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1 ksteube 1811
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4 ksteube 1811 % Copyright (c) 2003-2008 by University of Queensland
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6     % http://www.uq.edu.au/esscc
7 gross 625 %
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12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13 jgs 102
14 ksteube 1811
15 jgs 121 \section{The First Steps}
16 jgs 102 \label{FirstSteps}
17    
18    
19 artak 1971
20 ksteube 1316 In this chapter we give an introduction how to use \escript to solve
21     a partial differential equation \index{partial differential equation} (PDE \index{partial differential equation!PDE}). We assume you are at least a little familiar with Python. The knowledge presented at the Python tutorial at \url{http://docs.python.org/tut/tut.html}
22     is more than sufficient.
23 jgs 102
24 gross 2363 The PDE \index{partial differential equation} we wish to solve is the Poisson equation \index{Poisson equation}
25 jgs 102 \begin{equation}
26     -\Delta u =f
27     \label{eq:FirstSteps.1}
28     \end{equation}
29 ksteube 1316 for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$,
30 jgs 102 is the unit square
31     \begin{equation}
32     \Omega=[0,1]^2=\{ (x\hackscore 0;x\hackscore 1) | 0\le x\hackscore{0} \le 1 \mbox{ and } 0\le x\hackscore{1} \le 1 \}
33     \label{eq:FirstSteps.1b}
34     \end{equation}
35 jgs 107 The domain is shown in \fig{fig:FirstSteps.1}.
36 artak 1971 \begin{figure} [h!]
37     \centerline{\includegraphics[width=\figwidth]{figures/FirstStepDomain}}
38     \caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.}
39     \label{fig:FirstSteps.1}
40     \end{figure}
41 jgs 102
42 ksteube 1316 $\Delta$ denotes the Laplace operator\index{Laplace operator}, which is defined by
43 jgs 102 \begin{equation}
44     \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1}
45     \label{eq:FirstSteps.1.1}
46     \end{equation}
47 ksteube 1316 where, for any function $u$ and any direction $i$, $u\hackscore{,i}$
48 jgs 107 denotes the partial derivative \index{partial derivative} of $u$ with respect to $i$.
49 ksteube 1316 \footnote{You
50 jgs 102 may be more familiar with the Laplace operator\index{Laplace operator} being written
51     as $\nabla^2$, and written in the form
52     \begin{equation*}
53 jgs 110 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2}
54 jgs 102 + \frac{\partial^2 u}{\partial x\hackscore 1^2}
55     \end{equation*}
56     and \eqn{eq:FirstSteps.1} as
57     \begin{equation*}
58     -\nabla^2 u = f
59     \end{equation*}
60     }
61 jgs 107 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect
62 artak 1971 to the index. To get a more compact form we will write $u\hackscore{,ij}=(u\hackscore {,i})\hackscore{,j}$
63 jgs 102 which leads to
64     \begin{equation}
65     \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii}
66     \label{eq:FirstSteps.1.1b}
67     \end{equation}
68 ksteube 1316 We often find that use
69     of nested $\sum$ symbols makes formulas cumbersome, and we use the more
70     convenient Einstein summation convention \index{summation convention}. This
71     drops the $\sum$ sign and assumes that a summation is performed over any repeated index.
72     For instance we write
73 jgs 102 \begin{eqnarray}
74     x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\
75     x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\
76     u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\
77 jgs 107 x\hackscore{ij}u\hackscore{i,j}=\sum\hackscore{j=0}^2\sum\hackscore{i=0}^2 x\hackscore{ij}u\hackscore{i,j} \\
78 jgs 102 \label{eq:FirstSteps.1.1c}
79     \end{eqnarray}
80     With the summation convention we can write the Poisson equation \index{Poisson equation} as
81     \begin{equation}
82     - u\hackscore{,ii} =1
83     \label{eq:FirstSteps.1.sum}
84     \end{equation}
85 lkettle 575 where $f=1$ in this example.
86    
87 jgs 102 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$
88     of the solution $u$ shall be zero, ie. $u$ shall fulfill
89     the homogeneous Neumann boundary condition\index{Neumann
90     boundary condition!homogeneous}
91     \begin{equation}
92     n\hackscore{i} u\hackscore{,i}= 0 \;.
93     \label{eq:FirstSteps.2}
94     \end{equation}
95     $n=(n\hackscore{i})$ denotes the outer normal field
96     of the domain, see \fig{fig:FirstSteps.1}. Remember that we
97     are applying the Einstein summation convention \index{summation convention}, i.e
98 jgs 107 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} +
99     n\hackscore{1} u\hackscore{,1}$.
100 jgs 102 \footnote{Some readers may familiar with the notation
101 artak 1971 $
102 jgs 102 \frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i}
103 artak 1971 $
104 jgs 102 for the normal derivative.}
105     The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the
106     set $\Gamma^N$ which is the top and right edge of the domain:
107     \begin{equation}
108     \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \}
109     \label{eq:FirstSteps.2b}
110     \end{equation}
111 jgs 107 On the bottom and the left edge of the domain which is defined
112 jgs 102 as
113     \begin{equation}
114     \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \}
115     \label{eq:FirstSteps.2c}
116     \end{equation}
117     the solution shall be identically zero:
118     \begin{equation}
119     u=0 \; .
120     \label{eq:FirstSteps.2d}
121     \end{equation}
122 jgs 107 This kind of boundary condition is called a homogeneous Dirichlet boundary condition
123 jgs 102 \index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together
124     with the Neumann boundary condition \eqn{eq:FirstSteps.2} and
125     Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so
126     called boundary value
127 jgs 107 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for
128 artak 1971 the unknown function~$u$.
129 jgs 102
130    
131 artak 1971 \begin{figure}[h]
132 jfenwick 2335 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepMesh}}
133 jgs 102 \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here
134     each element is a quadrilateral and described by four nodes, namely
135     the corner points. The solution is interpolated by a bi-linear
136     polynomial.}
137     \label{fig:FirstSteps.2}
138     \end{figure}
139    
140     In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical
141     methods have to be used construct an approximation of the solution
142     $u$. Here we will use the finite element method\index{finite element
143     method} (FEM\index{finite element
144     method!FEM}). The basic idea is to fill the domain with a
145 jgs 107 set of points called nodes. The solution is approximated by its
146 jgs 102 values on the nodes\index{finite element
147 lkettle 573 method!nodes}. Moreover, the domain is subdivided into smaller
148     sub-domains called elements \index{finite element
149 jgs 102 method!element}. On each element the solution is
150     represented by a polynomial of a certain degree through its values at
151     the nodes located in the element. The nodes and its connection through
152     elements is called a mesh\index{finite element
153 jgs 107 method!mesh}. \fig{fig:FirstSteps.2} shows an
154 jgs 102 example of a FEM mesh with four elements in the $x_0$ and four elements
155     in the $x_1$ direction over the unit square.
156 gross 2370 For more details we refer the reader to the literature, for instance \Ref{Zienc,NumHand}.
157 jgs 102
158 gross 2370 The \escript solver we want to use to solve this problem is embedded into the python interpreter language. So you can solve the problem interactively but you will learn quickly
159     that is more efficient to use scripts which you can edit with your favourite editor.
160     To enter the escript environment you use \program{escript} command\footnote{\program{escript} is not available under Windows yet. If you run under windows you can just use the
161     \program{python} command and the \env{OMP_NUM_THREADS} environment variable to control the number
162     of threads.}:
163     \begin{verbatim}
164     escript
165     \end{verbatim}
166     which will pass you on to the python prompt
167     \begin{verbatim}
168     Python 2.5.2 (r252:60911, Oct 5 2008, 19:29:17)
169     [GCC 4.3.2] on linux2
170     Type "help", "copyright", "credits" or "license" for more information.
171     >>>
172     \end{verbatim}
173     Here you can use all available python commands and language features, for instance
174     \begin{python}
175     >>> x=2+3
176     >>> print "2+3=",x
177     2+3= 5
178     \end{python}
179     We refer to the python users guide if you not familar with python.
180    
181 jgs 102 \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}.
182     (We will discuss a more general form of a PDE \index{partial differential equation!PDE}
183     that can be defined through the \LinearPDE class later). The instantiation of
184     a \Poisson class object requires the specification of the domain $\Omega$. In \escript
185     the \Domain class objects are used to describe the geometry of a domain but it also
186     contains information about the discretization methods and the actual solver which is used
187     to solve the PDE. Here we are using the FEM library \finley \index{finite element
188     method}. The following statements create the \Domain object \var{mydomain} from the
189     \finley method \method{Rectangle}
190     \begin{python}
191 ksteube 1316 from esys.finley import Rectangle
192     mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
193 jgs 102 \end{python}
194     In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and
195     the right, upper corner at $(\var{l0},\var{l1})=(1,1)$.
196 jgs 107 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and
197 jgs 102 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and
198     other \Domain generators within the \finley module,
199     see \Chap{CHAPTER ON FINLEY}.
200    
201 jgs 107 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and
202 jgs 102 the right hand side $f$ of the PDE to constant $1$:
203     \begin{python}
204 ksteube 1316 from esys.escript.linearPDEs import Poisson
205     mypde = Poisson(mydomain)
206     mypde.setValue(f=1)
207 jgs 102 \end{python}
208     We have not specified any boundary condition but the
209     \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann
210     boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary
211     condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$
212     and any constant $C$ the function $u+C$ becomes a solution as well. We have to add
213     a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done
214 jgs 107 by defining a characteristic function \index{characteristic function}
215     which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set
216 jgs 102 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c},
217 gross 565 we need to construct a function \var{gammaD} which is positive for the cases $x\hackscore{0}=0$ or $x\hackscore{1}=0$. To get
218 ksteube 1316 an object \var{x} which contains the coordinates of the nodes in the domain use
219 jgs 102 \begin{python}
220 ksteube 1316 x=mydomain.getX()
221 jgs 102 \end{python}
222 gross 660 The method \method{getX} of the \Domain \var{mydomain}
223 jgs 107 gives access to locations
224 ksteube 1316 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which will be
225     discussed in \Chap{ESCRIPT CHAP} in more detail. What we need to know here is that
226 gross 660
227 ksteube 1316 \var{x} has \Rank (number of dimensions) and a \Shape (list of dimensions) which can be viewed by
228 gross 565 calling the \method{getRank} and \method{getShape} methods:
229     \begin{python}
230 ksteube 1316 print "rank ",x.getRank(),", shape ",x.getShape()
231 gross 565 \end{python}
232 ksteube 1316 This will print something like
233 gross 565 \begin{python}
234 ksteube 1316 rank 1, shape (2,)
235 gross 565 \end{python}
236     The \Data object also maintains type information which is represented by the
237     \FunctionSpace of the object. For instance
238     \begin{python}
239 ksteube 1316 print x.getFunctionSpace()
240 gross 565 \end{python}
241     will print
242     \begin{python}
243 ksteube 1316 Function space type: Finley_Nodes on FinleyMesh
244 gross 565 \end{python}
245 ksteube 1316 which tells us that the coordinates are stored on the nodes of (rather than on points in the interior of) a \finley mesh.
246 gross 565 To get the $x\hackscore{0}$ coordinates of the locations we use the
247     statement
248     \begin{python}
249 ksteube 1316 x0=x[0]
250 gross 565 \end{python}
251     Object \var{x0}
252     is again a \Data object now with \Rank $0$ and
253     \Shape $()$. It inherits the \FunctionSpace from \var{x}:
254     \begin{python}
255 ksteube 1316 print x0.getRank(),x0.getShape(),x0.getFunctionSpace()
256 gross 565 \end{python}
257     will print
258     \begin{python}
259 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh
260 gross 565 \end{python}
261 ksteube 1316 We can now construct a function \var{gammaD} which is only non-zero on the bottom and left edges
262     of the domain with
263 gross 565 \begin{python}
264 ksteube 1316 from esys.escript import whereZero
265     gammaD=whereZero(x[0])+whereZero(x[1])
266 gross 565 \end{python}
267 ksteube 1316
268     \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (almost) equal to zero
269 jgs 107 and $0$ elsewhere.
270 gross 565 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is
271 jgs 107 equal to zero and $0$ elsewhere.
272 gross 565 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])}
273 artak 1971 gives a function on the domain \var{mydomain} which is strictly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero.
274 gross 565 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from
275     \begin{python}
276 ksteube 1316 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace()
277 gross 565 \end{python}
278     one gets
279     \begin{python}
280 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh
281 gross 565 \end{python}
282 ksteube 1316 An additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the
283 jgs 102 characteristic function \index{characteristic function} of the locations
284     of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
285     are set. The complete definition of our example is now:
286     \begin{python}
287 ksteube 1316 from esys.linearPDEs import Poisson
288     x = mydomain.getX()
289     gammaD = whereZero(x[0])+whereZero(x[1])
290     mypde = Poisson(domain=mydomain)
291     mypde.setValue(f=1,q=gammaD)
292 jgs 102 \end{python}
293 lkettle 573 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package.
294 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its
295 jgs 102 \method{getSolution}.
296    
297 ksteube 1316 Now we can write the script to solve our Poisson problem
298 jgs 102 \begin{python}
299 ksteube 1316 from esys.escript import *
300     from esys.escript.linearPDEs import Poisson
301     from esys.finley import Rectangle
302     # generate domain:
303     mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
304     # define characteristic function of Gamma^D
305     x = mydomain.getX()
306     gammaD = whereZero(x[0])+whereZero(x[1])
307     # define PDE and get its solution u
308     mypde = Poisson(domain=mydomain)
309     mypde.setValue(f=1,q=gammaD)
310     u = mypde.getSolution()
311     # write u to an external file
312     saveVTK("u.xml",sol=u)
313 jgs 102 \end{python}
314 ksteube 1316 The entire code is available as \file{poisson.py} in the \ExampleDirectory
315 jgs 102
316 ksteube 1316 The last statement writes the solution (tagged with the name "sol") to a file named \file{u.xml} in
317     \VTK file format.
318 gross 2370 Now you may run the script using the \escript environment
319     and visualize the solution using \mayavi:
320 ksteube 1316 \begin{verbatim}
321 gross 2370 escript poisson.py
322 ksteube 1316 mayavi -d u.xml -m SurfaceMap
323     \end{verbatim}
324     See \fig{fig:FirstSteps.3}.
325    
326 jgs 102 \begin{figure}
327 jfenwick 2335 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepResult}}
328 lkettle 573 \caption{Visualization of the Poisson Equation Solution for $f=1$}
329 jgs 102 \label{fig:FirstSteps.3}
330     \end{figure}
331    

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