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1
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
11 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14 \chapter{The \escript Module}\label{ESCRIPT CHAP}
15
16 \section{Concepts}
17 \escript is a \PYTHON module that allows you to represent the values of
18 a function at points in a \Domain in such a way that the function will
19 be useful for the Finite Element Method (FEM) simulation. It also
20 provides what we call a function space that describes how the data is
21 used in the simulation. Stored along with the data is information
22 about the elements and nodes which will be used by \finley.
23
24 \subsection{Function spaces}
25 In order to understand what we mean by the term 'function space',
26 consider that the solution of a partial differential
27 equation\index{partial differential equation} (PDE) is a function on a domain
28 $\Omega$. When solving a PDE using FEM, the solution is
29 piecewise-differentiable but, in general, its gradient is discontinuous.
30 To reflect these different degrees of smoothness, different function spaces
31 are used.
32 For instance, in FEM, the displacement field is represented by its values at
33 the nodes of the mesh, and so is continuous.
34 The strain, which is the symmetric part of the gradient of the displacement
35 field, is stored on the element centers, and so is considered to be
36 discontinuous.
37
38 A function space is described by a \FunctionSpace object.
39 The following statement generates the object \var{solution_space} which is
40 a \FunctionSpace object and provides access to the function space of
41 PDE solutions on the \Domain \var{mydomain}:
42
43 \begin{python}
44 solution_space=Solution(mydomain)
45 \end{python}
46 The following generators for function spaces on a \Domain \var{mydomain} are commonly used:
47 \begin{itemize}
48 \item \var{Solution(mydomain)}: solutions of a PDE
49 \item \var{ReducedSolution(mydomain)}: solutions of a PDE with a reduced
50 smoothness requirement, e.g. using a lower order approximation on the same
51 element or using macro elements\index{macro elements}
52 \item \var{ContinuousFunction(mydomain)}: continuous functions, e.g. a temperature distribution
53 \item \var{Function(mydomain)}: general functions which are not necessarily continuous, e.g. a stress field
54 \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, e.g. 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 In some cases under-integration is used. For these cases the user may use a
59 \FunctionSpace from the following list:
60 \begin{itemize}
61 \item \var{ReducedFunction(mydomain)}
62 \item \var{ReducedFunctionOnBoundary(mydomain)}
63 \item \var{ReducedFunctionOnContact0(mydomain)}
64 \item \var{ReducedFunctionOnContact1(mydomain)}
65 \end{itemize}
66 In comparison to the corresponding full version they use a reduced number of
67 integration nodes (typically one only) to represent values.
68
69 \begin{figure}
70 \centering
71 \includegraphics{EscriptDiagram1}
72 \caption{\label{ESCRIPT DEP}Dependency of function spaces in \finley.
73 An arrow indicates that a function in the \FunctionSpace at the starting point
74 can be interpolated to the \FunctionSpace of the arrow target.
75 All function spaces above the dotted line can be interpolated to any of
76 the function spaces below the line. See also \Sec{SEC Projection}.}
77 \end{figure}
78
79 The reduced smoothness for a PDE solution is often used to fulfill the
80 Ladyzhenskaya-Babuska-Brezzi condition\cite{LBB} when solving saddle point
81 problems\index{saddle point problems}, e.g. the Stokes equation.
82 A discontinuity\index{discontinuity} is a region within the domain across
83 which functions may be discontinuous.
84 The location of a discontinuity is defined in the \Domain object.
85 \fig{ESCRIPT DEP} shows the dependency between the types of function spaces
86 in \finley (other libraries may have different relationships).
87
88 The solution of a PDE is a continuous function. Any continuous function can
89 be seen as a general function on the domain and can be restricted to the
90 boundary as well as to one side of a discontinuity (the result will be
91 different depending on which side is chosen). Functions on any side of the
92 discontinuity can be seen as a function on the corresponding other side.
93
94 A function on the boundary or on one side of the discontinuity cannot be seen
95 as a general function on the domain as there are no values defined for the
96 interior. For most PDE solver libraries the space of the solution and
97 continuous functions is identical, however in some cases, for example when
98 periodic boundary conditions are used in \finley, a solution fulfills periodic
99 boundary conditions while a continuous function does not have to be periodic.
100
101 The concept of function spaces describes the properties of functions and
102 allows abstraction from the actual representation of the function in the
103 context of a particular application. For instance, in the FEM context a
104 function of the \Function type (written as \emph{Function()} in \fig{ESCRIPT DEP})
105 is usually represented by its values at the element center,
106 but in a finite difference scheme the edge midpoint of cells is preferred.
107 By changing its function space you can use the same function in a Finite
108 Difference scheme instead of Finite Element scheme.
109 Changing the function space of a particular function will typically lead to
110 a change of its representation.
111 So, when seen as a general function, a continuous function which is typically
112 represented by its values on the nodes of the FEM mesh or finite difference
113 grid must be interpolated to the element centers or the cell edges,
114 respectively. Interpolation happens automatically in \escript whenever it is
115 required\index{interpolation}. The user needs to be aware that an
116 interpolation is not always possible, see \fig{ESCRIPT DEP} for \finley.
117 An alternative approach to change the representation (=\FunctionSpace) is
118 projection\index{projection}, see \Sec{SEC Projection}.
119
120 \subsection{\Data Objects}
121 In \escript the class that stores these functions is called \Data.
122 The function is represented through its values on \DataSamplePoints where
123 the \DataSamplePoints are chosen according to the function space of the
124 function.
125 \Data class objects are used to define the coefficients of the PDEs to be
126 solved by a PDE solver library and also to store the solutions of the PDE.
127
128 The values of the function have a rank which gives the number of indices,
129 and a \Shape defining the range of each index.
130 The rank in \escript is limited to the range 0 through 4 and it is assumed
131 that the rank and \Shape is the same for all \DataSamplePoints.
132 The \Shape of a \Data object is a tuple (list) \var{s} of integers.
133 The length of \var{s} is the rank of the \Data object and the \var{i}-th
134 index ranges between 0 and $\var{s[i]}-1$.
135 For instance, a stress field has rank 2 and \Shape $(d,d)$ where $d$ is the
136 spatial dimension.
137 The following statement creates the \Data object \var{mydat} representing a
138 continuous function with values of \Shape $(2,3)$ and rank $2$:
139 \begin{python}
140 mydat=Data(value=1, what=ContinuousFunction(myDomain), shape=(2,3))
141 \end{python}
142 The initial value is the constant 1 for all \DataSamplePoints and all
143 components.
144
145 \Data objects can also be created from any \numpy array or any object, such
146 as a list of floating point numbers, that can be converted into
147 a \numpyNDA\cite{NUMPY}.
148 The following two statements create objects which are equivalent
149 to \var{mydat}:
150 \begin{python}
151 mydat1=Data(value=numpy.ones((2,3)), what=ContinuousFunction(myDomain))
152 mydat2=Data(value=[[1,1], [1,1], [1,1]], what=ContinuousFunction(myDomain))
153 \end{python}
154 In the first case the initial value is \var{numpy.ones((2,3))} which generates
155 a $2 \times 3$ matrix as a \numpyNDA filled with ones.
156 The \Shape of the created \Data object is taken from the \Shape of the array.
157 In the second case, the creator converts the initial value, which is a list of
158 lists, into a \numpyNDA before creating the actual \Data object.
159
160 For convenience \escript provides creators for the most common types
161 of \Data objects in the following forms (\var{d} defines the spatial dimension):
162 \begin{itemize}
163 \item \code{Scalar(0, Function(mydomain))} is the same as \code{Data(0, Function(myDomain),(,))}
164 (each value is a scalar), e.g. a temperature field
165 \item \code{Vector(0, Function(mydomain))} is the same as \code{Data(0, Function(myDomain),(d))}
166 (each value is a vector), e.g. a velocity field
167 \item \code{Tensor(0, Function(mydomain))} is the same as \code{Data(0, Function(myDomain), (d,d))},
168 e.g. a stress field
169 \item \code{Tensor4(0,Function(mydomain))} is the same as \code{Data(0,Function(myDomain), (d,d,d,d))}
170 e.g. a Hook tensor field
171 \end{itemize}
172 Here the initial value is 0 but any object that can be converted into
173 a \numpyNDA and whose \Shape is consistent with \Shape of the \Data object to
174 be created can be used as the initial value.
175
176 \Data objects can be manipulated by applying unary operations (e.g. cos, sin,
177 log), and they can be combined point-wise by applying arithmetic operations
178 (e.g. +, - ,* , /).
179 We emphasize that \escript itself does not handle any spatial dependencies as
180 it does not know how values are interpreted by the processing PDE solver library.
181 However \escript invokes interpolation if this is needed during data manipulations.
182 Typically, this occurs in binary operations when both arguments belong to
183 different function spaces or when data are handed over to a PDE solver library
184 which requires functions to be represented in a particular way.
185
186 The following example shows the usage of \Data objects. Assume we have a
187 displacement field $u$ and we want to calculate the corresponding stress field
188 $\sigma$ using the linear-elastic isotropic material model
189 \begin{eqnarray}\label{eq: linear elastic stress}
190 \sigma_{ij}=\lambda u_{k,k} \delta_{ij} + \mu ( u_{i,j} + u_{j,i})
191 \end{eqnarray}
192 where $\delta_{ij}$ is the Kronecker symbol and
193 $\lambda$ and $\mu$ are the Lame coefficients. The following function
194 takes the displacement \var{u} and the Lame coefficients \var{lam} and \var{mu}
195 as arguments and returns the corresponding stress:
196 \begin{python}
197 from esys.escript import *
198 def getStress(u, lam, mu):
199 d=u.getDomain().getDim()
200 g=grad(u)
201 stress=lam*trace(g)*kronecker(d)+mu*(g+transpose(g))
202 return stress
203 \end{python}
204 The variable \var{d} gives the spatial dimension of the domain on which the
205 displacements are defined.
206 \var{kronecker} returns the Kronecker symbol with indexes $i$ and $j$ running
207 from 0 to \var{d}-1.
208 The call \var{grad(u)} requires the displacement field \var{u} to be in
209 the \var{Solution} or \ContinuousFunction.
210 The result \var{g} as well as the returned stress will be in the \Function.
211 If, for example, \var{u} is the solution of a PDE then \code{getStress} might
212 be called in the following way:
213 \begin{python}
214 s=getStress(u, 1., 2.)
215 \end{python}
216 However \code{getStress} can also be called with \Data objects as values for
217 \var{lam} and \var{mu} which, for instance in the case of a temperature
218 dependency, are calculated by an expression.
219 The following call is equivalent to the previous example:
220 \begin{python}
221 lam=Scalar(1., ContinuousFunction(mydomain))
222 mu=Scalar(2., Function(mydomain))
223 s=getStress(u, lam, mu)
224 \end{python}
225 %
226 The function \var{lam} belongs to the \ContinuousFunction but with \var{g} the
227 function \var{trace(g)} is in the \Function.
228 In the evaluation of the product \var{lam*trace(g)} we have different function
229 spaces (on the nodes versus in the centers) and at first glance we have incompatible data.
230 \escript converts the arguments into an appropriate function space according
231 to \fig{ESCRIPT DEP}.
232 In this example that means \escript sees \var{lam} as a function of the \Function.
233 In the context of FEM this means the nodal values of \var{lam} are
234 interpolated to the element centers.
235 The interpolation is automatic and requires no special handling.
236
237 \begin{figure}
238 \centering
239 \includegraphics{EscriptDiagram2}
240 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region
241 with two rock types {\it white} and {\it gray} is shown.
242 The number in each cell refers to the major rock type present in the cell
243 ($1$ for {\it white} and $2$ for {\it gray}).}
244 \end{figure}
245
246 \subsection{Tagged, Expanded and Constant Data}
247 Material parameters such as the Lame coefficients are typically dependent on
248 rock types present in the area of interest.
249 A common technique to handle these kinds of material parameters is
250 \emph{tagging}\index{tagging}, which uses storage efficiently.
251 \fig{Figure: tag} shows an example. In this case two rock types {\it white}
252 and {\it gray} can be found in the domain.
253 The domain is subdivided into triangular shaped cells.
254 Each cell has a tag indicating the rock type predominantly found in this cell.
255 Here $1$ is used to indicate rock type {\it white} and $2$ for rock type {\it gray}.
256 The tags are assigned at the time when the cells are generated and stored in
257 the \Domain class object. To allow easier usage of tags, names can be used
258 instead of numbers. These names are typically defined at the time when the
259 geometry is generated.
260
261 The following statements show how to use tagged values for \var{lam} as shown
262 in \fig{Figure: tag} for the stress calculation discussed above:
263 \begin{python}
264 lam=Scalar(value=2., what=Function(mydomain))
265 insertTaggedValue(lam, white=30., gray=5000.)
266 s=getStress(u, lam, 2.)
267 \end{python}
268 In this example \var{lam} is set to $30$ for those cells with tag {\it white}
269 (=$1$) and to $5000$ for cells with tag {\it gray} (=$2$).
270 The initial value $2$ of \var{lam} is used as a default value for the case
271 when a tag is encountered which has not been linked with a value.
272 The \code{getStress} method does not need to be changed now that we are using tags.
273 \escript resolves the tags when \var{lam*trace(g)} is calculated.
274
275 This brings us to a very important point about \escript.
276 You can develop a simulation with constant Lame coefficients, and then later
277 switch to tagged Lame coefficients without otherwise changing your \PYTHON script.
278 In short, you can use the same script for models with different domains and
279 different types of input data.
280
281 There are three main ways in which \Data objects are represented internally --
282 constant, tagged, and expanded.
283 In the constant case, the same value is used at each sample point while only a
284 single value is stored to save memory.
285 In the expanded case, each sample point has an individual value (such as for the solution of a PDE).
286 This is where your largest data sets will be created because the values are
287 stored as a complete array.
288 The tagged case has already been discussed above.
289 Expanded data is created when specifying \code{expanded=True} in the \Data
290 object constructor, while tagged data requires calling the \member{insertTaggedValue}
291 method as shown above.
292
293 Values are accessed through a sample reference number.
294 Operations on expanded \Data objects have to be performed for each sample
295 point individually.
296 When tagged values are used, the values are held in a dictionary.
297 Operations on tagged data require processing the set of tagged values only,
298 rather than processing the value for each individual sample point.
299 \escript allows any mixture of constant, tagged and expanded data in a single expression.
300
301 \subsection{Saving and Restoring Simulation Data}
302 \Data objects can be written to disk files with the \member{dump} method and
303 read back using the \member{load} method, both of which use the
304 \netCDF\cite{NETCDF} file format.
305 Use these to save data for checkpoint/restart or simply to save and reuse data
306 that was expensive to compute.
307 For instance, to save the coordinates of the data points of a
308 \ContinuousFunction to the file \file{x.nc} use
309 \begin{python}
310 x=ContinuousFunction(mydomain).getX()
311 x.dump("x.nc")
312 mydomain.dump("dom.nc")
313 \end{python}
314 To recover the object \var{x}, and you know that \var{mydomain} was an \finley
315 mesh, use
316 \begin{python}
317 from esys.finley import LoadMesh
318 mydomain=LoadMesh("dom.nc")
319 x=load("x.nc", mydomain)
320 \end{python}
321 Obviously, it is possible to execute the same steps that were originally used
322 to generate \var{mydomain} to recreate it. However, in most cases using
323 \member{dump} and \member{load} is faster, particularly if optimization has
324 been applied.
325 If \escript is running on more than one \MPI process \member{dump} will create
326 an individual file for each process containing the local data.
327 In order to avoid conflicts the file names are extended by the \MPI processor
328 rank, that is instead of one file \file{dom.nc} you would get
329 \file{dom.nc.0000}, \file{dom.nc.0001}, etc. You still call
330 \code{LoadMesh("dom.nc")} to load the domain but you have to make sure that
331 the appropriate file is accessible from the corresponding rank, and loading
332 will only succeed if you run with as many processes as were used when calling
333 \member{dump}.
334
335 The function space of the \Data is stored in \file{x.nc}.
336 If the \Data object is expanded, the number of data points in the file and of
337 the \Domain for the particular \FunctionSpace must match.
338 Moreover, the ordering of the values is checked using the reference
339 identifiers provided by \FunctionSpace on the \Domain.
340 In some cases, data points will be reordered so be aware and confirm that you
341 get what you wanted.
342
343 A newer, more flexible way of saving and restoring \escript simulation data
344 is through a \class{DataManager} class object.
345 It has the advantage of allowing to save and load not only a \Domain and
346 \Data objects but also other values\footnote{The \PYTHON \emph{pickle} module
347 is used for other types.} you compute in your simulation script.
348 Further, \class{DataManager} objects can simultaneously create files for
349 visualization so no extra calls to \code{saveVTK} etc. are needed.
350
351 The following example shows how the \class{DataManager} class can be used.
352 For an explanation of all member functions and options see the relevant
353 reference section.
354 \begin{python}
355 from esys.escript import DataManager, Scalar, Function
356 from esys.finley import Rectangle
357
358 dm = DataManager(formats=[DataManager.RESTART, DataManager.VTK])
359 if dm.hasData():
360 mydomain=dm.getDomain()
361 val=dm.getValue("val")
362 t=dm.getValue("t")
363 t_max=dm.getValue("t_max")
364 else:
365 mydomain=Rectangle()
366 val=Function(mydomain).getX()
367 t=0.
368 t_max=2.5
369
370 while t<t_max:
371 t+=.01
372 val=val+t/2
373 dm.addData(val=val, t=t, t_max=t_max)
374 dm.export()
375 \end{python}
376 In the constructor we specify that we want \code{RESTART} (i.e. dump) files
377 and \code{VTK} files to be saved.
378 By default, the constructor will look for previously saved \code{RESTART}
379 files under the current directory and load them.
380 We can then enquire if such files were found by calling the \member{hasData}
381 method. If it returns \True we retrieve the domain and values into local
382 variables. Otherwise the same variables are initialized with appropriate
383 values to start a new simulation.
384 Note, that \var{t} and \var{t_max} are regular floating point values and not
385 \Data objects. Yet they are treated the same way by the \class{DataManager}.
386
387 After this initialization step the script enters the main simulation loop
388 where calculations are performed.
389 When these are finalized for a time step we call the \member{addData} method
390 to let the manager know which variables to store on disk.
391 This does not actually save the data yet and it is allowed to call
392 \member{addData} more than once to add information incrementally, e.g. from
393 separate functions that have access to the \class{DataManager} instance.
394 Once all variables have been added the \member{export} method has to be called
395 to flush all data to disk and clear the manager.
396 In this example, this call dumps \var{mydomain} and \var{val} to files
397 in a restart directory and also stores \var{t} and \var{t_max} on disk.
398 Additionally, it generates a \VTK file for visualization of the data.
399 If the script would stop running before its completion for some reason (e.g.
400 because its runtime limit was exceeded in a multiuser environment), you could
401 simply run it again and it would resume at the point it stopped before.
402
403 \section{\escript Classes}
404
405 \subsection{The \Domain class}
406 \begin{classdesc}{Domain}{}
407 A \Domain object is used to describe a geometric region together with
408 a way of representing functions over this region.
409 The \Domain class provides an abstract interface to the domain of \FunctionSpace and \Data objects.
410 \Domain needs to be subclassed in order to provide a complete implementation.
411 \end{classdesc}
412
413 \noindent The following methods are available:
414 \begin{methoddesc}[Domain]{getDim}{}
415 returns the spatial dimension of the \Domain.
416 \end{methoddesc}
417 %
418 \begin{methoddesc}[Domain]{dump}{filename}
419 writes the \Domain to the file \var{filename} using the \netCDF file format.
420 \end{methoddesc}
421 %
422 \begin{methoddesc}[Domain]{getX}{}
423 returns the locations in the \Domain. The \FunctionSpace of the returned
424 \Data object is chosen by the \Domain implementation. Typically it will be
425 in the \Function.
426 \end{methoddesc}
427 %
428 \begin{methoddesc}[Domain]{setX}{newX}
429 assigns new locations to the \Domain. \var{newX} has to have \Shape $(d,)$
430 where $d$ is the spatial dimension of the domain. Typically \var{newX}
431 must be in the \ContinuousFunction but the space actually to be used
432 depends on the \Domain implementation.
433 \end{methoddesc}
434 %
435 \begin{methoddesc}[Domain]{getNormal}{}
436 returns the surface normals on the boundary of the \Domain as a \Data object.
437 \end{methoddesc}
438 %
439 \begin{methoddesc}[Domain]{getSize}{}
440 returns the local sample size, i.e. the element diameter, as a \Data object.
441 \end{methoddesc}
442 %
443 \begin{methoddesc}[Domain]{setTagMap}{tag_name, tag}
444 defines a mapping of the tag name \var{tag_name} to the \var{tag}.
445 \end{methoddesc}
446 %
447 \begin{methoddesc}[Domain]{getTag}{tag_name}
448 returns the tag associated with the tag name \var{tag_name}.
449 \end{methoddesc}
450 %
451 \begin{methoddesc}[Domain]{isValidTagName}{tag_name}
452 returns \True if \var{tag_name} is a valid tag name.
453 \end{methoddesc}
454 %
455 \begin{methoddesc}[Domain]{__eq__}{arg}
456 (\PYTHON \var{==} operator) returns \True if the \Domain \var{arg}
457 describes the same domain, \False otherwise.
458 \end{methoddesc}
459 %
460 \begin{methoddesc}[Domain]{__ne__}{arg}
461 (\PYTHON \var{!=} operator) returns \True if the \Domain \var{arg} does
462 not describe the same domain, \False otherwise.
463 \end{methoddesc}
464 %
465 \begin{methoddesc}[Domain]{__str__}{}
466 (\PYTHON \var{str()} function) returns a string representation of the
467 \Domain.
468 \end{methoddesc}
469 %
470 \begin{methoddesc}[Domain]{onMasterProcessor)}{}
471 returns \True if the processor is the master processor within the \MPI
472 processor group used by the \Domain. This is the processor with rank 0.
473 If \MPI support is not enabled the return value is always \True.
474 \end{methoddesc}
475 %
476 \begin{methoddesc}[Domain]{getMPISize}{}
477 returns the number of \MPI processors used for this \Domain. If \MPI
478 support is not enabled 1 is returned.
479 \end{methoddesc}
480 %
481 \begin{methoddesc}[Domain]{getMPIRank}{}
482 returns the rank of the processor executing the statement within the
483 \MPI processor group used by the \Domain. If \MPI support is not enabled
484 0 is returned.
485 \end{methoddesc}
486 %
487 \begin{methoddesc}[Domain]{MPIBarrier}{}
488 executes barrier synchronization within the \MPI processor group used by
489 the \Domain. If \MPI support is not enabled, this command does nothing.
490 \end{methoddesc}
491
492 \subsection{The \FunctionSpace class}
493 \begin{classdesc}{FunctionSpace}{}
494 \FunctionSpace objects are used to define properties of \Data objects such as continuity.
495 \FunctionSpace objects are instantiated by generator functions.
496 A \Data object in a particular \FunctionSpace is represented by its values at
497 \DataSamplePoints which are defined by the type and the \Domain of the \FunctionSpace.
498 \end{classdesc}
499 %
500 The following methods are available:
501 %
502 \begin{methoddesc}[FunctionSpace]{getDim}{}
503 returns the spatial dimension of the \Domain of the \FunctionSpace.
504 \end{methoddesc}
505 %
506 \begin{methoddesc}[FunctionSpace]{getX}{}
507 returns the location of the \DataSamplePoints.
508 \end{methoddesc}
509 %
510 \begin{methoddesc}[FunctionSpace]{getNormal}{}
511 If the domain of functions in the \FunctionSpace is a hyper-manifold (e.g.
512 the boundary of a domain) the method returns the outer normal at each of
513 the \DataSamplePoints. Otherwise an exception is raised.
514 \end{methoddesc}
515 %
516 \begin{methoddesc}[FunctionSpace]{getSize}{}
517 returns a \Data object measuring the spacing of the \DataSamplePoints.
518 The size may be zero.
519 \end{methoddesc}
520 %
521 \begin{methoddesc}[FunctionSpace]{getDomain}{}
522 returns the \Domain of the \FunctionSpace.
523 \end{methoddesc}
524 %
525 \begin{methoddesc}[FunctionSpace]{setTags}{new_tag, mask}
526 assigns a new tag \var{new_tag} to all data samples where \var{mask} is
527 positive for a least one data point.
528 \var{mask} must be defined on this \FunctionSpace.
529 Use the \var{setTagMap} to assign a tag name to \var{new_tag}.
530 \end{methoddesc}
531 %
532 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
533 (\PYTHON \var{==} operator) returns \True if the \FunctionSpace \var{arg}
534 describes the same function space, \False otherwise.
535 \end{methoddesc}
536 %
537 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
538 (\PYTHON \var{!=} operator) returns \True if the \FunctionSpace \var{arg}
539 does not describe the same function space, \False otherwise.
540 \end{methoddesc}
541
542 \begin{methoddesc}[Domain]{__str__}{}
543 (\PYTHON \var{str()} function) returns a string representation of the
544 \FunctionSpace.
545 \end{methoddesc}
546 %
547 The following functions provide generators for \FunctionSpace objects:
548 %
549 \begin{funcdesc}{Function}{domain}
550 returns the \Function on the \Domain \var{domain}. \Data objects in this
551 type of \Function are defined over the whole geometric region defined by
552 \var{domain}.
553 \end{funcdesc}
554 %
555 \begin{funcdesc}{ContinuousFunction}{domain}
556 returns the \ContinuousFunction on the \Domain domain. \Data objects in
557 this type of \Function are defined over the whole geometric region defined
558 by \var{domain} and assumed to represent a continuous function.
559 \end{funcdesc}
560 %
561 \begin{funcdesc}{FunctionOnBoundary}{domain}
562 returns the \FunctionOnBoundary on the \Domain domain. \Data objects in
563 this type of \Function are defined on the boundary of the geometric region
564 defined by \var{domain}.
565 \end{funcdesc}
566 %
567 \begin{funcdesc}{FunctionOnContactZero}{domain}
568 returns the \FunctionOnContactZero the \Domain domain. \Data objects in
569 this type of \Function are defined on side 0 of a discontinuity within
570 the geometric region defined by \var{domain}.
571 The discontinuity is defined when \var{domain} is instantiated.
572 \end{funcdesc}
573 %
574 \begin{funcdesc}{FunctionOnContactOne}{domain}
575 returns the \FunctionOnContactOne on the \Domain domain. \Data objects in
576 this type of \Function are defined on side 1 of a discontinuity within
577 the geometric region defined by \var{domain}.
578 The discontinuity is defined when \var{domain} is instantiated.
579 \end{funcdesc}
580 %
581 \begin{funcdesc}{Solution}{domain}
582 returns the \SolutionFS on the \Domain domain. \Data objects in this type
583 of \Function are defined on the geometric region defined by \var{domain}
584 and are solutions of partial differential equations\index{partial differential equation}.
585 \end{funcdesc}
586 %
587 \begin{funcdesc}{ReducedSolution}{domain}
588 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in
589 this type of \Function are defined on the geometric region defined by
590 \var{domain} and are solutions of partial differential
591 equations\index{partial differential equation} with a reduced smoothness
592 for the solution approximation.
593 \end{funcdesc}
594
595 \subsection{The \Data Class}
596 \label{SEC ESCRIPT DATA}
597
598 The following table shows arithmetic operations that can be performed
599 point-wise on \Data objects:
600 \begin{center}
601 \begin{tabular}{l|l}
602 \textbf{Expression} & \textbf{Description}\\
603 \hline
604 \code{+arg} & identical to \var{arg}\index{+}\\
605 \code{-arg} & negation of \var{arg}\index{-}\\
606 \code{arg0+arg1} & adds \var{arg0} and \var{arg1}\index{+}\\
607 \code{arg0*arg1} & multiplies \var{arg0} and \var{arg1}\index{*}\\
608 \code{arg0-arg1} & subtracts \var{arg1} from \var{arg0}\index{-}\\
609 \code{arg0/arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
610 \code{arg0**arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
611 \end{tabular}
612 \end{center}
613 At least one of the arguments \var{arg0} or \var{arg1} must be a \Data object.
614 Either of the arguments may be a \Data object, a \PYTHON number or a \numpy
615 object.
616 If \var{arg0} or \var{arg1} are not defined on the same \FunctionSpace, then
617 an attempt is made to convert \var{arg0} to the \FunctionSpace of \var{arg1}
618 or to convert \var{arg1} to the \FunctionSpace of \var{arg0}.
619 Both arguments must have the same \Shape or one of the arguments may be of
620 rank 0 (a constant).
621 The returned \Data object has the same \Shape and is defined on
622 the \DataSamplePoints as \var{arg0} or \var{arg1}.
623
624 The following table shows the update operations that can be applied to
625 \Data objects:
626 \begin{center}
627 \begin{tabular}{l|l}
628 \textbf{Expression} & \textbf{Description}\\
629 \hline
630 \code{arg0+=arg1} & adds \var{arg1} to \var{arg0}\index{+}\\
631 \code{arg0*=arg1} & multiplies \var{arg0} by \var{arg1}\index{*}\\
632 \code{arg0-=arg1} & subtracts \var{arg1} from\var{arg0}\index{-}\\
633 \code{arg0/=arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
634 \code{arg0**=arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
635 \end{tabular}
636 \end{center}
637 \var{arg0} must be a \Data object. \var{arg1} must be a \Data object or an
638 object that can be converted into a \Data object.
639 \var{arg1} must have the same \Shape as \var{arg0} or have rank 0.
640 In the latter case it is assumed that the values of \var{arg1} are constant
641 for all components. \var{arg1} must be defined in the same \FunctionSpace as
642 \var{arg0} or it must be possible to interpolate \var{arg1} onto the
643 \FunctionSpace of \var{arg0}.
644
645 The \Data class supports taking slices as well as assigning new values to a
646 slice of an existing \Data object\index{slicing}.
647 The following expressions for taking and setting slices are valid:
648 \begin{center}
649 \begin{tabular}{l|ll}
650 \textbf{Rank of \var{arg}} & \textbf{Slicing expression} & \textbf{\Shape of returned and assigned object}\\
651 \hline
652 0 & no slicing & N/A\\
653 1 & \var{arg[l0:u0]} & (\var{u0}-\var{l0},)\\
654 2 & \var{arg[l0:u0,l1:u1]} & (\var{u0}-\var{l0},\var{u1}-\var{l1})\\
655 3 & \var{arg[l0:u0,l1:u1,l2:u2]} & (\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})\\
656 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})\\
657 \end{tabular}
658 \end{center}
659 where \var{s} is the \Shape of \var{arg} and
660 \[0 \le \var{l0} \le \var{u0} \le \var{s[0]},\]
661 \[0 \le \var{l1} \le \var{u1} \le \var{s[1]},\]
662 \[0 \le \var{l2} \le \var{u2} \le \var{s[2]},\]
663 \[0 \le \var{l3} \le \var{u3} \le \var{s[3]}.\]
664 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case
665 $0$ is assumed.
666 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may be omitted, in which case, the upper limit for that dimension is assumed.
667 The lower and upper index may be identical, in which case the column and the lower or upper
668 index may be dropped. In the returned or in the object assigned to a slice, the corresponding component is dropped,
669 i.e. the rank is reduced by one in comparison to \var{arg}.
670 The following examples show slicing in action:
671 \begin{python}
672 t=Data(1.,(4,4,6,6),Function(mydomain))
673 t[1,1,1,0]=9.
674 s=t[:2,:,2:6,5] # s has rank 3
675 s[:,:,1]=1.
676 t[:2,:2,5,5]=s[2:4,1,:2]
677 \end{python}
678
679 \subsection{Generation of \Data objects}
680 \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expand=\False}
681 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
682 The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True
683 the \Data object is represented in expanded from.
684 \end{classdesc}
685
686 \begin{classdesc}{Data}{value,what=FunctionSpace(),expand=\False}
687 creates a \Data object in the \FunctionSpace \var{what}.
688 The value for each \DataSamplePoints is set to \var{value}, which could be a \numpy, \Data object \var{value} or a dictionary of
689 \numpy or floating point numbers. In the latter case the keys must be integers and are used
690 as tags.
691 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True
692 the \Data object is represented in expanded form.
693 \end{classdesc}
694
695 \begin{classdesc}{Data}{}
696 creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present
697 where a \Data object is required.
698 \end{classdesc}
699
700 \begin{funcdesc}{Scalar}{value=0.,what=FunctionSpace(),expand=\False}
701 returns a \Data object of rank 0 (a constant) in the \FunctionSpace \var{what}.
702 Values are initialized with \var{value}, a double precision quantity. If \var{expanded} is \True
703 the \Data object is represented in expanded from.
704 \end{funcdesc}
705
706 \begin{funcdesc}{Vector}{value=0.,what=FunctionSpace(),expand=\False}
707 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what},
708 where \var{d} is the spatial dimension of the \Domain of \var{what}.
709 Values are initialed with \var{value}, a double precision quantity. If \var{expanded} is \True
710 the \Data object is represented in expanded from.
711 \end{funcdesc}
712
713 \begin{funcdesc}{Tensor}{value=0.,what=FunctionSpace(),expand=\False}
714 returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what},
715 where \var{d} is the spatial dimension of the \Domain of \var{what}.
716 Values are initialed with \var{value}, a double precision quantity. If \var{expanded} is \True
717 the \Data object is represented in expanded from.
718 \end{funcdesc}
719
720 \begin{funcdesc}{Tensor3}{value=0.,what=FunctionSpace(),expand=\False}
721 returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what},
722 where \var{d} is the spatial dimension of the \Domain of \var{what}.
723 Values are initialed with \var{value}, a double precision quantity. If \var{expanded} is \True
724 the \Data object is re\var{arg}presented in expanded from.
725 \end{funcdesc}
726
727 \begin{funcdesc}{Tensor4}{value=0.,what=FunctionSpace(),expand=\False}
728 returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what},
729 where \var{d} is the spatial dimension of the \Domain of \var{what}.
730 Values are initialized with \var{value}, a double precision quantity. If \var{expanded} is \True
731 the \Data object is represented in expanded from.
732 \end{funcdesc}
733
734 \begin{funcdesc}{load}{filename,domain}
735 recovers a \Data object on \Domain \var{domain} from the file \var{filename}, which was created by \function{dump}.
736 \end{funcdesc}
737
738 \subsection{\Data methods}
739 These are the most frequently-used methods of the \Data class.
740 A complete list of methods can be found on \ReferenceGuide.
741 \begin{methoddesc}[Data]{getFunctionSpace}{}
742 returns the \FunctionSpace of the object.
743 \end{methoddesc}
744
745 \begin{methoddesc}[Data]{getDomain}{}
746 returns the \Domain of the object.
747 \end{methoddesc}
748
749 \begin{methoddesc}[Data]{getShape}{}
750 returns the \Shape of the object as a \class{tuple} of
751 integers.
752 \end{methoddesc}
753
754 \begin{methoddesc}[Data]{getRank}{}
755 returns the rank of the data on each data point. \index{rank}
756 \end{methoddesc}
757
758 \begin{methoddesc}[Data]{isEmpty}{}
759 returns \True id the \Data object is the \EmptyData object.
760 Otherwise \False is returned.
761 Note that this is not the same as asking if the object contains no \DataSamplePoints.
762 \end{methoddesc}
763
764 \begin{methoddesc}[Data]{setTaggedValue}{tag_name,value}
765 assigns the \var{value} to all \DataSamplePoints which have the tag
766 assigned to \var{tag_name}. \var{value} must be an object of class
767 \class{numpy.ndarray} or must be convertible into a
768 \class{numpy.ndarray} object. \var{value} (or the corresponding
769 \class{numpy.ndarray} object) must be of rank $0$ or must have the
770 same rank like the object.
771 If a value has already be defined for tag \var{tag_name} within the object
772 it is overwritten by the new \var{value}. If the object is expanded,
773 the value assigned to \DataSamplePoints with tag \var{tag_name} is replaced by
774 \var{value}. If no tag is assigned tag name \var{tag_name}, no value is set.
775 \end{methoddesc}
776
777 \begin{methoddesc}[Data]{dump}{filename}
778 dumps the \Data object to the file \var{filename}. The file stores the
779 function space but not the \Domain. It is in the responsibility of the user to
780 save the \Domain.
781 \end{methoddesc}
782
783 \begin{methoddesc}[Data]{__str__}{}
784 returns a string representation of the object.
785 \end{methoddesc}
786
787 \subsection{Functions of \Data objects}
788 This section lists the most important functions for \Data class objects \var{a}.
789 A complete list and a more detailed description of the functionality can be found on \ReferenceGuide.
790 \begin{funcdesc}{saveVTK}{filename,**kwdata}
791 writes \Data defined by keywords in the file with \var{filename} using the
792 vtk file format \VTK file format. The key word is used as an identifier. The statement
793 \begin{python}
794 saveVTK("out.xml",temperature=T,velocity=v)
795 \end{python}
796 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
797 file \file{out.xml}. Restrictions on the allowed combinations of \FunctionSpace apply.
798 \end{funcdesc}
799 \begin{funcdesc}{saveDX}{filename,**kwdata}
800 writes \Data defined by keywords in the file with \var{filename} using the
801 vtk file format \OpenDX file format. The key word is used as an identifier. The statement
802 \begin{python}
803 saveDX("out.dx",temperature=T,velocity=v)
804 \end{python}
805 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
806 file \file{out.dx}. Restrictions on the allowed combinations of \FunctionSpace apply.
807 \end{funcdesc}
808 \begin{funcdesc}{kronecker}{d}
809 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
810 \begin{equation}
811 \code{kronecker(d)}\left[ i,j\right] = \left\{
812 \begin{array}{cc}
813 1 & \mbox{ if } i=j \\
814 0 & \mbox{ otherwise }
815 \end{array}
816 \right.
817 \end{equation}
818 If \var{d} is an integer a $(d,d)$ \numpy array is returned.
819 \end{funcdesc}
820 \begin{funcdesc}{identityTensor}{d}
821 is a synonym for \code{kronecker} (see above).
822 % returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
823 % \begin{equation}
824 % \code{identityTensor(d)}\left[ i,j\right] = \left\{
825 % \begin{array}{cc}
826 % 1 & \mbox{ if } i=j \\
827 % 0 & \mbox{ otherwise }
828 % \end{array}
829 % \right.
830 % \end{equation}
831 % If \var{d} is an integer a $(d,d)$ \numpy array is returned.
832 \end{funcdesc}
833 \begin{funcdesc}{identityTensor4}{d}
834 returns a \RankFour \Data object in \FunctionSpace \var{d} such that
835 \begin{equation}
836 \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
837 \begin{array}{cc}
838 1 & \mbox{ if } i=k \mbox{ and } j=l\\
839 0 & \mbox{ otherwise }
840 \end{array}
841 \right.
842 \end{equation}
843 If \var{d} is an integer a $(d,d,d,d)$ \numpy array is returned.
844 \end{funcdesc}
845 \begin{funcdesc}{unitVector}{i,d}
846 returns a \RankOne \Data object in \FunctionSpace \var{d} such that
847 \begin{equation}
848 \code{identityTensor(d)}\left[ j \right] = \left\{
849 \begin{array}{cc}
850 1 & \mbox{ if } j=i\\
851 0 & \mbox{ otherwise }
852 \end{array}
853 \right.
854 \end{equation}
855 If \var{d} is an integer a $(d,)$ \numpy array is returned.
856
857 \end{funcdesc}
858
859 \begin{funcdesc}{Lsup}{a}
860 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values
861 over all components and all \DataSamplePoints of \var{a}.
862 \end{funcdesc}
863
864 \begin{funcdesc}{sup}{a}
865 returns the maximum value over all components and all \DataSamplePoints of \var{a}.
866 \end{funcdesc}
867
868 \begin{funcdesc}{inf}{a}
869 returns the minimum value over all components and all \DataSamplePoints of \var{a}
870 \end{funcdesc}
871
872
873
874 \begin{funcdesc}{minval}{a}
875 returns at each \DataSamplePoints the minimum value over all components.
876 \end{funcdesc}
877
878 \begin{funcdesc}{maxval}{a}
879 returns at each \DataSamplePoints the maximum value over all components.
880 \end{funcdesc}
881
882 \begin{funcdesc}{length}{a}
883 returns at Euclidean norm at each \DataSamplePoints. For a \RankFour \var{a} this is
884 \begin{equation}
885 \code{length(a)}=\sqrt{\sum_{ijkl} \var{a} \left[i,j,k,l\right]^2}
886 \end{equation}
887 \end{funcdesc}
888 \begin{funcdesc}{trace}{a\optional{,axis_offset=0}}
889 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
890 case of a \RankTwo function and this is
891 \begin{equation}
892 \code{trace(a)}=\sum_{i} \var{a} \left[i,i\right]
893 \end{equation}
894 and for a \RankFour function and \code{axis_offset=1} this is
895 \begin{equation}
896 \code{trace(a,1)}\left[i,j\right]=\sum_{k} \var{a} \left[i,k,k,j\right]
897 \end{equation}
898 \end{funcdesc}
899
900 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
901 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
902 present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
903 the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
904 case of a \RankTwo function and this is
905 \begin{equation}
906 \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
907 \end{equation}
908 and for a \RankFour function and \code{axis_offset=1} this is
909 \begin{equation}
910 \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
911 \end{equation}
912 \end{funcdesc}
913
914 \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
915 returns \var{a} but with swapped components \var{axis0} and \var{axis1}. The argument \var{a} must be
916 at least of \RankTwo. For instance in the
917 for a \RankFour argument, \code{axis0=1} and \code{axis1=2} this is
918 \begin{equation}
919 \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
920 \end{equation}
921 \end{funcdesc}
922
923 \begin{funcdesc}{symmetric}{a}
924 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
925 \end{funcdesc}
926 \begin{funcdesc}{nonsymmetric}{a}
927 returns the non--symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
928 \end{funcdesc}
929 \begin{funcdesc}{inverse}{a}
930 return the inverse of \var{a}. This is
931 \begin{equation}
932 \code{matrix_mult(inverse(a),a)=kronecker(d)}
933 \end{equation}
934 if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape
935 \code{(2,2)} and \code{(3,3)}.
936 \end{funcdesc}
937 \begin{funcdesc}{eigenvalues}{a}
938 return the eigenvalues of \var{a}. This is
939 \begin{equation}
940 \code{matrix_mult(a,V)=e[i]*V}
941 \end{equation}
942 where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.
943 The eigenvalues are ordered in increasing size.
944 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
945 The current implementation is restricted to arguments of shape
946 \code{(2,2)} and \code{(3,3)}.
947 \end{funcdesc}
948 \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
949 return the eigenvalues and eigenvectors of \var{a}. This is
950 \begin{equation}
951 \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
952 \end{equation}
953 where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.
954 \begin{equation}
955 \code{matrix_mult(transpose(V),V)=kronecker(d)}
956 \end{equation}
957 if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.
958 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
959 The current implementation is restricted to arguments of shape
960 \code{(2,2)} and \code{(3,3)}.
961 \end{funcdesc}
962 \begin{funcdesc}{maximum}{*a}
963 returns the maximum value over all arguments at all \DataSamplePoints and for each component.
964 For instance
965 \begin{equation}
966 \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
967 \end{equation}
968 at all \DataSamplePoints.
969 \end{funcdesc}
970 \begin{funcdesc}{minimum}{*a}
971 returns the minimum value over all arguments at all \DataSamplePoints and for each component.
972 For instance
973 \begin{equation}
974 \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
975 \end{equation}
976 at all \DataSamplePoints.
977 \end{funcdesc}
978
979 \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
980 cuts back \var{a} into the range between \var{minval} and \var{maxval}. A value in the returned object equals
981 \var{minval} if the corresponding value of \var{a} is less than \var{minval}, equals \var{maxval} if the
982 corresponding value of \var{a} is greater than \var{maxval}
983 or corresponding value of \var{a} otherwise.
984 \end{funcdesc}
985 \begin{funcdesc}{inner}{a0,a1}
986 returns the inner product of \var{a0} and \var{a1}. For instance in the
987 case of \RankTwo arguments and this is
988 \begin{equation}
989 \code{inner(a)}=\sum_{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
990 \end{equation}
991 and for a \RankFour arguments this is
992 \begin{equation}
993 \code{inner(a)}=\sum_{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
994 \end{equation}
995 \end{funcdesc}
996
997 \begin{funcdesc}{matrix_mult}{a0,a1}
998 returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is
999 \begin{equation}
1000 \code{matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
1001 \end{equation}
1002 and if \var{a1} is \RankTwo this is
1003 \begin{equation}
1004 \code{matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
1005 \end{equation}
1006 \end{funcdesc}
1007
1008 \begin{funcdesc}{transposed_matrix_mult}{a0,a1}
1009 returns the matrix product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
1010 \code{matrix_mult(transpose(a0),a1)}.
1011 If \var{a1} is \RankOne this is
1012 \begin{equation}
1013 \code{transposed_matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k\right]
1014 \end{equation}
1015 and if \var{a1} is \RankTwo this is
1016 \begin{equation}
1017 \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k,j\right]
1018 \end{equation}
1019 \end{funcdesc}
1020
1021 \begin{funcdesc}{matrix_transposed_mult}{a0,a1}
1022 returns the matrix product of \var{a0} and the transposed of \var{a1}.
1023 The function is equivalent to
1024 \code{matrix_mult(a0,transpose(a1))}.
1025 If \var{a1} is \RankTwo this is
1026 \begin{equation}
1027 \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[j,k\right]
1028 \end{equation}
1029 \end{funcdesc}
1030
1031 \begin{funcdesc}{outer}{a0,a1}
1032 returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then
1033 \begin{equation}
1034 \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
1035 \end{equation}
1036 and if \var{a0} is \RankOne and \var{a1} is \RankThree
1037 \begin{equation}
1038 \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
1039 \end{equation}
1040 \end{funcdesc}
1041
1042 \begin{funcdesc}{tensor_mult}{a0,a1}
1043 returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is
1044 \begin{equation}
1045 \code{tensor_mult(a)}\left[i,j\right]=\sum_{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[k,l\right]
1046 \end{equation}
1047 and if \var{a1} is \RankFour this is
1048 \begin{equation}
1049 \code{tensor_mult(a)}\left[i,j,k,l\right]=\sum_{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[m,n,k,l\right]
1050 \end{equation}
1051 \end{funcdesc}
1052
1053 \begin{funcdesc}{transposed_tensor_mult}{a0,a1}
1054 returns the tensor product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
1055 \code{tensor_mult(transpose(a0),a1)}.
1056 If \var{a1} is \RankTwo this is
1057 \begin{equation}
1058 \code{transposed_tensor_mult(a)}\left[i,j\right]=\sum_{kl}\var{a0}\left[k,l,i,j\right] \cdot \var{a1} \left[k,l\right]
1059 \end{equation}
1060 and if \var{a1} is \RankFour this is
1061 \begin{equation}
1062 \code{transposed_tensor_mult(a)}\left[i,j,k,l\right]=\sum_{mn}\var{a0} \left[m,n,i,j\right] \cdot \var{a1} \left[m,n,k,l\right]
1063 \end{equation}
1064 \end{funcdesc}
1065
1066 \begin{funcdesc}{tensor_transposed_mult}{a0,a1}
1067 returns the tensor product of \var{a0} and the transposed of \var{a1}.
1068 The function is equivalent to
1069 \code{tensor_mult(a0,transpose(a1))}.
1070 If \var{a1} is \RankTwo this is
1071 \begin{equation}
1072 \code{tensor_transposed_mult(a)}\left[i,j\right]=\sum_{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[l,k\right]
1073 \end{equation}
1074 and if \var{a1} is \RankFour this is
1075 \begin{equation}
1076 \code{tensor_transposed_mult(a)}\left[i,j,k,l\right]=\sum_{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[k,l,m,n\right]
1077 \end{equation}
1078 \end{funcdesc}
1079
1080 \begin{funcdesc}{grad}{a\optional{, where=None}}
1081 returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a
1082 default \FunctionSpace is used. In case that \var{a} has \RankTwo one has
1083 \begin{equation}
1084 \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x_{k}}
1085 \end{equation}
1086 \end{funcdesc}
1087 \begin{funcdesc}{integrate}{a\optional{ ,where=None}}
1088 returns the integral of \var{a} where the domain of integration is defined by the \FunctionSpace of \var{a}. If \var{where} is
1089 present the argument is interpolated into \FunctionSpace \var{where} before integration. For instance in the case of
1090 a \RankTwo argument in \ContinuousFunction it is
1091 \begin{equation}
1092 \code{integrate(a)}\left[i,j\right]=\int_{\Omega}\var{a} \left[i,j\right] \; d\Omega
1093 \end{equation}
1094 where $\Omega$ is the spatial domain and $d\Omega$ volume integration. To integrate over the boundary of the domain one uses
1095 \begin{equation}
1096 \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int_{\partial \Omega} a\left[i,j\right] \; ds
1097 \end{equation}
1098 where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or line integration.
1099 \end{funcdesc}
1100 \begin{funcdesc}{interpolate}{a,where}
1101 interpolates argument \var{a} into the \FunctionSpace \var{where}.
1102 \end{funcdesc}
1103 \begin{funcdesc}{div}{a\optional{ ,where=None}}
1104 returns the divergence of \var{a}. This
1105 \begin{equation}
1106 \code{div(a)}=trace(grad(a),where)
1107 \end{equation}
1108 \end{funcdesc}
1109 \begin{funcdesc}{jump}{a\optional{ ,domain=None}}
1110 returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present
1111 in \var{domain}.
1112 \begin{equation}
1113 \begin{array}{rcl}
1114 \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
1115 & & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
1116 \end{array}
1117 \end{equation}
1118 \end{funcdesc}
1119 \begin{funcdesc}{L2}{a}
1120 returns the $L^2$-norm of \var{a} in its function space. This is
1121 \begin{equation}
1122 \code{L2(a)=integrate(length(a)}^2\code{)} \; .
1123 \end{equation}
1124 \end{funcdesc}
1125
1126 The following functions operate ``point-wise''. That is, the operation is applied to each component of each point
1127 individually.
1128
1129 \begin{funcdesc}{sin}{a}
1130 applies sine function to \var{a}.
1131 \end{funcdesc}
1132
1133 \begin{funcdesc}{cos}{a}
1134 applies cosine function to \var{a}.
1135 \end{funcdesc}
1136
1137 \begin{funcdesc}{tan}{a}
1138 applies tangent function to \var{a}.
1139 \end{funcdesc}
1140
1141 \begin{funcdesc}{asin}{a}
1142 applies arc (inverse) sine function to \var{a}.
1143 \end{funcdesc}
1144
1145 \begin{funcdesc}{acos}{a}
1146 applies arc (inverse) cosine function to \var{a}.
1147 \end{funcdesc}
1148
1149 \begin{funcdesc}{atan}{a}
1150 applies arc (inverse) tangent function to \var{a}.
1151 \end{funcdesc}
1152
1153 \begin{funcdesc}{sinh}{a}
1154 applies hyperbolic sine function to \var{a}.
1155 \end{funcdesc}
1156
1157 \begin{funcdesc}{cosh}{a}
1158 applies hyperbolic cosine function to \var{a}.
1159 \end{funcdesc}
1160
1161 \begin{funcdesc}{tanh}{a}
1162 applies hyperbolic tangent function to \var{a}.
1163 \end{funcdesc}
1164
1165 \begin{funcdesc}{asinh}{a}
1166 applies arc (inverse) hyperbolic sine function to \var{a}.
1167 \end{funcdesc}
1168
1169 \begin{funcdesc}{acosh}{a}
1170 applies arc (inverse) hyperbolic cosine function to \var{a}.
1171 \end{funcdesc}
1172
1173 \begin{funcdesc}{atanh}{a}
1174 applies arc (inverse) hyperbolic tangent function to \var{a}.
1175 \end{funcdesc}
1176
1177 \begin{funcdesc}{exp}{a}
1178 applies exponential function to \var{a}.
1179 \end{funcdesc}
1180
1181 \begin{funcdesc}{sqrt}{a}
1182 applies square root function to \var{a}.
1183 \end{funcdesc}
1184
1185 \begin{funcdesc}{log}{a}
1186 applies the natural logarithm to \var{a}.
1187 \end{funcdesc}
1188
1189 \begin{funcdesc}{log10}{a}
1190 applies the base-$10$ logarithm to \var{a}.
1191 \end{funcdesc}
1192
1193 \begin{funcdesc}{sign}{a}
1194 applies the sign function to \var{a}, that is $1$ where \var{a} is positive,
1195 $-1$ where \var{a} is negative and $0$ otherwise.
1196 \end{funcdesc}
1197
1198 \begin{funcdesc}{wherePositive}{a}
1199 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
1200 \end{funcdesc}
1201
1202 \begin{funcdesc}{whereNegative}{a}
1203 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
1204 \end{funcdesc}
1205
1206 \begin{funcdesc}{whereNonNegative}{a}
1207 returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.
1208 \end{funcdesc}
1209
1210 \begin{funcdesc}{whereNonPositive}{a}
1211 returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.
1212 \end{funcdesc}
1213
1214 \begin{funcdesc}{whereZero}{a\optional{, tol=None, \optional{, rtol=1.e-8}}}
1215 returns a function which is $1$ where \var{a} equals zero with tolerance \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum value of C{a} times C{rtol} is used.
1216 \end{funcdesc}
1217
1218 \begin{funcdesc}{whereNonZero}{a, \optional{, tol=None, \optional{, rtol=1.e-8}}}
1219 returns a function which is $1$ where \var{a} different from zero with tolerance \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum value of C{a} times C{rtol} is used.
1220 \end{funcdesc}
1221
1222 \subsection{Interpolating Data}
1223 \index{interpolateTable}
1224 In some cases, it may be useful to produce Data objects which fit some user defined function.
1225 Manually modifying each value in the Data object is not a good idea since it depends on
1226 knowing the location and order of each datapoint in the domain.
1227 Instead \escript can use an interpolation table to produce a Data object.
1228
1229 The following example is available as \file{int_save.py} in the examples directory.
1230 We will produce a \Data object which aproximates a sine curve.
1231
1232 \begin{python}
1233 from esys.escript import saveDataCSV, sup
1234 import numpy
1235 from esys.finley import Rectangle
1236
1237 n=4
1238 r=Rectangle(n,n)
1239 x=r.getX()
1240 x0=x[0]
1241 x1=x[1] #we'll use this later
1242 toobig=100
1243 \end{python}
1244
1245 First we produce an interpolation table.
1246 \begin{python}
1247 sine_table=[0, 0.70710678118654746, 1, 0.70710678118654746, 0,
1248 -0.70710678118654746, -1, -0.70710678118654746, 0]
1249 \end{python}
1250
1251 We wish to identify $0$ and $1$ with the ends of the curve.
1252 That is, with the first and eighth values in the table.
1253
1254 \begin{python}
1255 numslices=len(sine_table)-1
1256
1257 minval=0
1258 maxval=1
1259
1260 step=sup(maxval-minval)/numslices
1261 \end{python}
1262
1263 So the values $v$ from the input lie in the interval minval$\leq v < $maxval.
1264 \var{step} represents the gap (in the input range) between entries in the table.
1265 By default values of $v$ outside the table argument range (minval, maxval) will
1266 be pushed back into the range, ie. if $v <$ minval the value minval will be used to
1267 evaluate the table. Similarly, for values $v>$ maxval the value maxval is used.
1268
1269 Now we produce our new \Data object.
1270
1271 \begin{python}
1272 result=x0.interpolateTable(sine_table, minval, step, toobig)
1273 \end{python}
1274 Any values which interpolate to larger than \var{toobig} will raise an exception. You can
1275 switch on boundary checking by adding ''check_boundaries=True`` the argument list.
1276
1277
1278 Now for a 2D example.
1279 We will interpolate a surface such that the bottom edge is the sine curve described above.
1280 The amplitude of the curve decreases as we move towards the top edge.
1281
1282 Our interpolation table will have three rows.
1283 \begin{python}
1284 st=numpy.array(sine_table)
1285
1286 table=[st, 0.5*st, 0*st ]
1287 \end{python}
1288
1289 The use of numpy and multiplication here is just to save typing.
1290
1291 \begin{python}
1292 result2=x1.interpolateTable(table, 0, 0.55, x0, minval, step, toobig)
1293 \end{python}
1294
1295 In the 2D case, the parameters for the x1 direction (min=0, step=0.55) come first followed by the x0 data object and
1296 its parameters.
1297 By default, if a point is specified which is outside the boundary, then \var{interpolateTable} will operate
1298 as if the point was on the boundary.
1299 Passing \var{check_boundaries}=\var{True} will \var{interpolateTable} to reject any points outside the boundaries.
1300
1301 \subsection{Saving Data as CSV}
1302 \index{saveDataCSV}
1303 \index{CSV}
1304 For simple post-processing, \Data objects can be saved in comma separated value format.
1305
1306 If \var{mydata1} and \var{mydata2} are scalar data, the following command:
1307 \begin{python}
1308 saveDataCSV('output.csv',U=mydata1, V=mydata2)
1309 \end{python}
1310 will record the values of mydata in \texttt{output.csv} in the following format:
1311 \begin{verbatim}
1312 U, V
1313 1.0000000e+0, 2.0000000e-1
1314 5.0000000e-0, 1.0000000e+1
1315 ...
1316 \end{verbatim}
1317
1318 The names of the keyword parameters form the names of columns in the output.
1319 If the data objects are over different function spaces, then saveDataCSV will attempt to
1320 interpolate to a common function space.
1321 If this is not possible, then an exception will be raised.
1322
1323 Output can be restricted using a scalar mask.
1324 \begin{python}
1325 saveDataCSV('outfile.csv', U=mydata1, V=mydata2, mask=myscalar)
1326 \end{python}
1327 Will only output those rows which correspond to to positive values of \var{myscalar}.
1328 Some aspects of the output can be tuned using additional parameters.
1329 \begin{python}
1330 saveDataCSV('data.csv', append=True, sep=' ', csep='/', mask=mymask, e=mat1)
1331 \end{python}
1332
1333 \begin{itemize}
1334 \item \var{append} - specifies that the output should be written to the end of an existing file.
1335 \item \var{sep} - defines the separator between fields.
1336 \item \var{csep} - defines the separator between components in the header line. For example between the components of a matrix.
1337 \end{itemize}
1338
1339 The above command would produce output like this:
1340 \begin{verbatim}
1341 e/0/0 e/1/0 e/0/1 e/1/1
1342 1.0000000000e+00 2.0000000000e+00 3.0000000000e+00 4.0000000000e+00
1343 ...
1344 \end{verbatim}
1345
1346 Note that while the order in which rows are output can vary, all the elements in a given row
1347 always correspond to the same input.
1348
1349
1350 \subsection{The \Operator Class}
1351 The \Operator class provides an abstract access to operators build
1352 within the \LinearPDE class. \Operator objects are created
1353 when a PDE is handed over to a PDE solver library and handled
1354 by the \LinearPDE object defining the PDE. The user can gain access
1355 to the \Operator of a \LinearPDE object through the \var{getOperator}
1356 method.
1357
1358 \begin{classdesc}{Operator}{}
1359 creates an empty \Operator object.
1360 \end{classdesc}
1361
1362 \begin{methoddesc}[Operator]{isEmpty}{fileName}
1363 returns \True is the object is empty. Otherwise \True is returned.
1364 \end{methoddesc}
1365
1366 \begin{methoddesc}[Operator]{setValue}{value}
1367 resets all entries in the object representation to \var{value}
1368 \end{methoddesc}
1369
1370 \begin{methoddesc}[Operator]{solves}{rhs}
1371 solves the operator equation with right hand side \var{rhs}
1372 \end{methoddesc}
1373
1374 \begin{methoddesc}[Operator]{of}{u}
1375 applies the operator to the \Data object \var{u}
1376 \end{methoddesc}
1377
1378 \begin{methoddesc}[Operator]{saveMM}{fileName}
1379 saves the object to a matrix market format file of name
1380 \var{fileName}, see
1381 \url{http://maths.nist.gov/MatrixMarket}
1382 % \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}.
1383 \index{Matrix Market}
1384 \end{methoddesc}
1385
1386 \section{Physical Units}
1387 \escript provides support for physical units in the SI system \index{SI units} including unit conversion. So the
1388 user can define variables in the form
1389 \begin{python}
1390 from esys.escript.unitsSI import *
1391 l=20*m
1392 w=30*kg
1393 w2=40*lb
1394 T=100*Celsius
1395 \end{python}
1396 In the two latter cases an conversion from pounds\index{pounds} and degree Celsius\index{Celsius} is performed into the appropriate SI units kg and Kelvin is performed. In addition
1397 composed units can be used, for instance
1398 \begin{python}
1399 from esys.escript.unitsSI import *
1400 rho=40*lb/cm**3
1401 \end{python}
1402 to define the density in the units of pounds per cubic centimeter. The value $40$ will be converted
1403 into SI units, in this case kg per cubic meter.
1404 Moreover unit prefixes are supported:
1405 \begin{python}
1406 from esys.escript.unitsSI import *
1407 p=40*Mega*Pa
1408 \end{python}
1409 to the the pressure to 40 Mega Pascal. Units can also be converted back from the SI system into
1410 a desired unit, e.g
1411 \begin{python}
1412 from esys.escript.unitsSI import *
1413 print p/atm
1414 \end{python}
1415 can be used print the pressure in units of atmosphere\index{atmosphere}.
1416
1417 This is an incomplete list of supported physical units:
1418
1419 \begin{datadesc}{km}
1420 unit of kilo meter
1421 \end{datadesc}
1422
1423 \begin{datadesc}{m}
1424 unit of meter
1425 \end{datadesc}
1426
1427 \begin{datadesc}{cm}
1428 unit of centi meter
1429 \end{datadesc}
1430
1431 \begin{datadesc}{mm}
1432 unit of milli meter
1433 \end{datadesc}
1434
1435 \begin{datadesc}{sec}
1436 unit of second
1437 \end{datadesc}
1438
1439 \begin{datadesc}{minute}
1440 unit of minute
1441 \end{datadesc}
1442
1443 \begin{datadesc}{h}
1444 unit of hour
1445 \end{datadesc}
1446 \begin{datadesc}{day}
1447 unit of day
1448 \end{datadesc}
1449 \begin{datadesc}{yr}
1450 unit of year
1451 \end{datadesc}
1452
1453 \begin{datadesc}{gram}
1454 unit of gram
1455 \end{datadesc}
1456 \begin{datadesc}{kg}
1457 unit of kilo gram
1458 \end{datadesc}
1459 \begin{datadesc}{lb}
1460 unit of pound
1461 \end{datadesc}
1462 \begin{datadesc}{ton}
1463 metric ton
1464 \end{datadesc}
1465
1466 \begin{datadesc}{A}
1467 unit of Ampere
1468 \end{datadesc}
1469
1470 \begin{datadesc}{Hz}
1471 unit of Hertz
1472 \end{datadesc}
1473
1474 \begin{datadesc}{N}
1475 unit of Newton
1476 \end{datadesc}
1477 \begin{datadesc}{Pa}
1478 unit of Pascal
1479 \end{datadesc}
1480 \begin{datadesc}{atm}
1481 unit of atmosphere
1482 \end{datadesc}
1483 \begin{datadesc}{J}
1484 unit of Joule
1485 \end{datadesc}
1486
1487 \begin{datadesc}{W}
1488 unit of Watt
1489 \end{datadesc}
1490
1491 \begin{datadesc}{C}
1492 unit of Coulomb
1493 \end{datadesc}
1494 \begin{datadesc}{V}
1495 unit of Volt
1496 \end{datadesc}
1497 \begin{datadesc}{F}
1498 unit of Farad
1499 \end{datadesc}
1500
1501 \begin{datadesc}{Ohm}
1502 unit of Ohm
1503 \end{datadesc}
1504 \begin{datadesc}{K}
1505 unit of Kelvin
1506 \end{datadesc}
1507 \begin{datadesc}{Celsius}
1508 unit of Celsius
1509 \end{datadesc}
1510
1511 \begin{datadesc}{Fahrenheit}
1512 unit of Fahrenheit
1513 \end{datadesc}
1514
1515 Moreover unit prefixes are supported:
1516
1517 \begin{datadesc}{Yotta}
1518 prefix yotta = $10^{24}$.
1519
1520 \end{datadesc}
1521
1522 \begin{datadesc}{Zetta}
1523 prefix zetta= $10^{21}$.
1524 \end{datadesc}
1525
1526 \begin{datadesc}{Exa}
1527 prefix exa= $10^{18}$.
1528 \end{datadesc}
1529
1530 \begin{datadesc}{Peta}
1531 prefix peta= $10^{15}$.
1532 \end{datadesc}
1533
1534 \begin{datadesc}{Tera}
1535 prefix tera= $10^{12}$.
1536 \end{datadesc}
1537
1538 \begin{datadesc}{Giga}
1539 prefix giga= $10^9$.
1540 \end{datadesc}
1541
1542 \begin{datadesc}{Mega}
1543 prefix mega= $10^6$.
1544 \end{datadesc}
1545
1546 \begin{datadesc}{Kilo}
1547 prefix kilo= $10^3$.
1548 \end{datadesc}
1549
1550 \begin{datadesc}{Hecto}
1551 prefix hecto= $10^2$.
1552 \end{datadesc}
1553
1554 \begin{datadesc}{Deca}
1555 prefix deca= $10^1$.
1556 \end{datadesc}
1557
1558 \begin{datadesc}{Deci}
1559 prefix deci= $10^{-1}$.
1560 \end{datadesc}
1561
1562 \begin{datadesc}{Centi}
1563 prefix centi= $10^{-2}$.
1564 \end{datadesc}
1565
1566 \begin{datadesc}{Milli}
1567 prefix milli= $10^{-3}$.
1568 \end{datadesc}
1569
1570 \begin{datadesc}{Micro}
1571 prefix micro= $10^{-6}$.
1572 \end{datadesc}
1573
1574 \begin{datadesc}{Nano}
1575 prefix nano= $10^{-9}$.
1576 \end{datadesc}
1577
1578 \begin{datadesc}{Pico}
1579 prefix pico= $10^{-12}$.
1580 \end{datadesc}
1581
1582 \begin{datadesc}{Femto}
1583 prefix femto= $10^{-15}$.
1584 \end{datadesc}
1585
1586 \begin{datadesc}{Atto}
1587 prefix atto= $10^{-18}$.
1588 \end{datadesc}
1589
1590 \begin{datadesc}{Zepto}
1591 prefix zepto= $10^{-21}$.
1592 \end{datadesc}
1593
1594 \begin{datadesc}{Yocto}
1595 prefix yocto= $10^{-24}$.
1596 \end{datadesc}
1597
1598
1599 \section{Utilities}
1600
1601 The \class{FileWriter} provides a mechanism to write data to a file.
1602 In essence, this class wraps the standard \class{file} class to write data
1603 that are global in MPI to a file. In fact, data are written on the processor
1604 with \MPI rank 0 only. It is recommended to use \class{FileWriter}
1605 rather than \class{open} in order to write code that is running
1606 with and without \MPI. It is save to use \class{open} under MPI to read data which are global under \MPI.
1607
1608 \begin{classdesc}{FileWriter}{fn\optional{,append=\False, \optional{createLocalFiles=\False}})}
1609 Opens a file of name \var{fn} for writing. If \var{append} is set to \True
1610 written data are append at the end of the file.
1611 If running under \MPI only the first processor with rank==0
1612 will open the file and write to it.
1613 If \var{createLocalFiles} is set each individual processor will create a file
1614 where for any processor with rank>0 the file name is extended by its rank. This option is normally used for debug purposes only.
1615 \end{classdesc}
1616
1617 The following methods are available:
1618 \begin{methoddesc}[FileWriter]{close}{}
1619 closes the file.
1620 \end{methoddesc}
1621 \begin{methoddesc}[FileWriter]{flush}{}
1622 flushes the internal buffer to disk.
1623 \end{methoddesc}
1624 \begin{methoddesc}[FileWriter]{write}{txt}
1625 Write string \var{txt} to file.
1626 Note that newline is not added.
1627 \end{methoddesc}
1628 \begin{methoddesc}[FileWriter]{writelines}{txts}
1629 Write the list \var{txts} of strings to the file..
1630 Note that newlines are not added.
1631 This method is equivalent to call write() for each string.
1632 \end{methoddesc}
1633 \begin{memberdesc}[FileWriter]{closed}
1634 \True if file is closed.
1635 \end{memberdesc}
1636 \begin{memberdesc}[FileWriter]{mode}
1637 access mode.
1638 \end{memberdesc}
1639 \begin{memberdesc}[FileWriter]{name}
1640 file name.
1641 \end{memberdesc}
1642 \begin{memberdesc}[FileWriter]{newlines}
1643 line separator
1644 \end{memberdesc}
1645
1646
1647 \begin{funcdesc}{setEscriptParamInt}{name,value}
1648 assigns the integer value \var{value} to the parameter \var{name}.
1649 If \var{name}="TOO_MANY_LINES" conversion of any \Data object to a string switches to a
1650 condensed format if more than \var{value} lines would be created.
1651 \end{funcdesc}
1652
1653 \begin{funcdesc}{getEscriptParamInt}{name}
1654 returns the current value of integer parameter \var{name}.
1655 \end{funcdesc}
1656
1657 \begin{funcdesc}{listEscriptParams}{a}
1658 returns a list of valid parameters and their description.
1659 \end{funcdesc}
1660
1661 \begin{funcdesc}{getMPISizeWorld}{}
1662 returns the number of \MPI processors in use in the \env{MPI_COMM_WORLD} processor group.
1663 If \MPI is not used 1 is returned.
1664 \end{funcdesc}
1665 \begin{funcdesc}{getMPIRankWorld}{}
1666 returns the rank of the process within the \env{MPI_COMM_WORLD} processor group.
1667 If \MPI is not used 0 is returned.
1668 \end{funcdesc}
1669 \begin{funcdesc}{MPIBarrierWorld}{}
1670 performs a barrier synchronization across all processors within \env{MPI_COMM_WORLD}
1671 processor group.
1672 \end{funcdesc}
1673 \begin{funcdesc}{getMPIWorldMax}{a}
1674 returns the maximum value of the integer \var{a} across all
1675 processors within \env{MPI_COMM_WORLD}.
1676 \end{funcdesc}
1677

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