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revision 2867 by gross, Fri Jan 22 06:28:02 2010 UTC revision 2870 by gross, Mon Jan 25 08:34:15 2010 UTC
# Line 112  u=T^{(n)}; Line 112  u=T^{(n)};
112  A = \kappa; D = \frac{\rho c \hackscore{p}}{h}; f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n-1)}  A = \kappa; D = \frac{\rho c \hackscore{p}}{h}; f = q \hackscore{H} + \frac{\rho c\hackscore p}{h} T^{(n-1)}
113  \end{equation}  \end{equation}
114    
115  Now that the general form has been established, it can be submitted to \esc. Note that it is necessary to establish the state of our system at time zero or $T^{(n=0)}$. This is due to the time derivative approximation we have used from \refEq{eqn:Tbeuler}. Our model stipulates a starting temperature in the iron bar of 0\textcelsius. Thus the temperature distribution is simply;  % Now that the general form has been established, it can be submitted to \esc. Note that it is necessary to establish the state of our system at time zero or $T^{(n=0)}$. This is due to the time derivative approximation we have used from \refEq{eqn:Tbeuler}. Our model stipulates a starting temperature in the iron bar of 0\textcelsius. Thus the temperature distribution is simply;
116  \begin{equation}  % \begin{equation}
117  T(x,0) = T\hackscore{ref} = 0  % T(x,0) = \left
118  \end{equation}  % \end{equation}
119  for all $x$ in the domain.  % for all $x$ in the domain.
120    
121  \subsection{Boundary Conditions}  \subsection{Boundary Conditions}
122  With the PDE sufficiently modified, consideration must now be given to the boundary conditions of our model. Typically there are two main types of boundary conditions known as \textbf{Neumann} and \textbf{Dirichlet} boundary conditions\footnote{More information on Boundary Conditions is available at Wikipedia \url{http://en.wikipedia.org/wiki/Boundary_conditions}}, respectively.  With the PDE sufficiently modified, consideration must now be given to the boundary conditions of our model. Typically there are two main types of boundary conditions known as \textbf{Neumann} and \textbf{Dirichlet} boundary conditions\footnote{More information on Boundary Conditions is available at Wikipedia \url{http://en.wikipedia.org/wiki/Boundary_conditions}}, respectively.
# Line 169  condition translates into Line 169  condition translates into
169  \end{equation}  \end{equation}
170  for the right hand side of the rod. This is identical to the Neuman boundary condition we want to set. \esc will take care of this condition for us. We will discuss the Dirichlet boundary condition later.  for the right hand side of the rod. This is identical to the Neuman boundary condition we want to set. \esc will take care of this condition for us. We will discuss the Dirichlet boundary condition later.
171    
172  \subsection{A \textit{1D} Clarification}  \subsection{Outline of the Implementation}
173    \label{sec:outline}
174    To solve the heat diffusion equation (equation \refEq{eqn:hd}) we will write a simple \pyt script. At this point we assume that you have some basic understanding of the \pyt programming language. If not there are some pointers and links available in Section \ref{sec:escpybas}. The script we will discuss later in details will have four major steps. Firstly we need to define the domain where we want to
175    calculate the temperature. For our problem this is the iron rod which has a rectangular shape. Secondly we need to define the PDE
176    we need to solve in each time step to get the updated temperature. Thirdly we need to define the the coefficients of the PDE and finally we need to solve the PDE. The last two steps need to be repeated until the final time marker has been reached.
177    
178    In the terminology of \pyt the domain and PDE are represented by \textbf{objects}. The nice feature of an object is that it defined by it usage and features
179    rather than its actual representation. So we will create a domain object to decribe the geometry of our iron rod. The main feature
180    of the object we will use is the fact that we can define PDEs and spatially distributed values such as the temperature
181    on a domain. In fact the domain object has many more features - most of them you will
182    never use and do not need to understand. Similar a PDE object is defined by the fact that we can define the coefficients of the PDE and solve the PDE. At a
183    later stage you may use more advanved features of the PDE class but you need to worry about them only at the point when you use them.
184    
185    
186    \begin{figure}[t]
187     \centering
188       \includegraphics[width=6in]{figures/functionspace.pdf}
189       \label{fig:fs}
190       \caption{\esc domain construction overview}
191    \end{figure}
192    
193    \subsection{The Domain Constructor in \esc}
194    \label{ss:domcon}
195    It is helpful to have a better understanding how spatially distributed value such as the temperature or PDE coefficients are interpreted in \esc. Again
196    from the user's point of view the representation of these spatially distributed values is not relevant.
197    
198    There are various ways to construct domain objects. The simplest form is as rectangular shaped region with a length and height. There is
199    a ready to use function call for this. Besides the spatial dimensions the function call will require you to specify the number
200    elements or cells to be used along the length and height, see Figure~\ref{fig:fs}. Any spacially distributed value
201    and the PDE is represented in discrete form using this element representation\footnote{We will use the finite element method (FEM), see \url{http://en.wikipedia.org/wiki/Finite_element_method}i for details.}. Therefore we will have access to an approximation of the true PDE solution only.
202    The quality of the approximation depends - besides other factors- mainly on the number of elements being used. In fact, the
203    approximation becomes better the more elements are used. However, computational costs and compute time grow with the number of
204    elements being used. It therefore important that you find the right balance between the demand in accuarcy and acceptable resource usage.
205    
206    In general, one can thinks about a domain object as a composition of nodes and elements.
207    As shown in Figure~\ref{fig:fs}, an element is defined by the nodes used to describe its vertices.
208    To represent spatial distributed values the user can use
209    the values at the nodes, at the elements in the interior of the domain or at elements located at the surface of the domain.
210    The different approach used to represent values is called \textbf{function space} and is attached to all objects
211    in \esc representing a spatial distributed value such as the solution of a PDE. The three
212    function spaces we will use at the moment are;
213    \begin{enumerate}
214    \item the nodes, called by \verb|ContinuousFunction(domain)| ;
215    \item the elements/cells, called by \verb|Function(domain)| ; and
216    \item the boundary, called by \verb|FunctionOnBoundary(domain)| .
217    \end{enumerate}
218    A function space object such as \verb|ContinuousFunction(domain)| has the method \verb|getX| attached to it. This method returns the
219    location of the so-called \textbf{sample points} used to represent values with the particular function space attached to it. So the
220    call \verb|ContinuousFunction(domain).getX()| will return the coordinates of the nodes used to describe the domain while
221    the  \verb|Function(domain).getX()| returns the coordinates of numerical integration points within elements, see
222    Figure~\ref{fig:fs}.
223    
224    This distiction between different representations of spatial distributed values
225    is important in order to be able to vary the degrees of smoothness in a PDE problem.
226    The coefficients of a PDE need not be continuous thus this qualifies as a \verb|Function()| type.
227    On the other hand a temperature distribution must be continuous and needs to be represented with a \verb|ContinuousFunction()| function space.
228    An influx may only be defined at the boundary and is therefore a \verb FunctionOnBoundary()  object.  
229    \esc allows certain transformations of the function spaces. A \verb ContinuousFunction()  can be transformed into a \verb|FunctionOnBoundary()|
230    or \verb|Function()|. On the other hand there is not enough information in a \verb FunctionOnBoundary()  to transform it to a \verb ContinuousFunction()  .
231    These transformations, which ar ecalled \textbf{interpolation} are invoked autmatically by \esc if needed.
232    
233    Later in this introduction we will discuss how
234    to define specific areas of geometry with different materials which are represented by different material coefficients such the
235    thermal conductivities $kappa$. A very powerful techique to define these types of PDE
236    coefficients is tagging. Blocks of materials and boundaries can be named and values can be defined on subregions based on their names.
237    This is simplifing PDE coefficient and flux definitions. It makes for much easier scripting. We will discuss this technique in Section~\ref{STEADY-STATE HEAT REFRACTION}.
238    
239    
240    \subsection{A Clarification for the 1D Case}
241  It is necessary for clarification that we revisit the general PDE from \refeq{eqn:commonform nabla} under the light of a two dimensional domain. \esc is inherently designed to solve problems that are greater than one dimension and so \refEq{eqn:commonform nabla} needs to be read as a higher dimensional problem. In the case of two spatial dimensions the \textit{Nabla operator} has in fact two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. In full, \refEq{eqn:commonform nabla} assuming a constant coefficient $A$, takes the form;  It is necessary for clarification that we revisit the general PDE from \refeq{eqn:commonform nabla} under the light of a two dimensional domain. \esc is inherently designed to solve problems that are greater than one dimension and so \refEq{eqn:commonform nabla} needs to be read as a higher dimensional problem. In the case of two spatial dimensions the \textit{Nabla operator} has in fact two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. In full, \refEq{eqn:commonform nabla} assuming a constant coefficient $A$, takes the form;
242  \begin{equation}\label{eqn:commonform2D}  \begin{equation}\label{eqn:commonform2D}
243  -A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}}  -A\hackscore{00}\frac{\partial^{2}u}{\partial x^{2}}
# Line 187  shown in \refEq{eqn:commonform} we need Line 255  shown in \refEq{eqn:commonform} we need
255  A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0  A\hackscore{00}=A; A\hackscore{01}=A\hackscore{10}=A\hackscore{11}=0
256  \end{equation}  \end{equation}
257    
 \subsection{Outline of the PDE Solution Script}  
   
   
258    
259  \subsection{Developing a PDE Solution Script}  \subsection{Developing a PDE Solution Script}
260  \label{sec:key}  \label{sec:key}
261  To solve the heat diffusion equation (equation \refEq{eqn:hd}) we will write a simple \pyt script which uses the \modescript, \modfinley and \modmpl modules. At this point we assume that you have some basic understanding of the \pyt programming language. If not there are some pointers and links available in Section \ref{sec:escpybas} .  We will write a simple \pyt script which uses the \modescript, \modfinley and \modmpl modules.
   
262  By developing a script for \esc, the heat diffusion equation can be solved at successive time steps for a predefined period using our general form \refEq{eqn:hdgenf}. Firstly it is necessary to import all the libraries\footnote{The libraries contain predefined scripts that are required to solve certain problems, these can be simple like $sine$ and $cosine$ functions or more complicated like those from our \esc library.}  By developing a script for \esc, the heat diffusion equation can be solved at successive time steps for a predefined period using our general form \refEq{eqn:hdgenf}. Firstly it is necessary to import all the libraries\footnote{The libraries contain predefined scripts that are required to solve certain problems, these can be simple like $sine$ and $cosine$ functions or more complicated like those from our \esc library.}
263  that we will require.  that we will require.
264  \begin{python}  \begin{python}

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