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some modifications on the pycad implementation to make it easier to build
interfaces for other mesh generators. The script statement generation is now
done by the Design and not the Primitive classes.


1 jgs 82 % $Id$
2 gross 625 %
3     % Copyright © 2006 by ACcESS MNRF
4     % http://www.access.edu.au
5     % Primary Business: Queensland, Australia.
6     % Licensed under the Open Software License version 3.0
7     % http://www.opensource.org/licenses/osl-3.0.php
8     %
9 jgs 82
10 gross 625
11 jgs 82 \chapter{The module \escript}
12 gross 660 \label{ESCRIPT CHAP}
13 jgs 82
14    
15 jgs 102 \begin{figure}
16 gross 599 \includegraphics[width=\textwidth]{figures/EscriptDiagram1.eps}
17 jgs 102 \caption{\label{ESCRIPT DEP}Dependency of Function Spaces. An arrow indicates that a function in the
18     function space at the starting point can be interpreted as a function in the function space of the arrow target.}
19     \end{figure}
20 jgs 82
21 jgs 102 \escript is an extension of Python to handle functions represented by their values on
22 jgs 107 \DataSamplePoints for the geometrical region on which
23 jgs 102 the function is defined. The region as well as the method which is used
24     to interpolate value on the \DataSamplePoints is defined by
25     \Domain class objects. For instance when using
26     the finite element method (FEM) \index{finite element method}
27     \Domain object holds the information about the FEM mesh, eg.
28     a table of nodes and a table of elements. Although \Domain contains
29     the discretization method to be used \escript does not use this information directly.
30     \Domain objects are created from a module which want to make use
31     \escript, e.g. \finley.
32 jgs 82
33 jgs 102 The solution of a PDE is a function of its location in the domain of interest $\Omega$.
34     When solving a partial differential equation \index{partial differential equation} (PDE) using FEM
35     the solution is (piecewise) differentiable but, in general, its gradient
36     is discontinuous. To reflect these different degrees of smoothness different
37     representations of the functions are used. For instance; in FEM
38     the displacement field is represented by its values at the nodes of the mesh, while the
39     strain, which is the symmetric part of the gradient of the displacement field, is stored on the
40     element centers. To be able to classify functions with respect to their smoothness, \escript has the
41     concept of the "function space". A function space is described by a \FunctionSpace object.
42     The following statement generates the object \var{solution_space} which is
43     a \FunctionSpace object and provides access to the function space of
44     PDE solutions on the \Domain \var{mydomain}:
45     \begin{python}
46     solution_space=Solution(mydomain)
47     \end{python}
48 jgs 107 The following generators for function spaces on a \Domain \var{mydomain} are available:
49 jgs 102 \begin{itemize}
50 gross 599 \item \var{Solution(mydomain)}: solutions of a PDE.
51     \item \var{ReducedSolution(mydomain)}: solutions of a PDE with a reduced smoothness requirement.
52 jgs 102 \item \var{ContinuousFunction(mydomain)}: continuous functions, eg. a temperature distribution.
53     \item \var{Function(mydomain)}: general functions which are not necessarily continuous, eg. a stress field.
54     \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, eg. a surface pressure.
55     \item \var{FunctionOnContact0(mydomain)}: functions on side $0$ of the discontinuity.
56     \item \var{FunctionOnContact1(mydomain)}: functions on side $1$ of the discontinuity.
57     \end{itemize}
58 gross 599 The reduced smoothness for PDE solution is often used to fulfill the Ladyzhenskaya–-Babuska–-Brezzi condition \cite{LBB} when
59     solving saddle point problems \index{saddle point problems}, eg. the Stokes equation.
60 jgs 102 A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous.
61     The location of discontinuity is defined in the \Domain object.
62     \fig{ESCRIPT DEP} shows the dependency between the types of function spaces.
63     The solution of a PDE is a continuous function. Any continuous function can be seen as a general function
64     on the domain and can be restricted to the boundary as well as to any side of the
65     discontinuity (the result will be different depending on
66     which side is chosen). Functions on any side of the
67     discontinuity can be seen as a function on the corresponding other side.
68     A function on the boundary or on one side of
69     the discontinuity cannot be seen as a general function on the domain as there are no values
70     defined for the interior. For most PDE solver libraries
71     the space of the solution and continuous functions is identical, however in some cases, eg.
72     when periodic boundary conditions are used in \finley, a solution
73     fulfils periodic boundary conditions while a continuous function does not have to be periodic.
74    
75     The concept of function spaces describes the properties of
76     functions and allows abstraction from the actual representation
77     of the function in the context of a particular application. For instance,
78     in the FEM context a
79     function in the \Function function space
80     is typically represented by its values at the element center,
81     but in a finite difference scheme the edge midpoint of cells is preferred.
82     Using the concept of function spaces
83     allows the user to run the same script on different
84     PDE solver libraries by just changing the creator of the \Domain object.
85     Changing the function space of a particular function
86     will typically lead to a change of its representation.
87     So, when seen as a general function,
88     a continuous function which is typically represented by its values
89     on the node of the FEM mesh or finite difference grid
90 jgs 107 must be interpolated to the element centers or the cell edges,
91 jgs 102 respectively.
92 jgs 82
93 jgs 102 \Data class objects store functions of the location in a domain.
94     The function is represented through its values on \DataSamplePoints where
95     the \DataSamplePoints are chosen according to the function space
96     of the function.
97     \Data class objects are used to define the coefficients
98     of the PDEs to be solved by a PDE solver library
99     and to store the returned solutions.
100 jgs 82
101 jgs 102 The values of the function have a rank which gives the
102 jgs 107 number of indices, and a \Shape defining the range of each index.
103 jgs 102 The rank in \escript is limited to the range $0$ through $4$ and
104     it is assumed that the rank and \Shape is the same for all \DataSamplePoints.
105     The \Shape of a \Data object is a tuple \var{s} of integers. The length
106     of \var{s} is the rank of the \Data object and \var{s[i]} is the maximum
107     value for the \var{i}-th index.
108     For instance, a stress field has rank $2$ and
109     \Shape $(d,d)$ where $d$ is the spatial dimension.
110     The following statement creates the \Data object
111     \var{mydat} representing a
112     continuous function with values
113     of \Shape $(2,3)$ and rank $2$:
114     \begin{python}
115     mydat=Data(value=1,what=ContinuousFunction(myDomain),shape=(2,3))
116     \end{python}
117     The initial value is the constant $1$ for all \DataSamplePoints and
118     all components.
119 jgs 82
120 jgs 102 \Data objects can also be created from any \numarray
121     array or any object, such as a list of floating point numbers,
122 gross 593 that can be converted into a \numarray.NumArray \Ref{NUMARRAY}.
123 jgs 102 The following two statements
124     create objects which are equivalent to \var{mydat}:
125     \begin{python}
126     mydat1=Data(value=numarray.ones((2,3)),what=ContinuousFunction(myDomain))
127     mydat2=Data(value=[[1,1],[1,1],[1,1]],what=ContinuousFunction(myDomain))
128     \end{python}
129     In the first case the initial value is \var{numarray.ones((2,3))}
130 gross 593 which generates a $2 \times 3$ matrix as a \numarray.NumArray
131 jgs 102 filled with ones. The \Shape of the created \Data object
132     it taken from the \Shape of the array. In the second
133     case, the creator converts the initial value, which is a list of lists,
134 gross 593 and converts it into a \numarray.NumArray before creating the actual
135 jgs 102 \Data object.
136 jgs 82
137 jgs 102 For convenience \escript provides creators for the most common types
138     of \Data objects in the following forms (\var{d} defines the
139     spatial dimension):
140     \begin{itemize}
141     \item \var{Scalar(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(,))},
142     e.g a temperature field.
143     \item \var{Vector(0,Function(mydomain))}is the same as \var{Data(0,Function(myDomain),(d))}, e.g
144     a velocity field.
145     \item \var{Tensor(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d))},
146     eg. a stress field.
147     \item \var{Tensor4(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d,d,d))}
148     eg. a Hook tensor field.
149     \end{itemize}
150 gross 593 Here the initial value is $0$ but any object that can be converted into a \numarray.NumArray and whose \Shape
151 jgs 102 is consistent with \Shape of the \Data object to be created can be used as the initial value.
152 jgs 82
153 jgs 102 \Data objects can be manipulated by applying unitary operations (eg. cos, sin, log)
154 jgs 107 and can be combined by applying binary operations (eg. +, - ,* , /).
155     It is to be emphasized that \escript itself does not handle any spatial dependencies as
156 jgs 102 it does not know how values are interpreted by the processing PDE solver library.
157     However \escript invokes interpolation if this is needed during data manipulations.
158     Typically, this occurs in binary operation when both arguments belong to different
159     function spaces or when data are handed over to a PDE solver library
160     which requires functions to be represented in a particular way.
161 jgs 82
162 jgs 102 The following example shows the usage of {\tt Data} objects: Assume we have a
163     displacement field $u$ and we want to calculate the corresponding stress field
164     $\sigma$ using the linear--elastic isotropic material model
165     \begin{eqnarray}\label{eq: linear elastic stress}
166     \sigma\hackscore {ij}=\lambda u\hackscore {k,k} \delta\hackscore {ij} + \mu ( u\hackscore {i,j} + u\hackscore {j,i})
167     \end{eqnarray}
168     where $\delta\hackscore {ij}$ is the Kronecker symbol and
169     $\lambda$ and $\mu$ are the Lame coefficients. The following function
170     takes the displacement {\tt u} and the Lame coefficients
171     \var{lam} and \var{mu} as arguments and returns the corresponding stress:
172     \begin{python}
173 gross 599 from esys.escript import *
174 jgs 102 def getStress(u,lam,mu):
175     d=u.getDomain().getDim()
176     g=grad(u)
177 gross 599 stress=lam*trace(g)*kronecker(d)+mu*(g+transpose(g))
178 jgs 102 return stress
179     \end{python}
180     The variable
181     \var{d} gives the spatial dimension of the
182     domain on which the displacements are defined.
183 gross 599 \var{kronecker} returns the Kronecker symbol with indexes
184 jgs 102 $i$ and $j$ running from $0$ to \var{d}-1. The call \var{grad(u)} requires
185     the displacement field \var{u} to be in the \var{Solution} or \ContinuousFunction
186     function space. The result \var{g} as well as the returned stress will be in the \Function function space.
187     If \var{u} is available, eg. by solving a PDE, \var{getStress} might be called
188     in the following way:
189     \begin{python}
190     s=getStress(u,1.,2.)
191     \end{python}
192     However \var{getStress} can also be called with \Data objects as values for
193     \var{lam} and \var{mu} which,
194     for instance in the case of a temperature dependency, are calculated by an expression.
195     The following call is equivalent to the previous example:
196     \begin{python}
197     lam=Scalar(1.,ContinuousFunction(mydomain))
198     mu=Scalar(2.,Function(mydomain))
199     s=getStress(u,lam,mu)
200     \end{python}
201     The function \var{lam} belongs to the \ContinuousFunction function space
202     but with \var{g} the function \var{trace(g)} is in the \Function function space.
203     Therefore the evaluation of the product \var{lam*trace(g)} in the stress calculation
204     produces a problem, as both functions are represented differently, eg. in FEM
205     \var{lam} by its values on the node, and in \var{trace(g)} by its values at the element centers.
206     In the case of inconsistent function spaces of arguments in a binary operation, \escript
207     interprets the arguments in the appropriate function space according to the inclusion
208     defined in Table~\ref{ESCRIPT DEP}. In this example that means
209     \escript sees \var{lam} as a function of the \Function function space.
210     In the context of FEM this means the nodal values of
211     \var{lam} are interpolated to the element centers. Behind the scenes
212     \escript calls the appropriate function from the PDE solver library.
213 jgs 82
214 jgs 102 \begin{figure}
215 gross 599 \includegraphics[width=\textwidth]{figures/EscriptDiagram2.eps}
216 jgs 102 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region with two rock types {\it white} and {\it gray}.
217     The number in each cell refers to the major rock type present in the cell ($1$ for {\it white} and $2$ for {\it gray}).
218     }
219     \end{figure}
220 jgs 82
221 jgs 102 Material parameters such as the Lame coefficients are typically dependent on rock types present in the
222     area of interest. A common technique to handle these kinds of material parameters is "tagging". \fig{Figure: tag}
223     shows an example. In this case two rock types {\it white} and {\it gray} can be found in the domain. The domain
224 gross 599 is subdivided into triangular shaped cells. Each
225 jgs 102 cell has a tag indicating the rock type predominately found in this cell. Here $1$ is used to indicate
226     rock type {\it white} and $2$ for rock type {\it gray}. The tags are assigned at the time when the cells are generated
227 gross 1044 and stored in the \Domain class object. To allow easier usage of tags names can be used. These names are typically defined
228     at the time when the geometry is generated.
229    
230     The following statements show how for the
231 jgs 102 example of \fig{Figure: tag} and the stress calculation discussed before tagged values are used for
232     \var{lam}:
233     \begin{python}
234     lam=Scalar(value=2.,what=Function(mydomain))
235 gross 1044 insertTaggedValue(lam,white=30.,gray=5000.)
236 jgs 102 s=getStress(u,lam,2.)
237     \end{python}
238 gross 1044 In this example \var{lam} is set to $30$ for those cells with tag {\it white} (=$1$) and to $5000.$ for those cells
239     with tag {\it gray} (=$2$_. The initial value $2$ of \var{lam} is used as a default value for the case when a tag
240 jgs 102 is encountered which has not been linked with a value. Note that the \var{getStress} method
241     is called without modification. \escript resolves the tags when \var{lam*trace(g)} is calculated.
242 jgs 82
243 jgs 102 The \Data class provides a transparent interface to various data representations and the
244     translations between them. As shown in the example of stress calculation, this allows the user to
245     develop and test algorithms for a simple case (for instance with the Lame coefficients as constants)
246     and then without further modifications of the program code to apply the algorithm in a
247     more complex application (for instance a definition of the Lame coefficients using tags).
248 gross 599 As described here, there are three ways in which \Data objects are represented internally, constant,
249     tagged, and expanded (other representations will become available in later versions of \escript):
250 jgs 102 In the constant case, if the same value is used at each sample point a single value is stored to save memory and compute time.
251     Any operation on this constant data will only be performed on the single value.
252     In the expanded case, each sample point has an individual value, eg. the solution of a PDE,
253     and the values are stored as a complete array. The tagged case has already been discussed above.
254    
255     Values are accessed through a sample reference number. Operations on expanded \Data
256     objects have to be performed for each sample point individually. If tagged values are used values are
257     held in a dictionary. Operations on tagged data require processing the set of tagged values only, rather than
258     processing the value for each individual sample point.
259     \escript allows use of constant, tagged and expanded data in a single expression.
260 jgs 82
261 gross 983 The \var{dump} method provides a possibility to save \Data objects to a file, for instance to restart a simuation
262 gross 999 or to save data for visualization. The file format uses \netCDF which commonly is using the file extension
263 gross 983 {\tt nc}. For instance to save the coordinates of the data points of the \FunctionSpace
264     \ContinuousFunction to the file {\tt x.nc} one uses:
265     \begin{python}
266     x=ContinuousFunction(mydomain).getX()
267     x.dump("x.nc")
268     \end{python}
269     In order to keep the dump files small {\tt x.nc} does not contain a representation of the \Domain. It has to be saved using
270     apropriated methods of \var{mydomain} to be loaded before \var{x}. Alternatively, the \Domain can be reconstructed.
271     To recover the object \var{x} one uses
272     \begin{python}
273     x=load("x.nc", mydomain)
274     \end{python}
275     The \Data object represented by {\tt x.nc} is tight to a \FunctionSpace - in this case \ContinuousFunction - but not
276     o a \Domain. That means that \Data objects that are constant or tagged can be recovered with any \Domain. If the \Data object
277     is expanded, the number of data points in the file and of the \Domain for the particular \FunctionSpace must match.
278     Moreover, the ordering of the value is checked using the reference identifiers provided by
279     \FunctionSpace on the \Domain. In some cases, data points will be reordered.
280    
281    
282 gross 999 \section{\escript Classes}
283     \declaremodule{extension}{esys.escript}
284     \modulesynopsis{Data manipulation}
285    
286     \subsection{\Domain class}
287 jgs 102 \begin{classdesc}{Domain}{}
288 jgs 107 A \Domain object is used to describe a geometrical region together with
289 jgs 102 a way of representing functions over this region.
290     The \Domain class provides an abstract access to the domain of \FunctionSpace and \Data objects.
291     \Domain itself has no initialization but implementations of \Domain are
292     instantiated by numerical libraries making use of \Data objects.
293 jgs 82 \end{classdesc}
294 gross 593 The following methds are available:
295 jgs 102 \begin{methoddesc}[Domain]{getDim}{}
296     returns the spatial dimension of the \Domain.
297     \end{methoddesc}
298    
299     \begin{methoddesc}[Domain]{getX}{}
300     returns the locations in the \Domain. The \FunctionSpace of the returned
301     \Data object is chosen by the \Domain implementation. Typically it will be
302     in the \Function.
303     \end{methoddesc}
304    
305     \begin{methoddesc}[Domain]{setX}{newX}
306     assigns a new location to the \Domain. \var{newX} has to have \Shape $(d,)$
307     where $d$ is the spatial dimension of the domain. Typically \var{newX} must be
308     in the \ContinuousFunction but the space actually to be used depends on the \Domain implementation.
309     \end{methoddesc}
310    
311     \begin{methoddesc}[Domain]{getNormal}{}
312     returns the surface normals on the boundary of the \Domain as \Data object.
313     \end{methoddesc}
314    
315     \begin{methoddesc}[Domain]{getSize}{}
316     returns the local sample size, e.g. the element diameter, as \Data object.
317     \end{methoddesc}
318    
319 gross 1044 \begin{methoddesc}[Domain]{setTagMap}{tag_name, tag}
320     defines a mapping of the tag name \var{tag_name} to the \var{tag}.
321     \end{methoddesc}
322     \begin{methoddesc}[Domain]{getTag}{tag_name}
323     returns the tag associated with the tag name \var{tag_name}.
324     \end{methoddesc}
325     \begin{methoddesc}[Domain]{isValidTagName}{tag_name}
326     return \True if \var{tag_name} is a valid tag name.
327     \end{methoddesc}
328    
329 jgs 102 \begin{methoddesc}[Domain]{__eq__}{arg}
330     returns \True of the \Domain \var{arg} describes the same domain. Otherwise
331     \False is returned.
332     \end{methoddesc}
333    
334     \begin{methoddesc}[Domain]{__ne__}{arg}
335     returns \True of the \Domain \var{arg} does not describe the same domain.
336     Otherwise \False is returned.
337     \end{methoddesc}
338    
339 gross 593 \begin{methoddesc}[Domain]{__str__}{g}
340     returns string represention of the \Domain.
341     \end{methoddesc}
342    
343 gross 999 \subsection{\FunctionSpace class}
344 jgs 102 \begin{classdesc}{FunctionSpace}{}
345     \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects
346     are instantiated by generator functions. \Data objects in particular \FunctionSpace are
347     represented by their values at \DataSamplePoints which are defined by the type and the \Domain of the
348     \FunctionSpace.
349 jgs 82 \end{classdesc}
350 gross 1044 The following methods are available:
351 jgs 102 \begin{methoddesc}[FunctionSpace]{getDim}{}
352     returns the spatial dimension of the \Domain of the \FunctionSpace.
353     \end{methoddesc}
354 jgs 82
355 gross 1044
356    
357 jgs 102 \begin{methoddesc}[FunctionSpace]{getX}{}
358     returns the location of the \DataSamplePoints.
359     \end{methoddesc}
360 jgs 82
361 jgs 102 \begin{methoddesc}[FunctionSpace]{getNormal}{}
362     If the domain of functions in the \FunctionSpace
363     is a hypermanifold (e.g. the boundary of a domain)
364     the method returns the outer normal at each of the
365     \DataSamplePoints. Otherwise an exception is raised.
366     \end{methoddesc}
367 jgs 82
368 jgs 102 \begin{methoddesc}[FunctionSpace]{getSize}{}
369     returns a \Data objects measuring the spacing of the \DataSamplePoints.
370     The size may be zero.
371     \end{methoddesc}
372 jgs 82
373 jgs 102 \begin{methoddesc}[FunctionSpace]{getDomain}{}
374     returns the \Domain of the \FunctionSpace.
375     \end{methoddesc}
376 jgs 82
377 gross 1044 \begin{methoddesc}[FunctionSpace]{setTags}{new_tag, mask}
378     assigns a new tag \var{new_tag} to all data sample
379     where \var{mask} is positive for a least one data point.
380     \var{mask} must be defined on the this \FunctionSpace.
381     Use the \var{setTagMap} to assign a tage name to \var{new_tag}.
382     \end{methoddesc}
383    
384 jgs 102 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
385     returns \True of the \Domain \var{arg} describes the same domain. Otherwise
386     \False is returned.
387     \end{methoddesc}
388 jgs 82
389 jgs 102 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
390     returns \True of the \Domain \var{arg} describes the note same domain.
391     Otherwise \False is returned.
392     \end{methoddesc}
393 jgs 82
394 gross 593 \begin{methoddesc}[Domain]{__str__}{g}
395     returns string represention of the \Domain.
396     \end{methoddesc}
397    
398 jgs 102 The following function provide generators for \FunctionSpace objects:
399     \begin{funcdesc}{Function}{domain}
400     returns the \Function on the \Domain \var{domain}. \Data objects in this type of \Function
401     are defined over the whole geometrical region defined by \var{domain}.
402 jgs 82 \end{funcdesc}
403    
404 jgs 102 \begin{funcdesc}{ContinuousFunction}{domain}
405     returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
406     are defined over the whole geometrical region defined by \var{domain} and assumed to represent
407     a continuous function.
408 jgs 82 \end{funcdesc}
409    
410 jgs 102 \begin{funcdesc}{FunctionOnBoundary}{domain}
411     returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
412     are defined on the boundary of the geometrical region defined by \var{domain}.
413 jgs 82 \end{funcdesc}
414    
415 jgs 102 \begin{funcdesc}{FunctionOnContactZero}{domain}
416     returns the \FunctionOnContactZero the \Domain domain. \Data objects in this type of \Function
417     are defined on side 0 of a discontinutiy within the geometrical region defined by \var{domain}.
418     The discontinutiy is defined when \var{domain} is instantiated.
419 jgs 82 \end{funcdesc}
420    
421 jgs 102 \begin{funcdesc}{FunctionOnContactOne}{domain}
422     returns the \FunctionOnContactOne on the \Domain domain.
423     \Data objects in this type of \Function
424     are defined on side 1 of a discontinutiy within the geometrical region defined by \var{domain}.
425     The discontinutiy is defined when \var{domain} is instantiated.
426 jgs 82 \end{funcdesc}
427    
428 jgs 102 \begin{funcdesc}{Solution}{domain}
429     returns the \SolutionFS on the \Domain domain. \Data objects in this type of \Function
430     are defined on geometrical region defined by \var{domain} and are solutions of
431     partial differential equations \index{partial differential equation}.
432 jgs 82 \end{funcdesc}
433    
434 jgs 102 \begin{funcdesc}{ReducedSolution}{domain}
435     returns the \ReducedSolutionFS on the \Domain domain. \Data objects in this type of \Function
436     are defined on geometrical region defined by \var{domain} and are solutions of
437     partial differential equations \index{partial differential equation} with a reduced smoothness
438     for the solution approximation.
439 jgs 82 \end{funcdesc}
440    
441 gross 999 \subsection{\Data Class}
442 jgs 107 \label{SEC ESCRIPT DATA}
443 jgs 82
444 jgs 102 The following table shows binary and unitary operations that can be applied to
445     \Data objects:
446     \begin{tableii}{l|l}{textrm}{expression}{Description}
447 gross 625 \lineii{+\var{arg0}} {just \var{arg} \index{+}}
448     \lineii{-\var{arg0}} {swapping the sign\index{-}}
449     \lineii{\var{arg0}+\var{arg1}} {adds \var{arg0} and \var{arg1} \index{+}}
450     \lineii{\var{arg0}*\var{arg1}} {multiplies \var{arg0} and \var{arg1} \index{*}}
451     \lineii{\var{arg0}-\var{arg1}} {difference \var{arg1} from\var{arg1} \index{-}}
452     \lineii{\var{arg0}/\var{arg1}} {ratio \var{arg0} by \var{arg1} \index{/}}
453     \lineii{\var{arg0}**\var{arg1}} {raises \var{arg0} to the power of \var{arg1} \index{**}}
454 jgs 102 \end{tableii}
455 gross 625 At least one of the arguments \var{arg0} or \var{arg1} must be a
456 jgs 102 \Data object. One of the arguments may be an object that can be
457 gross 625 converted into a \Data object. If \var{arg0} or \var{arg1} are
458     defined on different \FunctionSpace an attempt is made to embed \var{arg0}
459     into the \FunctionSpace of \var{arg1} or to embed \var{arg1} into
460     the \FunctionSpace of \var{arg0}. Boths arguments must have the same
461     \Shape or one of the arguments my be of rank 0. In the
462 jgs 102 latter case it is assumed that the particular argument is of the same
463 jgs 107 \Shape as the other argument but constant over all components.
464 jgs 82
465 jgs 102 The returned \Data object has the same \Shape and is defined on
466 gross 625 the \DataSamplePoints as \var{arg0} or \var{arg1}.
467 jgs 82
468 jgs 102 The following table shows the update operations that can be applied to
469     \Data objects:
470     \begin{tableii}{l|l}{textrm}{expression}{Description}
471 gross 625 \lineii{\var{arg0}+=\var{arg2}} {adds \var{arg0} to \var{arg2} \index{+}}
472     \lineii{\var{arg0}*=\var{arg2}} {multiplies \var{arg0} with \var{arg2} \index{*}}
473     \lineii{\var{arg0}-=\var{arg2}} {subtracts \var{arg2} from\var{arg2} \index{-}}
474     \lineii{\var{arg0}/=\var{arg2}} {divides \var{arg0} by \var{arg2} \index{/}}
475     \lineii{\var{arg0}**=\var{arg2}} {raises \var{arg0} by \var{arg2} \index{**}}
476 jgs 102 \end{tableii}
477 gross 625 \var{arg0} must be a \Data object. \var{arg1} must be a
478 jgs 102 \Data object or an object that can be converted into a
479     \Data object. \var{arg1} must have the same \Shape like
480 gross 625 \var{arg1} or has rank 0. In the latter case it is
481 jgs 102 assumed that the values of \var{arg1} are constant for all
482 gross 625 components. \var{arg1} must be defined in the same \FunctionSpace as
483     \var{arg0} or it must be possible to interpolate \var{arg1} onto the
484     \FunctionSpace of \var{arg1}.
485 jgs 82
486 jgs 102 The \Data class supports getting slices as well as assigning new values to components in an existing
487 jgs 107 \Data object. \index{slicing}
488 jgs 102 The following expression for getting (expression on the right hand side of the
489     equal sign) and setting slices (expression on the left hand side of the
490     equal sign) are valid:
491     \begin{tableiii}{l|ll}{textrm}{rank of \var{arg}}{slicing expression}{\Shape of returned and assigned object}
492     \lineiii{0}{ no slicing } {-}
493     \lineiii{1}{\var{arg[l0:u0]}} {(\var{u0}-\var{l0},)}
494     \lineiii{2}{\var{arg[l0:u0,l1:u1]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1})}
495     \lineiii{3}{\var{arg[l0:u0,l1:u1,l2:u2]} } {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})}
496     \lineiii{4}{\var{arg[l0:u0,l1:u1,l2:u2,l3:u3]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2},\var{u3}-\var{l3})}
497     \end{tableiii}
498     where
499     $0 \le \var{l0} \le \var{u0} \le \var{s[0]}$,
500     $0 \le \var{l1} \le \var{u1} \le \var{s[1]}$,
501     $0 \le \var{l2} \le \var{u2} \le \var{s[2]}$,
502     $0 \le \var{l3} \le \var{u3} \le \var{s[3]}$ and \var{s} the \Shape if \var{arg}.
503     Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case
504     $0$ is assumed.
505     Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may not be present in which case
506     \var{s} is assumed. The lower and upper index may be identical, in which case the column and the lower or upper
507     index may be dropped. In the returned or in the object assigned to a slice the corresponding component is dropped,
508     i.e. the rank is reduced by one in comparison to \var{arg}.
509 jgs 107 The following examples show slicing usage:
510 jgs 102 \begin{python}
511 gross 593 t=Data(1.,(4,4,6,6),Function(mydomain))
512 jgs 102 t[1,1,1,0]=9.
513     s=t[:2,:,2:6,5] # s has rank 3
514     s[:,:,1]=1.
515     t[:2,:2,5,5]=s[2:4,1,:2]
516     \end{python}
517    
518 gross 593 \subsection{Generation of \Data class objects}
519     \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expand=\False}
520 jgs 102 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
521     The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True
522     the \Data object is represented in expanded from.
523 jgs 82 \end{classdesc}
524    
525 gross 593 \begin{classdesc}{Data}{value,what=FunctionSpace(),expand=\False}
526 jgs 102 creates a \Data object in the \FunctionSpace \var{what}.
527 gross 593 The value for each \DataSamplePoints is set to \numarray, \Data object \var{value} or a dictionary of
528     \numarray or floating point numbers. In the latter case the keys muts be integers and are used
529     as tags.
530 jgs 102 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True
531     the \Data object is represented in expanded from.
532     \end{classdesc}
533    
534     \begin{classdesc}{Data}{}
535     creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present
536     where a \Data object is required.
537     \end{classdesc}
538    
539 gross 593 \begin{funcdesc}{Scalar}{value=0.,what=escript::FunctionSpace(),expand=\False}
540     returns a \Data object of rank 0 in the \FunctionSpace \var{what}.
541     Values are initialed with the double \var{value}. If \var{expanded} is \True
542     the \Data object is represented in expanded from.
543     \end{funcdesc}
544    
545     \begin{funcdesc}{Vector}{value=0.,what=escript::FunctionSpace(),expand=\False}
546     returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what}
547     where \var{d} is the spatial dimension of the \Domain of \var{what}.
548     Values are initialed with the double \var{value}. If \var{expanded} is \True
549     the \Data object is represented in expanded from.
550     \end{funcdesc}
551    
552     \begin{funcdesc}{Tensor}{value=0.,what=escript::FunctionSpace(),expand=\False}
553     returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what}
554     where \var{d} is the spatial dimension of the \Domain of \var{what}.
555     Values are initialed with the double \var{value}. If \var{expanded} is \True
556     the \Data object is represented in expanded from.
557     \end{funcdesc}
558    
559     \begin{funcdesc}{Tensor3}{value=0.,what=escript::FunctionSpace(),expand=\False}
560     returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what}
561     where \var{d} is the spatial dimension of the \Domain of \var{what}.
562     Values are initialed with the double \var{value}. If \var{expanded} is \True
563     the \Data object is re\var{arg}presented in expanded from.
564     \end{funcdesc}
565    
566     \begin{funcdesc}{Tensor4}{value=0.,what=escript::FunctionSpace(),expand=\False}
567     returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what}
568     where \var{d} is the spatial dimension of the \Domain of \var{what}.
569     Values are initialed with the double \var{value}. If \var{expanded} is \True
570     the \Data object is represented in expanded from.
571     \end{funcdesc}
572    
573 gross 983 \begin{funcdesc}{load}{filename,domain}
574     recovers a \Data object on \Domain \var{domain} from the dump file \var{filename}.
575     \end{funcdesc}
576    
577 gross 593 \subsection{\Data class methods}
578     This is a list of frequently used methods of the
579     \Data class. A complete list can be fond on \ReferenceGuide.
580 jgs 102 \begin{methoddesc}[Data]{getFunctionSpace}{}
581     returns the \FunctionSpace of the object.
582 jgs 82 \end{methoddesc}
583    
584 gross 593 \begin{methoddesc}[Data]{getDomain}{}
585 jgs 102 returns the \Domain of the object.
586     \end{methoddesc}
587    
588 jgs 82 \begin{methoddesc}[Data]{getShape}{}
589 jgs 102 returns the \Shape of the object as a \class{tuple} of
590     integers.
591 jgs 82 \end{methoddesc}
592    
593     \begin{methoddesc}[Data]{getRank}{}
594     returns the rank of the data on each data point. \index{rank}
595     \end{methoddesc}
596    
597 jgs 102 \begin{methoddesc}[Data]{isEmpty}{}
598     returns \True id the \Data object is the \EmptyData object.
599     Otherwise \False is returned.
600 jgs 82 \end{methoddesc}
601    
602 gross 1044 \begin{methoddesc}[Data]{setTaggedValue}{tag_name,value}
603 jgs 102 assigns the \var{value} to all \DataSamplePoints which have the tag
604 gross 1044 assigned to \var{tag_name}. \var{value} must be an object of class
605 gross 593 \class{numarray.NumArray} or must be convertible into a
606     \class{numarray.NumArray} object. \var{value} (or the corresponding
607     \class{numarray.NumArray} object) must be of rank $0$ or must have the
608 jgs 102 same rank like the object.
609 gross 1044 If a value has already be defined for tag \var{tag_name} within the object
610 jgs 102 it is overwritten by the new \var{value}. If the object is expanded,
611 gross 1044 the value assigned to \DataSamplePoints with tag \var{tag_name} is replaced by
612 gross 1045 \var{value}. If no tag is assigned tag name \var{tag_name}, no value is set.
613 jgs 82 \end{methoddesc}
614    
615 gross 983 \begin{methoddesc}[Data]{dump}{filename}
616     dumps the \Data object to the file \var{filename}. The file stores the
617     function space but not the \Domain. It is in the responsibilty of the user to
618     save the \Domain.
619     \end{methoddesc}
620    
621 gross 593 \begin{methoddesc}[Data]{__str__}{}
622     returns a string representation of the object.
623     \end{methoddesc}
624    
625 gross 999 \subsection{Functions of \Data class objects}
626 gross 593 This section lists the most important functions for \Data class objects \var{a}.
627     A complete list and a more detailed description of the functionality can be fond on \ReferenceGuide.
628 gross 599 \begin{funcdesc}{saveVTK}{filename,**kwdata}
629     writes \Data defined by keywords in the file with \var{filename} using the
630     vtk file format \VTK file format. The key word is used as an identifier. The statement
631     \begin{python}
632     saveVTK("out.xml",temperature=T,velocity=v)
633     \end{python}
634     will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
635     file \file{out.xml}. Restrictions on the allowed combinations of \FunctionSpace apply.
636 gross 593 \end{funcdesc}
637 gross 599 \begin{funcdesc}{saveDX}{filename,**kwdata}
638     writes \Data defined by keywords in the file with \var{filename} using the
639     vtk file format \OpenDX file format. The key word is used as an identifier. The statement
640     \begin{python}
641     saveDX("out.dx",temperature=T,velocity=v)
642     \end{python}
643     will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
644     file \file{out.dx}. Restrictions on the allowed combinations of \FunctionSpace apply.
645 gross 593 \end{funcdesc}
646     \begin{funcdesc}{kronecker}{d}
647 gross 599 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
648 gross 593 \begin{equation}
649 gross 599 \code{kronecker(d)}\left[ i,j\right] = \left\{
650 gross 593 \begin{array}{cc}
651     1 & \mbox{ if } i=j \\
652     0 & \mbox{ otherwise }
653     \end{array}
654     \right.
655     \end{equation}
656 gross 599 If \var{d} is an integer a $(d,d)$ \numarray array is returned.
657 gross 593 \end{funcdesc}
658     \begin{funcdesc}{identityTensor}{d}
659 gross 599 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
660     \begin{equation}
661     \code{identityTensor(d)}\left[ i,j\right] = \left\{
662     \begin{array}{cc}
663     1 & \mbox{ if } i=j \\
664     0 & \mbox{ otherwise }
665     \end{array}
666     \right.
667     \end{equation}
668     If \var{d} is an integer a $(d,d)$ \numarray array is returned.
669 gross 593 \end{funcdesc}
670     \begin{funcdesc}{identityTensor4}{d}
671 gross 599 returns a \RankFour \Data object in \FunctionSpace \var{d} such that
672     \begin{equation}
673     \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
674     \begin{array}{cc}
675     1 & \mbox{ if } i=k \mbox{ and } j=l\\
676     0 & \mbox{ otherwise }
677     \end{array}
678     \right.
679     \end{equation}
680     If \var{d} is an integer a $(d,d,d,d)$ \numarray array is returned.
681 gross 593 \end{funcdesc}
682     \begin{funcdesc}{unitVector}{i,d}
683 gross 599 returns a \RankOne \Data object in \FunctionSpace \var{d} such that
684     \begin{equation}
685     \code{identityTensor(d)}\left[ j \right] = \left\{
686     \begin{array}{cc}
687     1 & \mbox{ if } j=i\\
688     0 & \mbox{ otherwise }
689     \end{array}
690     \right.
691     \end{equation}
692     If \var{d} is an integer a $(d,)$ \numarray array is returned.
693    
694 gross 593 \end{funcdesc}
695    
696     \begin{funcdesc}{Lsup}{a}
697     returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values
698     over all components and all \DataSamplePoints of \var{a}.
699     \end{funcdesc}
700    
701     \begin{funcdesc}{sup}{a}
702     returns the maximum value over all components and all \DataSamplePoints of \var{a}.
703     \end{funcdesc}
704    
705     \begin{funcdesc}{inf}{a}
706     returns the minimum value over all components and all \DataSamplePoints of \var{a}
707     \end{funcdesc}
708    
709     \begin{funcdesc}{sin}{a}
710     applies sine function to \var{a}.
711     \end{funcdesc}
712    
713     \begin{funcdesc}{cos}{a}
714     applies cosine function to \var{a}.
715     \end{funcdesc}
716    
717     \begin{funcdesc}{tan}{a}
718     applies tangent function to \var{a}.
719     \end{funcdesc}
720    
721     \begin{funcdesc}{asin}{a}
722     applies arc (inverse) sine function to \var{a}.
723     \end{funcdesc}
724    
725     \begin{funcdesc}{acos}{a}
726     applies arc (inverse) cosine function to \var{a}.
727     \end{funcdesc}
728    
729     \begin{funcdesc}{atan}{a}
730     applies arc (inverse) tangent function to \var{a}.
731     \end{funcdesc}
732    
733     \begin{funcdesc}{sinh}{a}
734     applies hyperbolic sine function to \var{a}.
735     \end{funcdesc}
736    
737     \begin{funcdesc}{cosh}{a}
738     applies hyperbolic cosine function to \var{a}.
739     \end{funcdesc}
740    
741     \begin{funcdesc}{tanh}{a}
742     applies hyperbolic tangent function to \var{a}.
743     \end{funcdesc}
744    
745     \begin{funcdesc}{asinh}{a}
746     applies arc (inverse) hyperbolic sine function to \var{a}.
747     \end{funcdesc}
748    
749     \begin{funcdesc}{acosh}{a}
750     applies arc (inverse) hyperbolic cosine function to \var{a}.
751     \end{funcdesc}
752    
753     \begin{funcdesc}{atanh}{a}
754     applies arc (inverse) hyperbolic tangent function to \var{a}.
755     \end{funcdesc}
756    
757     \begin{funcdesc}{exp}{a}
758     applies exponential function to \var{a}.
759     \end{funcdesc}
760    
761     \begin{funcdesc}{sqrt}{a}
762     applies square root function to \var{a}.
763     \end{funcdesc}
764    
765     \begin{funcdesc}{log}{a}
766 gross 599 applies the natural logarithm to \var{a}.
767 gross 593 \end{funcdesc}
768    
769     \begin{funcdesc}{log10}{a}
770 gross 599 applies the base-$10$ logarithm to \var{a}.
771 gross 593 \end{funcdesc}
772    
773     \begin{funcdesc}{sign}{a}
774 gross 599 applies the sign function to \var{a}, that is $1$ where \var{a} is positive,
775     $-1$ where \var{a} is negative and $0$ otherwise.
776 gross 593 \end{funcdesc}
777    
778     \begin{funcdesc}{wherePositive}{a}
779 gross 599 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
780 gross 593 \end{funcdesc}
781 gross 599
782 gross 593 \begin{funcdesc}{whereNegative}{a}
783 gross 599 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
784 gross 593 \end{funcdesc}
785 gross 599
786 gross 593 \begin{funcdesc}{whereNonNegative}{a}
787 gross 599 returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.
788 gross 593 \end{funcdesc}
789 gross 599
790 gross 593 \begin{funcdesc}{whereNonPositive}{a}
791 gross 599 returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.
792 gross 593 \end{funcdesc}
793 gross 599
794     \begin{funcdesc}{whereZero}{a\optional{, tol=0.}}
795     returns a function which is $1$ where \var{a} equals zero with tolerance \var{tol} and $0$ otherwise.
796 gross 593 \end{funcdesc}
797 gross 599
798     \begin{funcdesc}{whereNonZero}{a\optional{, tol=0.}}
799     returns a function which is $1$ where \var{a} different from zero with tolerance \var{tol} and $0$ otherwise.
800 gross 593 \end{funcdesc}
801 gross 599
802 gross 593 \begin{funcdesc}{minval}{a}
803 gross 599 returns at each \DataSamplePoints the minumum value over all components.
804 gross 593 \end{funcdesc}
805 gross 599
806 gross 593 \begin{funcdesc}{maxval}{a}
807 gross 599 returns at each \DataSamplePoints the maximum value over all components.
808 gross 593 \end{funcdesc}
809 gross 599
810 gross 593 \begin{funcdesc}{length}{a}
811 gross 599 returns at Euclidean norm at each \DataSamplePoints. For a \RankFour function \var{a} this is
812     \begin{equation}
813     \code{length(a)}=\sqrt{\sum\hackscore{ijkl} \var{a} \left[i,j,k,l\right]^2}
814     \end{equation}
815 gross 593 \end{funcdesc}
816 gross 599 \begin{funcdesc}{trace}{a\optional{,axis_offset=0}}
817     returns the trace of \var{a}. This is the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
818     case of a \RankTwo function and this is
819     \begin{equation}
820     \code{trace(a)}=\sum\hackscore{i} \var{a} \left[i,i\right]
821     \end{equation}
822     and for a \RankFour function and \code{axis_offset=1} this is
823     \begin{equation}
824     \code{trace(a,1)}\left[i,j\right]=\sum\hackscore{k} \var{a} \left[i,k,k,j\right]
825     \end{equation}
826 gross 593 \end{funcdesc}
827 gross 804
828 gross 599 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
829     returns the transpose of \var{a}. This swaps the first \var{axis_offset} components of \var{a} with the rest. If \var{axis_offset} is not
830     present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
831     the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
832     case of a \RankTwo function and this is
833     \begin{equation}
834     \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
835     \end{equation}
836     and for a \RankFour function and \code{axis_offset=1} this is
837     \begin{equation}
838     \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
839     \end{equation}
840 gross 593 \end{funcdesc}
841 gross 804
842     \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
843     returns \var{a} but with swapped componets \var{axis0} and \var{axis1}. The argument \var{a} must be
844     at least of \RankTwo. For instance in the
845     for a \RankFour argument, \code{axis0=1} and \code{axis1=2} this is
846     \begin{equation}
847     \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
848     \end{equation}
849     \end{funcdesc}
850    
851 gross 593 \begin{funcdesc}{symmetric}{a}
852 gross 599 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
853 gross 593 \end{funcdesc}
854     \begin{funcdesc}{nonsymmetric}{a}
855 gross 599 returns the non--symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
856 gross 593 \end{funcdesc}
857     \begin{funcdesc}{inverse}{a}
858 gross 599 return the inverse of \var{a}. This is
859     \begin{equation}
860 gross 809 \code{matrix_mult(inverse(a),a)=kronecker(d)}
861 gross 599 \end{equation}
862     if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape
863     \code{(2,2)} and \code{(3,3)}.
864 gross 593 \end{funcdesc}
865     \begin{funcdesc}{eigenvalues}{a}
866 gross 599 return the eigenvalues of \var{a}. This is
867     \begin{equation}
868 gross 809 \code{matrix_mult(a,V)=e[i]*V}
869 gross 599 \end{equation}
870     where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.
871     The eigenvalues are ordered in increasing size.
872     The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
873     The current implementation is restricted to arguments of shape
874     \code{(2,2)} and \code{(3,3)}.
875 gross 593 \end{funcdesc}
876     \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
877 gross 599 return the eigenvalues and eigenvectors of \var{a}. This is
878     \begin{equation}
879 gross 809 \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
880 gross 599 \end{equation}
881     where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.
882     \begin{equation}
883 gross 809 \code{matrix_mult(transpose(V),V)=kronecker(d)}
884 gross 599 \end{equation}
885     if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.
886     The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
887     The current implementation is restricted to arguments of shape
888     \code{(2,2)} and \code{(3,3)}.
889 gross 593 \end{funcdesc}
890 gross 599 \begin{funcdesc}{maximum}{*a}
891     returns the maximum value over all arguments at all \DataSamplePoints and for each component.
892     For instance
893     \begin{equation}
894     \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
895     \end{equation}
896     at all \DataSamplePoints.
897 gross 593 \end{funcdesc}
898 gross 599 \begin{funcdesc}{minimum}{*a}
899     returns the minimum value over all arguments at all \DataSamplePoints and for each component.
900     For instance
901     \begin{equation}
902     \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
903     \end{equation}
904     at all \DataSamplePoints.
905 gross 593 \end{funcdesc}
906 gross 599
907     \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
908     cuts back \var{a} into the range between \var{minval} and \var{maxval}. A value in the returned object equals
909     \var{minval} if the corresponding value of \var{a} is less than \var{minval}, equals \var{maxval} if the
910     corresponding value of \var{a} is greater than \var{maxval}
911     or corresponding value of \var{a} otherwise.
912 gross 593 \end{funcdesc}
913     \begin{funcdesc}{inner}{a0,a1}
914 gross 599 returns the inner product of \var{a0} and \var{a1}. For instance in the
915     case of \RankTwo arguments and this is
916     \begin{equation}
917     \code{inner(a)}=\sum\hackscore{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
918     \end{equation}
919     and for a \RankFour arguments this is
920     \begin{equation}
921     \code{inner(a)}=\sum\hackscore{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
922     \end{equation}
923 gross 593 \end{funcdesc}
924 gross 809
925     \begin{funcdesc}{matrix_mult}{a0,a1}
926 gross 599 returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is
927     \begin{equation}
928 gross 809 \code{matrix_mult(a)}\left[i\right]=\sum\hackscore{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
929 gross 599 \end{equation}
930     and if \var{a1} is \RankTwo this is
931     \begin{equation}
932 gross 809 \code{matrix_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
933 gross 599 \end{equation}
934 gross 593 \end{funcdesc}
935 gross 809
936     \begin{funcdesc}{transposed_matrix_mult}{a0,a1}
937     returns the matrix product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
938     \code{matrix_mult(transpose(a0),a1)}.
939     If \var{a1} is \RankOne this is
940     \begin{equation}
941     \code{transposed_matrix_mult(a)}\left[i\right]=\sum\hackscore{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k\right]
942     \end{equation}
943     and if \var{a1} is \RankTwo this is
944     \begin{equation}
945     \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k,j\right]
946     \end{equation}
947     \end{funcdesc}
948    
949     \begin{funcdesc}{matrix_transposed_mult}{a0,a1}
950     returns the matrix product of \var{a0} and the transposed of \var{a1}.
951     The function is equivalent to
952     \code{matrix_mult(a0,transpose(a1))}.
953     If \var{a1} is \RankTwo this is
954     \begin{equation}
955     \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[j,k\right]
956     \end{equation}
957     \end{funcdesc}
958    
959 gross 593 \begin{funcdesc}{outer}{a0,a1}
960 gross 599 returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then
961     \begin{equation}
962     \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
963     \end{equation}
964     and if \var{a0} is \RankOne and \var{a1} is \RankThree
965     \begin{equation}
966     \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
967     \end{equation}
968 gross 593 \end{funcdesc}
969 gross 809
970     \begin{funcdesc}{tensor_mult}{a0,a1}
971 gross 599 returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is
972     \begin{equation}
973 gross 809 \code{tensor_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[k,l\right]
974 gross 599 \end{equation}
975     and if \var{a1} is \RankFour this is
976     \begin{equation}
977 gross 809 \code{tensor_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[m,n,k,l\right]
978 gross 599 \end{equation}
979 gross 593 \end{funcdesc}
980 gross 809
981     \begin{funcdesc}{transposed_tensor_mult}{a0,a1}
982     returns the tensor product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
983     \code{tensor_mult(transpose(a0),a1)}.
984     If \var{a1} is \RankTwo this is
985     \begin{equation}
986     \code{transposed_tensor_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[k,l,i,j\right] \cdot \var{a1} \left[k,l\right]
987     \end{equation}
988     and if \var{a1} is \RankFour this is
989     \begin{equation}
990     \code{transposed_tensor_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[m,n,i,j\right] \cdot \var{a1} \left[m,n,k,l\right]
991     \end{equation}
992     \end{funcdesc}
993    
994     \begin{funcdesc}{tensor_transposed_mult}{a0,a1}
995     returns the tensor product of \var{a0} and the transposed of \var{a1}.
996     The function is equivalent to
997     \code{tensor_mult(a0,transpose(a1))}.
998     If \var{a1} is \RankTwo this is
999     \begin{equation}
1000     \code{tensor_transposed_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[l,k\right]
1001     \end{equation}
1002     and if \var{a1} is \RankFour this is
1003     \begin{equation}
1004     \code{tensor_transposed_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[k,l,m,n\right]
1005     \end{equation}
1006     \end{funcdesc}
1007    
1008 gross 599 \begin{funcdesc}{grad}{a\optional{, where=None}}
1009     returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a
1010     default \FunctionSpace is used. In case that \var{a} has \RankTwo one has
1011     \begin{equation}
1012     \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x\hackscore{k}}
1013     \end{equation}
1014 gross 593 \end{funcdesc}
1015 gross 599 \begin{funcdesc}{integrate}{a\optional{ ,where=None}}
1016     returns the integral of \var{a} where the domain of integration is defined by the \FunctionSpace of \var{a}. If \var{where} is
1017     present the argument is interpolated into \FunctionSpace \var{where} before integration. For instance in the case of
1018     a \RankTwo argument in \ContinuousFunction it is
1019     \begin{equation}
1020     \code{integrate(a)}\left[i,j\right]=\int\hackscore{\Omega}\var{a} \left[i,j\right] \; d\Omega
1021     \end{equation}
1022     where $\Omega$ is the spatial domain and $d\Omega$ volume integration. To integrate over the boundary of the domain one uses
1023     \begin{equation}
1024     \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int\hackscore{\partial \Omega} a\left[i,j\right] \; ds
1025     \end{equation}
1026     where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or line integration.
1027 gross 593 \end{funcdesc}
1028     \begin{funcdesc}{interpolate}{a,where}
1029 gross 599 interpolates argument \var{a} into the \FunctionSpace \var{where}.
1030 gross 593 \end{funcdesc}
1031 gross 599 \begin{funcdesc}{div}{a\optional{ ,where=None}}
1032     returns the divergence of \var{a}. This
1033     \begin{equation}
1034     \code{div(a)}=trace(grad(a),where)
1035     \end{equation}
1036 gross 593 \end{funcdesc}
1037 gross 599 \begin{funcdesc}{jump}{a\optional{ ,domain=None}}
1038     returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present
1039     in \var{domain}.
1040     \begin{equation}
1041 gross 809 \begin{array}{rcl}
1042     \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
1043     & & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
1044     \end{array}
1045 gross 599 \end{equation}
1046 gross 593 \end{funcdesc}
1047     \begin{funcdesc}{L2}{a}
1048 gross 599 returns the $L^2$-norm of \var{a} in its function space. This is
1049     \begin{equation}
1050 gross 809 \code{L2(a)=integrate(length(a)}^2\code{)} \; .
1051 gross 599 \end{equation}
1052 gross 593 \end{funcdesc}
1053    
1054 gross 999 \subsection{\Operator Class}
1055 jgs 102 The \Operator class provides an abstract access to operators build
1056     within the \LinearPDE class. \Operator objects are created
1057     when a PDE is handed over to a PDE solver library and handled
1058     by the \LinearPDE class defining the PDE. The user can gain access
1059     to the \Operator of a \LinearPDE object through the \var{getOperator}
1060     method.
1061    
1062     \begin{classdesc}{Operator}{}
1063     creates an empty \Operator object.
1064     \end{classdesc}
1065    
1066     \begin{methoddesc}[Operator]{isEmpty}{fileName}
1067     returns \True is the object is empty. Otherwise \True is returned.
1068 jgs 82 \end{methoddesc}
1069    
1070 jgs 102 \begin{methoddesc}[Operator]{setValue}{value}
1071     resets all entires in the obeject representation to \var{value}
1072 jgs 82 \end{methoddesc}
1073    
1074 jgs 102 \begin{methoddesc}[Operator]{solves}{rhs}
1075     solves the operator equation with right hand side \var{rhs}
1076 jgs 82 \end{methoddesc}
1077    
1078 jgs 102 \begin{methoddesc}[Operator]{of}{u}
1079     applies the operator to the \Data object \var{u}
1080 jgs 82 \end{methoddesc}
1081    
1082 jgs 102 \begin{methoddesc}[Operator]{saveMM}{fileName}
1083 jgs 82 saves the object to a matrix market format file of name
1084     \var{fileName}, see
1085     \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}.
1086     \index{Matrix Market}
1087     \end{methoddesc}
1088    

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