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% $Id$ |
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|
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\chapter{The First Steps} |
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\label{FirstSteps} |
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|
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{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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In this chapter we will show the basics of how to use \escript to solve |
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a partial differential equation \index{partial differential equation} (PDE \index{partial differential equation!PDE}). The reader should be familiar with |
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the basics of Python. The knowledge presented the Python tutorial at \url{http://docs.python.org/tut/tut.html} |
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is sufficient. It is helpful if the reader has some basic knowledge on PDEs \index{PDE}. |
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|
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The \index{PDE} we want to solve is 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 domain of interest which we denote 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 Figure~\fig{fig:FirstSteps.1}. |
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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 $w$ and any direction $i$ $u\hackscore{,i}$ |
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denotes the partial derivative \index{partial derivative} of $w$ with respect to $i$. |
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\footnote{Some readers |
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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 = \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 left to the comma denotes a spatial derivative with respect |
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to the index. To get a more compact form we will write $w\hackscore{,ij}=(w\hackscore {,i})\hackscore{,j}$ |
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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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In some cases, and we will see examples for this in the next chapter, |
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the usage of the nested $\sum$ symbols blows up the formulas and therefore |
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it is convenient to use Einstein summation convention \index{summation convention} which |
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says that $\sum$ sign is dropped and a summation over a repeated index is performed |
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("repeated index means summation"). 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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\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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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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$n\hackscore{i} u\hackscore{,i}= n\hackscore1 u\hackscore{,1} + |
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n\hackscore2 u\hackscore{,2}$. |
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\footnote{Some readers may familiar with the notation |
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\begin{equation*} |
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\frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i} |
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\end{equation*} |
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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 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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The 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 unknown |
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function $u$. |
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|
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|
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{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, so 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 subdivide into small |
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sub-domain, so-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}. Figure~\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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import finley |
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mydomain = finley.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{l0} and \var{l1} 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 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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import escript |
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mypde = escript.Poisson(domain=mydomain,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 defines a characteristic function \index{characteristic function} |
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which has a positive values at locations $(x_0,x_1)$ 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 a function which is positive for the cases $x_0=0$ or $x_1=0$: |
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\begin{python} |
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x=mydomain.getX() |
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gammaD=x[0].whereZero()+x[1].whereZero() |
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\end{python} |
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In first statement returns, the method \method{getX} of the \Domain \var{mydomain} access to the locations |
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in the domain defined by \var{mydomain}. The object \var{x} is actually an \Data object |
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which we will learn more about later. \code{x[0]} returns the $x_0$ coordinates of the locations and |
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\code{x[0].whereZero()} creates function which equals $1$ where \code{x[0]} is (nearly) equal to zero |
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and $0$ elsewhere. The sum of the results of \code{x[0].whereZero()} and \code{x[1].whereZero()} gives a function on the domain \var{mydomain} which is exactly positive where $x_0$ or $x_1$ is equal to zero. |
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|
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The additional parameter \var{q} of the \Poisson object creater 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 linearPDEs import Poisson |
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x = mydomain.getX() |
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gammaD = x[0].whereZero()+x[1].whereZero() |
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mypde = Poisson(domain=mydomain,f=1,q=gammaD) |
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\end{python} |
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The first statement imports the \Poisson class definition form the \linearPDEsPack module which is part of the \escript module. |
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To get the solution of the Poisson equation defines by \var{mypde} we just have to call its |
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\method{getSolution}. |
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|
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Now we can write the script to solve our test problem (Remember that |
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lines starting with '\#' are commend lines in Python) (available as \file{mypoisson.py} |
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in the \ExampleDirectory): |
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\begin{python} |
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import esys.finley |
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from esys.linearPDEs import Poisson |
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# generate domain: |
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mydomain = esys.finley.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 = x[0].whereZero()+x[1].whereZero() |
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# define PDE and get its solution u |
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mypde = Poisson(domain=mydomain,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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u.saveDX("u.dx") |
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\end{python} |
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The last statement writes the solution to the external file \file{u.dx} in |
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\OpenDX file format. \OpenDX is a software package |
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for the visualization of scientific, engineering and analytical data and is freely available |
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from \url{http://www.opendx.org}. |
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|
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\begin{figure} |
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\centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}} |
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\caption{\OpenDX visualization of the Possion equation soluition for $f=1$} |
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\label{fig:FirstSteps.3} |
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\end{figure} |
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|
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You can edit this script using your favourite text editor (or the Integrated DeveLopment Environment IDLE |
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for Python). If the script file has the name \file{mypoisson.py} \index{scripts!\file{mypoisson.py}} you can run the |
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script from any shell using the command: |
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\begin{verbatim} |
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python mypoisson.py |
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\end{verbatim} |
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After the script has (hopefully successfully) been completed you will find the file \file{u.dx} in the current |
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directory. An easy way to visualize the results is the command |
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\begin{verbatim} |
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dx -prompter |
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\end{verbatim} |
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to start the generic data visualization interface of \OpenDX. \fig{fig:FirstSteps.3} shows the result. |
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