tempT","blockspyplot%03d.png") %i)
546 \end{python}
547 Some results are shown in \reffig{fig:onedheatout}.
548
549 \subsection{Make a video}
550 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.
551 \begin{python}
552 # compile the *.png files to create a *.avi videos that show T change
553 # with time. This operation uses Linux mencoder. For other operating
554 # systems it is possible to use your favourite video compiler to
555 # convert image files to videos.
556
557 os.system("mencoder mf://"+save_path+"/tempT"+"/*.png -mf type=png:\
558 w=800:h=600:fps=25 -ovc lavc -lavcopts vcodec=mpeg4 -oac copy -o \
559 onedheatdiff001tempT.avi")
560 \end{python}
561

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