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 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 jgs 107 method!nodes}. Moreover, the domain is subdivided into small, 134 sub-domain 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 gross 565 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 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 jgs 107 and $0$ elsewhere. 234 gross 565 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is 235 jgs 107 equal to zero and $0$ elsewhere. 236 gross 565 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 jgs 107 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the 247 jgs 102 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 jgs 107 from esys.linearPDEs import Poisson 252 jgs 102 x = mydomain.getX() 253 gross 565 gammaD = whereZero(x[0])+whereZero(x[1]) 254 jgs 107 mypde = Poisson(domain=mydomain) 255 mypde = setValue(f=1,q=gammaD) 256 jgs 102 \end{python} 257 gross 565 The first statement imports the \Poisson class definition form the \linearPDEs module \escript package. 258 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its 259 jgs 102 \method{getSolution}. 260 261 Now we can write the script to solve our test problem (Remember that 262 gross 569 lines starting with '\#' are comment lines in Python) (available as \file{poisson.py} 263 jgs 102 in the \ExampleDirectory): 264 \begin{python} 265 gross 565 from esys.escript import * 266 gross 569 from esys.escript.linearPDEs import Poisson 267 jgs 107 from esys.finley import Rectangle 268 jgs 102 # generate domain: 269 jgs 107 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 270 jgs 102 # define characteristic function of Gamma^D 271 x = mydomain.getX() 272 gross 565 gammaD = whereZero(x[0])+whereZero(x[1]) 273 jgs 102 # define PDE and get its solution u 274 gross 565 mypde = Poisson(domain=mydomain) 275 mypde.setValue(f=1,q=gammaD) 276 jgs 102 u = mypde.getSolution() 277 # write u to an external file 278 gross 565 saveVTK("u.xml",sol=u) 279 jgs 102 \end{python} 280 gross 565 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 jgs 102 for the visualization of scientific, engineering and analytical data and is freely available 283 gross 565 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. 286 jgs 102 287 \begin{figure} 288 \centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}} 289 gross 565 \caption{Visualization of the Possion Equation Solution for $f=1$} 290 jgs 102 \label{fig:FirstSteps.3} 291 \end{figure} 292 293 jgs 107 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE 294 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 295 jgs 102 script from any shell using the command: 296 gross 565 \begin{python} 297 gross 569 python poisson.py 298 gross 565 \end{python} 299 gross 569 After the script has (hopefully successfully) been completed you will find the file \file{u.xml} in the current 300 jgs 102 directory. An easy way to visualize the results is the command 301 gross 565 \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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