 # Contents of /trunk/doc/user/firststep.tex

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Fri Mar 3 03:33:07 2006 UTC (14 years, 9 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 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)+whereZero(x) 233 \end{python} 234 where 235 \code{whereZero(x)} creates function which equals $1$ where \code{x} is (allmost) equal to zero 236 and $0$ elsewhere. 237 Similarly, \code{whereZero(x)} creates function which equals $1$ where \code{x} is 238 equal to zero and $0$ elsewhere. 239 The sum of the results of \code{whereZero(x)} and \code{whereZero(x)} 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)+whereZero(x) 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)+whereZero(x) 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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