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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% |
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% Copyright (c) 2003-2008 by University of Queensland |
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% Earth Systems Science Computational Center (ESSCC) |
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% http://www.uq.edu.au/esscc |
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% |
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% Primary Business: Queensland, Australia |
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% Licensed under the Open Software License version 3.0 |
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% http://www.opensource.org/licenses/osl-3.0.php |
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% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{The First Steps} |
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\label{FirstSteps} |
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In this chapter we give an introduction how to use \escript to solve |
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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} |
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is more than sufficient. |
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|
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The PDE \index{partial differential equation} we wish to solve i s the Poisson equation \index{Poisson equation} |
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\begin{equation} |
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-\Delta u =f |
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\label{eq:FirstSteps.1} |
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\end{equation} |
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for the solution $u$. The function $f$ is the given right hand side. The domain of interest, denoted by $\Omega$, |
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is the unit square |
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\begin{equation} |
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\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 \} |
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\label{eq:FirstSteps.1b} |
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\end{equation} |
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The domain is shown in \fig{fig:FirstSteps.1}. |
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\begin{figure} [h!] |
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\centerline{\includegraphics[width=\figwidth]{figures/FirstStepDomain}} |
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\caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.} |
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\label{fig:FirstSteps.1} |
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\end{figure} |
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|
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$\Delta$ denotes the Laplace operator\index{Laplace operator}, which is defined by |
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\begin{equation} |
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\Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1} |
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\label{eq:FirstSteps.1.1} |
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\end{equation} |
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where, for any function $u$ and any direction $i$, $u\hackscore{,i}$ |
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denotes the partial derivative \index{partial derivative} of $u$ with respect to $i$. |
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\footnote{You |
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may be more familiar with the Laplace operator\index{Laplace operator} being written |
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as $\nabla^2$, and written in the form |
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\begin{equation*} |
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\nabla^2 u = \nabla^t \cdot \nabla u = \frac{\partial^2 u}{\partial x\hackscore 0^2} |
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+ \frac{\partial^2 u}{\partial x\hackscore 1^2} |
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\end{equation*} |
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and \eqn{eq:FirstSteps.1} as |
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\begin{equation*} |
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-\nabla^2 u = f |
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\end{equation*} |
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} |
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Basically, in the subindex of a function, any index to the left of the comma denotes a spatial derivative with respect |
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to the index. To get a more compact form we will write $u\hackscore{,ij}=(u\hackscore {,i})\hackscore{,j}$ |
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which leads to |
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\begin{equation} |
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\Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii} |
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\label{eq:FirstSteps.1.1b} |
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\end{equation} |
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We often find that use |
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of nested $\sum$ symbols makes formulas cumbersome, and we use the more |
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convenient Einstein summation convention \index{summation convention}. This |
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drops the $\sum$ sign and assumes that a summation is performed over any repeated index. |
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For instance we write |
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\begin{eqnarray} |
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x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\ |
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x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\ |
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u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\ |
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x\hackscore{ij}u\hackscore{i,j}=\sum\hackscore{j=0}^2\sum\hackscore{i=0}^2 x\hackscore{ij}u\hackscore{i,j} \\ |
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\label{eq:FirstSteps.1.1c} |
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\end{eqnarray} |
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With the summation convention we can write the Poisson equation \index{Poisson equation} as |
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\begin{equation} |
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- u\hackscore{,ii} =1 |
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\label{eq:FirstSteps.1.sum} |
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\end{equation} |
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where $f=1$ in this example. |
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|
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On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$ |
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of the solution $u$ shall be zero, ie. $u$ shall fulfill |
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the homogeneous Neumann boundary condition\index{Neumann |
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boundary condition!homogeneous} |
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\begin{equation} |
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n\hackscore{i} u\hackscore{,i}= 0 \;. |
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\label{eq:FirstSteps.2} |
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\end{equation} |
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$n=(n\hackscore{i})$ denotes the outer normal field |
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of the domain, see \fig{fig:FirstSteps.1}. Remember that we |
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are applying the Einstein summation convention \index{summation convention}, i.e |
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$n\hackscore{i} u\hackscore{,i}= n\hackscore{0} u\hackscore{,0} + |
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n\hackscore{1} u\hackscore{,1}$. |
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\footnote{Some readers may familiar with the notation |
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$ |
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\frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i} |
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$ |
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for the normal derivative.} |
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The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the |
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set $\Gamma^N$ which is the top and right edge of the domain: |
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\begin{equation} |
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\Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \} |
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\label{eq:FirstSteps.2b} |
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\end{equation} |
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On the bottom and the left edge of the domain which is defined |
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as |
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\begin{equation} |
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\Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega | x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \} |
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\label{eq:FirstSteps.2c} |
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\end{equation} |
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the solution shall be identically zero: |
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\begin{equation} |
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u=0 \; . |
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\label{eq:FirstSteps.2d} |
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\end{equation} |
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This kind of boundary condition is called a homogeneous Dirichlet boundary condition |
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\index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together |
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with the Neumann boundary condition \eqn{eq:FirstSteps.2} and |
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Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so |
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called boundary value |
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problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for |
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the unknown function~$u$. |
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|
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\begin{figure}[h] |
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\centerline{\includegraphics[width=\figwidth]{figures/FirstStepMesh}} |
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\caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here |
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each element is a quadrilateral and described by four nodes, namely |
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the corner points. The solution is interpolated by a bi-linear |
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polynomial.} |
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\label{fig:FirstSteps.2} |
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\end{figure} |
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|
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In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical |
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methods have to be used construct an approximation of the solution |
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$u$. Here we will use the finite element method\index{finite element |
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method} (FEM\index{finite element |
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method!FEM}). The basic idea is to fill the domain with a |
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set of points called nodes. The solution is approximated by its |
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values on the nodes\index{finite element |
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method!nodes}. Moreover, the domain is subdivided into smaller |
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sub-domains called elements \index{finite element |
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method!element}. On each element the solution is |
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represented by a polynomial of a certain degree through its values at |
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the nodes located in the element. The nodes and its connection through |
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elements is called a mesh\index{finite element |
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method!mesh}. \fig{fig:FirstSteps.2} shows an |
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example of a FEM mesh with four elements in the $x_0$ and four elements |
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in the $x_1$ direction over the unit square. |
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For more details we refer the reader to the literature, for instance |
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\Ref{Zienc,NumHand}. |
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|
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\escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}. |
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(We will discuss a more general form of a PDE \index{partial differential equation!PDE} |
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that can be defined through the \LinearPDE class later). The instantiation of |
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a \Poisson class object requires the specification of the domain $\Omega$. In \escript |
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the \Domain class objects are used to describe the geometry of a domain but it also |
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contains information about the discretization methods and the actual solver which is used |
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to solve the PDE. Here we are using the FEM library \finley \index{finite element |
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method}. The following statements create the \Domain object \var{mydomain} from the |
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\finley method \method{Rectangle} |
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\begin{python} |
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from esys.finley import Rectangle |
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mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) |
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\end{python} |
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In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and |
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the right, upper corner at $(\var{l0},\var{l1})=(1,1)$. |
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The arguments \var{n0} and \var{n1} define the number of elements in $x\hackscore{0}$ and |
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$x\hackscore{1}$-direction respectively. For more details on \method{Rectangle} and |
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other \Domain generators within the \finley module, |
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see \Chap{CHAPTER ON FINLEY}. |
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|
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The following statements define the \Poisson class object \var{mypde} with domain \var{mydomain} and |
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the right hand side $f$ of the PDE to constant $1$: |
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\begin{python} |
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from esys.escript.linearPDEs import Poisson |
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mypde = Poisson(mydomain) |
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mypde.setValue(f=1) |
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\end{python} |
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We have not specified any boundary condition but the |
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\Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann |
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boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary |
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condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$ |
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and any constant $C$ the function $u+C$ becomes a solution as well. We have to add |
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a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done |
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by defining a characteristic function \index{characteristic function} |
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which has positive values at locations $x=(x\hackscore{0},x\hackscore{1})$ where Dirichlet boundary condition is set |
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and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c}, |
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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 |
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an object \var{x} which contains the coordinates of the nodes in the domain use |
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\begin{python} |
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x=mydomain.getX() |
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\end{python} |
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The method \method{getX} of the \Domain \var{mydomain} |
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gives access to locations |
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in the domain defined by \var{mydomain}. The object \var{x} is actually a \Data object which will be |
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discussed in \Chap{ESCRIPT CHAP} in more detail. What we need to know here is that |
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\var{x} has \Rank (number of dimensions) and a \Shape (list of dimensions) which can be viewed by |
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calling the \method{getRank} and \method{getShape} methods: |
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\begin{python} |
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print "rank ",x.getRank(),", shape ",x.getShape() |
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\end{python} |
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This will print something like |
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\begin{python} |
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rank 1, shape (2,) |
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\end{python} |
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The \Data object also maintains type information which is represented by the |
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\FunctionSpace of the object. For instance |
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\begin{python} |
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print x.getFunctionSpace() |
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\end{python} |
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will print |
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\begin{python} |
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Function space type: Finley_Nodes on FinleyMesh |
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\end{python} |
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which tells us that the coordinates are stored on the nodes of (rather than on points in the interior of) a \finley mesh. |
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To get the $x\hackscore{0}$ coordinates of the locations we use the |
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statement |
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\begin{python} |
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x0=x[0] |
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\end{python} |
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Object \var{x0} |
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is again a \Data object now with \Rank $0$ and |
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\Shape $()$. It inherits the \FunctionSpace from \var{x}: |
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\begin{python} |
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print x0.getRank(),x0.getShape(),x0.getFunctionSpace() |
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\end{python} |
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will print |
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\begin{python} |
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0 () Function space type: Finley_Nodes on FinleyMesh |
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\end{python} |
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We can now construct a function \var{gammaD} which is only non-zero on the bottom and left edges |
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of the domain with |
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\begin{python} |
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from esys.escript import whereZero |
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gammaD=whereZero(x[0])+whereZero(x[1]) |
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\end{python} |
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\code{whereZero(x[0])} creates function which equals $1$ where \code{x[0]} is (almost) equal to zero |
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and $0$ elsewhere. |
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Similarly, \code{whereZero(x[1])} creates function which equals $1$ where \code{x[1]} is |
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equal to zero and $0$ elsewhere. |
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The sum of the results of \code{whereZero(x[0])} and \code{whereZero(x[1])} |
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gives a function on the domain \var{mydomain} which is strictly positive where $x\hackscore{0}$ or $x\hackscore{1}$ is equal to zero. |
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Note that \var{gammaD} has the same \Rank, \Shape and \FunctionSpace like \var{x0} used to define it. So from |
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\begin{python} |
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print gammaD.getRank(),gammaD.getShape(),gammaD.getFunctionSpace() |
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\end{python} |
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one gets |
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\begin{python} |
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0 () Function space type: Finley_Nodes on FinleyMesh |
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\end{python} |
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An additional parameter \var{q} of the \code{setValue} method of the \Poisson class defines the |
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characteristic function \index{characteristic function} of the locations |
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of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous} |
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are set. The complete definition of our example is now: |
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\begin{python} |
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from esys.linearPDEs import Poisson |
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x = mydomain.getX() |
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gammaD = whereZero(x[0])+whereZero(x[1]) |
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mypde = Poisson(domain=mydomain) |
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mypde.setValue(f=1,q=gammaD) |
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\end{python} |
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The first statement imports the \Poisson class definition from the \linearPDEs module \escript package. |
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To get the solution of the Poisson equation defined by \var{mypde} we just have to call its |
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\method{getSolution}. |
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Now we can write the script to solve our Poisson problem |
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\begin{python} |
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from esys.escript import * |
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from esys.escript.linearPDEs import Poisson |
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from esys.finley import Rectangle |
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# generate domain: |
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mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) |
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# define characteristic function of Gamma^D |
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x = mydomain.getX() |
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gammaD = whereZero(x[0])+whereZero(x[1]) |
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# define PDE and get its solution u |
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mypde = Poisson(domain=mydomain) |
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mypde.setValue(f=1,q=gammaD) |
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u = mypde.getSolution() |
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# write u to an external file |
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saveVTK("u.xml",sol=u) |
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\end{python} |
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The entire code is available as \file{poisson.py} in the \ExampleDirectory |
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|
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The last statement writes the solution (tagged with the name "sol") to a file named \file{u.xml} in |
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\VTK file format. |
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Now you may run the script and visualize the solution using \mayavi: |
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\begin{verbatim} |
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python poisson.py |
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mayavi -d u.xml -m SurfaceMap |
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\end{verbatim} |
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See \fig{fig:FirstSteps.3}. |
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|
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{figures/FirstStepResult}} |
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\caption{Visualization of the Poisson Equation Solution for $f=1$} |
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\label{fig:FirstSteps.3} |
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\end{figure} |
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