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2    
3     \chapter{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 show the basics of 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
14     the basics of Python. The knowledge presented the Python tutorial at \url{http://docs.python.org/tut/tut.html}
15     is sufficient. It is helpful if the reader has some basic knowledge on PDEs \index{PDE}.
16    
17     The \index{PDE} we want to solve is the Poisson equation \index{Poisson equation}
18     \begin{equation}
19     -\Delta u =f
20     \label{eq:FirstSteps.1}
21     \end{equation}
22     for the solution $u$. The domain of interest which we denote by $\Omega$
23     is the unit square
24     \begin{equation}
25     \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 \}
26     \label{eq:FirstSteps.1b}
27     \end{equation}
28     The domain is shown in Figure~\fig{fig:FirstSteps.1}.
29    
30     $\Delta$ denotes the Laplace operator\index{Laplace operator} which is defined by
31     \begin{equation}
32     \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1}
33     \label{eq:FirstSteps.1.1}
34     \end{equation}
35     where for any function $w$ and any direction $i$ $u\hackscore{,i}$
36     denotes the partial derivative \index{partial derivative} of $w$ with respect to $i$.
37     \footnote{Some readers
38     may be more familiar with the Laplace operator\index{Laplace operator} being written
39     as $\nabla^2$, and written in the form
40     \begin{equation*}
41     \nabla^2 u = \frac{\partial^2 u}{\partial x\hackscore 0^2}
42     + \frac{\partial^2 u}{\partial x\hackscore 1^2}
43     \end{equation*}
44     and \eqn{eq:FirstSteps.1} as
45     \begin{equation*}
46     -\nabla^2 u = f
47     \end{equation*}
48     }
49     Basically in the subindex of a function any index left to the comma denotes a spatial derivative with respect
50     to the index. To get a more compact form we will write $w\hackscore{,ij}=(w\hackscore {,i})\hackscore{,j}$
51     which leads to
52     \begin{equation}
53     \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii}
54     \label{eq:FirstSteps.1.1b}
55     \end{equation}
56     In some cases, and we will see examples for this in the next chapter,
57     the usage of the nested $\sum$ symbols blows up the formulas and therefore
58     it is convenient to use Einstein summation convention \index{summation convention} which
59     says that $\sum$ sign is dropped and a summation over a repeated index is performed
60     ("repeated index means summation"). For instance we write
61     \begin{eqnarray}
62     x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\
63     x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\
64     u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\
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     $n\hackscore{i} u\hackscore{,i}= n\hackscore1 u\hackscore{,1} +
84     n\hackscore2 u\hackscore{,2}$.
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 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     The 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 unknown
113     function $u$.
114    
115    
116     \begin{figure}
117     \centerline{\includegraphics[width=\figwidth]{FirstStepMesh}}
118     \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here
119     each element is a quadrilateral and described by four nodes, namely
120     the corner points. The solution is interpolated by a bi-linear
121     polynomial.}
122     \label{fig:FirstSteps.2}
123     \end{figure}
124    
125     In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical
126     methods have to be used construct an approximation of the solution
127     $u$. Here we will use the finite element method\index{finite element
128     method} (FEM\index{finite element
129     method!FEM}). The basic idea is to fill the domain with a
130     set of points, so called nodes. The solution is approximated by its
131     values on the nodes\index{finite element
132     method!nodes}. Moreover, the domain is subdivide into small
133     sub-domain, so-called elements \index{finite element
134     method!element}. On each element the solution is
135     represented by a polynomial of a certain degree through its values at
136     the nodes located in the element. The nodes and its connection through
137     elements is called a mesh\index{finite element
138     method!mesh}. Figure~\fig{fig:FirstSteps.2} shows an
139     example of a FEM mesh with four elements in the $x_0$ and four elements
140     in the $x_1$ direction over the unit square.
141     For more details we refer the reader to the literature, for instance
142     \Ref{Zienc,NumHand}.
143    
144     \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}.
145     (We will discuss a more general form of a PDE \index{partial differential equation!PDE}
146     that can be defined through the \LinearPDE class later). The instantiation of
147     a \Poisson class object requires the specification of the domain $\Omega$. In \escript
148     the \Domain class objects are used to describe the geometry of a domain but it also
149     contains information about the discretization methods and the actual solver which is used
150     to solve the PDE. Here we are using the FEM library \finley \index{finite element
151     method}. The following statements create the \Domain object \var{mydomain} from the
152     \finley method \method{Rectangle}
153     \begin{python}
154     import finley
155     mydomain = finley.Rectangle(l0=1.,l1=1.,n0=40, n1=20)
156     \end{python}
157     In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and
158     the right, upper corner at $(\var{l0},\var{l1})=(1,1)$.
159     The arguments \var{l0} and \var{l1} define the number of elements in $x\hackscore{0}$ and
160     $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and
161     other \Domain generators within the \finley module,
162     see \Chap{CHAPTER ON FINLEY}.
163    
164     The following statements define the \Poisson object \var{mypde} with domain var{mydomain} and
165     the right hand side $f$ of the PDE to constant $1$:
166     \begin{python}
167     import escript
168     mypde = escript.Poisson(domain=mydomain,f=1)
169     \end{python}
170     We have not specified any boundary condition but the
171     \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann
172     boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary
173     condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$
174     and any constant $C$ the function $u+C$ becomes a solution as well. We have to add
175     a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done
176     by defines a characteristic function \index{characteristic function}
177     which has a positive values at locations $(x_0,x_1)$ where Dirichlet boundary condition is set
178     and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c},
179     we need a function which is positive for the cases $x_0=0$ or $x_1=0$:
180     \begin{python}
181     x=mydomain.getX()
182     gammaD=x[0].whereZero()+x[1].whereZero()
183     \end{python}
184     In first statement returns, the method \method{getX} of the \Domain \var{mydomain} access to the locations
185     in the domain defined by \var{mydomain}. The object \var{x} is actually an \Data object
186     which we will learn more about later. \code{x[0]} returns the $x_0$ coordinates of the locations and
187     \code{x[0].whereZero()} creates function which equals $1$ where \code{x[0]} is (nearly) equal to zero
188     and $0$ elsewhere. 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 $x_0$ or $x_1$ is equal to zero.
189    
190     The additional parameter \var{q} of the \Poisson object creater defines the
191     characteristic function \index{characteristic function} of the locations
192     of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous}
193     are set. The complete definition of our example is now:
194     \begin{python}
195     from linearPDEs import Poisson
196     x = mydomain.getX()
197     gammaD = x[0].whereZero()+x[1].whereZero()
198     mypde = Poisson(domain=mydomain,f=1,q=gammaD)
199     \end{python}
200     The first statement imports the \Poisson class definition form the \linearPDEsPack module which is part of the \escript module.
201     To get the solution of the Poisson equation defines by \var{mypde} we just have to call its
202     \method{getSolution}.
203    
204     Now we can write the script to solve our test problem (Remember that
205     lines starting with '\#' are commend lines in Python) (available as \file{mypoisson.py}
206     in the \ExampleDirectory):
207     \begin{python}
208     import esys.finley
209     from esys.linearPDEs import Poisson
210     # generate domain:
211     mydomain = esys.finley.Rectangle(l0=1.,l1=1.,n0=40, n1=20)
212     # define characteristic function of Gamma^D
213     x = mydomain.getX()
214     gammaD = x[0].whereZero()+x[1].whereZero()
215     # define PDE and get its solution u
216     mypde = Poisson(domain=mydomain,f=1,q=gammaD)
217     u = mypde.getSolution()
218     # write u to an external file
219     u.saveDX("u.dx")
220     \end{python}
221     The last statement writes the solution to the external file \file{u.dx} in
222     \OpenDX file format. \OpenDX is a software package
223     for the visualization of scientific, engineering and analytical data and is freely available
224     from \url{http://www.opendx.org}.
225    
226     \begin{figure}
227     \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}}
228     \caption{\OpenDX visualization of the Possion equation soluition for $f=1$}
229     \label{fig:FirstSteps.3}
230     \end{figure}
231    
232     You can edit this script using your favourite text editor (or the Integrated DeveLopment Environment IDLE
233     for Python). If the script file has the name \file{mypoisson.py} \index{scripts!\file{mypoisson.py}} you can run the
234     script from any shell using the command:
235     \begin{verbatim}
236     python mypoisson.py
237     \end{verbatim}
238     After the script has (hopefully successfully) been completed you will find the file \file{u.dx} in the current
239     directory. An easy way to visualize the results is the command
240     \begin{verbatim}
241     dx -prompter
242     \end{verbatim}
243     to start the generic data visualization interface of \OpenDX. \fig{fig:FirstSteps.3} shows the result.
244    
245    
246    
247    

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