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1 % $Id$
3 \section{The First Steps}
4 \label{FirstSteps}
6 \begin{figure}
7 \centerline{\includegraphics[width=\figwidth]{FirstStepDomain}}
8 \caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.}
9 \label{fig:FirstSteps.1}
10 \end{figure}
12 In this chapter we will give an introduction how to use \escript to solve
13 a partial differential equation \index{partial differential equation} (PDE \index{partial differential equation!PDE}). The reader should be familiar with Python. The knowledge presented at the Python tutorial at \url{http://docs.python.org/tut/tut.html}
14 is sufficient. It is helpful if the reader has some basic knowledge of PDEs \index{partial differential equation}.
16 The PDE \index{partial differential equation} we wish to solve is the Poisson equation \index{Poisson equation}
17 \begin{equation}
18 -\Delta u =f
19 \label{eq:FirstSteps.1}
20 \end{equation}
21 for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$
22 is the unit square
23 \begin{equation}
24 \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 \}
25 \label{eq:FirstSteps.1b}
26 \end{equation}
27 The domain is shown in \fig{fig:FirstSteps.1}.
29 $\Delta$ denotes the Laplace operator\index{Laplace operator} which is defined by
30 \begin{equation}
31 \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1}
32 \label{eq:FirstSteps.1.1}
33 \end{equation}
34 where, for any function $w$ and any direction $i$, $u\hackscore{,i}$
35 denotes the partial derivative \index{partial derivative} of $u$ with respect to $i$.
36 \footnote{Some readers
37 may be more familiar with the Laplace operator\index{Laplace operator} being written
38 as $\nabla^2$, and written in the form
39 \begin{equation*}
40 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2}
41 + \frac{\partial^2 u}{\partial x\hackscore 1^2}
42 \end{equation*}
43 and \eqn{eq:FirstSteps.1} as
44 \begin{equation*}
45 -\nabla^2 u = f
46 \end{equation*}
47 }
48 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect
49 to the index. To get a more compact form we will write $w\hackscore{,ij}=(w\hackscore {,i})\hackscore{,j}$
50 which leads to
51 \begin{equation}
52 \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii}
53 \label{eq:FirstSteps.1.1b}
54 \end{equation}
55 In some cases, and we will see examples for this in the next chapter,
56 the usage of the nested $\sum$ symbols blows up the formulas and therefore
57 it is convenient to use the Einstein summation convention \index{summation convention}. This
58 drops the $\sum$ sign and assumes that a summation over a repeated index is performed
59 ("repeated index means summation"). For instance we write
60 \begin{eqnarray}
61 x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\
62 x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\
63 u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\
64 x\hackscore{ij}u\hackscore{i,j}=\sum\hackscore{j=0}^2\sum\hackscore{i=0}^2 x\hackscore{ij}u\hackscore{i,j} \\
65 \label{eq:FirstSteps.1.1c}
66 \end{eqnarray}
67 With the summation convention we can write the Poisson equation \index{Poisson equation} as
68 \begin{equation}
69 - u\hackscore{,ii} =1
70 \label{eq:FirstSteps.1.sum}
71 \end{equation}
72 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$
73 of the solution $u$ shall be zero, ie. $u$ shall fulfill
74 the homogeneous Neumann boundary condition\index{Neumann
75 boundary condition!homogeneous}
76 \begin{equation}
77 n\hackscore{i} u\hackscore{,i}= 0 \;.
78 \label{eq:FirstSteps.2}
79 \end{equation}
80 $n=(n\hackscore{i})$ denotes the outer normal field
81 of the domain, see \fig{fig:FirstSteps.1}. Remember that we
82 are applying the Einstein summation convention \index{summation convention}, i.e
83 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} +
84 n\hackscore{1} u\hackscore{,1}$.
85 \footnote{Some readers may familiar with the notation
86 \begin{equation*}
87 \frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i}
88 \end{equation*}
89 for the normal derivative.}
90 The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the
91 set $\Gamma^N$ which is the top and right edge of the domain:
92 \begin{equation}
93 \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \}
94 \label{eq:FirstSteps.2b}
95 \end{equation}
96 On the bottom and the left edge of the domain which is defined
97 as
98 \begin{equation}
99 \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \}
100 \label{eq:FirstSteps.2c}
101 \end{equation}
102 the solution shall be identically zero:
103 \begin{equation}
104 u=0 \; .
105 \label{eq:FirstSteps.2d}
106 \end{equation}
107 This kind of boundary condition is called a homogeneous Dirichlet boundary condition
108 \index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together
109 with the Neumann boundary condition \eqn{eq:FirstSteps.2} and
110 Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so
111 called boundary value
112 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for
113 the unknown
114 function $u$.
117 \begin{figure}
118 \centerline{\includegraphics[width=\figwidth]{FirstStepMesh}}
119 \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here
120 each element is a quadrilateral and described by four nodes, namely
121 the corner points. The solution is interpolated by a bi-linear
122 polynomial.}
123 \label{fig:FirstSteps.2}
124 \end{figure}
126 In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical
127 methods have to be used construct an approximation of the solution
128 $u$. Here we will use the finite element method\index{finite element
129 method} (FEM\index{finite element
130 method!FEM}). The basic idea is to fill the domain with a
131 set of points called nodes. The solution is approximated by its
132 values on the nodes\index{finite element
133 method!nodes}. Moreover, the domain is subdivided into small,
134 sub-domain called elements \index{finite element
135 method!element}. On each element the solution is
136 represented by a polynomial of a certain degree through its values at
137 the nodes located in the element. The nodes and its connection through
138 elements is called a mesh\index{finite element
139 method!mesh}. \fig{fig:FirstSteps.2} shows an
140 example of a FEM mesh with four elements in the $x_0$ and four elements
141 in the $x_1$ direction over the unit square.
142 For more details we refer the reader to the literature, for instance
143 \Ref{Zienc,NumHand}.
145 \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}.
146 (We will discuss a more general form of a PDE \index{partial differential equation!PDE}
147 that can be defined through the \LinearPDE class later). The instantiation of
148 a \Poisson class object requires the specification of the domain $\Omega$. In \escript
149 the \Domain class objects are used to describe the geometry of a domain but it also
150 contains information about the discretization methods and the actual solver which is used
151 to solve the PDE. Here we are using the FEM library \finley \index{finite element
152 method}. The following statements create the \Domain object \var{mydomain} from the
153 \finley method \method{Rectangle}
154 \begin{python}
155 from esys.finley import Rectangle
156 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
157 \end{python}
158 In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and
159 the right, upper corner at $(\var{l0},\var{l1})=(1,1)$.
160 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and
161 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and
162 other \Domain generators within the \finley module,
163 see \Chap{CHAPTER ON FINLEY}.
165 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and
166 the right hand side $f$ of the PDE to constant $1$:
167 \begin{python}
168 from esys.escript.linearPDEs import Poisson
169 mypde = Poisson(mydomain)
170 mypde.setValue(f=1)
171 \end{python}
172 We have not specified any boundary condition but the
173 \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann
174 boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary
175 condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$
176 and any constant $C$ the function $u+C$ becomes a solution as well. We have to add
177 a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done
178 by defining a characteristic function \index{characteristic function}
179 which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set
180 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c},
181 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
182 an object \var{x} which represents locations in the domain one uses
183 \begin{python}
184 x=mydomain.getX() \;.
185 \end{python}
186 In fact \var{x} is a \Data object which we will learn more about in Chapter~\ref{X}. At this stage we only have to know
187 that \var{x} has a
189 In the first statement, the method \method{getX} of the \Domain \var{mydomain}
190 gives access to locations
191 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which is
192 discussed in Chpater\ref{X} in more details. What we need to know here is that
193 \var{x} has \Rank (=number of dimensions) and a \Shape (=tuple of dimensions) which can be checked by
194 calling the \method{getRank} and \method{getShape} methods:
195 \begin{python}
196 print "rank ",x.getRank(),", shape ",x.getShape()
197 \end{python}
198 will print something like
199 \begin{python}
200 rank 1, shape (2,)
201 \end{python}
202 The \Data object also maintains type information which is represented by the
203 \FunctionSpace of the object. For instance
204 \begin{python}
205 print x.getFunctionSpace()
206 \end{python}
207 will print
208 \begin{python}
209 Function space type: Finley_Nodes on FinleyMesh
210 \end{python}
211 which tells us that the coordinates are stored on the nodes of a \finley mesh.
212 To get the $x\hackscore{0}$ coordinates of the locations we use the
213 statement
214 \begin{python}
215 x0=x[0]
216 \end{python}
217 Object \var{x0}
218 is again a \Data object now with \Rank $0$ and
219 \Shape $()$. It inherits the \FunctionSpace from \var{x}:
220 \begin{python}
221 print x0.getRank(),x0.getShape(),x0.getFunctionSpace()
222 \end{python}
223 will print
224 \begin{python}
225 0 () Function space type: Finley_Nodes on FinleyMesh
226 \end{python}
227 We can now construct the function \var{gammaD} by
228 \begin{python}
229 gammaD=whereZero(x[0])+whereZero(x[1])
230 \end{python}
231 where
232 \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (allmost) equal to zero
233 and $0$ elsewhere.
234 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is
235 equal to zero and $0$ elsewhere.
236 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])}
237 gives a function on the domain \var{mydomain} which is exactly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero.
238 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from
239 \begin{python}
240 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace()
241 \end{python}
242 one gets
243 \begin{python}
244 0 () Function space type: Finley_Nodes on FinleyMesh
245 \end{python}
246 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the
247 characteristic function \index{characteristic function} of the locations
248 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
249 are set. The complete definition of our example is now:
250 \begin{python}
251 from esys.linearPDEs import Poisson
252 x = mydomain.getX()
253 gammaD = whereZero(x[0])+whereZero(x[1])
254 mypde = Poisson(domain=mydomain)
255 mypde = setValue(f=1,q=gammaD)
256 \end{python}
257 The first statement imports the \Poisson class definition form the \linearPDEs module \escript package.
258 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its
259 \method{getSolution}.
261 Now we can write the script to solve our test problem (Remember that
262 lines starting with '\#' are comment lines in Python) (available as \file{poisson.py}
263 in the \ExampleDirectory):
264 \begin{python}
265 from esys.escript import *
266 from esys.escript.linearPDEs import Poisson
267 from esys.finley import Rectangle
268 # generate domain:
269 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
270 # define characteristic function of Gamma^D
271 x = mydomain.getX()
272 gammaD = whereZero(x[0])+whereZero(x[1])
273 # define PDE and get its solution u
274 mypde = Poisson(domain=mydomain)
275 mypde.setValue(f=1,q=gammaD)
276 u = mypde.getSolution()
277 # write u to an external file
278 saveVTK("u.xml",sol=u)
279 \end{python}
280 The last statement writes the solution tagged with the name "sol" to the external file \file{u.xml} in
281 \VTK file format. \VTK is a software library
282 for the visualization of scientific, engineering and analytical data and is freely available
283 from \url{http://www.vtk.org}. There are a variaty of graphical user interfaces
284 for \VTK available, for instance \mayavi which can be downloaded from \url{http://mayavi.sourceforge.net/} but is also available on most
285 \LINUX distributions.
287 \begin{figure}
288 \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}}
289 \caption{Visualization of the Possion Equation Solution for $f=1$}
290 \label{fig:FirstSteps.3}
291 \end{figure}
293 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE
294 for Python, see \url{http://idlefork.sourceforge.net}). If the script file has the name \file{poisson.py} \index{scripts!\file{poisson.py}} you can run the
295 script from any shell using the command:
296 \begin{python}
297 python poisson.py
298 \end{python}
299 After the script has (hopefully successfully) been completed you will find the file \file{u.xml} in the current
300 directory. An easy way to visualize the results is the command
301 \begin{python}
302 mayavi -d u.xml -m SurfaceMap &
303 \end{python}
304 to show the results, see \fig{fig:FirstSteps.3}.


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