/[escript]/trunk/doc/user/firststep.tex
ViewVC logotype

Annotation of /trunk/doc/user/firststep.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 573 - (hide annotations)
Thu Mar 2 00:42:53 2006 UTC (13 years, 8 months ago) by lkettle
File MIME type: application/x-tex
File size: 14277 byte(s)
I have made a few changes to the documentation for the online reference
guide for the Poisson and diffusion examples.

1 jgs 102 % $Id$
2    
3 jgs 121 \section{The First Steps}
4 jgs 102 \label{FirstSteps}
5    
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}
11    
12 jgs 107 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}.
15 jgs 102
16 jgs 107 The PDE \index{partial differential equation} we wish to solve is the Poisson equation \index{Poisson equation}
17 jgs 102 \begin{equation}
18     -\Delta u =f
19     \label{eq:FirstSteps.1}
20     \end{equation}
21 jgs 107 for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$
22 jgs 102 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 jgs 107 The domain is shown in \fig{fig:FirstSteps.1}.
28 jgs 102
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 jgs 107 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 jgs 102 \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 jgs 110 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2}
41 jgs 102 + \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 jgs 107 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect
49 jgs 102 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 jgs 107 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 jgs 102 ("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 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} \\
65 jgs 102 \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 jgs 107 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} +
84     n\hackscore{1} u\hackscore{,1}$.
85 jgs 102 \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 jgs 107 On the bottom and the left edge of the domain which is defined
97 jgs 102 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 jgs 107 This kind of boundary condition is called a homogeneous Dirichlet boundary condition
108 jgs 102 \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 jgs 107 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for
113     the unknown
114 jgs 102 function $u$.
115    
116    
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}
125    
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 jgs 107 set of points called nodes. The solution is approximated by its
132 jgs 102 values on the nodes\index{finite element
133 lkettle 573 method!nodes}. Moreover, the domain is subdivided into smaller
134     sub-domains called elements \index{finite element
135 jgs 102 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 jgs 107 method!mesh}. \fig{fig:FirstSteps.2} shows an
140 jgs 102 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}.
144    
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 jgs 107 from esys.finley import Rectangle
156     mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
157 jgs 102 \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 jgs 107 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and
161 jgs 102 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and
162     other \Domain generators within the \finley module,
163     see \Chap{CHAPTER ON FINLEY}.
164    
165 jgs 107 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and
166 jgs 102 the right hand side $f$ of the PDE to constant $1$:
167     \begin{python}
168 gross 565 from esys.escript.linearPDEs import Poisson
169 jgs 107 mypde = Poisson(mydomain)
170     mypde.setValue(f=1)
171 jgs 102 \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 jgs 107 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 jgs 102 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c},
181 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
182     an object \var{x} which represents locations in the domain one uses
183 jgs 102 \begin{python}
184 lkettle 573 x=mydomain.getX()
185 jgs 102 \end{python}
186 gross 565 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
188    
189 jgs 107 In the first statement, the method \method{getX} of the \Domain \var{mydomain}
190     gives access to locations
191 gross 565 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 lkettle 573 from esys.escript import whereZero
230 gross 565 gammaD=whereZero(x[0])+whereZero(x[1])
231     \end{python}
232     where
233     \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (allmost) equal to zero
234 jgs 107 and $0$ elsewhere.
235 gross 565 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is
236 jgs 107 equal to zero and $0$ elsewhere.
237 gross 565 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])}
238     gives a function on the domain \var{mydomain} which is exactly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero.
239     Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from
240     \begin{python}
241     print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace()
242     \end{python}
243     one gets
244     \begin{python}
245     0 () Function space type: Finley_Nodes on FinleyMesh
246     \end{python}
247 jgs 107 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the
248 jgs 102 characteristic function \index{characteristic function} of the locations
249     of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
250     are set. The complete definition of our example is now:
251     \begin{python}
252 jgs 107 from esys.linearPDEs import Poisson
253 jgs 102 x = mydomain.getX()
254 gross 565 gammaD = whereZero(x[0])+whereZero(x[1])
255 jgs 107 mypde = Poisson(domain=mydomain)
256 lkettle 573 mypde.setValue(f=1,q=gammaD)
257 jgs 102 \end{python}
258 lkettle 573 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package.
259 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its
260 jgs 102 \method{getSolution}.
261    
262     Now we can write the script to solve our test problem (Remember that
263 gross 569 lines starting with '\#' are comment lines in Python) (available as \file{poisson.py}
264 jgs 102 in the \ExampleDirectory):
265     \begin{python}
266 gross 565 from esys.escript import *
267 gross 569 from esys.escript.linearPDEs import Poisson
268 jgs 107 from esys.finley import Rectangle
269 jgs 102 # generate domain:
270 jgs 107 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
271 jgs 102 # define characteristic function of Gamma^D
272     x = mydomain.getX()
273 gross 565 gammaD = whereZero(x[0])+whereZero(x[1])
274 jgs 102 # define PDE and get its solution u
275 gross 565 mypde = Poisson(domain=mydomain)
276     mypde.setValue(f=1,q=gammaD)
277 jgs 102 u = mypde.getSolution()
278     # write u to an external file
279 gross 565 saveVTK("u.xml",sol=u)
280 jgs 102 \end{python}
281 gross 565 The last statement writes the solution tagged with the name "sol" to the external file \file{u.xml} in
282     \VTK file format. \VTK is a software library
283 jgs 102 for the visualization of scientific, engineering and analytical data and is freely available
284 lkettle 573 from \url{http://www.vtk.org}. There are a variety of graphical user interfaces
285 gross 565 for \VTK available, for instance \mayavi which can be downloaded from \url{http://mayavi.sourceforge.net/} but is also available on most
286     \LINUX distributions.
287 jgs 102
288     \begin{figure}
289     \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}}
290 lkettle 573 \caption{Visualization of the Poisson Equation Solution for $f=1$}
291 jgs 102 \label{fig:FirstSteps.3}
292     \end{figure}
293    
294 jgs 107 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE
295 gross 569 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
296 jgs 102 script from any shell using the command:
297 gross 565 \begin{python}
298 gross 569 python poisson.py
299 gross 565 \end{python}
300 gross 569 After the script has (hopefully successfully) been completed you will find the file \file{u.xml} in the current
301 jgs 102 directory. An easy way to visualize the results is the command
302 gross 565 \begin{python}
303     mayavi -d u.xml -m SurfaceMap &
304     \end{python}
305     to show the results, see \fig{fig:FirstSteps.3}.

Properties

Name Value
svn:eol-style native
svn:keywords Author Date Id Revision

  ViewVC Help
Powered by ViewVC 1.1.26