ViewVC logotype

Contents of /trunk/doc/cookbook/example01.tex

Parent Directory Parent Directory | Revision Log Revision Log

Revision 2979 - (show annotations)
Tue Mar 9 02:54:32 2010 UTC (9 years, 10 months ago) by ahallam
File MIME type: application/x-tex
File size: 37735 byte(s)
cookbook review final final 3.1 - artak and tony corrections
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
11 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14 \begin{figure}[h!]
15 \centerline{\includegraphics[width=4.in]{figures/onedheatdiff001}}
16 \caption{Example 1: Temperature differential along a single interface between
17 two granite blocks.}
18 \label{fig:onedgbmodel}
19 \end{figure}
21 \section{Example 1: One Dimensional Heat Diffusion in Granite}
22 \label{Sec:1DHDv00}
24 The first model consists of two blocks of isotropic material, for instance
25 granite, sitting next to each other.
26 Initial temperature in \textit{Block 1} is \verb|T1| and in \textit{Block 2} is
27 \verb|T2|.
28 We assume that the system is insulated.
29 What would happen to the temperature distribution in each block over time?
30 Intuition tells us that heat will be transported from the hotter block to the
31 cooler one until both
32 blocks have the same temperature.
34 \subsection{1D Heat Diffusion Equation}
35 We can model the heat distribution of this problem over time using one
36 dimensional heat diffusion equation\footnote{A detailed discussion on how the
37 heat diffusion equation is derived can be found at
38 \url{
39 http://online.redwoods.edu/instruct/darnold/DEProj/sp02/AbeRichards/paper.pdf}};
40 which is defined as:
41 \begin{equation}
42 \rho c\hackscore p \frac{\partial T}{\partial t} - \kappa \frac{\partial^{2}
43 T}{\partial x^{2}} = q\hackscore H
44 \label{eqn:hd}
45 \end{equation}
46 where $\rho$ is the material density, $c\hackscore p$ is the specific heat and
47 $\kappa$ is the thermal
48 conductivity\footnote{A list of some common thermal conductivities is available
49 from Wikipedia
50 \url{http://en.wikipedia.org/wiki/List_of_thermal_conductivities}}. Here we
51 assume that these material
52 parameters are \textbf{constant}.
53 The heat source is defined by the right hand side of \refEq{eqn:hd} as
54 $q\hackscore{H}$; this can take the form of a constant or a function of time and
55 space. For example $q\hackscore{H} = q\hackscore{0}e^{-\gamma t}$ where we have
56 the output of our heat source decaying with time. There are also two partial
57 derivatives in \refEq{eqn:hd}; $\frac{\partial T}{\partial t}$ describes the
58 change in temperature with time while $\frac{\partial ^2 T}{\partial x^2}$ is
59 the spatial change of temperature. As there is only a single spatial dimension
60 to our problem, our temperature solution $T$ is only dependent on the time $t$
61 and our signed distance from the the block-block interface $x$.
63 \subsection{PDEs and the General Form}
64 It is possible to solve PDE \refEq{eqn:hd} analytically and obtain an exact
65 solution to our problem. However, it is not always practical to solve the
66 problem this way. Alternatively, computers can be used to find the solution. To
67 do this, a numerical approach is required to discretise
68 the PDE \refEq{eqn:hd} across time and space, this reduces the problem to a
69 finite number of equations for a finite number of spatial points and time steps.
70 These parameters together define the model. While discretisation introduces
71 approximations and a degree of error, a sufficiently sampled model is generally
72 accurate enough to satisfy the accuracy requirements for the final solution.
74 Firstly, we discretise the PDE \refEq{eqn:hd} in time. This leaves us with a
75 steady linear PDE which involves spatial derivatives only and needs to be solved
76 in each time step to progress in time. \esc can help us here.
78 For time discretization we use the Backwards Euler approximation
79 scheme\footnote{see \url{http://en.wikipedia.org/wiki/Euler_method}}. It is
80 based on the
81 approximation
82 \begin{equation}
83 \frac{\partial T(t)}{\partial t} \approx \frac{T(t)-T(t-h)}{h}
84 \label{eqn:beuler}
85 \end{equation}
86 for $\frac{\partial T}{\partial t}$ at time $t$
87 where $h$ is the time step size. This can also be written as;
88 \begin{equation}
89 \frac{\partial T}{\partial t}(t^{(n)}) \approx \frac{T^{(n)} - T^{(n-1)}}{h}
90 \label{eqn:Tbeuler}
91 \end{equation}
92 where the upper index $n$ denotes the n\textsuperscript{th} time step. So one
93 has
94 \begin{equation}
95 \begin{array}{rcl}
96 t^{(n)} & = & t^{(n-1)}+h \\
97 T^{(n)} & = & T(t^{(n-1)}) \\
98 \end{array}
99 \label{eqn:Neuler}
100 \end{equation}
101 Substituting \refEq{eqn:Tbeuler} into \refEq{eqn:hd} we get;
102 \begin{equation}
103 \frac{\rho c\hackscore p}{h} (T^{(n)} - T^{(n-1)}) - \kappa \frac{\partial^{2}
104 T^{(n)}}{\partial x^{2}} = q\hackscore H
105 \label{eqn:hddisc}
106 \end{equation}
107 Notice that we evaluate the spatial derivative term at the current time
108 $t^{(n)}$ - therefore the name \textbf{backward Euler} scheme. Alternatively,
109 one can evaluate the spatial derivative term at the previous time $t^{(n-1)}$.
110 This
111 approach is called the \textbf{forward Euler} scheme. This scheme can provide
112 some computational advantages, which
113 are not discussed here. However, the \textbf{forward Euler} scheme has a major
114 disadvantage. Namely, depending on the
115 material parameters as well as the domain discretization of the spatial
116 derivative term, the time step size $h$ needs to be chosen sufficiently small to
117 achieve a stable temperature when progressing in time. Stabiliy is achieved if
118 the temperature does not grow beyond its initial bounds and become
119 non-physical.
120 The backward Euler scheme, which we use here, is unconditionally stable meaning
121 that under the assumption of a
122 physically correct problem set-up the temperature approximation remains physical
123 for all time steps.
124 The user needs to keep in mind that the discretization error introduced by
125 \refEq{eqn:beuler}
126 is sufficiently small, thus a good approximation of the true temperature is
127 computed. It is
128 therefore very important that any results are viewed with caution. For example,
129 one may compare the results for different time and spatial step sizes.
131 To get the temperature $T^{(n)}$ at time $t^{(n)}$ we need to solve the linear
132 differential equation \refEq{eqn:hddisc} which only includes spatial
133 derivatives. To solve this problem
134 we want to use \esc.
136 In \esc any given PDE can be described by the general form. For the purpose of
137 this introduction we illustrate a simpler version of the general form for full
138 linear PDEs which is available in the \esc user's guide. A simplified form that
139 suits our heat diffusion problem\footnote{The form in the \esc users guide which
140 uses the Einstein convention is written as
141 $-(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}+D u =Y$}
142 is described by;
143 \begin{equation}\label{eqn:commonform nabla}
144 -\nabla\cdot(A\cdot\nabla u) + Du = f
145 \end{equation}
146 where $A$, $D$ and $f$ are known values and $u$ is the unknown solution. The
147 symbol $\nabla$ which is called the \textit{Nabla operator} or \textit{del
148 operator} represents
149 the spatial derivative of its subject - in this case $u$. Lets assume for a
150 moment that we deal with a one-dimensional problem then ;
151 \begin{equation}
152 \nabla = \frac{\partial}{\partial x}
153 \end{equation}
154 and we can write \refEq{eqn:commonform nabla} as;
155 \begin{equation}\label{eqn:commonform}
156 -A\frac{\partial^{2}u}{\partial x^{2}} + Du = f
157 \end{equation}
158 if $A$ is constant. To match this simplified general form to our problem
159 \refEq{eqn:hddisc}
160 we rearrange \refEq{eqn:hddisc};
161 \begin{equation}
162 \frac{\rho c\hackscore p}{h} T^{(n)} - \kappa \frac{\partial^2 T^{(n)}}{\partial
163 x^2} = q\hackscore H + \frac{\rho c\hackscore p}{h} T^{(n-1)}
164 \label{eqn:hdgenf}
165 \end{equation}
166 The PDE is now in a form that satisfies \refEq{eqn:commonform nabla} which is
167 required for \esc to solve our PDE. This can be done by generating a solution
168 for successive increments in the time nodes $t^{(n)}$ where
169 $t^{(0)}=0$ and $t^{(n)}=t^{(n-1)}+h$ where $h>0$ is the step size and assumed
170 to be constant.
171 In the following the upper index ${(n)}$ refers to a value at time $t^{(n)}$.
172 Finally, by comparing \refEq{eqn:hdgenf} with \refEq{eqn:commonform} one can see
173 that;
174 \begin{equation}\label{ESCRIPT SET}
175 u=T^{(n)};
176 A = \kappa; D = \frac{\rho c \hackscore{p}}{h}; f = q \hackscore{H} + \frac{\rho
177 c\hackscore p}{h} T^{(n-1)}
178 \end{equation}
180 \subsection{Boundary Conditions}
182 With the PDE sufficiently modified, consideration must now be given to the
183 boundary conditions of our model. Typically there are two main types of boundary
184 conditions known as \textbf{Neumann} and \textbf{Dirichlet} boundary
185 conditions\footnote{More information on Boundary Conditions is available at
186 Wikipedia \url{http://en.wikipedia.org/wiki/Boundary_conditions}},
187 respectively.
188 A \textbf{Dirichlet boundary condition} is conceptually simpler and is used to
189 prescribe a known value to the unknown solution (in our example the temperature)
190 on parts of the boundary or on the entire boundary of the region of interest.
191 We discuss Dirichlet boundary condition in our second example presented in
192 Section~\ref{Sec:1DHDv0}.
194 However, for this example we have made the model assumption that the system is
195 insulated, so we need
196 to add an appropriate boundary condition to prevent
197 any loss or inflow of energy at the boundary of our domain. Mathematically this
198 is expressed by prescribing
199 the heat flux $\kappa \frac{\partial T}{\partial x}$ to zero. In our simplified
200 one dimensional model this is expressed
201 in the form;
202 \begin{equation}
203 \kappa \frac{\partial T}{\partial x} = 0
204 \end{equation}
205 or in a more general case as
206 \begin{equation}\label{NEUMAN 1}
207 \kappa \nabla T \cdot n = 0
208 \end{equation}
209 where $n$ is the outer normal field \index{outer normal field} at the surface
210 of the domain.
211 The $\cdot$ (dot) refers to the dot product of the vectors $\nabla T$ and $n$.
212 In fact, the term $\nabla T \cdot n$ is the normal derivative of
213 the temperature $T$. Other notations used here are\footnote{The \esc notation
214 for the normal
215 derivative is $T\hackscore{,i} n\hackscore i$.};
216 \begin{equation}
217 \nabla T \cdot n = \frac{\partial T}{\partial n} \; .
218 \end{equation}
219 A condition of the type \refEq{NEUMAN 1} defines a \textbf{Neuman boundary
220 condition} for the PDE.
222 The PDE \refEq{eqn:hdgenf}
223 and the Neuman boundary condition~\ref{eqn:hdgenf} (potentially together with
224 the Dirichlet boundary conditions) define a \textbf{boundary value problem}.
225 It is the nature of a boundary value problem to allow making statements about
226 the solution in the
227 interior of the domain from information known on the boundary only. In most
228 cases
229 we use the term partial differential equation but in fact it is a boundary value
230 problem.
231 It is important to keep in mind that boundary conditions need to be complete and
232 consistent in the sense that
233 at any point on the boundary either a Dirichlet or a Neuman boundary condition
234 must be set.
236 Conveniently, \esc makes default assumption on the boundary conditions which the
237 user may modify where appropriate.
238 For a problem of the form in~\refEq{eqn:commonform nabla} the default
239 condition\footnote{In the form of the \esc users guide which is using the
240 Einstein convention is written as
241 $n\hackscore{j}A\hackscore{jl} u\hackscore{,l}=0$.} is;
242 \begin{equation}\label{NEUMAN 2}
243 -n\cdot A \cdot\nabla u = 0
244 \end{equation}
245 which is used everywhere on the boundary. Again $n$ denotes the outer normal
246 field.
247 Notice that the coefficient $A$ is the same as in the \esc
248 PDE~\ref{eqn:commonform nabla}.
249 With the settings for the coefficients we have already identified in
250 \refEq{ESCRIPT SET} this
251 condition translates into
252 \begin{equation}\label{NEUMAN 2b}
253 \kappa \frac{\partial T}{\partial x} = 0
254 \end{equation}
255 for the boundary of the domain. This is identical to the Neuman boundary
256 condition we want to set. \esc will take care of this condition for us. We
257 discuss the Dirichlet boundary condition later.
259 \subsection{Outline of the Implementation}
260 \label{sec:outline}
261 To solve the heat diffusion equation (\refEq{eqn:hd}) we write a simple \pyt
262 script. At this point we assume that you have some basic understanding of the
263 \pyt programming language. If not, there are some pointers and links available
264 in Section \ref{sec:escpybas}. The script (discussed in \refSec{sec:key}) has
265 four major steps. Firstly, we need to define the domain where we want to
266 calculate the temperature. For our problem this is the joint blocks of granite
267 which has a rectangular shape. Secondly, we need to define the PDE to solve in
268 each time step to get the updated temperature. Thirdly, we need to define the
269 coefficients of the PDE and finally we need to solve the PDE. The last two steps
270 need to be repeated until the final time marker has been reached. The work flow
271 described in \reffig{fig:wf}.
272 % \begin{enumerate}
273 % \item create domain
274 % \item create PDE
275 % \item while end time not reached:
276 % \begin{enumerate}
277 % \item set PDE coefficients
278 % \item solve PDE
279 % \item update time marker
280 % \end{enumerate}
281 % \item end of calculation
282 % \end{enumerate}
284 \begin{figure}[h!]
285 \centering
286 \includegraphics[width=1in]{figures/workflow.png}
287 \caption{Workflow for developing an \esc model and solution.}
288 \label{fig:wf}
289 \end{figure}
291 In the terminology of \pyt, the domain and PDE are represented by
292 \textbf{objects}. The nice feature of an object is that it is defined by its
293 usage and features
294 rather than its actual representation. So we will create a domain object to
295 describe the geometry of the two
296 granite blocks. Then we define PDEs and spatially distributed values such as the
297 temperature
298 on this domain. Similarly, to define a PDE object we use the fact that one needs
299 only to define the coefficients of the PDE and solve the PDE. The PDE object has
300 advanced features, but these are not required in simple cases.
303 \begin{figure}[t]
304 \centering
305 \includegraphics[width=6in]{figures/functionspace.pdf}
306 \caption{\esc domain construction overview}
307 \label{fig:fs}
308 \end{figure}
310 \subsection{The Domain Constructor in \esc}
311 \label{ss:domcon}
312 Whilst it is not strictly relevant or necessary, a better understanding of
313 how values are spatially distributed (\textit{e.g.} Temperature) and how PDE
314 coefficients are interpreted in \esc can be helpful.
316 There are various ways to construct domain objects. The simplest form is a
317 rectangular shaped region with a length and height. There is
318 a ready to use function for this named \verb rectangle(). Besides the spatial
319 dimensions this function requires to specify the number of
320 elements or cells to be used along the length and height, see \reffig{fig:fs}.
321 Any spatially distributed value
322 and the PDE is represented in discrete form using this element
323 representation\footnote{We use the finite element method (FEM), see
324 \url{http://en.wikipedia.org/wiki/Finite_element_method} for details.}.
325 Therefore we will have access to an approximation of the true PDE solution
326 only.
327 The quality of the approximation depends - besides other factors- mainly on the
328 number of elements being used. In fact, the
329 approximation becomes better when more elements are used. However, computational
330 cost grows with the number of
331 elements being used. It is therefore important that you find the right balance
332 between the demand in accuracy and acceptable resource usage.
334 In general, one can think about a domain object as a composition of nodes and
335 elements.
336 As shown in \reffig{fig:fs}, an element is defined by the nodes that are used to
337 describe its vertices.
338 To represent spatial distributed values the user can use
339 the values at the nodes, at the elements in the interior of the domain or at the
340 elements located at the surface of the domain.
341 The different approach used to represent values is called \textbf{function
342 space} and is attached to all objects
343 in \esc representing a spatial distributed value such as the solution of a PDE.
344 The three
345 function spaces we use at the moment are;
346 \begin{enumerate}
347 \item the nodes, called by \verb|ContinuousFunction(domain)| ;
348 \item the elements/cells, called by \verb|Function(domain)| ; and
349 \item the boundary, called by \verb|FunctionOnBoundary(domain)| .
350 \end{enumerate}
351 A function space object such as \verb|ContinuousFunction(domain)| has the method
352 \verb|getX| attached to it. This method returns the
353 location of the so-called \textbf{sample points} used to represent values of the
354 particular function space. So the
355 call \verb|ContinuousFunction(domain).getX()| will return the coordinates of the
356 nodes used to describe the domain while
357 the \verb|Function(domain).getX()| returns the coordinates of numerical
358 integration points within elements, see
359 \reffig{fig:fs}.
361 This distinction between different representations of spatially distributed
362 values
363 is important in order to be able to vary the degrees of smoothness in a PDE
364 problem.
365 The coefficients of a PDE do not need to be continuous, thus this qualifies as a
366 \verb|Function()| type.
367 On the other hand a temperature distribution must be continuous and needs to be
368 represented with a \verb|ContinuousFunction()| function space.
369 An influx may only be defined at the boundary and is therefore a \verb
370 FunctionOnBoundary() object.
371 \esc allows certain transformations of the function spaces. A \verb
372 ContinuousFunction() can be transformed into a \verb|FunctionOnBoundary()|
373 or \verb|Function()|. On the other hand there is not enough information in a
374 \verb FunctionOnBoundary() to transform it to a \verb ContinuousFunction() .
375 These transformations, which are called \textbf{interpolation} are invoked
376 automatically by \esc if needed.
378 Later in this introduction we discuss how
379 to define specific areas of geometry with different materials which are
380 represented by different material coefficients such the
381 thermal conductivities $\kappa$. A very powerful technique to define these types
382 of PDE
383 coefficients is tagging. Blocks of materials and boundaries can be named and
384 values can be defined on subregions based on their names.
385 This is a method for simplifying PDE coefficient and flux definitions. It makes
386 scripting much easier and we will discuss this technique in
390 \subsection{A Clarification for the 1D Case}
392 It is necessary for clarification that we revisit our general PDE from
393 \refeq{eqn:commonform nabla} for two dimensional domain. \esc is inherently
394 designed to solve problems that are greater than one dimension and so
395 \refEq{eqn:commonform nabla} needs to be read as a higher dimensional problem.
396 In the case of two spatial dimensions the \textit{Nabla operator} has in fact
397 two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial
398 y})$. Assuming the coefficient $A$ is constant, the \refEq{eqn:commonform nabla}
399 takes the following form;
400 \begin{equation}\label{eqn:commonform2D}
401 -A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}}
402 -A\hackscore{01}\frac{\partial^{2}u}{\partial x\partial y}
403 -A\hackscore{10}\frac{\partial^{2}u}{\partial y\partial x}
404 -A\hackscore{11}\frac{\partial^{2}u}{\partial y^{2}}
405 + Du = f
406 \end{equation}
407 Notice that for the higher dimensional case $A$ becomes a matrix. It is also
408 important to notice that the usage of the Nabla operator creates
409 a compact formulation which is also independent from the spatial dimension.
410 To make the general PDE \refEq{eqn:commonform2D} one dimensional as
411 shown in \refEq{eqn:commonform} we need to set
412 \begin{equation}
413 A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0
414 \end{equation}
417 \subsection{Developing a PDE Solution Script}
418 \label{sec:key}
419 \sslist{example01a.py}
420 We write a simple \pyt script which uses the \modescript, \modfinley and \modmpl
421 modules.
422 By developing a script for \esc, the heat diffusion equation can be solved at
423 successive time steps for a predefined period using our general form
424 \refEq{eqn:hdgenf}. Firstly it is necessary to import all the
425 libraries\footnote{The libraries contain predefined scripts that are required to
426 solve certain problems, these can be simple like sine and cosine functions or
427 more complicated like those from our \esc library.}
428 that we will require.
429 \begin{python}
430 from esys.escript import *
431 # This defines the LinearPDE module as LinearPDE
432 from esys.escript.linearPDEs import LinearPDE
433 # This imports the rectangle domain function from finley.
434 from esys.finley import Rectangle
435 # A useful unit handling package which will make sure all our units
436 # match up in the equations under SI.
437 from esys.escript.unitsSI import *
438 \end{python}
439 It is generally a good idea to import all of the \modescript library, although
440 if the functions and classes required are known they can be specified
441 individually. The function \verb|LinearPDE| has been imported explicitly for
442 ease of use later in the script. \verb|Rectangle| is going to be our type of
443 domain. The module \verb unitsSI provides support for SI unit definitions with
444 our variables.
446 Once our library dependencies have been established, defining the problem
447 specific variables is the next step. In general the number of variables needed
448 will vary between problems. These variables belong to two categories. They are
449 either directly related to the PDE and can be used as inputs into the \esc
450 solver, or they are script variables used to control internal functions and
451 iterations in our problem. For this PDE there are a number of constants which
452 need values. Firstly, the domain upon which we wish to solve our problem needs
453 to be defined. There are different types of domains in \modescript which we
454 demonstrate in later tutorials but for our granite blocks, we simply use a
455 rectangular domain.
457 Using a rectangular domain simplifies our granite blocks (which would in reality
458 be a \textit{3D} object) into a single dimension. The granite blocks will have a
459 lengthways cross section that looks like a rectangle. As a result we do not
460 need to model the volume of the block due to symmetry. There are four arguments
461 we must consider when we decide to create a rectangular domain, the domain
462 \textit{length}, \textit{width} and \textit{step size} in each direction. When
463 defining the size of our problem it will help us determine appropriate values
464 for our model arguments. If we make our dimensions large but our step sizes very
465 small we increase the accuracy of our solution. Unfortunately we also increase
466 the number of calculations that must be solved per time step. This means more
467 computational time is required to produce a solution. In this \textit{1D}
468 problem, the bar is defined as being 1 metre long. An appropriate step size
469 \verb|ndx| would be 1 to 10\% of the length. Our \verb|ndy| need only be 1, this
470 is because our problem stipulates no partial derivatives in the $y$ direction.
471 Thus the temperature does not vary with $y$. Hence, the model parameters can be
472 defined as follows; note we have used the \verb unitsSI convention to make sure
473 all our input units are converted to SI.
474 \begin{python}
475 mx = 500.*m #meters - model length
476 my = 100.*m #meters - model width
477 ndx = 50 # mesh steps in x direction
478 ndy = 1 # mesh steps in y direction
479 boundloc = mx/2 # location of boundary between the two blocks
480 \end{python}
481 The material constants and the temperature variables must also be defined. For
482 the granite in the model they are defined as:
483 \begin{python}
484 #PDE related
485 rho = 2750. *kg/m**3 #kg/m^{3} density of iron
486 cp = 790.*J/(kg*K) # J/Kg.K thermal capacity
487 rhocp = rho*cp
488 kappa = 2.2*W/m/K # watts/m.Kthermal conductivity
489 qH=0 * J/(sec*m**3) # J/(sec.m^{3}) no heat source
490 T1=20 * Celsius # initial temperature at Block 1
491 T2=2273. * Celsius # base temperature at Block 2
492 \end{python}
493 Finally, to control our script we will have to specify our timing controls and
494 where we would like to save the output from the solver. This is simple enough:
495 \begin{python}
496 t=0 * day #our start time, usually zero
497 tend=1. * day # - time to end simulation
498 outputs = 200 # number of time steps required.
499 h=(tend-t)/outputs #size of time step
500 #user warning statement
501 print "Expected Number of time outputs is: ", (tend-t)/h
502 i=0 #loop counter
503 \end{python}
504 Now that we know our inputs we will build a domain using the
505 \verb Rectangle() function from \verb finley . The four arguments allow us to define our domain
506 \verb model as:
507 \begin{python}
508 #generate domain using rectangle
509 blocks = Rectangle(l0=mx,l1=my,n0=ndx, n1=ndy)
510 \end{python}
511 \verb blocks now describes a domain in the manner of Section \ref{ss:domcon}.
513 With a domain and all our required variables established, it is now possible to
514 set up our PDE so that it can be solved by \esc. The first step is to define the
515 type of PDE that we are trying to solve in each time step. In this example it is
516 a single linear PDE\footnote{in comparison to a system of PDEs which we discuss
517 later.}. We also need to state the values of our general form variables.
518 \begin{python}
519 mypde=LinearPDE(blocks)
520 A=zeros((2,2)))
521 A[0,0]=kappa
522 mypde.setValue(A=A, D=rhocp/h)
523 \end{python}
524 In a many cases it may be possible to decrease the computational time of the
525 solver if the PDE is symmetric.
526 Symmetry of a PDE is defined by;
527 \begin{equation}\label{eqn:symm}
528 A\hackscore{jl}=A\hackscore{lj}
529 \end{equation}
530 Symmetry is only dependent on the $A$ coefficient in the general form and the
531 other coefficients $D$ as well as the right hand side $Y$. From the above
532 definition we can see that our PDE is symmetric. The \verb LinearPDE class
533 provides the method \method{checkSymmetry} to check if the given PDE is
534 symmetric. As our PDE is symmetrical we enable symmetry via;
535 \begin{python}
536 myPDE.setSymmetryOn()
537 \end{python}
538 Next we need to establish the initial temperature distribution \verb|T|. We need
539 to
540 assign the value \verb|T1| to all sample points left to the contact interface at
541 $x\hackscore{0}=\frac{mx}{2}$
542 and the value \verb|T2| right to the contact interface. \esc
543 provides the \verb|whereNegative| function to construct this. In fact,
544 \verb|whereNegative| returns the value $1$ at those sample points where the
545 argument
546 has a negative value. Otherwise zero is returned. If \verb|x| are the
547 $x\hackscore{0}$
548 coordinates of the sample points used to represent the temperature distribution
549 then \verb|x[0]-boundloc| gives us a negative value for
550 all sample points left to the interface and non-negative value to
551 the right of the interface. So with;
552 \begin{python}
553 # ... set initial temperature ....
554 T= T1*whereNegative(x[0]-boundloc)+T2*(1-whereNegative(x[0]-boundloc))
555 \end{python}
556 we get the desired temperature distribution. To get the actual sample points
557 \verb|x| we use
558 the \verb|getX()| method of the function space \verb|Solution(blocks)|
559 which is used to represent the solution of a PDE;
560 \begin{python}
561 x=Solution(blocks).getX()
562 \end{python}
563 As \verb|x| are the sample points for the function space
564 \verb|Solution(blocks)|
565 the initial temperature \verb|T| is using these sample points for
566 representation.
567 Although \esc is trying to be forgiving with the choice of sample points and to
568 convert
569 where necessary the adjustment of the function space is not always possible. So
570 it is
571 advisable to make a careful choice on the function space used.
573 Finally we initialise an iteration loop to solve our PDE for all the time steps
574 we specified in the variable section. As the right hand side of the general form
575 is dependent on the previous values for temperature \verb T across the bar this
576 must be updated in the loop. Our output at each time step is \verb T the heat
577 distribution and \verb totT the total heat in the system.
578 \begin{python}
579 while t < tend:
580 i+=1 #increment the counter
581 t+=h #increment the current time
582 mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients
583 T=mypde.getSolution() #get the PDE solution
584 totE = integrate(rhocp*T) #get the total heat (energy) in the system
585 \end{python}
586 The last statement in this script calculates the total energy in the system as
587 volume integral
588 of $\rho c\hackscore{p} T$ over the block. As the blocks are insulated no energy
589 should be get lost or added.
590 The total energy should stay constant for the example discussed here.
592 \subsection{Running the Script}
593 The script presented so far is available under
594 \verb|example01a.py|. You can edit this file with your favourite text editor.
595 On most operating systems\footnote{The you can use \texttt{run-escript} launcher
596 is not supported under {\it MS Windows} yet.} you can use the
597 \program{run-escript} command
598 to launch {\it escript} scripts. For the example script use;
599 \begin{verbatim}
600 run-escript example01a.py
601 \end{verbatim}
602 The program will print a progress report. Alternatively, you can use
603 the python interpreter directly;
604 \begin{verbatim}
605 python example01a.py
606 \end{verbatim}
607 if the system is configured correctly (Please talk to your system
608 administrator).
610 \begin{figure}
611 \begin{center}
612 \includegraphics[width=4in]{figures/ttblockspyplot150}
613 \caption{Example 1b: Total Energy in the Blocks over Time (in seconds).}
614 \label{fig:onedheatout1}
615 \end{center}
616 \end{figure}
618 \subsection{Plotting the Total Energy}
619 \sslist{example01b.py}
621 \esc does not include its own plotting capabilities. However, it is possible to
622 use a variety of free \pyt packages for visualisation.
623 Two types will be demonstrated in this cookbook;
624 \mpl\footnote{\url{http://matplotlib.sourceforge.net/}} and
625 \verb VTK \footnote{\url{http://www.vtk.org/}} visualisation.
626 The \mpl package is a component of SciPy\footnote{\url{http://www.scipy.org}}
627 and is good for basic graphs and plots.
628 For more complex visualisation tasks in particular, two and three dimensional
629 problems we recommend the use of more advanced tools. For instance, \mayavi
630 \footnote{\url{http://code.enthought.com/projects/mayavi/}}
631 which is based upon the \verb|VTK| toolkit. The usage of \verb|VTK| based
632 visualization is discussed in Chapter~\ref{Sec:2DHD} which focusses on a two
633 dimensional PDE.
635 For our simple granite block problem, we have two plotting tasks. Firstly, we
636 are interested in showing the
637 behaviour of the total energy over time and secondly, how the temperature
638 distribution within the block is
639 developing over time. Lets start with the first task.
641 The trick is to create a record of the time marks and the corresponding total
642 energies observed.
643 \pyt provides the concept of lists for this. Before
644 the time loop is opened we create empty lists for the time marks \verb|t_list|
645 and the total energies \verb|E_list|.
646 After the new temperature has been calculated by solving the PDE we append the
647 new time marker and the total energy value for that time
648 to the corresponding list using the \verb|append| method. With these
649 modifications our script looks as follows:
650 \begin{python}
651 t_list=[]
652 E_list=[]
653 # ... start iteration:
654 while t<tend:
655 t+=h
656 mypde.setValue(Y=qH+rhocp/h*T) # set variable PDE coefficients
657 T=mypde.getSolution() #get the PDE solution
658 totE=integrate(rhocp*T)
659 t_list.append(t) # add current time mark to record
660 E_list.append(totE) # add current total energy to record
661 \end{python}
662 To plot $t$ over $totE$ we use \mpl a module contained within \pylab which needs
663 to be loaded before used;
664 \begin{python}
665 import pylab as pl # plotting package.
666 \end{python}
667 Here we are not using the \verb|from pylab import *| in order to avoid name
668 clashes for function names
669 within \esc.
671 The following statements are added to the script after the time loop has been
672 completed;
673 \begin{python}
674 pl.plot(t_list,E_list)
675 pl.title("Total Energy")
676 pl.axis([0,max(t_list),0,max(E_list)*1.1])
677 pl.savefig("totE.png")
678 \end{python}
679 The first statement hands over the time marks and corresponding total energies
680 to the plotter.
681 The second statment is setting the title for the plot. The third statement
682 sets the axis ranges. In most cases these are set appropriately by the plotter.
684 The last statement renders the plot and writes the
685 result into the file \verb|totE.png| which can be displayed by (almost) any
686 image viewer.
687 As expected the total energy is constant over time, see
688 \reffig{fig:onedheatout1}.
690 \subsection{Plotting the Temperature Distribution}
691 \label{sec: plot T}
692 \sslist{example01c.py}
693 For plotting the spatial distribution of the temperature we need to modify the
694 strategy we have used
695 for the total energy. Instead of producing a final plot at the end we will
696 generate a
697 picture at each time step which can be browsed as a slide show or composed into
698 a movie.
699 The first problem we encounter is that if we produce an image at each time step
700 we need
701 to make sure that the images previously generated are not overwritten.
703 To develop an incrementing file name we can use the following convention. It is
704 convenient to
705 put all image file showing the same variable - in our case the temperature
706 distribution -
707 into a separate directory. As part of the \verb|os| module\footnote{The
708 \texttt{os} module provides
709 a powerful interface to interact with the operating system, see
710 \url{http://docs.python.org/library/os.html}.} \pyt
711 provides the \verb|os.path.join| command to build file and
712 directory names in a platform independent way. Assuming that
713 \verb|save_path| is name of directory we want to put the results the command
714 is;
715 \begin{python}
716 import os
717 os.path.join(save_path, "tempT%03d.png"%i )
718 \end{python}
719 where \verb|i| is the time step counter.
720 There are two arguments to the \verb join command. The
721 \verb save_path variable is a predefined string pointing to the directory we want to save our
722 data, for example a single sub-folder called \verb data would be defined by;
723 \begin{verbatim}
724 save_path = "data"
725 \end{verbatim}
726 while a sub-folder of \verb data called \verb example01 would be defined by;
727 \begin{verbatim}
728 save_path = os.path.join("data","example01")
729 \end{verbatim}
730 The second argument of \verb join \xspace contains a string which is the file
731 name or subdirectory name. We can use the operator \verb|%| to use the value of
732 \verb|i| as part of our filename. The sub-string \verb %03d indicates that we
733 want to substitude a value into the name;
734 \begin{itemize}
735 \item \verb 0 means that small numbers should have leading zeroes;
736 \item \verb 3 means that numbers should be written using at least 3 digits;
737 and
738 \item \verb d means that the value to substitute will be an integer.
739 \end{itemize}
740 To actually substitute the value of \verb|i| into the name write \verb %i after
741 the string.
742 When done correctly, the output files from this command would be place in the
743 directory defined by \verb save_path as;
744 \begin{verbatim}
745 blockspyplot001.png
746 blockspyplot002.png
747 blockspyplot003.png
748 ...
749 \end{verbatim}
750 and so on.
752 A sub-folder check/constructor is available in \esc. The command;
753 \begin{verbatim}
754 mkDir(save_path)
755 \end{verbatim}
756 will check for the existence of \verb save_path and if missing, make the
757 required directories.
759 We start by modifying our solution script.
760 Prior to the \verb|while| loop we will need to extract our finite solution
761 points to a data object that is compatible with \mpl. First we create the node
762 coordinates of the sample points used to represent
763 the temperature as a \pyt list of tuples or a \numpy array as requested by the
764 plotting function.
765 We need to convert the array \verb|x| previously set as
766 \verb|Solution(blocks).getX()| into a \pyt list
767 and then to a \numpy array. The $x\hackscore{0}$ component is then extracted via
768 an array slice to the variable \verb|plx|;
769 \begin{python}
770 import numpy as np # array package.
771 #convert solution points for plotting
772 plx = x.toListOfTuples()
773 plx = np.array(plx) # convert to tuple to numpy array
774 plx = plx[:,0] # extract x locations
775 \end{python}
777 \begin{figure}
778 \begin{center}
779 \includegraphics[width=4in]{figures/blockspyplot001}
780 \includegraphics[width=4in]{figures/blockspyplot050}
781 \includegraphics[width=4in]{figures/blockspyplot200}
782 \caption{Example 1c: Temperature ($T$) distribution in the blocks at time steps
783 $1$, $50$ and $200$.}
784 \label{fig:onedheatout}
785 \end{center}
786 \end{figure}
788 We use the same techniques provided by \mpl as we have used to plot the total
789 energy over time.
790 For each time step we generate a plot of the temperature distribution and save
791 each to a file.
792 The following is appended to the end of the \verb while loop and creates one
793 figure of the temperature distribution. We start by converting the solution to a
794 tuple and then plotting this against our \textit{x coordinates} \verb plx we
795 have generated before. We add a title to the diagram before it is rendered into
796 a file.
797 Finally, the figure is saved to a \verb|*.png| file and cleared for the
798 following iteration.
799 \begin{python}
800 # ... start iteration:
801 while t<tend:
802 ....
803 T=mypde.getSolution() #get the PDE solution
804 tempT = T.toListOfTuples() # convert to a tuple
805 pl.plot(plx,tempT) # plot solution
806 # set scale (Temperature should be between Tref and T0)
807 pl.axis([0,mx,Tref*.9,T0*1.1])
808 # add title
809 pl.title("Temperature across the blocks at time %e minutes"%(t/day))
810 #save figure to file
811 pl.savefig(os.path.join(save_path,"tempT","blockspyplot%03d.png") %i)
812 \end{python}
813 Some results are shown in \reffig{fig:onedheatout}.
815 \subsection{Make a video}
816 Our saved plots from the previous section can be cast into a video using the
817 following command appended to the end of the script. The \verb mencoder command
818 is Linux only, so other platform users need to use an alternative video encoder.
819 \begin{python}
820 # compile the *.png files to create a *.avi videos that show T change
821 # with time. This operation uses Linux mencoder. For other operating
822 # systems it is possible to use your favourite video compiler to
823 # convert image files to videos.
825 os.system("mencoder mf://"+save_path+"/tempT"+"/*.png -mf type=png:\
826 w=800:h=600:fps=25 -ovc lavc -lavcopts vcodec=mpeg4 -oac copy -o \
827 example01tempT.avi")
828 \end{python}

  ViewVC Help
Powered by ViewVC 1.1.26