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

Revision 3274 - (hide annotations)
Thu Oct 14 06:14:30 2010 UTC (9 years ago) by caltinay
File MIME type: application/x-tex
File size: 21135 byte(s)
Some initial corrections to the user's guide.


 1 ksteube 1811 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 ksteube 1316 % 4 jfenwick 2881 % Copyright (c) 2003-2010 by University of Queensland 5 ksteube 1811 % Earth Systems Science Computational Center (ESSCC) 6 7 gross 625 % 8 ksteube 1811 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 gross 625 % 12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 jgs 102 14 ksteube 1811 15 caltinay 3274 \section{The First Steps}\label{FirstSteps} 16 ksteube 1316 In this chapter we give an introduction how to use \escript to solve 17 caltinay 3274 a partial differential equation\index{partial differential equation} (PDE\index{partial differential equation!PDE}). 18 We assume you are at least a little familiar with Python. 19 The knowledge presented in the Python tutorial at \url{http://docs.python.org/tut/tut.html} is more than sufficient. 20 jgs 102 21 caltinay 3274 The PDE\index{partial differential equation} we wish to solve is the Poisson equation\index{Poisson equation} 22 jgs 102 \begin{equation} 23 caltinay 3274 -\Delta u=f 24 \label{eq:FirstSteps.1} 25 jgs 102 \end{equation} 26 ksteube 1316 for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$, 27 jgs 102 is the unit square 28 \begin{equation} 29 \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 \} 30 \label{eq:FirstSteps.1b} 31 \end{equation} 32 jgs 107 The domain is shown in \fig{fig:FirstSteps.1}. 33 caltinay 3274 \begin{figure}[ht] 34 \centerline{\includegraphics{figures/FirstStepDomain}} 35 \caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.} 36 \label{fig:FirstSteps.1} 37 artak 1971 \end{figure} 38 jgs 102 39 ksteube 1316 $\Delta$ denotes the Laplace operator\index{Laplace operator}, which is defined by 40 jgs 102 \begin{equation} 41 \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1} 42 \label{eq:FirstSteps.1.1} 43 \end{equation} 44 ksteube 1316 where, for any function $u$ and any direction $i$, $u\hackscore{,i}$ 45 caltinay 3274 denotes the partial derivative \index{partial derivative} of $u$ with respect 46 to $i$.\footnote{You may be more familiar with the Laplace 47 operator\index{Laplace operator} being written as $\nabla^2$, and written in 48 the form 49 jgs 102 \begin{equation*} 50 caltinay 3274 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2} 51 + \frac{\partial^2 u}{\partial x\hackscore 1^2} 52 jgs 102 \end{equation*} 53 and \eqn{eq:FirstSteps.1} as 54 \begin{equation*} 55 caltinay 3274 -\nabla^2 u = f 56 jgs 102 \end{equation*} 57 } 58 jgs 107 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect 59 artak 1971 to the index. To get a more compact form we will write $u\hackscore{,ij}=(u\hackscore {,i})\hackscore{,j}$ 60 jgs 102 which leads to 61 \begin{equation} 62 \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii} 63 \label{eq:FirstSteps.1.1b} 64 \end{equation} 65 ksteube 1316 We often find that use 66 of nested $\sum$ symbols makes formulas cumbersome, and we use the more 67 caltinay 3274 convenient Einstein summation convention\index{summation convention}. This 68 ksteube 1316 drops the $\sum$ sign and assumes that a summation is performed over any repeated index. 69 For instance we write 70 jgs 102 \begin{eqnarray} 71 x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\ 72 x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\ 73 u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\ 74 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} \\ 75 jgs 102 \label{eq:FirstSteps.1.1c} 76 \end{eqnarray} 77 With the summation convention we can write the Poisson equation \index{Poisson equation} as 78 \begin{equation} 79 - u\hackscore{,ii} =1 80 \label{eq:FirstSteps.1.sum} 81 \end{equation} 82 lkettle 575 where $f=1$ in this example. 83 84 jgs 102 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$ 85 caltinay 3274 of the solution $u$ shall be zero, i.e. $u$ shall fulfill 86 jgs 102 the homogeneous Neumann boundary condition\index{Neumann 87 boundary condition!homogeneous} 88 \begin{equation} 89 n\hackscore{i} u\hackscore{,i}= 0 \;. 90 \label{eq:FirstSteps.2} 91 \end{equation} 92 $n=(n\hackscore{i})$ denotes the outer normal field 93 of the domain, see \fig{fig:FirstSteps.1}. Remember that we 94 caltinay 3274 are applying the Einstein summation convention \index{summation convention}, i.e. $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} +% 95 n\hackscore{1} u\hackscore{,1}$.\footnote{Some readers may familiar with the 96 notation $\frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i}$ 97 jgs 102 for the normal derivative.} 98 The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the 99 set $\Gamma^N$ which is the top and right edge of the domain: 100 \begin{equation} 101 caltinay 3274 \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \} 102 \label{eq:FirstSteps.2b} 103 jgs 102 \end{equation} 104 jgs 107 On the bottom and the left edge of the domain which is defined 105 jgs 102 as 106 \begin{equation} 107 caltinay 3274 \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \} 108 \label{eq:FirstSteps.2c} 109 jgs 102 \end{equation} 110 caltinay 3274 the solution shall be identical to zero: 111 jgs 102 \begin{equation} 112 caltinay 3274 u=0 \; . 113 \label{eq:FirstSteps.2d} 114 jgs 102 \end{equation} 115 caltinay 3274 This kind of boundary condition is called a homogeneous Dirichlet boundary 116 condition\index{Dirichlet boundary condition!homogeneous}. 117 The partial differential equation in \eqn{eq:FirstSteps.1.sum} together 118 jgs 102 with the Neumann boundary condition \eqn{eq:FirstSteps.2} and 119 caltinay 3274 Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so-called 120 boundary value 121 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) 122 for the unknown function~$u$. 123 jgs 102 124 gross 2371 \begin{figure}[ht] 125 caltinay 3274 \centerline{\includegraphics{figures/FirstStepMesh}} 126 \caption{Mesh of $4 \times 4$ elements on a rectangular domain. Here 127 each element is a quadrilateral and described by four nodes, namely 128 the corner points. The solution is interpolated by a bi-linear 129 polynomial.} 130 \label{fig:FirstSteps.2} 131 jgs 102 \end{figure} 132 133 caltinay 3274 In general the BVP\index{boundary value problem!BVP} cannot be solved 134 analytically and numerical methods have to be used to construct an 135 approximation of the solution $u$. 136 Here we will use the finite element method\index{finite element method} 137 (FEM\index{finite element method!FEM}). 138 The basic idea is to fill the domain with a set of points called nodes. 139 The solution is approximated by its values on the nodes\index{finite element method!nodes}. 140 Moreover, the domain is subdivided into smaller sub-domains called 141 elements\index{finite element method!element}. 142 On each element the solution is represented by a polynomial of a certain 143 degree through its values at the nodes located in the element. 144 The nodes and their connection through elements is called a 145 mesh\index{finite element method!mesh}. \fig{fig:FirstSteps.2} shows an 146 jgs 102 example of a FEM mesh with four elements in the $x_0$ and four elements 147 caltinay 3274 in the $x_1$ direction over the unit square. 148 gross 2370 For more details we refer the reader to the literature, for instance \Ref{Zienc,NumHand}. 149 jgs 102 150 caltinay 3274 The \escript solver we want to use to solve this problem is embedded into the python interpreter language. 151 So you can solve the problem interactively but you will learn quickly that it 152 is more efficient to use scripts which you can edit with your favorite editor. 153 To enter the escript environment, use the \program{run-escript} 154 command\footnote{\program{run-escript} is not available under Windows yet. 155 If you run under Windows you can just use the \program{python} command and the 156 \env{OMP_NUM_THREADS} environment variable to control the number of threads.}: 157 gross 2370 \begin{verbatim} 158 caltinay 3274 run-escript 159 gross 2370 \end{verbatim} 160 which will pass you on to the python prompt 161 \begin{verbatim} 162 Python 2.5.2 (r252:60911, Oct 5 2008, 19:29:17) 163 [GCC 4.3.2] on linux2 164 Type "help", "copyright", "credits" or "license" for more information. 165 >>> 166 \end{verbatim} 167 Here you can use all available python commands and language features, for instance 168 \begin{python} 169 caltinay 3274 >>> x=2+3 170 >>> print "2+3=",x 171 2+3= 5 172 gross 2370 \end{python} 173 caltinay 3274 We refer to the python user's guide if you not familiar with python. 174 gross 2370 175 caltinay 3274 \escript provides the class \Poisson to define a Poisson equation\index{Poisson equation}. 176 (We will discuss a more general form of a PDE\index{partial differential equation!PDE} 177 that can be defined through the \LinearPDE class later.) 178 The instantiation of a \Poisson class object requires the specification of the domain $\Omega$. 179 In \escript the \Domain class objects are used to describe the geometry of a 180 domain but it also contains information about the discretization methods and 181 the actual solver which is used to solve the PDE. 182 Here we are using the FEM library \finley\index{finite element method}. 183 The following statements create the \Domain object \var{mydomain} from the 184 \finley method \method{Rectangle}: 185 jgs 102 \begin{python} 186 ksteube 1316 from esys.finley import Rectangle 187 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 188 jgs 102 \end{python} 189 In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and 190 the right, upper corner at $(\var{l0},\var{l1})=(1,1)$. 191 jgs 107 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and 192 jgs 102 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and 193 other \Domain generators within the \finley module, 194 see \Chap{CHAPTER ON FINLEY}. 195 196 jgs 107 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and 197 jgs 102 the right hand side $f$ of the PDE to constant $1$: 198 \begin{python} 199 ksteube 1316 from esys.escript.linearPDEs import Poisson 200 mypde = Poisson(mydomain) 201 mypde.setValue(f=1) 202 jgs 102 \end{python} 203 We have not specified any boundary condition but the 204 \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann 205 boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary 206 condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$ 207 and any constant $C$ the function $u+C$ becomes a solution as well. We have to add 208 a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done 209 jgs 107 by defining a characteristic function \index{characteristic function} 210 which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set 211 jgs 102 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c}, 212 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 213 ksteube 1316 an object \var{x} which contains the coordinates of the nodes in the domain use 214 jgs 102 \begin{python} 215 ksteube 1316 x=mydomain.getX() 216 jgs 102 \end{python} 217 gross 660 The method \method{getX} of the \Domain \var{mydomain} 218 jgs 107 gives access to locations 219 ksteube 1316 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which will be 220 discussed in \Chap{ESCRIPT CHAP} in more detail. What we need to know here is that 221 gross 660 222 ksteube 1316 \var{x} has \Rank (number of dimensions) and a \Shape (list of dimensions) which can be viewed by 223 gross 565 calling the \method{getRank} and \method{getShape} methods: 224 \begin{python} 225 ksteube 1316 print "rank ",x.getRank(),", shape ",x.getShape() 226 gross 565 \end{python} 227 ksteube 1316 This will print something like 228 gross 565 \begin{python} 229 ksteube 1316 rank 1, shape (2,) 230 gross 565 \end{python} 231 The \Data object also maintains type information which is represented by the 232 \FunctionSpace of the object. For instance 233 \begin{python} 234 ksteube 1316 print x.getFunctionSpace() 235 gross 565 \end{python} 236 will print 237 \begin{python} 238 ksteube 1316 Function space type: Finley_Nodes on FinleyMesh 239 gross 565 \end{python} 240 ksteube 1316 which tells us that the coordinates are stored on the nodes of (rather than on points in the interior of) a \finley mesh. 241 gross 565 To get the $x\hackscore{0}$ coordinates of the locations we use the 242 statement 243 \begin{python} 244 ksteube 1316 x0=x[0] 245 gross 565 \end{python} 246 Object \var{x0} 247 is again a \Data object now with \Rank $0$ and 248 \Shape $()$. It inherits the \FunctionSpace from \var{x}: 249 \begin{python} 250 ksteube 1316 print x0.getRank(),x0.getShape(),x0.getFunctionSpace() 251 gross 565 \end{python} 252 will print 253 \begin{python} 254 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh 255 gross 565 \end{python} 256 ksteube 1316 We can now construct a function \var{gammaD} which is only non-zero on the bottom and left edges 257 of the domain with 258 gross 565 \begin{python} 259 ksteube 1316 from esys.escript import whereZero 260 gammaD=whereZero(x[0])+whereZero(x[1]) 261 gross 565 \end{python} 262 ksteube 1316 263 \code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (almost) equal to zero 264 jgs 107 and $0$ elsewhere. 265 gross 565 Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is 266 jgs 107 equal to zero and $0$ elsewhere. 267 gross 565 The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])} 268 artak 1971 gives a function on the domain \var{mydomain} which is strictly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero. 269 gross 565 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from 270 \begin{python} 271 ksteube 1316 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace() 272 gross 565 \end{python} 273 one gets 274 \begin{python} 275 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh 276 gross 565 \end{python} 277 ksteube 1316 An additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the 278 jgs 102 characteristic function \index{characteristic function} of the locations 279 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous} 280 are set. The complete definition of our example is now: 281 \begin{python} 282 ksteube 1316 from esys.linearPDEs import Poisson 283 x = mydomain.getX() 284 gammaD = whereZero(x[0])+whereZero(x[1]) 285 mypde = Poisson(domain=mydomain) 286 mypde.setValue(f=1,q=gammaD) 287 jgs 102 \end{python} 288 lkettle 573 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package. 289 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its 290 jgs 102 \method{getSolution}. 291 292 ksteube 1316 Now we can write the script to solve our Poisson problem 293 jgs 102 \begin{python} 294 ksteube 1316 from esys.escript import * 295 from esys.escript.linearPDEs import Poisson 296 from esys.finley import Rectangle 297 # generate domain: 298 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 299 # define characteristic function of Gamma^D 300 x = mydomain.getX() 301 gammaD = whereZero(x[0])+whereZero(x[1]) 302 # define PDE and get its solution u 303 mypde = Poisson(domain=mydomain) 304 mypde.setValue(f=1,q=gammaD) 305 u = mypde.getSolution() 306 jgs 102 \end{python} 307 gross 2574 The question is what we do with the calculated solution \var{u}. Besides postprocessing, eg. calculating the gradient or the average value, which will be discussed later, plotting the solution is one one things you might want to do. \escript offers two ways to do this, both base on external modules or packages and so data need to converted 308 to hand over the solution. The first option is using the \MATPLOTLIB module which allows plotting 2D results relatively quickly, see~\cite{matplotlib}. However, there are limitations when using this tool, eg. in problem size and when solving 3D problems. Therefore \escript provides a second options based on \VTK files which is especially 309 designed for large scale and 3D problem and which can be read by a variety of software packages such as \mayavi \cite{mayavi}, \VisIt~\cite{VisIt}. 310 jgs 102 311 gross 2574 \subsection{Plotting Using \MATPLOTLIB} 312 gross 2580 The \MATPLOTLIB module provides a simple and easy to use way to visualize PDE solutions (or other \Data objects). 313 To hand over data from \escript to \MATPLOTLIB the values need to mapped onto a rectangular grid 314 \footnote{Users of Debian 5(Lenny) please note: this example makes use of the \function{griddata} method in \module{matplotlib.mlab}. 315 jfenwick 2575 This method is not part of version 0.98.1 which is available with Lenny. 316 jfenwick 2583 If you wish to use contour plots, you may need to install a later version. 317 Users of Ubuntu 8.10 or later should be fine.}. We will make use 318 gross 2574 of the \numpy module. 319 320 First we need to create a rectangular grid. We use the following statements: 321 \begin{python} 322 import numpy 323 x_grid = numpy.linspace(0.,1.,50) 324 y_grid = numpy.linspace(0.,1.,50) 325 \end{python} 326 \var{x_grid} is an array defining the x coordinates of the grids while 327 \var{y_grid} defines the y coordinates of the grid. In this case we use $50$ points over the interval $[0,1]$ 328 in both directions. 329 330 jfenwick 2575 Now the values created by \escript need to be interpolated to this grid. We will use the \MATPLOTLIB 331 gross 2574 \function{mlab.griddata} function to do this. We can easily extract spatial coordinates as a \var{list} by 332 \begin{python} 333 x=mydomain.getX()[0].toListOfTuples() 334 y=mydomain.getX()[1].toListOfTuples() 335 \end{python} 336 In principle we can apply the same \member{toListOfTuples} method to extract the values from the 337 PDE solution \var{u}. However, we have to make sure that the \Data object we extract the values from 338 gross 2580 uses the same \FunctionSpace as we have us when extracting \var{x} and \var{y}. We apply the 339 \function{interpolation} to \var{u} before extraction to achieve this: 340 gross 2574 \begin{python} 341 z=interpolate(u,mydomain.getX().getFunctionSpace()) 342 \end{python} 343 The values in \var{z} are now the values at the points with the coordinates given by \var{x} and \var{y}. These 344 values are now interpolated to the grid defined by \var{x_grid} and \var{y_grid} by using 345 \begin{python} 346 import matplotlib 347 z_grid = matplotlib.mlab.griddata(x,y,z,xi=x_grid,yi=y_grid ) 348 \end{python} 349 \var{z_grid} gives now the values of the PDE solution \var{u} at the grid. The values can be plotted now 350 using the \function{contourf}: 351 \begin{python} 352 matplotlib.pyplot.contourf(x_grid, y_grid, z_grid, 5) 353 matplotlib.pyplot.savefig("u.png") 354 \end{python} 355 Here we use $5$ contours. The last statement writes the plot to the file \file{u.png} in the PNG format. Alternatively, one can use 356 \begin{python} 357 matplotlib.pyplot.contourf(x_grid, y_grid, z_grid, 5) 358 matplotlib.pyplot.show() 359 \end{python} 360 which gives an interactive browser window. 361 362 \begin{figure} 363 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepResultMATPLOTLIB}} 364 \caption{Visualization of the Poisson Equation Solution for $f=1$ using \MATPLOTLIB.} 365 \label{fig:FirstSteps.3b} 366 \end{figure} 367 368 Now we can write the script to solve our Poisson problem 369 \begin{python} 370 from esys.escript import * 371 from esys.escript.linearPDEs import Poisson 372 from esys.finley import Rectangle 373 import numpy 374 import matplotlib 375 import pylab 376 # generate domain: 377 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 378 # define characteristic function of Gamma^D 379 x = mydomain.getX() 380 gammaD = whereZero(x[0])+whereZero(x[1]) 381 # define PDE and get its solution u 382 mypde = Poisson(domain=mydomain) 383 mypde.setValue(f=1,q=gammaD) 384 u = mypde.getSolution() 385 # interpolate u to a matplotlib grid: 386 x_grid = numpy.linspace(0.,1.,50) 387 y_grid = numpy.linspace(0.,1.,50) 388 x=mydomain.getX()[0].toListOfTuples() 389 y=mydomain.getX()[1].toListOfTuples() 390 z=interpolate(u,mydomain.getX().getFunctionSpace()) 391 z_grid = matplotlib.mlab.griddata(x,y,z,xi=x_grid,yi=y_grid ) 392 # interpolate u to a rectangular grid: 393 matplotlib.pyplot.contourf(x_grid, y_grid, z_grid, 5) 394 matplotlib.pyplot.savefig("u.png") 395 \end{python} 396 The entire code is available as \file{poisson\hackscore matplotlib.py} in the \ExampleDirectory. 397 You can run the script using the {\it escript} environment 398 ksteube 1316 \begin{verbatim} 399 jfenwick 2923 run-escript poisson_matplotlib.py 400 ksteube 1316 \end{verbatim} 401 gross 2574 This will create the \file{u.png}, see Figure~\fig{fig:FirstSteps.3b}. 402 For details on the usage of the \MATPLOTLIB module we refer to the documentation~\cite{matplotlib}. 403 ksteube 1316 404 gross 2574 As pointed out, \MATPLOTLIB is restricted to the two-dimensional case and 405 should be used for small problems only. It can not be used under \MPI as the \member{toListOfTuples} method is 406 jfenwick 2575 not safe under \MPI\footnote{The phrase 'safe under \MPI' means that a program will produce correct results when run on more than one processor under \MPI.}. 407 gross 2574 408 gross 2580 \begin{figure} 409 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepResult}} 410 \caption{Visualization of the Poisson Equation Solution for $f=1$} 411 \label{fig:FirstSteps.3} 412 \end{figure} 413 414 gross 2574 \subsection{Visualization using \VTK} 415 416 As an alternative {\it escript} supports the usage of visualization tools which base on \VTK, eg. mayavi \cite{mayavi}, \VisIt~\cite{VisIt}. In this case the solution is written to a file in the \VTK format. This file the can read by the tool of choice. Using \VTK file is \MPI safe. 417 418 To write the solution \var{u} in Poisson problem to the file \file{u.xml} one need to add the line 419 \begin{python} 420 saveVTK("u.xml",sol=u) 421 \end{python} 422 The solution \var{u} is now available in the \file{u.xml} tagged with the name "sol". 423 424 The Poisson problem script is now 425 \begin{python} 426 from esys.escript import * 427 from esys.escript.linearPDEs import Poisson 428 from esys.finley import Rectangle 429 # generate domain: 430 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 431 # define characteristic function of Gamma^D 432 x = mydomain.getX() 433 gammaD = whereZero(x[0])+whereZero(x[1]) 434 # define PDE and get its solution u 435 mypde = Poisson(domain=mydomain) 436 mypde.setValue(f=1,q=gammaD) 437 u = mypde.getSolution() 438 # write u to an external file 439 saveVTK("u.xml",sol=u) 440 \end{python} 441 gross 2683 The entire code is available as \file{poisson\hackscore VTK.py} in the \ExampleDirectory. 442 gross 2574 443 You can run the script using the {\it escript} environment 444 and visualize the solution using \mayavi: 445 \begin{verbatim} 446 jfenwick 2955 run-escript poisson_VTK.py 447 gross 2574 mayavi2 -d u.xml -m SurfaceMap 448 \end{verbatim} 449 The result is shown in Figure~\fig{fig:FirstSteps.3}.

## Properties

Name Value
svn:eol-style native
svn:keywords Author Date Id Revision