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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
290 Expanded data is created when you create a \Data object with \code{expanded=True}.
291 Tagged data sets are created when you use the \member{insertTaggedValue}
292 method as shown above.
293
294 Values are accessed through a sample reference number.
295 Operations on expanded \Data objects have to be performed for each sample
296 point individually.
297 When tagged values are used, the values are held in a dictionary.
298 Operations on tagged data require processing the set of tagged values only,
299 rather than processing the value for each individual sample point.
300 \escript allows any mixture of constant, tagged and expanded data in a single expression.
301
302 \subsection{Saving and Restoring Simulation Data}
303 \Data objects can be written to disk files with the \member{dump} method and
304 read back using the \member{load} method, both of which use the
305 \netCDF\cite{NETCDF} file format.
306 Use these to save data for checkpoint/restart or simply to save and reuse data
307 that was expensive to compute.
308
309 For instance, to save the coordinates of the data points of a
310 \ContinuousFunction to the file \file{x.nc} use
311 \begin{python}
312 x=ContinuousFunction(mydomain).getX()
313 x.dump("x.nc")
314 mydomain.dump("dom.nc")
315 \end{python}
316 To recover the object \var{x}, and \var{mydomain} was a \finley mesh use
317 \begin{python}
318 from esys.finley import LoadMesh
319 mydomain=LoadMesh("dom.nc")
320 x=load("x.nc", mydomain)
321 \end{python}
322 Obviously, it is possible to execute the same steps that were originally used
323 to generate \var{mydomain} to recreate it. However, in most cases using
324 \member{dump} and \member{load} is faster, particularly if optimization has
325 been applied.
326 If \escript is running on more than one \MPI process \member{dump} will create
327 an individual file for each process containing the local data.
328 In order to avoid conflicts the file names are extended by the \MPI processor
329 rank, that is instead of one file \file{dom.nc} you would get
330 \file{dom.nc.0000}, \file{dom.nc.0001}, etc. You still call
331 \code{LoadMesh('dom.nc')} to load the domain but you have to make sure that
332 the appropriate file is accessible from the corresponding rank, and loading
333 will only succeed if you run with as many processes as were used when calling
334 \member{dump}.
335
336 The function space of the \Data is stored in \file{x.nc}.
337 If the \Data object is expanded, the number of data points in the file and of
338 the \Domain for the particular \FunctionSpace must match.
339 Moreover, the ordering of the values is checked using the reference
340 identifiers provided by \FunctionSpace on the \Domain.
341 In some cases, data points will be reordered so be aware and confirm that you
342 get what you wanted.
343
344 A newer, more flexible way of saving and restoring \escript simulation data
345 is through a \class{DataManager} class object.
346 It has the advantage of allowing to save and load not only a \Domain and
347 \Data objects but also other values\footnote{The \PYTHON \emph{pickle} module
348 is used for other types.} you compute in your simulation script.
349 Further, \class{DataManager} objects can simultaneously create files for
350 visualization so no extra calls to, e.g. \code{saveVTK} are needed.
351
352 The following example shows how the \class{DataManager} class can be used.
353 For an explanation of all member functions and options see the relevant
354 reference section.
355 \begin{python}
356 from esys.escript import DataManager, Scalar, Function
357 from esys.finley import Rectangle
358
359 dm = DataManager(formats=[DataManager.RESTART, DataManager.VTK])
360 if dm.hasData():
361 mydomain=dm.getDomain()
362 val=dm.getValue("val")
363 t=dm.getValue("t")
364 t_max=dm.getValue("t_max")
365 else:
366 mydomain=Rectangle()
367 val=Function(mydomain).getX()
368 t=0.
369 t_max=2.5
370
371 while t<t_max:
372 t+=.01
373 val=val+t/2
374 dm.addData(val=val, t=t, t_max=t_max)
375 dm.export()
376 \end{python}
377 In the constructor we specify that we want \code{RESTART} (i.e. dump) files
378 and \code{VTK} files to be saved.
379 By default, the constructor will look for previously saved \code{RESTART}
380 files under the current directory and load them.
381 We can then enquire if such files were found by calling the \member{hasData}
382 method. If it returns \True we retrieve the domain and values into local
383 variables. Otherwise the same variables are initialized with appropriate
384 values to start a new simulation.
385 Note, that \var{t} and \var{t_max} are regular floating point values and not
386 \Data objects but are treated the same way by the \class{DataManager}.
387
388 After this initialization step the script enters the main simulation loop
389 where calculations are performed.
390 When these are finalized for a time step we call the \member{addData} method
391 to let the manager know which variables to store on disk.
392 Note, that this does not actually save the data yet and it is allowed to call
393 \member{addData} more than once to add information incrementally, e.g. from
394 separate functions that have access to the \class{DataManager} instance.
395 Once all variables have been added the \member{export} method has to be called
396 to flush all data to disk and clear the manager.
397 In this example, this call dumps \var{mydomain} and \var{val} to files
398 in a restart directory and also stores \var{t} and \var{t_max} on disk.
399 Additionally, it generates a \VTK file for visualization of the data.
400
401 \section{\escript Classes}
402
403 \subsection{The \Domain class}
404 \begin{classdesc}{Domain}{}
405 A \Domain object is used to describe a geometric region together with
406 a way of representing functions over this region.
407 The \Domain class provides an abstract interface to the domain of \FunctionSpace and \Data objects.
408 \Domain needs to be subclassed in order to provide a complete implementation.
409 \end{classdesc}
410 The following methods are available:
411 \begin{methoddesc}[Domain]{getDim}{}
412 returns the spatial dimension of the \Domain.
413 \end{methoddesc}
414 \begin{methoddesc}[Domain]{dump}{filename}
415 dumps the \Domain into the file \var{filename}.
416 \end{methoddesc}
417 \begin{methoddesc}[Domain]{getX}{}
418 returns the locations in the \Domain. The \FunctionSpace of the returned
419 \Data object is chosen by the \Domain implementation. Typically it will be
420 in the \Function.
421 \end{methoddesc}
422
423 \begin{methoddesc}[Domain]{setX}{newX}
424 assigns a new location to the \Domain. \var{newX} has to have \Shape $(d,)$
425 where $d$ is the spatial dimension of the domain. Typically \var{newX} must be
426 in the \ContinuousFunction but the space actually to be used depends on the \Domain implementation.
427 \end{methoddesc}
428
429 \begin{methoddesc}[Domain]{getNormal}{}
430 returns the surface normals on the boundary of the \Domain as \Data object.
431 \end{methoddesc}
432
433 \begin{methoddesc}[Domain]{getSize}{}
434 returns the local sample size, e.g. the element diameter, as \Data object.
435 \end{methoddesc}
436
437 \begin{methoddesc}[Domain]{setTagMap}{tag_name, tag}
438 defines a mapping of the tag name \var{tag_name} to the \var{tag}.
439 \end{methoddesc}
440 \begin{methoddesc}[Domain]{getTag}{tag_name}
441 returns the tag associated with the tag name \var{tag_name}.
442 \end{methoddesc}
443 \begin{methoddesc}[Domain]{isValidTagName}{tag_name}
444 return \True if \var{tag_name} is a valid tag name.
445 \end{methoddesc}
446
447 \begin{methoddesc}[Domain]{__eq__}{arg}
448 (python == operator) returns \True if the \Domain \var{arg} describes the same domain. Otherwise
449 \False is returned.
450 \end{methoddesc}
451
452 \begin{methoddesc}[Domain]{__ne__}{arg}
453 (python != operator) returns \True if the \Domain \var{arg} does not describe the same domain.
454 Otherwise \False is returned.
455 \end{methoddesc}
456
457 \begin{methoddesc}[Domain]{__str__}{arg}
458 (python str() function) returns string representation of the \Domain.
459 \end{methoddesc}
460
461 \begin{methoddesc}[Domain]{onMasterProcessor)}{}
462 returns \True if the processor is the master processor within
463 the \MPI processor group used by the \Domain. This is the processor with rank 0.
464 If \MPI support is not enabled the return value is always \True.
465 \end{methoddesc}
466
467 \begin{methoddesc}[Domain]{getMPISize}{}
468 returns the number of \MPI processors used for this \Domain. If \MPI support is not enabled
469 1 is returned.
470 \end{methoddesc}
471
472 \begin{methoddesc}[Domain]{getMPIRank}{}
473 returns the rank of the processor executing the statement
474 within the \MPI processor group used by the \Domain.
475 If \MPI support is not enabled 0 is returned.
476 \end{methoddesc}
477
478 \begin{methoddesc}[Domain]{MPIBarrier}{}
479 executes barrier synchronization within
480 the \MPI processor group used by the \Domain.
481 If \MPI support is not enabled, this command does nothing.
482 \end{methoddesc}
483
484 \subsection{The \FunctionSpace class}
485 \begin{classdesc}{FunctionSpace}{}
486 \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects
487 are instantiated by generator functions. A \Data object in a particular \FunctionSpace is
488 represented by its values at \DataSamplePoints which are defined by the type and the \Domain of the
489 \FunctionSpace.
490 \end{classdesc}
491 The following methods are available:
492 \begin{methoddesc}[FunctionSpace]{getDim}{}
493 returns the spatial dimension of the \Domain of the \FunctionSpace.
494 \end{methoddesc}
495
496
497
498 \begin{methoddesc}[FunctionSpace]{getX}{}
499 returns the location of the \DataSamplePoints.
500 \end{methoddesc}
501
502 \begin{methoddesc}[FunctionSpace]{getNormal}{}
503 If the domain of functions in the \FunctionSpace
504 is a hyper-manifold (e.g. the boundary of a domain)
505 the method returns the outer normal at each of the
506 \DataSamplePoints. Otherwise an exception is raised.
507 \end{methoddesc}
508
509 \begin{methoddesc}[FunctionSpace]{getSize}{}
510 returns a \Data objects measuring the spacing of the \DataSamplePoints.
511 The size may be zero.
512 \end{methoddesc}
513
514 \begin{methoddesc}[FunctionSpace]{getDomain}{}
515 returns the \Domain of the \FunctionSpace.
516 \end{methoddesc}
517
518 \begin{methoddesc}[FunctionSpace]{setTags}{new_tag, mask}
519 assigns a new tag \var{new_tag} to all data sample
520 where \var{mask} is positive for a least one data point.
521 \var{mask} must be defined on the this \FunctionSpace.
522 Use the \var{setTagMap} to assign a tag name to \var{new_tag}.
523 \end{methoddesc}
524
525 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
526 (python == operator) returns \True if the \Domain \var{arg} describes the same domain. Otherwise
527 \False is returned.
528 \end{methoddesc}
529
530 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
531 (python != operator) returns \True if the \Domain \var{arg} do not describe the same domain.
532 Otherwise \False is returned.
533 \end{methoddesc}
534
535 \begin{methoddesc}[Domain]{__str__}{g}
536 (python str() function) returns string representation of the \Domain.
537 \end{methoddesc}
538
539 The following function provide generators for \FunctionSpace objects:
540 \begin{funcdesc}{Function}{domain}
541 returns the \Function on the \Domain \var{domain}. \Data objects in this type of \Function
542 are defined over the whole geometric region defined by \var{domain}.
543 \end{funcdesc}
544
545 \begin{funcdesc}{ContinuousFunction}{domain}
546 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
547 are defined over the whole geometric region defined by \var{domain} and assumed to represent
548 a continuous function.
549 \end{funcdesc}
550
551 \begin{funcdesc}{FunctionOnBoundary}{domain}
552 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
553 are defined on the boundary of the geometric region defined by \var{domain}.
554 \end{funcdesc}
555
556 \begin{funcdesc}{FunctionOnContactZero}{domain}
557 returns the \FunctionOnContactZero the \Domain domain. \Data objects in this type of \Function
558 are defined on side 0 of a discontinuity within the geometric region defined by \var{domain}.
559 The discontinuity is defined when \var{domain} is instantiated.
560 \end{funcdesc}
561
562 \begin{funcdesc}{FunctionOnContactOne}{domain}
563 returns the \FunctionOnContactOne on the \Domain domain.
564 \Data objects in this type of \Function
565 are defined on side 1 of a discontinuity within the geometric region defined by \var{domain}.
566 The discontinuity is defined when \var{domain} is instantiated.
567 \end{funcdesc}
568
569 \begin{funcdesc}{Solution}{domain}
570 returns the \SolutionFS on the \Domain domain. \Data objects in this type of \Function
571 are defined on geometric region defined by \var{domain} and are solutions of
572 partial differential equations \index{partial differential equation}.
573 \end{funcdesc}
574
575 \begin{funcdesc}{ReducedSolution}{domain}
576 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in this type of \Function
577 are defined on geometric region defined by \var{domain} and are solutions of
578 partial differential equations \index{partial differential equation} with a reduced smoothness
579 for the solution approximation.
580 \end{funcdesc}
581
582 \subsection{The \Data Class}
583 \label{SEC ESCRIPT DATA}
584
585 The following table shows arithmetic operations that can be performed point-wise on
586 \Data objects.
587 \begin{table}
588 \centering
589 \begin{tabular}{l|l}
590 \bfseries expression & Description\\
591 \hline
592 +\var{arg0} & identical to \var{arg} \index{+}\\
593 -\var{arg0} & negation\index{-}\\
594 \var{arg0}+\var{arg1} & adds \var{arg0} and \var{arg1} \index{+}\\
595 \var{arg0}*\var{arg1} & multiplies \var{arg0} and \var{arg1} \index{*}\\
596 \var{arg0}-\var{arg1} & difference \var{arg1} from\var{arg1} \index{-}\\
597 \var{arg0}/\var{arg1} & divide \var{arg0} by \var{arg1} \index{/}\\
598 \var{arg0}**\var{arg1} & raises \var{arg0} to the power of \var{arg1} \index{**}\\
599 \end{tabular}
600 \end{table}
601 At least one of the arguments \var{arg0} or \var{arg1} must be a
602 \Data object.
603 Either of the arguments may be a \Data object, a python number or a \numpy object.
604
605 If \var{arg0} or \var{arg1} are
606 not defined on the same \FunctionSpace, then an attempt is made to convert \var{arg0}
607 to the \FunctionSpace of \var{arg1} or to convert \var{arg1} to
608 the \FunctionSpace of \var{arg0}. Both arguments must have the same
609 \Shape or one of the arguments may be of rank 0 (a constant).
610
611 The returned \Data object has the same \Shape and is defined on
612 the \DataSamplePoints as \var{arg0} or \var{arg1}.
613
614 The following table shows the update operations that can be applied to
615 \Data objects:
616 \begin{table}
617 \centering
618 \begin{tabular}{l|l}
619 \bfseries Expression & Description\\
620 \hline
621 \var{arg0}+=\var{arg2} & adds \var{arg0} to \var{arg2}\index{+}\\
622 \var{arg0}*=\var{arg2} & multiplies \var{arg0} with \var{arg2}\index{*}\\
623 \var{arg0}-=\var{arg2} & subtracts \var{arg2} from\var{arg2}\index{-}\\
624 \var{arg0}/=\var{arg2} & divides \var{arg0} by \var{arg2}\index{/}\\
625 \var{arg0}**=\var{arg2} & raises \var{arg0} by \var{arg2}\index{**}\\
626 \end{tabular}
627 \end{table}
628 \var{arg0} must be a \Data object. \var{arg1} must be a
629 \Data object or an object that can be converted into a
630 \Data object. \var{arg1} must have the same \Shape as
631 \var{arg0} or have rank 0. In the latter case it is
632 assumed that the values of \var{arg1} are constant for all
633 components. \var{arg1} must be defined in the same \FunctionSpace as
634 \var{arg0} or it must be possible to interpolate \var{arg1} onto the
635 \FunctionSpace of \var{arg0}.
636
637 The \Data class supports taking slices from a \Data object as well as assigning new values to a slice of an existing
638 \Data object. \index{slicing}
639 The following expressions for taking and setting slices are valid:
640 \begin{table}
641 \centering
642 \begin{tabular}{l|ll}
643 \bfseries rank of \var{arg} & slicing expression & \Shape of returned and assigned object\\
644 \hline
645 0 & no slicing & -\\
646 1 & \var{arg[l0:u0]} & (\var{u0}-\var{l0},)\\
647 2 & \var{arg[l0:u0,l1:u1]} & (\var{u0}-\var{l0},\var{u1}-\var{l1})\\
648 3 & \var{arg[l0:u0,l1:u1,l2:u2]} & (\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})\\
649 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})\\
650 \end{tabular}
651 \end{table}
652 where \var{s} is the \Shape of \var{arg} and
653 \[0 \le \var{l0} \le \var{u0} \le \var{s[0]},\]
654 \[0 \le \var{l1} \le \var{u1} \le \var{s[1]},\]
655 \[0 \le \var{l2} \le \var{u2} \le \var{s[2]},\]
656 \[0 \le \var{l3} \le \var{u3} \le \var{s[3]}.\]
657 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case
658 $0$ is assumed.
659 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.
660 The lower and upper index may be identical, in which case the column and the lower or upper
661 index may be dropped. In the returned or in the object assigned to a slice, the corresponding component is dropped,
662 i.e. the rank is reduced by one in comparison to \var{arg}.
663 The following examples show slicing in action:
664 \begin{python}
665 t=Data(1.,(4,4,6,6),Function(mydomain))
666 t[1,1,1,0]=9.
667 s=t[:2,:,2:6,5] # s has rank 3
668 s[:,:,1]=1.
669 t[:2,:2,5,5]=s[2:4,1,:2]
670 \end{python}
671
672 \subsection{Generation of \Data objects}
673 \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expand=\False}
674 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
675 The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True
676 the \Data object is represented in expanded from.
677 \end{classdesc}
678
679 \begin{classdesc}{Data}{value,what=FunctionSpace(),expand=\False}
680 creates a \Data object in the \FunctionSpace \var{what}.
681 The value for each \DataSamplePoints is set to \var{value}, which could be a \numpy, \Data object \var{value} or a dictionary of
682 \numpy or floating point numbers. In the latter case the keys must be integers and are used
683 as tags.
684 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True
685 the \Data object is represented in expanded form.
686 \end{classdesc}
687
688 \begin{classdesc}{Data}{}
689 creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present
690 where a \Data object is required.
691 \end{classdesc}
692
693 \begin{funcdesc}{Scalar}{value=0.,what=FunctionSpace(),expand=\False}
694 returns a \Data object of rank 0 (a constant) in the \FunctionSpace \var{what}.
695 Values are initialized with \var{value}, a double precision quantity. If \var{expanded} is \True
696 the \Data object is represented in expanded from.
697 \end{funcdesc}
698
699 \begin{funcdesc}{Vector}{value=0.,what=FunctionSpace(),expand=\False}
700 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what},
701 where \var{d} is the spatial dimension of the \Domain of \var{what}.
702 Values are initialed 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}{Tensor}{value=0.,what=FunctionSpace(),expand=\False}
707 returns a \Data object of \Shape \var{(d,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}{Tensor3}{value=0.,what=FunctionSpace(),expand=\False}
714 returns a \Data object of \Shape \var{(d,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 re\var{arg}presented in expanded from.
718 \end{funcdesc}
719
720 \begin{funcdesc}{Tensor4}{value=0.,what=FunctionSpace(),expand=\False}
721 returns a \Data object of \Shape \var{(d,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 initialized with \var{value}, a double precision quantity. If \var{expanded} is \True
724 the \Data object is represented in expanded from.
725 \end{funcdesc}
726
727 \begin{funcdesc}{load}{filename,domain}
728 recovers a \Data object on \Domain \var{domain} from the file \var{filename}, which was created by \function{dump}.
729 \end{funcdesc}
730
731 \subsection{\Data methods}
732 These are the most frequently-used methods of the
733 \Data class. A complete list of methods can be found on \ReferenceGuide.
734 \begin{methoddesc}[Data]{getFunctionSpace}{}
735 returns the \FunctionSpace of the object.
736 \end{methoddesc}
737
738 \begin{methoddesc}[Data]{getDomain}{}
739 returns the \Domain of the object.
740 \end{methoddesc}
741
742 \begin{methoddesc}[Data]{getShape}{}
743 returns the \Shape of the object as a \class{tuple} of
744 integers.
745 \end{methoddesc}
746
747 \begin{methoddesc}[Data]{getRank}{}
748 returns the rank of the data on each data point. \index{rank}
749 \end{methoddesc}
750
751 \begin{methoddesc}[Data]{isEmpty}{}
752 returns \True id the \Data object is the \EmptyData object.
753 Otherwise \False is returned.
754 Note that this is not the same as asking if the object contains no \DataSamplePoints.
755 \end{methoddesc}
756
757 \begin{methoddesc}[Data]{setTaggedValue}{tag_name,value}
758 assigns the \var{value} to all \DataSamplePoints which have the tag
759 assigned to \var{tag_name}. \var{value} must be an object of class
760 \class{numpy.ndarray} or must be convertible into a
761 \class{numpy.ndarray} object. \var{value} (or the corresponding
762 \class{numpy.ndarray} object) must be of rank $0$ or must have the
763 same rank like the object.
764 If a value has already be defined for tag \var{tag_name} within the object
765 it is overwritten by the new \var{value}. If the object is expanded,
766 the value assigned to \DataSamplePoints with tag \var{tag_name} is replaced by
767 \var{value}. If no tag is assigned tag name \var{tag_name}, no value is set.
768 \end{methoddesc}
769
770 \begin{methoddesc}[Data]{dump}{filename}
771 dumps the \Data object to the file \var{filename}. The file stores the
772 function space but not the \Domain. It is in the responsibility of the user to
773 save the \Domain.
774 \end{methoddesc}
775
776 \begin{methoddesc}[Data]{__str__}{}
777 returns a string representation of the object.
778 \end{methoddesc}
779
780 \subsection{Functions of \Data objects}
781 This section lists the most important functions for \Data class objects \var{a}.
782 A complete list and a more detailed description of the functionality can be found on \ReferenceGuide.
783 \begin{funcdesc}{saveVTK}{filename,**kwdata}
784 writes \Data defined by keywords in the file with \var{filename} using the
785 vtk file format \VTK file format. The key word is used as an identifier. The statement
786 \begin{python}
787 saveVTK("out.xml",temperature=T,velocity=v)
788 \end{python}
789 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
790 file \file{out.xml}. Restrictions on the allowed combinations of \FunctionSpace apply.
791 \end{funcdesc}
792 \begin{funcdesc}{saveDX}{filename,**kwdata}
793 writes \Data defined by keywords in the file with \var{filename} using the
794 vtk file format \OpenDX file format. The key word is used as an identifier. The statement
795 \begin{python}
796 saveDX("out.dx",temperature=T,velocity=v)
797 \end{python}
798 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
799 file \file{out.dx}. Restrictions on the allowed combinations of \FunctionSpace apply.
800 \end{funcdesc}
801 \begin{funcdesc}{kronecker}{d}
802 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
803 \begin{equation}
804 \code{kronecker(d)}\left[ i,j\right] = \left\{
805 \begin{array}{cc}
806 1 & \mbox{ if } i=j \\
807 0 & \mbox{ otherwise }
808 \end{array}
809 \right.
810 \end{equation}
811 If \var{d} is an integer a $(d,d)$ \numpy array is returned.
812 \end{funcdesc}
813 \begin{funcdesc}{identityTensor}{d}
814 is a synonym for \code{kronecker} (see above).
815 % returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
816 % \begin{equation}
817 % \code{identityTensor(d)}\left[ i,j\right] = \left\{
818 % \begin{array}{cc}
819 % 1 & \mbox{ if } i=j \\
820 % 0 & \mbox{ otherwise }
821 % \end{array}
822 % \right.
823 % \end{equation}
824 % If \var{d} is an integer a $(d,d)$ \numpy array is returned.
825 \end{funcdesc}
826 \begin{funcdesc}{identityTensor4}{d}
827 returns a \RankFour \Data object in \FunctionSpace \var{d} such that
828 \begin{equation}
829 \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
830 \begin{array}{cc}
831 1 & \mbox{ if } i=k \mbox{ and } j=l\\
832 0 & \mbox{ otherwise }
833 \end{array}
834 \right.
835 \end{equation}
836 If \var{d} is an integer a $(d,d,d,d)$ \numpy array is returned.
837 \end{funcdesc}
838 \begin{funcdesc}{unitVector}{i,d}
839 returns a \RankOne \Data object in \FunctionSpace \var{d} such that
840 \begin{equation}
841 \code{identityTensor(d)}\left[ j \right] = \left\{
842 \begin{array}{cc}
843 1 & \mbox{ if } j=i\\
844 0 & \mbox{ otherwise }
845 \end{array}
846 \right.
847 \end{equation}
848 If \var{d} is an integer a $(d,)$ \numpy array is returned.
849
850 \end{funcdesc}
851
852 \begin{funcdesc}{Lsup}{a}
853 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values
854 over all components and all \DataSamplePoints of \var{a}.
855 \end{funcdesc}
856
857 \begin{funcdesc}{sup}{a}
858 returns the maximum value over all components and all \DataSamplePoints of \var{a}.
859 \end{funcdesc}
860
861 \begin{funcdesc}{inf}{a}
862 returns the minimum value over all components and all \DataSamplePoints of \var{a}
863 \end{funcdesc}
864
865
866
867 \begin{funcdesc}{minval}{a}
868 returns at each \DataSamplePoints the minimum value over all components.
869 \end{funcdesc}
870
871 \begin{funcdesc}{maxval}{a}
872 returns at each \DataSamplePoints the maximum value over all components.
873 \end{funcdesc}
874
875 \begin{funcdesc}{length}{a}
876 returns at Euclidean norm at each \DataSamplePoints. For a \RankFour \var{a} this is
877 \begin{equation}
878 \code{length(a)}=\sqrt{\sum_{ijkl} \var{a} \left[i,j,k,l\right]^2}
879 \end{equation}
880 \end{funcdesc}
881 \begin{funcdesc}{trace}{a\optional{,axis_offset=0}}
882 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
883 case of a \RankTwo function and this is
884 \begin{equation}
885 \code{trace(a)}=\sum_{i} \var{a} \left[i,i\right]
886 \end{equation}
887 and for a \RankFour function and \code{axis_offset=1} this is
888 \begin{equation}
889 \code{trace(a,1)}\left[i,j\right]=\sum_{k} \var{a} \left[i,k,k,j\right]
890 \end{equation}
891 \end{funcdesc}
892
893 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
894 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
895 present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
896 the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
897 case of a \RankTwo function and this is
898 \begin{equation}
899 \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
900 \end{equation}
901 and for a \RankFour function and \code{axis_offset=1} this is
902 \begin{equation}
903 \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
904 \end{equation}
905 \end{funcdesc}
906
907 \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
908 returns \var{a} but with swapped components \var{axis0} and \var{axis1}. The argument \var{a} must be
909 at least of \RankTwo. For instance in the
910 for a \RankFour argument, \code{axis0=1} and \code{axis1=2} this is
911 \begin{equation}
912 \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
913 \end{equation}
914 \end{funcdesc}
915
916 \begin{funcdesc}{symmetric}{a}
917 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
918 \end{funcdesc}
919 \begin{funcdesc}{nonsymmetric}{a}
920 returns the non--symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
921 \end{funcdesc}
922 \begin{funcdesc}{inverse}{a}
923 return the inverse of \var{a}. This is
924 \begin{equation}
925 \code{matrix_mult(inverse(a),a)=kronecker(d)}
926 \end{equation}
927 if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape
928 \code{(2,2)} and \code{(3,3)}.
929 \end{funcdesc}
930 \begin{funcdesc}{eigenvalues}{a}
931 return the eigenvalues of \var{a}. This is
932 \begin{equation}
933 \code{matrix_mult(a,V)=e[i]*V}
934 \end{equation}
935 where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.
936 The eigenvalues are ordered in increasing size.
937 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
938 The current implementation is restricted to arguments of shape
939 \code{(2,2)} and \code{(3,3)}.
940 \end{funcdesc}
941 \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
942 return the eigenvalues and eigenvectors of \var{a}. This is
943 \begin{equation}
944 \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
945 \end{equation}
946 where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.
947 \begin{equation}
948 \code{matrix_mult(transpose(V),V)=kronecker(d)}
949 \end{equation}
950 if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.
951 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
952 The current implementation is restricted to arguments of shape
953 \code{(2,2)} and \code{(3,3)}.
954 \end{funcdesc}
955 \begin{funcdesc}{maximum}{*a}
956 returns the maximum value over all arguments at all \DataSamplePoints and for each component.
957 For instance
958 \begin{equation}
959 \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
960 \end{equation}
961 at all \DataSamplePoints.
962 \end{funcdesc}
963 \begin{funcdesc}{minimum}{*a}
964 returns the minimum value over all arguments at all \DataSamplePoints and for each component.
965 For instance
966 \begin{equation}
967 \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
968 \end{equation}
969 at all \DataSamplePoints.
970 \end{funcdesc}
971
972 \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
973 cuts back \var{a} into the range between \var{minval} and \var{maxval}. A value in the returned object equals
974 \var{minval} if the corresponding value of \var{a} is less than \var{minval}, equals \var{maxval} if the
975 corresponding value of \var{a} is greater than \var{maxval}
976 or corresponding value of \var{a} otherwise.
977 \end{funcdesc}
978 \begin{funcdesc}{inner}{a0,a1}
979 returns the inner product of \var{a0} and \var{a1}. For instance in the
980 case of \RankTwo arguments and this is
981 \begin{equation}
982 \code{inner(a)}=\sum_{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
983 \end{equation}
984 and for a \RankFour arguments this is
985 \begin{equation}
986 \code{inner(a)}=\sum_{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
987 \end{equation}
988 \end{funcdesc}
989
990 \begin{funcdesc}{matrix_mult}{a0,a1}
991 returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is
992 \begin{equation}
993 \code{matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
994 \end{equation}
995 and if \var{a1} is \RankTwo this is
996 \begin{equation}
997 \code{matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
998 \end{equation}
999 \end{funcdesc}
1000
1001 \begin{funcdesc}{transposed_matrix_mult}{a0,a1}
1002 returns the matrix product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
1003 \code{matrix_mult(transpose(a0),a1)}.
1004 If \var{a1} is \RankOne this is
1005 \begin{equation}
1006 \code{transposed_matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k\right]
1007 \end{equation}
1008 and if \var{a1} is \RankTwo this is
1009 \begin{equation}
1010 \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k,j\right]
1011 \end{equation}
1012 \end{funcdesc}
1013
1014 \begin{funcdesc}{matrix_transposed_mult}{a0,a1}
1015 returns the matrix product of \var{a0} and the transposed of \var{a1}.
1016 The function is equivalent to
1017 \code{matrix_mult(a0,transpose(a1))}.
1018 If \var{a1} is \RankTwo this is
1019 \begin{equation}
1020 \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[j,k\right]
1021 \end{equation}
1022 \end{funcdesc}
1023
1024 \begin{funcdesc}{outer}{a0,a1}
1025 returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then
1026 \begin{equation}
1027 \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
1028 \end{equation}
1029 and if \var{a0} is \RankOne and \var{a1} is \RankThree
1030 \begin{equation}
1031 \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
1032 \end{equation}
1033 \end{funcdesc}
1034
1035 \begin{funcdesc}{tensor_mult}{a0,a1}
1036 returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is
1037 \begin{equation}
1038 \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]
1039 \end{equation}
1040 and if \var{a1} is \RankFour this is
1041 \begin{equation}
1042 \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]
1043 \end{equation}
1044 \end{funcdesc}
1045
1046 \begin{funcdesc}{transposed_tensor_mult}{a0,a1}
1047 returns the tensor product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
1048 \code{tensor_mult(transpose(a0),a1)}.
1049 If \var{a1} is \RankTwo this is
1050 \begin{equation}
1051 \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]
1052 \end{equation}
1053 and if \var{a1} is \RankFour this is
1054 \begin{equation}
1055 \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]
1056 \end{equation}
1057 \end{funcdesc}
1058
1059 \begin{funcdesc}{tensor_transposed_mult}{a0,a1}
1060 returns the tensor product of \var{a0} and the transposed of \var{a1}.
1061 The function is equivalent to
1062 \code{tensor_mult(a0,transpose(a1))}.
1063 If \var{a1} is \RankTwo this is
1064 \begin{equation}
1065 \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]
1066 \end{equation}
1067 and if \var{a1} is \RankFour this is
1068 \begin{equation}
1069 \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]
1070 \end{equation}
1071 \end{funcdesc}
1072
1073 \begin{funcdesc}{grad}{a\optional{, where=None}}
1074 returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a
1075 default \FunctionSpace is used. In case that \var{a} has \RankTwo one has
1076 \begin{equation}
1077 \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x_{k}}
1078 \end{equation}
1079 \end{funcdesc}
1080 \begin{funcdesc}{integrate}{a\optional{ ,where=None}}
1081 returns the integral of \var{a} where the domain of integration is defined by the \FunctionSpace of \var{a}. If \var{where} is
1082 present the argument is interpolated into \FunctionSpace \var{where} before integration. For instance in the case of
1083 a \RankTwo argument in \ContinuousFunction it is
1084 \begin{equation}
1085 \code{integrate(a)}\left[i,j\right]=\int_{\Omega}\var{a} \left[i,j\right] \; d\Omega
1086 \end{equation}
1087 where $\Omega$ is the spatial domain and $d\Omega$ volume integration. To integrate over the boundary of the domain one uses
1088 \begin{equation}
1089 \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int_{\partial \Omega} a\left[i,j\right] \; ds
1090 \end{equation}
1091 where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or line integration.
1092 \end{funcdesc}
1093 \begin{funcdesc}{interpolate}{a,where}
1094 interpolates argument \var{a} into the \FunctionSpace \var{where}.
1095 \end{funcdesc}
1096 \begin{funcdesc}{div}{a\optional{ ,where=None}}
1097 returns the divergence of \var{a}. This
1098 \begin{equation}
1099 \code{div(a)}=trace(grad(a),where)
1100 \end{equation}
1101 \end{funcdesc}
1102 \begin{funcdesc}{jump}{a\optional{ ,domain=None}}
1103 returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present
1104 in \var{domain}.
1105 \begin{equation}
1106 \begin{array}{rcl}
1107 \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
1108 & & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
1109 \end{array}
1110 \end{equation}
1111 \end{funcdesc}
1112 \begin{funcdesc}{L2}{a}
1113 returns the $L^2$-norm of \var{a} in its function space. This is
1114 \begin{equation}
1115 \code{L2(a)=integrate(length(a)}^2\code{)} \; .
1116 \end{equation}
1117 \end{funcdesc}
1118
1119 The following functions operate ``point-wise''. That is, the operation is applied to each component of each point
1120 individually.
1121
1122 \begin{funcdesc}{sin}{a}
1123 applies sine function to \var{a}.
1124 \end{funcdesc}
1125
1126 \begin{funcdesc}{cos}{a}
1127 applies cosine function to \var{a}.
1128 \end{funcdesc}
1129
1130 \begin{funcdesc}{tan}{a}
1131 applies tangent function to \var{a}.
1132 \end{funcdesc}
1133
1134 \begin{funcdesc}{asin}{a}
1135 applies arc (inverse) sine function to \var{a}.
1136 \end{funcdesc}
1137
1138 \begin{funcdesc}{acos}{a}
1139 applies arc (inverse) cosine function to \var{a}.
1140 \end{funcdesc}
1141
1142 \begin{funcdesc}{atan}{a}
1143 applies arc (inverse) tangent function to \var{a}.
1144 \end{funcdesc}
1145
1146 \begin{funcdesc}{sinh}{a}
1147 applies hyperbolic sine function to \var{a}.
1148 \end{funcdesc}
1149
1150 \begin{funcdesc}{cosh}{a}
1151 applies hyperbolic cosine function to \var{a}.
1152 \end{funcdesc}
1153
1154 \begin{funcdesc}{tanh}{a}
1155 applies hyperbolic tangent function to \var{a}.
1156 \end{funcdesc}
1157
1158 \begin{funcdesc}{asinh}{a}
1159 applies arc (inverse) hyperbolic sine function to \var{a}.
1160 \end{funcdesc}
1161
1162 \begin{funcdesc}{acosh}{a}
1163 applies arc (inverse) hyperbolic cosine function to \var{a}.
1164 \end{funcdesc}
1165
1166 \begin{funcdesc}{atanh}{a}
1167 applies arc (inverse) hyperbolic tangent function to \var{a}.
1168 \end{funcdesc}
1169
1170 \begin{funcdesc}{exp}{a}
1171 applies exponential function to \var{a}.
1172 \end{funcdesc}
1173
1174 \begin{funcdesc}{sqrt}{a}
1175 applies square root function to \var{a}.
1176 \end{funcdesc}
1177
1178 \begin{funcdesc}{log}{a}
1179 applies the natural logarithm to \var{a}.
1180 \end{funcdesc}
1181
1182 \begin{funcdesc}{log10}{a}
1183 applies the base-$10$ logarithm to \var{a}.
1184 \end{funcdesc}
1185
1186 \begin{funcdesc}{sign}{a}
1187 applies the sign function to \var{a}, that is $1$ where \var{a} is positive,
1188 $-1$ where \var{a} is negative and $0$ otherwise.
1189 \end{funcdesc}
1190
1191 \begin{funcdesc}{wherePositive}{a}
1192 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
1193 \end{funcdesc}
1194
1195 \begin{funcdesc}{whereNegative}{a}
1196 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
1197 \end{funcdesc}
1198
1199 \begin{funcdesc}{whereNonNegative}{a}
1200 returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.
1201 \end{funcdesc}
1202
1203 \begin{funcdesc}{whereNonPositive}{a}
1204 returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.
1205 \end{funcdesc}
1206
1207 \begin{funcdesc}{whereZero}{a\optional{, tol=None, \optional{, rtol=1.e-8}}}
1208 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.
1209 \end{funcdesc}
1210
1211 \begin{funcdesc}{whereNonZero}{a, \optional{, tol=None, \optional{, rtol=1.e-8}}}
1212 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.
1213 \end{funcdesc}
1214
1215 \subsection{Interpolating Data}
1216 \index{interpolateTable}
1217 In some cases, it may be useful to produce Data objects which fit some user defined function.
1218 Manually modifying each value in the Data object is not a good idea since it depends on
1219 knowing the location and order of each datapoint in the domain.
1220 Instead \escript can use an interpolation table to produce a Data object.
1221
1222 The following example is available as \file{int_save.py} in the examples directory.
1223 We will produce a \Data object which aproximates a sine curve.
1224
1225 \begin{python}
1226 from esys.escript import saveDataCSV, sup
1227 import numpy
1228 from esys.finley import Rectangle
1229
1230 n=4
1231 r=Rectangle(n,n)
1232 x=r.getX()
1233 x0=x[0]
1234 x1=x[1] #we'll use this later
1235 toobig=100
1236 \end{python}
1237
1238 First we produce an interpolation table.
1239 \begin{python}
1240 sine_table=[0, 0.70710678118654746, 1, 0.70710678118654746, 0,
1241 -0.70710678118654746, -1, -0.70710678118654746, 0]
1242 \end{python}
1243
1244 We wish to identify $0$ and $1$ with the ends of the curve.
1245 That is, with the first and eighth values in the table.
1246
1247 \begin{python}
1248 numslices=len(sine_table)-1
1249
1250 minval=0
1251 maxval=1
1252
1253 step=sup(maxval-minval)/numslices
1254 \end{python}
1255
1256 So the values $v$ from the input lie in the interval minval$\leq v < $maxval.
1257 \var{step} represents the gap (in the input range) between entries in the table.
1258 By default values of $v$ outside the table argument range (minval, maxval) will
1259 be pushed back into the range, ie. if $v <$ minval the value minval will be used to
1260 evaluate the table. Similarly, for values $v>$ maxval the value maxval is used.
1261
1262 Now we produce our new \Data object.
1263
1264 \begin{python}
1265 result=x0.interpolateTable(sine_table, minval, step, toobig)
1266 \end{python}
1267 Any values which interpolate to larger than \var{toobig} will raise an exception. You can
1268 switch on boundary checking by adding ''check_boundaries=True`` the argument list.
1269
1270
1271 Now for a 2D example.
1272 We will interpolate a surface such that the bottom edge is the sine curve described above.
1273 The amplitude of the curve decreases as we move towards the top edge.
1274
1275 Our interpolation table will have three rows.
1276 \begin{python}
1277 st=numpy.array(sine_table)
1278
1279 table=[st, 0.5*st, 0*st ]
1280 \end{python}
1281
1282 The use of numpy and multiplication here is just to save typing.
1283
1284 \begin{python}
1285 result2=x1.interpolateTable(table, 0, 0.55, x0, minval, step, toobig)
1286 \end{python}
1287
1288 In the 2D case, the parameters for the x1 direction (min=0, step=0.55) come first followed by the x0 data object and
1289 its parameters.
1290 By default, if a point is specified which is outside the boundary, then \var{interpolateTable} will operate
1291 as if the point was on the boundary.
1292 Passing \var{check_boundaries}=\var{True} will \var{interpolateTable} to reject any points outside the boundaries.
1293
1294 \subsection{Saving Data as CSV}
1295 \index{saveDataCSV}
1296 \index{CSV}
1297 For simple post-processing, \Data objects can be saved in comma separated value format.
1298
1299 If \var{mydata1} and \var{mydata2} are scalar data, the following command:
1300 \begin{python}
1301 saveDataCSV('output.csv',U=mydata1, V=mydata2)
1302 \end{python}
1303 will record the values of mydata in \texttt{output.csv} in the following format:
1304 \begin{verbatim}
1305 U, V
1306 1.0000000e+0, 2.0000000e-1
1307 5.0000000e-0, 1.0000000e+1
1308 ...
1309 \end{verbatim}
1310
1311 The names of the keyword parameters form the names of columns in the output.
1312 If the data objects are over different function spaces, then saveDataCSV will attempt to
1313 interpolate to a common function space.
1314 If this is not possible, then an exception will be raised.
1315
1316 Output can be restricted using a scalar mask.
1317 \begin{python}
1318 saveDataCSV('outfile.csv', U=mydata1, V=mydata2, mask=myscalar)
1319 \end{python}
1320 Will only output those rows which correspond to to positive values of \var{myscalar}.
1321 Some aspects of the output can be tuned using additional parameters.
1322 \begin{python}
1323 saveDataCSV('data.csv', append=True, sep=' ', csep='/', mask=mymask, e=mat1)
1324 \end{python}
1325
1326 \begin{itemize}
1327 \item \var{append} - specifies that the output should be written to the end of an existing file.
1328 \item \var{sep} - defines the separator between fields.
1329 \item \var{csep} - defines the separator between components in the header line. For example between the components of a matrix.
1330 \end{itemize}
1331
1332 The above command would produce output like this:
1333 \begin{verbatim}
1334 e/0/0 e/1/0 e/0/1 e/1/1
1335 1.0000000000e+00 2.0000000000e+00 3.0000000000e+00 4.0000000000e+00
1336 ...
1337 \end{verbatim}
1338
1339 Note that while the order in which rows are output can vary, all the elements in a given row
1340 always correspond to the same input.
1341
1342
1343 \subsection{The \Operator Class}
1344 The \Operator class provides an abstract access to operators build
1345 within the \LinearPDE class. \Operator objects are created
1346 when a PDE is handed over to a PDE solver library and handled
1347 by the \LinearPDE object defining the PDE. The user can gain access
1348 to the \Operator of a \LinearPDE object through the \var{getOperator}
1349 method.
1350
1351 \begin{classdesc}{Operator}{}
1352 creates an empty \Operator object.
1353 \end{classdesc}
1354
1355 \begin{methoddesc}[Operator]{isEmpty}{fileName}
1356 returns \True is the object is empty. Otherwise \True is returned.
1357 \end{methoddesc}
1358
1359 \begin{methoddesc}[Operator]{setValue}{value}
1360 resets all entries in the object representation to \var{value}
1361 \end{methoddesc}
1362
1363 \begin{methoddesc}[Operator]{solves}{rhs}
1364 solves the operator equation with right hand side \var{rhs}
1365 \end{methoddesc}
1366
1367 \begin{methoddesc}[Operator]{of}{u}
1368 applies the operator to the \Data object \var{u}
1369 \end{methoddesc}
1370
1371 \begin{methoddesc}[Operator]{saveMM}{fileName}
1372 saves the object to a matrix market format file of name
1373 \var{fileName}, see
1374 \url{http://maths.nist.gov/MatrixMarket}
1375 % \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}.
1376 \index{Matrix Market}
1377 \end{methoddesc}
1378
1379 \section{Physical Units}
1380 \escript provides support for physical units in the SI system \index{SI units} including unit conversion. So the
1381 user can define variables in the form
1382 \begin{python}
1383 from esys.escript.unitsSI import *
1384 l=20*m
1385 w=30*kg
1386 w2=40*lb
1387 T=100*Celsius
1388 \end{python}
1389 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
1390 composed units can be used, for instance
1391 \begin{python}
1392 from esys.escript.unitsSI import *
1393 rho=40*lb/cm**3
1394 \end{python}
1395 to define the density in the units of pounds per cubic centimeter. The value $40$ will be converted
1396 into SI units, in this case kg per cubic meter.
1397 Moreover unit prefixes are supported:
1398 \begin{python}
1399 from esys.escript.unitsSI import *
1400 p=40*Mega*Pa
1401 \end{python}
1402 to the the pressure to 40 Mega Pascal. Units can also be converted back from the SI system into
1403 a desired unit, e.g
1404 \begin{python}
1405 from esys.escript.unitsSI import *
1406 print p/atm
1407 \end{python}
1408 can be used print the pressure in units of atmosphere\index{atmosphere}.
1409
1410 This is an incomplete list of supported physical units:
1411
1412 \begin{datadesc}{km}
1413 unit of kilo meter
1414 \end{datadesc}
1415
1416 \begin{datadesc}{m}
1417 unit of meter
1418 \end{datadesc}
1419
1420 \begin{datadesc}{cm}
1421 unit of centi meter
1422 \end{datadesc}
1423
1424 \begin{datadesc}{mm}
1425 unit of milli meter
1426 \end{datadesc}
1427
1428 \begin{datadesc}{sec}
1429 unit of second
1430 \end{datadesc}
1431
1432 \begin{datadesc}{minute}
1433 unit of minute
1434 \end{datadesc}
1435
1436 \begin{datadesc}{h}
1437 unit of hour
1438 \end{datadesc}
1439 \begin{datadesc}{day}
1440 unit of day
1441 \end{datadesc}
1442 \begin{datadesc}{yr}
1443 unit of year
1444 \end{datadesc}
1445
1446 \begin{datadesc}{gram}
1447 unit of gram
1448 \end{datadesc}
1449 \begin{datadesc}{kg}
1450 unit of kilo gram
1451 \end{datadesc}
1452 \begin{datadesc}{lb}
1453 unit of pound
1454 \end{datadesc}
1455 \begin{datadesc}{ton}
1456 metric ton
1457 \end{datadesc}
1458
1459 \begin{datadesc}{A}
1460 unit of Ampere
1461 \end{datadesc}
1462
1463 \begin{datadesc}{Hz}
1464 unit of Hertz
1465 \end{datadesc}
1466
1467 \begin{datadesc}{N}
1468 unit of Newton
1469 \end{datadesc}
1470 \begin{datadesc}{Pa}
1471 unit of Pascal
1472 \end{datadesc}
1473 \begin{datadesc}{atm}
1474 unit of atmosphere
1475 \end{datadesc}
1476 \begin{datadesc}{J}
1477 unit of Joule
1478 \end{datadesc}
1479
1480 \begin{datadesc}{W}
1481 unit of Watt
1482 \end{datadesc}
1483
1484 \begin{datadesc}{C}
1485 unit of Coulomb
1486 \end{datadesc}
1487 \begin{datadesc}{V}
1488 unit of Volt
1489 \end{datadesc}
1490 \begin{datadesc}{F}
1491 unit of Farad
1492 \end{datadesc}
1493
1494 \begin{datadesc}{Ohm}
1495 unit of Ohm
1496 \end{datadesc}
1497 \begin{datadesc}{K}
1498 unit of Kelvin
1499 \end{datadesc}
1500 \begin{datadesc}{Celsius}
1501 unit of Celsius
1502 \end{datadesc}
1503
1504 \begin{datadesc}{Fahrenheit}
1505 unit of Fahrenheit
1506 \end{datadesc}
1507
1508 Moreover unit prefixes are supported:
1509
1510 \begin{datadesc}{Yotta}
1511 prefix yotta = $10^{24}$.
1512
1513 \end{datadesc}
1514
1515 \begin{datadesc}{Zetta}
1516 prefix zetta= $10^{21}$.
1517 \end{datadesc}
1518
1519 \begin{datadesc}{Exa}
1520 prefix exa= $10^{18}$.
1521 \end{datadesc}
1522
1523 \begin{datadesc}{Peta}
1524 prefix peta= $10^{15}$.
1525 \end{datadesc}
1526
1527 \begin{datadesc}{Tera}
1528 prefix tera= $10^{12}$.
1529 \end{datadesc}
1530
1531 \begin{datadesc}{Giga}
1532 prefix giga= $10^9$.
1533 \end{datadesc}
1534
1535 \begin{datadesc}{Mega}
1536 prefix mega= $10^6$.
1537 \end{datadesc}
1538
1539 \begin{datadesc}{Kilo}
1540 prefix kilo= $10^3$.
1541 \end{datadesc}
1542
1543 \begin{datadesc}{Hecto}
1544 prefix hecto= $10^2$.
1545 \end{datadesc}
1546
1547 \begin{datadesc}{Deca}
1548 prefix deca= $10^1$.
1549 \end{datadesc}
1550
1551 \begin{datadesc}{Deci}
1552 prefix deci= $10^{-1}$.
1553 \end{datadesc}
1554
1555 \begin{datadesc}{Centi}
1556 prefix centi= $10^{-2}$.
1557 \end{datadesc}
1558
1559 \begin{datadesc}{Milli}
1560 prefix milli= $10^{-3}$.
1561 \end{datadesc}
1562
1563 \begin{datadesc}{Micro}
1564 prefix micro= $10^{-6}$.
1565 \end{datadesc}
1566
1567 \begin{datadesc}{Nano}
1568 prefix nano= $10^{-9}$.
1569 \end{datadesc}
1570
1571 \begin{datadesc}{Pico}
1572 prefix pico= $10^{-12}$.
1573 \end{datadesc}
1574
1575 \begin{datadesc}{Femto}
1576 prefix femto= $10^{-15}$.
1577 \end{datadesc}
1578
1579 \begin{datadesc}{Atto}
1580 prefix atto= $10^{-18}$.
1581 \end{datadesc}
1582
1583 \begin{datadesc}{Zepto}
1584 prefix zepto= $10^{-21}$.
1585 \end{datadesc}
1586
1587 \begin{datadesc}{Yocto}
1588 prefix yocto= $10^{-24}$.
1589 \end{datadesc}
1590
1591
1592 \section{Utilities}
1593
1594 The \class{FileWriter} provides a mechanism to write data to a file.
1595 In essence, this class wraps the standard \class{file} class to write data
1596 that are global in MPI to a file. In fact, data are written on the processor
1597 with \MPI rank 0 only. It is recommended to use \class{FileWriter}
1598 rather than \class{open} in order to write code that is running
1599 with and without \MPI. It is save to use \class{open} under MPI to read data which are global under \MPI.
1600
1601 \begin{classdesc}{FileWriter}{fn\optional{,append=\False, \optional{createLocalFiles=\False}})}
1602 Opens a file of name \var{fn} for writing. If \var{append} is set to \True
1603 written data are append at the end of the file.
1604 If running under \MPI only the first processor with rank==0
1605 will open the file and write to it.
1606 If \var{createLocalFiles} is set each individual processor will create a file
1607 where for any processor with rank>0 the file name is extended by its rank. This option is normally used for debug purposes only.
1608 \end{classdesc}
1609
1610 The following methods are available:
1611 \begin{methoddesc}[FileWriter]{close}{}
1612 closes the file.
1613 \end{methoddesc}
1614 \begin{methoddesc}[FileWriter]{flush}{}
1615 flushes the internal buffer to disk.
1616 \end{methoddesc}
1617 \begin{methoddesc}[FileWriter]{write}{txt}
1618 Write string \var{txt} to file.
1619 Note that newline is not added.
1620 \end{methoddesc}
1621 \begin{methoddesc}[FileWriter]{writelines}{txts}
1622 Write the list \var{txts} of strings to the file..
1623 Note that newlines are not added.
1624 This method is equivalent to call write() for each string.
1625 \end{methoddesc}
1626 \begin{memberdesc}[FileWriter]{closed}
1627 \True if file is closed.
1628 \end{memberdesc}
1629 \begin{memberdesc}[FileWriter]{mode}
1630 access mode.
1631 \end{memberdesc}
1632 \begin{memberdesc}[FileWriter]{name}
1633 file name.
1634 \end{memberdesc}
1635 \begin{memberdesc}[FileWriter]{newlines}
1636 line separator
1637 \end{memberdesc}
1638
1639
1640 \begin{funcdesc}{setEscriptParamInt}{name,value}
1641 assigns the integer value \var{value} to the parameter \var{name}.
1642 If \var{name}="TOO_MANY_LINES" conversion of any \Data object to a string switches to a
1643 condensed format if more than \var{value} lines would be created.
1644 \end{funcdesc}
1645
1646 \begin{funcdesc}{getEscriptParamInt}{name}
1647 returns the current value of integer parameter \var{name}.
1648 \end{funcdesc}
1649
1650 \begin{funcdesc}{listEscriptParams}{a}
1651 returns a list of valid parameters and their description.
1652 \end{funcdesc}
1653
1654 \begin{funcdesc}{getMPISizeWorld}{}
1655 returns the number of \MPI processors in use in the \env{MPI_COMM_WORLD} processor group.
1656 If \MPI is not used 1 is returned.
1657 \end{funcdesc}
1658 \begin{funcdesc}{getMPIRankWorld}{}
1659 returns the rank of the process within the \env{MPI_COMM_WORLD} processor group.
1660 If \MPI is not used 0 is returned.
1661 \end{funcdesc}
1662 \begin{funcdesc}{MPIBarrierWorld}{}
1663 performs a barrier synchronization across all processors within \env{MPI_COMM_WORLD}
1664 processor group.
1665 \end{funcdesc}
1666 \begin{funcdesc}{getMPIWorldMax}{a}
1667 returns the maximum value of the integer \var{a} across all
1668 processors within \env{MPI_COMM_WORLD}.
1669 \end{funcdesc}
1670

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