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 1 jgs 102 % $Id$ 2 gross 625 % 3 % Copyright © 2006 by ACcESS MNRF 4 % \url{http://www.access.edu.au 5 % Primary Business: Queensland, Australia. 6 % Licensed under the Open Software License version 3.0 7 8 % 9 jgs 102 10 gross 625 11 jgs 121 \section{The First Steps} 12 jgs 102 \label{FirstSteps} 13 14 \begin{figure} 15 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepDomain}} 16 jgs 102 \caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.} 17 \label{fig:FirstSteps.1} 18 \end{figure} 19 20 jgs 107 In this chapter we will give an introduction how to use \escript to solve 21 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} 22 is sufficient. It is helpful if the reader has some basic knowledge of PDEs \index{partial differential equation}. 23 jgs 102 24 jgs 107 The PDE \index{partial differential equation} we wish to solve is the Poisson equation \index{Poisson equation} 25 jgs 102 \begin{equation} 26 -\Delta u =f 27 \label{eq:FirstSteps.1} 28 \end{equation} 29 jgs 107 for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$ 30 jgs 102 is the unit square 31 \begin{equation} 32 \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 \} 33 \label{eq:FirstSteps.1b} 34 \end{equation} 35 jgs 107 The domain is shown in \fig{fig:FirstSteps.1}. 36 jgs 102 37 $\Delta$ denotes the Laplace operator\index{Laplace operator} which is defined by 38 \begin{equation} 39 \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1} 40 \label{eq:FirstSteps.1.1} 41 \end{equation} 42 jgs 107 where, for any function $w$ and any direction $i$, $u\hackscore{,i}$ 43 denotes the partial derivative \index{partial derivative} of $u$ with respect to $i$. 44 jgs 102 \footnote{Some readers 45 may be more familiar with the Laplace operator\index{Laplace operator} being written 46 as $\nabla^2$, and written in the form 47 \begin{equation*} 48 jgs 110 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2} 49 jgs 102 + \frac{\partial^2 u}{\partial x\hackscore 1^2} 50 \end{equation*} 51 and \eqn{eq:FirstSteps.1} as 52 \begin{equation*} 53 -\nabla^2 u = f 54 \end{equation*} 55 } 56 jgs 107 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect 57 jgs 102 to the index. To get a more compact form we will write $w\hackscore{,ij}=(w\hackscore {,i})\hackscore{,j}$ 58 which leads to 59 \begin{equation} 60 \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii} 61 \label{eq:FirstSteps.1.1b} 62 \end{equation} 63 In some cases, and we will see examples for this in the next chapter, 64 the usage of the nested $\sum$ symbols blows up the formulas and therefore 65 jgs 107 it is convenient to use the Einstein summation convention \index{summation convention}. This 66 drops the $\sum$ sign and assumes that a summation over a repeated index is performed 67 jgs 102 ("repeated index means summation"). For instance we write 68 \begin{eqnarray} 69 x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\ 70 x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\ 71 u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\ 72 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} \\ 73 jgs 102 \label{eq:FirstSteps.1.1c} 74 \end{eqnarray} 75 With the summation convention we can write the Poisson equation \index{Poisson equation} as 76 \begin{equation} 77 - u\hackscore{,ii} =1 78 \label{eq:FirstSteps.1.sum} 79 \end{equation} 80 lkettle 575 where $f=1$ in this example. 81 82 jgs 102 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$ 83 of the solution $u$ shall be zero, ie. $u$ shall fulfill 84 the homogeneous Neumann boundary condition\index{Neumann 85 boundary condition!homogeneous} 86 \begin{equation} 87 n\hackscore{i} u\hackscore{,i}= 0 \;. 88 \label{eq:FirstSteps.2} 89 \end{equation} 90 $n=(n\hackscore{i})$ denotes the outer normal field 91 of the domain, see \fig{fig:FirstSteps.1}. Remember that we 92 are applying the Einstein summation convention \index{summation convention}, i.e 93 jgs 107 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} + 94 n\hackscore{1} u\hackscore{,1}$. 95 jgs 102 \footnote{Some readers may familiar with the notation 96 \begin{equation*} 97 \frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i} 98 \end{equation*} 99 for the normal derivative.} 100 The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the 101 set $\Gamma^N$ which is the top and right edge of the domain: 102 \begin{equation} 103 \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \} 104 \label{eq:FirstSteps.2b} 105 \end{equation} 106 jgs 107 On the bottom and the left edge of the domain which is defined 107 jgs 102 as 108 \begin{equation} 109 \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \} 110 \label{eq:FirstSteps.2c} 111 \end{equation} 112 the solution shall be identically zero: 113 \begin{equation} 114 u=0 \; . 115 \label{eq:FirstSteps.2d} 116 \end{equation} 117 jgs 107 This kind of boundary condition is called a homogeneous Dirichlet boundary condition 118 jgs 102 \index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together 119 with the Neumann boundary condition \eqn{eq:FirstSteps.2} and 120 Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so 121 called boundary value 122 jgs 107 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for 123 the unknown 124 jgs 102 function $u$. 125 126 127 \begin{figure} 128 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepMesh.eps}} 129 jgs 102 \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here 130 each element is a quadrilateral and described by four nodes, namely 131 the corner points. The solution is interpolated by a bi-linear 132 polynomial.} 133 \label{fig:FirstSteps.2} 134 \end{figure} 135 136 In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical 137 methods have to be used construct an approximation of the solution 138 $u$. Here we will use the finite element method\index{finite element 139 method} (FEM\index{finite element 140 method!FEM}). The basic idea is to fill the domain with a 141 jgs 107 set of points called nodes. The solution is approximated by its 142 jgs 102 values on the nodes\index{finite element 143 lkettle 573 method!nodes}. Moreover, the domain is subdivided into smaller 144 sub-domains called elements \index{finite element 145 jgs 102 method!element}. On each element the solution is 146 represented by a polynomial of a certain degree through its values at 147 the nodes located in the element. The nodes and its connection through 148 elements is called a mesh\index{finite element 149 jgs 107 method!mesh}. \fig{fig:FirstSteps.2} shows an 150 jgs 102 example of a FEM mesh with four elements in the $x_0$ and four elements 151 in the $x_1$ direction over the unit square. 152 For more details we refer the reader to the literature, for instance 153 \Ref{Zienc,NumHand}. 154 155 \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}. 156 (We will discuss a more general form of a PDE \index{partial differential equation!PDE} 157 that can be defined through the \LinearPDE class later). The instantiation of 158 a \Poisson class object requires the specification of the domain $\Omega$. In \escript 159 the \Domain class objects are used to describe the geometry of a domain but it also 160 contains information about the discretization methods and the actual solver which is used 161 to solve the PDE. Here we are using the FEM library \finley \index{finite element 162 method}. The following statements create the \Domain object \var{mydomain} from the 163 \finley method \method{Rectangle} 164 \begin{python} 165 jgs 107 from esys.finley import Rectangle 166 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 167 jgs 102 \end{python} 168 In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and 169 the right, upper corner at $(\var{l0},\var{l1})=(1,1)$. 170 jgs 107 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and 171 jgs 102 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and 172 other \Domain generators within the \finley module, 173 see \Chap{CHAPTER ON FINLEY}. 174 175 jgs 107 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and 176 jgs 102 the right hand side $f$ of the PDE to constant $1$: 177 \begin{python} 178 gross 565 from esys.escript.linearPDEs import Poisson 179 jgs 107 mypde = Poisson(mydomain) 180 mypde.setValue(f=1) 181 jgs 102 \end{python} 182 We have not specified any boundary condition but the 183 \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann 184 boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary 185 condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$ 186 and any constant $C$ the function $u+C$ becomes a solution as well. We have to add 187 a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done 188 jgs 107 by defining a characteristic function \index{characteristic function} 189 which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set 190 jgs 102 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c}, 191 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 192 an object \var{x} which represents locations in the domain one uses 193 jgs 102 \begin{python} 194 lkettle 573 x=mydomain.getX() 195 jgs 102 \end{python} 196 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 197 that \var{x} has a 198 199 jgs 107 In the first statement, the method \method{getX} of the \Domain \var{mydomain} 200 gives access to locations 201 gross 565 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which is 202 discussed in Chpater\ref{X} in more details. What we need to know here is that 203 \var{x} has \Rank (=number of dimensions) and a \Shape (=tuple of dimensions) which can be checked by 204 calling the \method{getRank} and \method{getShape} methods: 205 \begin{python} 206 print "rank ",x.getRank(),", shape ",x.getShape() 207 \end{python} 208 will print something like 209 \begin{python} 210 rank 1, shape (2,) 211 \end{python} 212 The \Data object also maintains type information which is represented by the 213 \FunctionSpace of the object. For instance 214 \begin{python} 215 print x.getFunctionSpace() 216 \end{python} 217 will print 218 \begin{python} 219 Function space type: Finley_Nodes on FinleyMesh 220 \end{python} 221 which tells us that the coordinates are stored on the nodes of a \finley mesh. 222 To get the $x\hackscore{0}$ coordinates of the locations we use the 223 statement 224 \begin{python} 225 x0=x[0] 226 \end{python} 227 Object \var{x0} 228 is again a \Data object now with \Rank $0$ and 229 \Shape $()$. It inherits the \FunctionSpace from \var{x}: 230 \begin{python} 231 print x0.getRank(),x0.getShape(),x0.getFunctionSpace() 232 \end{python} 233 will print 234 \begin{python} 235 0 () Function space type: Finley_Nodes on FinleyMesh 236 \end{python} 237 We can now construct the function \var{gammaD} by 238 \begin{python} 239 lkettle 573 from esys.escript import whereZero 240 gross 565 gammaD=whereZero(x[0])+whereZero(x[1]) 241 \end{python} 242 where 243 \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (allmost) equal to zero 244 jgs 107 and $0$ elsewhere. 245 gross 565 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is 246 jgs 107 equal to zero and $0$ elsewhere. 247 gross 565 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])} 248 gives a function on the domain \var{mydomain} which is exactly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero. 249 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from 250 \begin{python} 251 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace() 252 \end{python} 253 one gets 254 \begin{python} 255 0 () Function space type: Finley_Nodes on FinleyMesh 256 \end{python} 257 jgs 107 The additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the 258 jgs 102 characteristic function \index{characteristic function} of the locations 259 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous} 260 are set. The complete definition of our example is now: 261 \begin{python} 262 jgs 107 from esys.linearPDEs import Poisson 263 jgs 102 x = mydomain.getX() 264 gross 565 gammaD = whereZero(x[0])+whereZero(x[1]) 265 jgs 107 mypde = Poisson(domain=mydomain) 266 lkettle 573 mypde.setValue(f=1,q=gammaD) 267 jgs 102 \end{python} 268 lkettle 573 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package. 269 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its 270 jgs 102 \method{getSolution}. 271 272 Now we can write the script to solve our test problem (Remember that 273 gross 569 lines starting with '\#' are comment lines in Python) (available as \file{poisson.py} 274 jgs 102 in the \ExampleDirectory): 275 \begin{python} 276 gross 565 from esys.escript import * 277 gross 569 from esys.escript.linearPDEs import Poisson 278 jgs 107 from esys.finley import Rectangle 279 jgs 102 # generate domain: 280 jgs 107 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 281 jgs 102 # define characteristic function of Gamma^D 282 x = mydomain.getX() 283 gross 565 gammaD = whereZero(x[0])+whereZero(x[1]) 284 jgs 102 # define PDE and get its solution u 285 gross 565 mypde = Poisson(domain=mydomain) 286 mypde.setValue(f=1,q=gammaD) 287 jgs 102 u = mypde.getSolution() 288 # write u to an external file 289 gross 565 saveVTK("u.xml",sol=u) 290 jgs 102 \end{python} 291 gross 565 The last statement writes the solution tagged with the name "sol" to the external file \file{u.xml} in 292 \VTK file format. \VTK is a software library 293 jgs 102 for the visualization of scientific, engineering and analytical data and is freely available 294 lkettle 573 from \url{http://www.vtk.org}. There are a variety of graphical user interfaces 295 gross 565 for \VTK available, for instance \mayavi which can be downloaded from \url{http://mayavi.sourceforge.net/} but is also available on most 296 \LINUX distributions. 297 jgs 102 298 \begin{figure} 299 gross 599 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepResult.eps}} 300 lkettle 573 \caption{Visualization of the Poisson Equation Solution for $f=1$} 301 jgs 102 \label{fig:FirstSteps.3} 302 \end{figure} 303 304 jgs 107 You can edit the script file using your favourite text editor (or the Integrated DeveLopment Environment IDLE 305 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 306 jgs 102 script from any shell using the command: 307 gross 565 \begin{python} 308 gross 569 python poisson.py 309 gross 565 \end{python} 310 gross 569 After the script has (hopefully successfully) been completed you will find the file \file{u.xml} in the current 311 jgs 102 directory. An easy way to visualize the results is the command 312 gross 565 \begin{python} 313 mayavi -d u.xml -m SurfaceMap & 314 \end{python} 315 to show the results, see \fig{fig:FirstSteps.3}.

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