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1 % $Id$
3 \chapter{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 \cot \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 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 a function which is positive for the cases $x\hackscore{0}=0$ or $x\hackscore{1}=0$:
182 \begin{python}
183 x=mydomain.getX()
184 gammaD=x[0].whereZero()+x[1].whereZero()
185 \end{python}
186 In the first statement, the method \method{getX} of the \Domain \var{mydomain}
187 gives access to locations
188 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object
189 which we will learn more about later. \code{x[0]} returns the $x\hackscore{0}$ coordinates of the locations and
190 \code{x[0].whereZero()} creates function which equals $1$ where \code{x[0]} is (nearly) equal to zero
191 and $0$ elsewhere.
192 Similarly, \code{x[1].whereZero()} creates function which equals $1$ where \code{x[1]} is
193 equal to zero and $0$ elsewhere.
194 The sum of the results of \code{x[0].whereZero()} and \code{x[1].whereZero()} gives a function on the domain \var{mydomain} which is exactly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero.
196 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the
197 characteristic function \index{characteristic function} of the locations
198 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
199 are set. The complete definition of our example is now:
200 \begin{python}
201 from esys.linearPDEs import Poisson
202 x = mydomain.getX()
203 gammaD = x[0].whereZero()+x[1].whereZero()
204 mypde = Poisson(domain=mydomain)
205 mypde = setValue(f=1,q=gammaD)
206 \end{python}
207 The first statement imports the \Poisson class definition form the \linearPDEsPack module which is part of the \ESyS package.
208 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its
209 \method{getSolution}.
211 Now we can write the script to solve our test problem (Remember that
212 lines starting with '\#' are comment lines in Python) (available as \file{mypoisson.py}
213 in the \ExampleDirectory):
214 \begin{python}
215 from esys.finley import Rectangle
216 from esys.linearPDEs import Poisson
217 # generate domain:
218 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
219 # define characteristic function of Gamma^D
220 x = mydomain.getX()
221 gammaD = x[0].whereZero()+x[1].whereZero()
222 # define PDE and get its solution u
223 mypde = Poisson(domain=mydomain,f=1,q=gammaD)
224 u = mypde.getSolution()
225 # write u to an external file
226 u.saveDX("u.dx")
227 \end{python}
228 The last statement writes the solution to the external file \file{u.dx} in
229 \OpenDX file format. \OpenDX is a software package
230 for the visualization of scientific, engineering and analytical data and is freely available
231 from \url{http://www.opendx.org}.
233 \begin{figure}
234 \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}}
235 \caption{\OpenDX Visualization of the Possion Equation Solution for $f=1$}
236 \label{fig:FirstSteps.3}
237 \end{figure}
239 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE
240 for Python, see \url{http://idlefork.sourceforge.net}). If the script file has the name \file{mypoisson.py} \index{scripts!\file{mypoisson.py}} you can run the
241 script from any shell using the command:
242 \begin{verbatim}
243 python mypoisson.py
244 \end{verbatim}
245 After the script has (hopefully successfully) been completed you will find the file \file{u.dx} in the current
246 directory. An easy way to visualize the results is the command
247 \begin{verbatim}
248 dx -prompter &
249 \end{verbatim}
250 to start the generic data visualization interface of \OpenDX. \fig{fig:FirstSteps.3} shows the result.


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