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Fri Mar 3 03:33:07 2006 UTC (13 years, 7 months ago) by lkettle
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I have changed some of the documentation and added more explanations for
the online reference guide for esys13. I have modified two of the
example source codes to write out the results for Helmholtz problem and
changed one variable name in the diffusion.py code to avoid confusion.

1 % $Id$
2
3 \section{The First Steps}
4 \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 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
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}.
28
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 where $f=1$ in this example.
73
74 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$
75 of the solution $u$ shall be zero, ie. $u$ shall fulfill
76 the homogeneous Neumann boundary condition\index{Neumann
77 boundary condition!homogeneous}
78 \begin{equation}
79 n\hackscore{i} u\hackscore{,i}= 0 \;.
80 \label{eq:FirstSteps.2}
81 \end{equation}
82 $n=(n\hackscore{i})$ denotes the outer normal field
83 of the domain, see \fig{fig:FirstSteps.1}. Remember that we
84 are applying the Einstein summation convention \index{summation convention}, i.e
85 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} +
86 n\hackscore{1} u\hackscore{,1}$.
87 \footnote{Some readers may familiar with the notation
88 \begin{equation*}
89 \frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i}
90 \end{equation*}
91 for the normal derivative.}
92 The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the
93 set $\Gamma^N$ which is the top and right edge of the domain:
94 \begin{equation}
95 \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \}
96 \label{eq:FirstSteps.2b}
97 \end{equation}
98 On the bottom and the left edge of the domain which is defined
99 as
100 \begin{equation}
101 \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \}
102 \label{eq:FirstSteps.2c}
103 \end{equation}
104 the solution shall be identically zero:
105 \begin{equation}
106 u=0 \; .
107 \label{eq:FirstSteps.2d}
108 \end{equation}
109 This kind of boundary condition is called a homogeneous Dirichlet boundary condition
110 \index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together
111 with the Neumann boundary condition \eqn{eq:FirstSteps.2} and
112 Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so
113 called boundary value
114 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for
115 the unknown
116 function $u$.
117
118
119 \begin{figure}
120 \centerline{\includegraphics[width=\figwidth]{FirstStepMesh}}
121 \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here
122 each element is a quadrilateral and described by four nodes, namely
123 the corner points. The solution is interpolated by a bi-linear
124 polynomial.}
125 \label{fig:FirstSteps.2}
126 \end{figure}
127
128 In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical
129 methods have to be used construct an approximation of the solution
130 $u$. Here we will use the finite element method\index{finite element
131 method} (FEM\index{finite element
132 method!FEM}). The basic idea is to fill the domain with a
133 set of points called nodes. The solution is approximated by its
134 values on the nodes\index{finite element
135 method!nodes}. Moreover, the domain is subdivided into smaller
136 sub-domains called elements \index{finite element
137 method!element}. On each element the solution is
138 represented by a polynomial of a certain degree through its values at
139 the nodes located in the element. The nodes and its connection through
140 elements is called a mesh\index{finite element
141 method!mesh}. \fig{fig:FirstSteps.2} shows an
142 example of a FEM mesh with four elements in the $x_0$ and four elements
143 in the $x_1$ direction over the unit square.
144 For more details we refer the reader to the literature, for instance
145 \Ref{Zienc,NumHand}.
146
147 \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}.
148 (We will discuss a more general form of a PDE \index{partial differential equation!PDE}
149 that can be defined through the \LinearPDE class later). The instantiation of
150 a \Poisson class object requires the specification of the domain $\Omega$. In \escript
151 the \Domain class objects are used to describe the geometry of a domain but it also
152 contains information about the discretization methods and the actual solver which is used
153 to solve the PDE. Here we are using the FEM library \finley \index{finite element
154 method}. The following statements create the \Domain object \var{mydomain} from the
155 \finley method \method{Rectangle}
156 \begin{python}
157 from esys.finley import Rectangle
158 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
159 \end{python}
160 In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and
161 the right, upper corner at $(\var{l0},\var{l1})=(1,1)$.
162 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and
163 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and
164 other \Domain generators within the \finley module,
165 see \Chap{CHAPTER ON FINLEY}.
166
167 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and
168 the right hand side $f$ of the PDE to constant $1$:
169 \begin{python}
170 from esys.escript.linearPDEs import Poisson
171 mypde = Poisson(mydomain)
172 mypde.setValue(f=1)
173 \end{python}
174 We have not specified any boundary condition but the
175 \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann
176 boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary
177 condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$
178 and any constant $C$ the function $u+C$ becomes a solution as well. We have to add
179 a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done
180 by defining a characteristic function \index{characteristic function}
181 which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set
182 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c},
183 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
184 an object \var{x} which represents locations in the domain one uses
185 \begin{python}
186 x=mydomain.getX()
187 \end{python}
188 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
189 that \var{x} has a
190
191 In the first statement, the method \method{getX} of the \Domain \var{mydomain}
192 gives access to locations
193 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which is
194 discussed in Chpater\ref{X} in more details. What we need to know here is that
195 \var{x} has \Rank (=number of dimensions) and a \Shape (=tuple of dimensions) which can be checked by
196 calling the \method{getRank} and \method{getShape} methods:
197 \begin{python}
198 print "rank ",x.getRank(),", shape ",x.getShape()
199 \end{python}
200 will print something like
201 \begin{python}
202 rank 1, shape (2,)
203 \end{python}
204 The \Data object also maintains type information which is represented by the
205 \FunctionSpace of the object. For instance
206 \begin{python}
207 print x.getFunctionSpace()
208 \end{python}
209 will print
210 \begin{python}
211 Function space type: Finley_Nodes on FinleyMesh
212 \end{python}
213 which tells us that the coordinates are stored on the nodes of a \finley mesh.
214 To get the $x\hackscore{0}$ coordinates of the locations we use the
215 statement
216 \begin{python}
217 x0=x[0]
218 \end{python}
219 Object \var{x0}
220 is again a \Data object now with \Rank $0$ and
221 \Shape $()$. It inherits the \FunctionSpace from \var{x}:
222 \begin{python}
223 print x0.getRank(),x0.getShape(),x0.getFunctionSpace()
224 \end{python}
225 will print
226 \begin{python}
227 0 () Function space type: Finley_Nodes on FinleyMesh
228 \end{python}
229 We can now construct the function \var{gammaD} by
230 \begin{python}
231 from esys.escript import whereZero
232 gammaD=whereZero(x[0])+whereZero(x[1])
233 \end{python}
234 where
235 \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (allmost) equal to zero
236 and $0$ elsewhere.
237 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is
238 equal to zero and $0$ elsewhere.
239 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])}
240 gives a function on the domain \var{mydomain} which is exactly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero.
241 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from
242 \begin{python}
243 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace()
244 \end{python}
245 one gets
246 \begin{python}
247 0 () Function space type: Finley_Nodes on FinleyMesh
248 \end{python}
249 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the
250 characteristic function \index{characteristic function} of the locations
251 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
252 are set. The complete definition of our example is now:
253 \begin{python}
254 from esys.linearPDEs import Poisson
255 x = mydomain.getX()
256 gammaD = whereZero(x[0])+whereZero(x[1])
257 mypde = Poisson(domain=mydomain)
258 mypde.setValue(f=1,q=gammaD)
259 \end{python}
260 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package.
261 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its
262 \method{getSolution}.
263
264 Now we can write the script to solve our test problem (Remember that
265 lines starting with '\#' are comment lines in Python) (available as \file{poisson.py}
266 in the \ExampleDirectory):
267 \begin{python}
268 from esys.escript import *
269 from esys.escript.linearPDEs import Poisson
270 from esys.finley import Rectangle
271 # generate domain:
272 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
273 # define characteristic function of Gamma^D
274 x = mydomain.getX()
275 gammaD = whereZero(x[0])+whereZero(x[1])
276 # define PDE and get its solution u
277 mypde = Poisson(domain=mydomain)
278 mypde.setValue(f=1,q=gammaD)
279 u = mypde.getSolution()
280 # write u to an external file
281 saveVTK("u.xml",sol=u)
282 \end{python}
283 The last statement writes the solution tagged with the name "sol" to the external file \file{u.xml} in
284 \VTK file format. \VTK is a software library
285 for the visualization of scientific, engineering and analytical data and is freely available
286 from \url{http://www.vtk.org}. There are a variety of graphical user interfaces
287 for \VTK available, for instance \mayavi which can be downloaded from \url{http://mayavi.sourceforge.net/} but is also available on most
288 \LINUX distributions.
289
290 \begin{figure}
291 \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}}
292 \caption{Visualization of the Poisson Equation Solution for $f=1$}
293 \label{fig:FirstSteps.3}
294 \end{figure}
295
296 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE
297 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
298 script from any shell using the command:
299 \begin{python}
300 python poisson.py
301 \end{python}
302 After the script has (hopefully successfully) been completed you will find the file \file{u.xml} in the current
303 directory. An easy way to visualize the results is the command
304 \begin{python}
305 mayavi -d u.xml -m SurfaceMap &
306 \end{python}
307 to show the results, see \fig{fig:FirstSteps.3}.

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