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revision 1965 by jfenwick, Tue Nov 4 01:49:19 2008 UTC revision 1966 by jfenwick, Wed Nov 5 04:59:22 2008 UTC
# Line 18  Line 18 
18    
19  \begin{figure}  \begin{figure}
20  \includegraphics[width=\textwidth]{figures/EscriptDiagram1}  \includegraphics[width=\textwidth]{figures/EscriptDiagram1}
21  \caption{\label{ESCRIPT DEP}Dependency of Function Spaces. An arrow indicates that a function in the  \caption{\label{ESCRIPT DEP}Dependency of Function Spaces in Finley. An arrow indicates that a function in the
22  function space at the starting point can be interpreted as a function in the function space of the arrow target.}  function space at the starting point can be interpolated to the function space of the arrow target.
23    All functionspaces on the left side can be interpolated to any of the functionspaces on the right.}
24  \end{figure}  \end{figure}
25    
26  \escript is a Python module that allows you to represent the values of  \escript is a Python module that allows you to represent the values of
# Line 64  The reduced smoothness for PDE solution Line 65  The reduced smoothness for PDE solution
65  solving saddle point problems \index{saddle point problems}, eg. the Stokes equation.  solving saddle point problems \index{saddle point problems}, eg. the Stokes equation.
66  A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous.    A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous.  
67  The location of discontinuity is defined in the \Domain object.  The location of discontinuity is defined in the \Domain object.
68  \fig{ESCRIPT DEP} shows the dependency between the types of function spaces.  \fig{ESCRIPT DEP} shows the dependency between the types of function spaces in Finley (other libraries may have different relationships).
69    
70  The solution of a PDE is a continuous function. Any continuous function can be seen as a general function  The solution of a PDE is a continuous function. Any continuous function can be seen as a general function
71  on the domain and can be restricted to the boundary as well as to one side of  on the domain and can be restricted to the boundary as well as to one side of
# Line 83  The concept of function spaces describes Line 84  The concept of function spaces describes
84  functions and allows abstraction from the actual representation  functions and allows abstraction from the actual representation
85  of the function in the context of a particular application. For instance,  of the function in the context of a particular application. For instance,
86  in the FEM context a  in the FEM context a
87  function of the \Function type  function of the \Function type (written as \emph{Function()} in Figure~\ref{ESCRIPT DEP})
88  is usually represented by its values at the element center,  is usually represented by its values at the element center,
89  but in a finite difference scheme the edge midpoint of cells is preferred.  but in a finite difference scheme the edge midpoint of cells is preferred.
90  By changing its function space you can use the same function in a Finite Difference  By changing its function space you can use the same function in a Finite Difference
# Line 270  processing the value for each individual Line 271  processing the value for each individual
271  \Data objects can be written to disk files and read with \var{dump} and \var{load}, both of which use \netCDF.  \Data objects can be written to disk files and read with \var{dump} and \var{load}, both of which use \netCDF.
272  Use these to save data for visualization, checkpoint/restart or simply to save and reuse data that was expensive to compute.  Use these to save data for visualization, checkpoint/restart or simply to save and reuse data that was expensive to compute.
273    
274  For instance to save the coordinates of the data points of the \FunctionSpace  For instance to save the coordinates of the data points of the
275  \ContinuousFunction to the file {\tt x.nc} use  \ContinuousFunction to the file {\tt x.nc} use
276  \begin{python}  \begin{python}
277    x=ContinuousFunction(mydomain).getX()    x=ContinuousFunction(mydomain).getX()
# Line 285  It is common to simply recreate the \Dom Line 286  It is common to simply recreate the \Dom
286  \var{domain=ReadMesh(fileName)} and \var{domain.write(fileName)}.  \var{domain=ReadMesh(fileName)} and \var{domain.write(fileName)}.
287    
288  The function space of the \Data is stored in {\tt x.nc}, though.  The function space of the \Data is stored in {\tt x.nc}, though.
 That means that \Data objects that are constant or tagged can be recovered with a different \Domain\footnote{This is provided that the domains use the same \FunctionSpace types.}.  
289  If the \Data object  If the \Data object
290  is expanded, the number of data points in the file and of the \Domain for the particular \FunctionSpace must match.  is expanded, the number of data points in the file and of the \Domain for the particular \FunctionSpace must match.
291  Moreover, the ordering of the values is checked using the reference identifiers provided by  Moreover, the ordering of the values is checked using the reference identifiers provided by
# Line 348  return \True if \var{tag_name} is a vali Line 348  return \True if \var{tag_name} is a vali
348  Otherwise \False is returned.  Otherwise \False is returned.
349  \end{methoddesc}  \end{methoddesc}
350    
351  \begin{methoddesc}[Domain]{__str__}{g}  \begin{methoddesc}[Domain]{__str__}{arg}
352  (python str() function) returns string representation of the \Domain.  (python str() function) returns string representation of the \Domain.
353  \end{methoddesc}  \end{methoddesc}
354    
355    \begin{methoddesc}[Domain]{onMasterProcessor}{}
356     returns \True if executed on the MPI master processor, \False otherwise.
357    This can be used in conjunction with MPIBarrier to ensure commands only run once.
358    \end{methoddesc}
359    
360    \begin{methoddesc}[Domain]{MPIBarrier}{}
361     executes an MPIBarrier command. If MPI support is not enabled, this command does nothing.
362    \end{methoddesc}
363    
364    \begin{methoddesc}[Domain]{MPIBarrier}{}
365     executes an MPIBarrier command. If MPI support is not enabled, this command does nothing.
366    \end{methoddesc}
367    
368    \begin{methoddesc}[Domain]{getMPISize}{}
369     returns the number of MPI processors used for this domain.
370    \end{methoddesc}
371    
372    \begin{methoddesc}[Domain]{getMPIRank}{}
373     returns the rank of the processor executing the statement.
374    \end{methoddesc}
375    
376  \subsection{\FunctionSpace class}  \subsection{\FunctionSpace class}
377  \begin{classdesc}{FunctionSpace}{}  \begin{classdesc}{FunctionSpace}{}
378  \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects  \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects
379  are instantiated by generator functions. \Data objects in particular \FunctionSpace are  are instantiated by generator functions. A \Data object in a particular \FunctionSpace is
380  represented by their values at \DataSamplePoints which are defined by the type and the \Domain of the  represented by its values at \DataSamplePoints which are defined by the type and the \Domain of the
381  \FunctionSpace.  \FunctionSpace.
382  \end{classdesc}  \end{classdesc}
383  The following methods are available:  The following methods are available:
# Line 718  returns the maximum value over all compo Line 739  returns the maximum value over all compo
739  returns the minimum value over all components and all \DataSamplePoints of \var{a}  returns the minimum value over all components and all \DataSamplePoints of \var{a}
740  \end{funcdesc}  \end{funcdesc}
741    
 \begin{funcdesc}{sin}{a}  
 applies sine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{cos}{a}  
 applies cosine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{tan}{a}  
 applies tangent function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{asin}{a}  
 applies arc (inverse) sine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{acos}{a}  
 applies arc (inverse) cosine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{atan}{a}  
 applies arc (inverse) tangent function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{sinh}{a}  
 applies hyperbolic sine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{cosh}{a}  
 applies hyperbolic cosine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{tanh}{a}  
 applies hyperbolic tangent function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{asinh}{a}  
 applies arc (inverse) hyperbolic sine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{acosh}{a}  
 applies arc (inverse) hyperbolic cosine function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{atanh}{a}  
 applies arc (inverse) hyperbolic tangent function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{exp}{a}  
 applies exponential function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{sqrt}{a}  
 applies square root function to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{log}{a}  
 applies the natural logarithm to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{log10}{a}  
 applies the base-$10$ logarithm to \var{a}.  
 \end{funcdesc}  
   
 \begin{funcdesc}{sign}{a}  
 applies the sign function to \var{a}, that is $1$ where \var{a} is positive,  
 $-1$ where \var{a} is negative and $0$ otherwise.  
 \end{funcdesc}  
   
 \begin{funcdesc}{wherePositive}{a}  
 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.  
 \end{funcdesc}  
   
 \begin{funcdesc}{whereNegative}{a}  
 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.  
 \end{funcdesc}  
   
 \begin{funcdesc}{whereNonNegative}{a}  
 returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.  
 \end{funcdesc}  
   
 \begin{funcdesc}{whereNonPositive}{a}  
 returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.  
 \end{funcdesc}  
   
 \begin{funcdesc}{whereZero}{a\optional{, tol=0.}}  
 returns a function which is $1$ where \var{a} equals zero with tolerance \var{tol} and $0$ otherwise.  
 \end{funcdesc}  
742    
 \begin{funcdesc}{whereNonZero}{a\optional{, tol=0.}}  
 returns a function which is $1$ where \var{a} different from zero with tolerance \var{tol} and $0$ otherwise.  
 \end{funcdesc}  
743    
744  \begin{funcdesc}{minval}{a}  \begin{funcdesc}{minval}{a}
745  returns at each \DataSamplePoints the minimum value over all components.  returns at each \DataSamplePoints the minimum value over all components.
# Line 1063  returns the $L^2$-norm of \var{a} in its Line 993  returns the $L^2$-norm of \var{a} in its
993  \end{equation}  \end{equation}
994  \end{funcdesc}  \end{funcdesc}
995    
996    The following functions operate ``point-wise''. That is, the operation is applied to each component of each point
997    individually.
998    
999    \begin{funcdesc}{sin}{a}
1000    applies sine function to \var{a}.
1001    \end{funcdesc}
1002    
1003    \begin{funcdesc}{cos}{a}
1004    applies cosine function to \var{a}.
1005    \end{funcdesc}
1006    
1007    \begin{funcdesc}{tan}{a}
1008    applies tangent function to \var{a}.
1009    \end{funcdesc}
1010    
1011    \begin{funcdesc}{asin}{a}
1012    applies arc (inverse) sine function to \var{a}.
1013    \end{funcdesc}
1014    
1015    \begin{funcdesc}{acos}{a}
1016    applies arc (inverse) cosine function to \var{a}.
1017    \end{funcdesc}
1018    
1019    \begin{funcdesc}{atan}{a}
1020    applies arc (inverse) tangent function to \var{a}.
1021    \end{funcdesc}
1022    
1023    \begin{funcdesc}{sinh}{a}
1024    applies hyperbolic sine function to \var{a}.
1025    \end{funcdesc}
1026    
1027    \begin{funcdesc}{cosh}{a}
1028    applies hyperbolic cosine function to \var{a}.
1029    \end{funcdesc}
1030    
1031    \begin{funcdesc}{tanh}{a}
1032    applies hyperbolic tangent function to \var{a}.
1033    \end{funcdesc}
1034    
1035    \begin{funcdesc}{asinh}{a}
1036    applies arc (inverse) hyperbolic sine function to \var{a}.
1037    \end{funcdesc}
1038    
1039    \begin{funcdesc}{acosh}{a}
1040    applies arc (inverse) hyperbolic cosine function to \var{a}.
1041    \end{funcdesc}
1042    
1043    \begin{funcdesc}{atanh}{a}
1044    applies arc (inverse) hyperbolic tangent function to \var{a}.
1045    \end{funcdesc}
1046    
1047    \begin{funcdesc}{exp}{a}
1048    applies exponential function to \var{a}.
1049    \end{funcdesc}
1050    
1051    \begin{funcdesc}{sqrt}{a}
1052    applies square root function to \var{a}.
1053    \end{funcdesc}
1054    
1055    \begin{funcdesc}{log}{a}
1056    applies the natural logarithm to \var{a}.
1057    \end{funcdesc}
1058    
1059    \begin{funcdesc}{log10}{a}
1060    applies the base-$10$ logarithm to \var{a}.
1061    \end{funcdesc}
1062    
1063    \begin{funcdesc}{sign}{a}
1064    applies the sign function to \var{a}, that is $1$ where \var{a} is positive,
1065    $-1$ where \var{a} is negative and $0$ otherwise.
1066    \end{funcdesc}
1067    
1068    \begin{funcdesc}{wherePositive}{a}
1069    returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
1070    \end{funcdesc}
1071    
1072    \begin{funcdesc}{whereNegative}{a}
1073    returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
1074    \end{funcdesc}
1075    
1076    \begin{funcdesc}{whereNonNegative}{a}
1077    returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.
1078    \end{funcdesc}
1079    
1080    \begin{funcdesc}{whereNonPositive}{a}
1081    returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.
1082    \end{funcdesc}
1083    
1084    \begin{funcdesc}{whereZero}{a\optional{, tol=0.}}
1085    returns a function which is $1$ where \var{a} equals zero with tolerance \var{tol} and $0$ otherwise.
1086    \end{funcdesc}
1087    
1088    \begin{funcdesc}{whereNonZero}{a\optional{, tol=0.}}
1089    returns a function which is $1$ where \var{a} different from zero with tolerance \var{tol} and $0$ otherwise.
1090    \end{funcdesc}
1091    
1092  \subsection{\Operator Class}  \subsection{\Operator Class}
1093  The \Operator class provides an abstract access to operators build  The \Operator class provides an abstract access to operators build
1094  within the \LinearPDE class. \Operator objects are created  within the \LinearPDE class. \Operator objects are created
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