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 1 ksteube 1811 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 ksteube 1316 % 4 jfenwick 2548 % Copyright (c) 2003-2009 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 jgs 121 \section{The First Steps} 16 jgs 102 \label{FirstSteps} 17 18 19 artak 1971 20 ksteube 1316 In this chapter we give an introduction how to use \escript to solve 21 a partial differential equation \index{partial differential equation} (PDE \index{partial differential equation!PDE}). We assume you are at least a little familiar with Python. The knowledge presented at the Python tutorial at \url{http://docs.python.org/tut/tut.html} 22 is more than sufficient. 23 jgs 102 24 gross 2363 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 ksteube 1316 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 gross 2371 \begin{figure} [ht] 37 artak 1971 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepDomain}} 38 \caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.} 39 \label{fig:FirstSteps.1} 40 \end{figure} 41 jgs 102 42 ksteube 1316 $\Delta$ denotes the Laplace operator\index{Laplace operator}, which is defined by 43 jgs 102 \begin{equation} 44 \Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1} 45 \label{eq:FirstSteps.1.1} 46 \end{equation} 47 ksteube 1316 where, for any function $u$ and any direction $i$, $u\hackscore{,i}$ 48 jgs 107 denotes the partial derivative \index{partial derivative} of $u$ with respect to $i$. 49 ksteube 1316 \footnote{You 50 jgs 102 may be more familiar with the Laplace operator\index{Laplace operator} being written 51 as $\nabla^2$, and written in the form 52 \begin{equation*} 53 jgs 110 \nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2} 54 jgs 102 + \frac{\partial^2 u}{\partial x\hackscore 1^2} 55 \end{equation*} 56 and \eqn{eq:FirstSteps.1} as 57 \begin{equation*} 58 -\nabla^2 u = f 59 \end{equation*} 60 } 61 jgs 107 Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect 62 artak 1971 to the index. To get a more compact form we will write $u\hackscore{,ij}=(u\hackscore {,i})\hackscore{,j}$ 63 jgs 102 which leads to 64 \begin{equation} 65 \Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii} 66 \label{eq:FirstSteps.1.1b} 67 \end{equation} 68 ksteube 1316 We often find that use 69 of nested $\sum$ symbols makes formulas cumbersome, and we use the more 70 convenient Einstein summation convention \index{summation convention}. This 71 drops the $\sum$ sign and assumes that a summation is performed over any repeated index. 72 For instance we write 73 jgs 102 \begin{eqnarray} 74 x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\ 75 x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\ 76 u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\ 77 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} \\ 78 jgs 102 \label{eq:FirstSteps.1.1c} 79 \end{eqnarray} 80 With the summation convention we can write the Poisson equation \index{Poisson equation} as 81 \begin{equation} 82 - u\hackscore{,ii} =1 83 \label{eq:FirstSteps.1.sum} 84 \end{equation} 85 lkettle 575 where $f=1$ in this example. 86 87 jgs 102 On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$ 88 of the solution $u$ shall be zero, ie. $u$ shall fulfill 89 the homogeneous Neumann boundary condition\index{Neumann 90 boundary condition!homogeneous} 91 \begin{equation} 92 n\hackscore{i} u\hackscore{,i}= 0 \;. 93 \label{eq:FirstSteps.2} 94 \end{equation} 95 $n=(n\hackscore{i})$ denotes the outer normal field 96 of the domain, see \fig{fig:FirstSteps.1}. Remember that we 97 are applying the Einstein summation convention \index{summation convention}, i.e 98 jgs 107 $n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} + 99 n\hackscore{1} u\hackscore{,1}$. 100 jgs 102 \footnote{Some readers may familiar with the notation 101 artak 1971 $102 jgs 102 \frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i} 103 artak 1971$ 104 jgs 102 for the normal derivative.} 105 The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the 106 set $\Gamma^N$ which is the top and right edge of the domain: 107 \begin{equation} 108 \Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \} 109 \label{eq:FirstSteps.2b} 110 \end{equation} 111 jgs 107 On the bottom and the left edge of the domain which is defined 112 jgs 102 as 113 \begin{equation} 114 \Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \} 115 \label{eq:FirstSteps.2c} 116 \end{equation} 117 the solution shall be identically zero: 118 \begin{equation} 119 u=0 \; . 120 \label{eq:FirstSteps.2d} 121 \end{equation} 122 jgs 107 This kind of boundary condition is called a homogeneous Dirichlet boundary condition 123 jgs 102 \index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together 124 with the Neumann boundary condition \eqn{eq:FirstSteps.2} and 125 Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so 126 called boundary value 127 jgs 107 problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for 128 artak 1971 the unknown function~$u$. 129 jgs 102 130 131 gross 2371 \begin{figure}[ht] 132 jfenwick 2335 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepMesh}} 133 jgs 102 \caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here 134 each element is a quadrilateral and described by four nodes, namely 135 the corner points. The solution is interpolated by a bi-linear 136 polynomial.} 137 \label{fig:FirstSteps.2} 138 \end{figure} 139 140 In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical 141 methods have to be used construct an approximation of the solution 142 $u$. Here we will use the finite element method\index{finite element 143 method} (FEM\index{finite element 144 method!FEM}). The basic idea is to fill the domain with a 145 jgs 107 set of points called nodes. The solution is approximated by its 146 jgs 102 values on the nodes\index{finite element 147 lkettle 573 method!nodes}. Moreover, the domain is subdivided into smaller 148 sub-domains called elements \index{finite element 149 jgs 102 method!element}. On each element the solution is 150 represented by a polynomial of a certain degree through its values at 151 the nodes located in the element. The nodes and its connection through 152 elements is called a mesh\index{finite element 153 jgs 107 method!mesh}. \fig{fig:FirstSteps.2} shows an 154 jgs 102 example of a FEM mesh with four elements in the $x_0$ and four elements 155 in the $x_1$ direction over the unit square. 156 gross 2370 For more details we refer the reader to the literature, for instance \Ref{Zienc,NumHand}. 157 jgs 102 158 gross 2370 The \escript solver we want to use to solve this problem is embedded into the python interpreter language. So you can solve the problem interactively but you will learn quickly 159 gross 2375 that is more efficient to use scripts which you can edit with your favorite editor. 160 gross 2370 To enter the escript environment you use \program{escript} command\footnote{\program{escript} is not available under Windows yet. If you run under windows you can just use the 161 \program{python} command and the \env{OMP_NUM_THREADS} environment variable to control the number 162 of threads.}: 163 \begin{verbatim} 164 escript 165 \end{verbatim} 166 which will pass you on to the python prompt 167 \begin{verbatim} 168 Python 2.5.2 (r252:60911, Oct 5 2008, 19:29:17) 169 [GCC 4.3.2] on linux2 170 Type "help", "copyright", "credits" or "license" for more information. 171 >>> 172 \end{verbatim} 173 Here you can use all available python commands and language features, for instance 174 \begin{python} 175 >>> x=2+3 176 >>> print "2+3=",x 177 2+3= 5 178 \end{python} 179 gross 2375 We refer to the python users guide if you not familiar with python. 180 gross 2370 181 jgs 102 \escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}. 182 (We will discuss a more general form of a PDE \index{partial differential equation!PDE} 183 that can be defined through the \LinearPDE class later). The instantiation of 184 a \Poisson class object requires the specification of the domain $\Omega$. In \escript 185 the \Domain class objects are used to describe the geometry of a domain but it also 186 contains information about the discretization methods and the actual solver which is used 187 to solve the PDE. Here we are using the FEM library \finley \index{finite element 188 method}. The following statements create the \Domain object \var{mydomain} from the 189 \finley method \method{Rectangle} 190 \begin{python} 191 ksteube 1316 from esys.finley import Rectangle 192 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 193 jgs 102 \end{python} 194 In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and 195 the right, upper corner at $(\var{l0},\var{l1})=(1,1)$. 196 jgs 107 The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and 197 jgs 102 $x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and 198 other \Domain generators within the \finley module, 199 see \Chap{CHAPTER ON FINLEY}. 200 201 jgs 107 The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and 202 jgs 102 the right hand side $f$ of the PDE to constant $1$: 203 \begin{python} 204 ksteube 1316 from esys.escript.linearPDEs import Poisson 205 mypde = Poisson(mydomain) 206 mypde.setValue(f=1) 207 jgs 102 \end{python} 208 We have not specified any boundary condition but the 209 \Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann 210 boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary 211 condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$ 212 and any constant $C$ the function $u+C$ becomes a solution as well. We have to add 213 a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done 214 jgs 107 by defining a characteristic function \index{characteristic function} 215 which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set 216 jgs 102 and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c}, 217 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 218 ksteube 1316 an object \var{x} which contains the coordinates of the nodes in the domain use 219 jgs 102 \begin{python} 220 ksteube 1316 x=mydomain.getX() 221 jgs 102 \end{python} 222 gross 660 The method \method{getX} of the \Domain \var{mydomain} 223 jgs 107 gives access to locations 224 ksteube 1316 in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which will be 225 discussed in \Chap{ESCRIPT CHAP} in more detail. What we need to know here is that 226 gross 660 227 ksteube 1316 \var{x} has \Rank (number of dimensions) and a \Shape (list of dimensions) which can be viewed by 228 gross 565 calling the \method{getRank} and \method{getShape} methods: 229 \begin{python} 230 ksteube 1316 print "rank ",x.getRank(),", shape ",x.getShape() 231 gross 565 \end{python} 232 ksteube 1316 This will print something like 233 gross 565 \begin{python} 234 ksteube 1316 rank 1, shape (2,) 235 gross 565 \end{python} 236 The \Data object also maintains type information which is represented by the 237 \FunctionSpace of the object. For instance 238 \begin{python} 239 ksteube 1316 print x.getFunctionSpace() 240 gross 565 \end{python} 241 will print 242 \begin{python} 243 ksteube 1316 Function space type: Finley_Nodes on FinleyMesh 244 gross 565 \end{python} 245 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. 246 gross 565 To get the $x\hackscore{0}$ coordinates of the locations we use the 247 statement 248 \begin{python} 249 ksteube 1316 x0=x 250 gross 565 \end{python} 251 Object \var{x0} 252 is again a \Data object now with \Rank $0$ and 253 \Shape $()$. It inherits the \FunctionSpace from \var{x}: 254 \begin{python} 255 ksteube 1316 print x0.getRank(),x0.getShape(),x0.getFunctionSpace() 256 gross 565 \end{python} 257 will print 258 \begin{python} 259 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh 260 gross 565 \end{python} 261 ksteube 1316 We can now construct a function \var{gammaD} which is only non-zero on the bottom and left edges 262 of the domain with 263 gross 565 \begin{python} 264 ksteube 1316 from esys.escript import whereZero 265 gammaD=whereZero(x)+whereZero(x) 266 gross 565 \end{python} 267 ksteube 1316 268 \code{whereZero(x)} creates function which equals $1$ where \code{x} is (almost) equal to zero 269 jgs 107 and $0$ elsewhere. 270 gross 565 Similarly, \code{whereZero(x)} creates function which equals $1$ where \code{x} is 271 jgs 107 equal to zero and $0$ elsewhere. 272 gross 565 The sum of the results of \code{whereZero(x)} and \code{whereZero(x)} 273 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. 274 gross 565 Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from 275 \begin{python} 276 ksteube 1316 print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace() 277 gross 565 \end{python} 278 one gets 279 \begin{python} 280 ksteube 1316 0 () Function space type: Finley_Nodes on FinleyMesh 281 gross 565 \end{python} 282 ksteube 1316 An additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the 283 jgs 102 characteristic function \index{characteristic function} of the locations 284 of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous} 285 are set. The complete definition of our example is now: 286 \begin{python} 287 ksteube 1316 from esys.linearPDEs import Poisson 288 x = mydomain.getX() 289 gammaD = whereZero(x)+whereZero(x) 290 mypde = Poisson(domain=mydomain) 291 mypde.setValue(f=1,q=gammaD) 292 jgs 102 \end{python} 293 lkettle 573 The first statement imports the \Poisson class definition from the \linearPDEs module \escript package. 294 jgs 107 To get the solution of the Poisson equation defined by \var{mypde} we just have to call its 295 jgs 102 \method{getSolution}. 296 297 ksteube 1316 Now we can write the script to solve our Poisson problem 298 jgs 102 \begin{python} 299 ksteube 1316 from esys.escript import * 300 from esys.escript.linearPDEs import Poisson 301 from esys.finley import Rectangle 302 # generate domain: 303 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 304 # define characteristic function of Gamma^D 305 x = mydomain.getX() 306 gammaD = whereZero(x)+whereZero(x) 307 # define PDE and get its solution u 308 mypde = Poisson(domain=mydomain) 309 mypde.setValue(f=1,q=gammaD) 310 u = mypde.getSolution() 311 # write u to an external file 312 saveVTK("u.xml",sol=u) 313 jgs 102 \end{python} 314 ksteube 1316 The entire code is available as \file{poisson.py} in the \ExampleDirectory 315 jgs 102 316 ksteube 1316 The last statement writes the solution (tagged with the name "sol") to a file named \file{u.xml} in 317 \VTK file format. 318 gross 2370 Now you may run the script using the \escript environment 319 and visualize the solution using \mayavi: 320 ksteube 1316 \begin{verbatim} 321 gross 2370 escript poisson.py 322 ksteube 1316 mayavi -d u.xml -m SurfaceMap 323 \end{verbatim} 324 See \fig{fig:FirstSteps.3}. 325 326 jgs 102 \begin{figure} 327 jfenwick 2335 \centerline{\includegraphics[width=\figwidth]{figures/FirstStepResult}} 328 lkettle 573 \caption{Visualization of the Poisson Equation Solution for $f=1$} 329 jgs 102 \label{fig:FirstSteps.3} 330 \end{figure} 331