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15
16 \chapter{The \escript Module}\label{ESCRIPT CHAP}
17
18 \section{Concepts}
19 \escript is a \PYTHON module that allows you to represent the values of
20 a function at points in a \Domain in such a way that the function will
21 be useful for the Finite Element Method (FEM) simulation. It also
22 provides what we call a function space that describes how the data is
23 used in the simulation. Stored along with the data is information
24 about the elements and nodes which will be used by the domain (e.g. \finley).
25
26 \subsection{Function spaces}
27 In order to understand what we mean by the term 'function space',
28 consider that the solution of a partial differential
29 equation\index{partial differential equation} (PDE) is a function on a domain
30 $\Omega$. When solving a PDE using FEM, the solution is
31 piecewise-differentiable but, in general, its gradient is discontinuous.
32 To reflect these different degrees of smoothness, different function spaces
33 are used.
34 For instance, in FEM, the displacement field is represented by its values at
35 the nodes of the mesh, and so is continuous.
36 The strain, which is the symmetric part of the gradient of the displacement
37 field, is stored on the element centers, and so is considered to be
38 discontinuous.
39
40 A function space is described by a \FunctionSpace object.
41 The following statement generates the object \var{solution_space} which is
42 a \FunctionSpace object and provides access to the function space of
43 PDE solutions on the \Domain \var{mydomain}:
44
45 \begin{python}
46 solution_space=Solution(mydomain)
47 \end{python}
48 The following generators for function spaces on a \Domain \var{mydomain} are commonly used:
49 \begin{itemize}
50 \item \var{Solution(mydomain)}: solutions of a PDE
51 \item \var{ReducedSolution(mydomain)}: solutions of a PDE with a reduced
52 smoothness requirement, e.g. using a lower order approximation on the same
53 element or using macro elements\index{macro elements}
54 \item \var{ContinuousFunction(mydomain)}: continuous functions, e.g. a temperature distribution
55 \item \var{Function(mydomain)}: general functions which are not necessarily continuous, e.g. a stress field
56 \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, e.g. a surface pressure
57 \item \var{DiracDeltaFunctions(mydomain)}: functions defined on a set of points
58 \item \var{FunctionOnContact0(mydomain)}: functions on side $0$ of a discontinuity
59 \item \var{FunctionOnContact1(mydomain)}: functions on side $1$ of a discontinuity
60 \end{itemize}
61 In some cases under-integration is used. For these cases the user may use a
62 \FunctionSpace from the following list:
63 \begin{itemize}
64 \item \var{ReducedFunction(mydomain)}
65 \item \var{ReducedFunctionOnBoundary(mydomain)}
66 \item \var{ReducedFunctionOnContact0(mydomain)}
67 \item \var{ReducedFunctionOnContact1(mydomain)}
68 \end{itemize}
69 In comparison to the corresponding full version they use a reduced number of
70 integration nodes (typically one only) to represent values.
71
72 \begin{figure}
73 \centering
74 \includegraphics{EscriptDiagram1}
75 \caption{\label{ESCRIPT DEP}Dependency of function spaces in \finley.
76 An arrow indicates that a function in the \FunctionSpace at the starting point
77 can be interpolated to the \FunctionSpace of the arrow target.
78 All function spaces above the dotted line can be interpolated to any of
79 the function spaces below the line. See also \Sec{SEC Projection}.}
80 \end{figure}
81
82 The reduced smoothness for a PDE solution is often used to fulfill the
83 Ladyzhenskaya-Babuska-Brezzi condition~\cite{LBB} when solving saddle point
84 problems\index{saddle point problems}, e.g. the Stokes equation.
85 A discontinuity\index{discontinuity} is a region within the domain across
86 which functions may be discontinuous.
87 The location of a discontinuity is defined in the \Domain object.
88 \fig{ESCRIPT DEP} shows the dependency between the types of function spaces
89 in \finley (other libraries may have different relationships).
90
91 The solution of a PDE is a continuous function. Any continuous function can
92 be seen as a general function on the domain and can be restricted to the
93 boundary as well as to one side of a discontinuity (the result will be
94 different depending on which side is chosen). Functions on any side of the
95 discontinuity can be seen as a function on the corresponding other side.
96
97 A function on the boundary or on one side of the discontinuity cannot be seen
98 as a general function on the domain as there are no values defined for the
99 interior. For most PDE solver libraries the space of the solution and
100 continuous functions is identical, however in some cases, for example when
101 periodic boundary conditions are used in \finley, a solution fulfills periodic
102 boundary conditions while a continuous function does not have to be periodic.
103
104 The concept of function spaces describes the properties of functions and
105 allows abstraction from the actual representation of the function in the
106 context of a particular application. For instance, in the FEM context a
107 function of the \Function type (written as \emph{Function()} in \fig{ESCRIPT DEP})
108 is usually represented by its values at the element center,
109 but in a finite difference scheme the edge midpoint of cells is preferred.
110 By changing its function space you can use the same function in a Finite
111 Difference scheme instead of Finite Element scheme.
112 Changing the function space of a particular function will typically lead to
113 a change of its representation.
114 So, when seen as a general function, a continuous function which is typically
115 represented by its values on the nodes of the FEM mesh or finite difference
116 grid must be interpolated to the element centers or the cell edges,
117 respectively. Interpolation happens automatically in \escript whenever it is
118 required\index{interpolation}. The user needs to be aware that an
119 interpolation is not always possible, see \fig{ESCRIPT DEP} for \finley.
120 An alternative approach to change the representation (=\FunctionSpace) is
121 projection\index{projection}, see \Sec{SEC Projection}.
122
123 \subsection{\Data Objects}
124 In \escript the class that stores these functions is called \Data.
125 The function is represented through its values on \DataSamplePoints where
126 the \DataSamplePoints are chosen according to the function space of the
127 function.
128 \Data class objects are used to define the coefficients of the PDEs to be
129 solved by a PDE solver library and also to store the solutions of the PDE.
130
131 The values of the function have a rank which gives the number of indices,
132 and a \Shape defining the range of each index.
133 The rank in \escript is limited to the range 0 through 4 and it is assumed
134 that the rank and \Shape is the same for all \DataSamplePoints.
135 The \Shape of a \Data object is a tuple (list) \var{s} of integers.
136 The length of \var{s} is the rank of the \Data object and the \var{i}-th
137 index ranges between 0 and $\var{s[i]}-1$.
138 For instance, a stress field has rank 2 and \Shape $(d,d)$ where $d$ is the
139 number of spatial dimensions.
140 The following statement creates the \Data object \var{mydat} representing a
141 continuous function with values of \Shape $(2,3)$ and rank $2$:
142 \begin{python}
143 mydat=Data(value=1, what=ContinuousFunction(myDomain), shape=(2,3))
144 \end{python}
145 The initial value is the constant 1 for all \DataSamplePoints and all
146 components.
147
148 \Data objects can also be created from any \numpy array or any object, such
149 as a list of floating point numbers, that can be converted into
150 a \numpyNDA\cite{NUMPY}.
151 The following two statements create objects which are equivalent
152 to \var{mydat}:
153 \begin{python}
154 mydat1=Data(value=numpy.ones((2,3)), what=ContinuousFunction(myDomain))
155 mydat2=Data(value=[[1,1], [1,1], [1,1]], what=ContinuousFunction(myDomain))
156 \end{python}
157 In the first case the initial value is \var{numpy.ones((2,3))} which generates
158 a $2 \times 3$ matrix as an instance of \numpyNDA filled with ones.
159 The \Shape of the created \Data object is taken from the \Shape of the array.
160 In the second case, the creator converts the initial value, which is a list of
161 lists, into a \numpyNDA before creating the actual \Data object.
162
163 For convenience \escript provides creators for the most common types
164 of \Data objects in the following forms (\var{d} defines the spatial
165 dimensionality):
166 \begin{itemize}
167 \item \code{Scalar(0, Function(mydomain))} is the same as \code{Data(0, Function(myDomain),(,))}\\
168 (each value is a scalar), e.g. a temperature field
169 \item \code{Vector(0, Function(mydomain))} is the same as \code{Data(0, Function(myDomain),(d,))}\\
170 (each value is a vector), e.g. a velocity field
171 \item \code{Tensor(0, Function(mydomain))} equals \code{Data(0, Function(myDomain), (d,d))},
172 e.g. a stress field
173 \item \code{Tensor4(0,Function(mydomain))} equals \code{Data(0,Function(myDomain), (d,d,d,d))},
174 e.g. a Hook tensor field
175 \end{itemize}
176 Here the initial value is 0 but any object that can be converted into
177 a \numpyNDA and whose \Shape is consistent with \Shape of the \Data object to
178 be created can be used as the initial value.
179
180 \Data objects can be manipulated by applying unary operations (e.g. cos, sin,
181 log), and they can be combined point-wise by applying arithmetic operations
182 (e.g. +, - ,* , /).
183 We emphasize that \escript itself does not handle any spatial dependencies as
184 it does not know how values are interpreted by the processing PDE solver library.
185 However \escript invokes interpolation if this is needed during data manipulations.
186 Typically, this occurs in binary operations when the arguments belong to
187 different function spaces or when data are handed over to a PDE solver library
188 which requires functions to be represented in a particular way.
189
190 The following example shows the usage of \Data objects. Assume we have a
191 displacement field $u$ and we want to calculate the corresponding stress field
192 $\sigma$ using the linear-elastic isotropic material model
193 \begin{eqnarray}\label{eq: linear elastic stress}
194 \sigma_{ij}=\lambda u_{k,k} \delta_{ij} + \mu ( u_{i,j} + u_{j,i})
195 \end{eqnarray}
196 where $\delta_{ij}$ is the Kronecker symbol and
197 $\lambda$ and $\mu$ are the Lam\'e coefficients. The following function
198 takes the displacement \var{u} and the Lam\'e coefficients \var{lam} and \var{mu}
199 as arguments and returns the corresponding stress:
200 \begin{python}
201 from esys.escript import *
202 def getStress(u, lam, mu):
203 d=u.getDomain().getDim()
204 g=grad(u)
205 stress=lam*trace(g)*kronecker(d)+mu*(g+transpose(g))
206 return stress
207 \end{python}
208 The variable \var{d} gives the spatial dimensionality of the domain on which
209 the displacements are defined.
210 \var{kronecker} returns the Kronecker symbol with indices $i$ and $j$ running
211 from 0 to \var{d}-1.
212 The call \var{grad(u)} requires the displacement field \var{u} to be in
213 the \var{Solution} or \ContinuousFunction.
214 The result \var{g} as well as the returned stress will be in the \Function.
215 If, for example, \var{u} is the solution of a PDE then \code{getStress} might
216 be called in the following way:
217 \begin{python}
218 s=getStress(u, 1., 2.)
219 \end{python}
220 However \code{getStress} can also be called with \Data objects as values for
221 \var{lam} and \var{mu} which, for instance in the case of a temperature
222 dependency, are calculated by an expression.
223 The following call is equivalent to the previous example:
224 \begin{python}
225 lam=Scalar(1., ContinuousFunction(mydomain))
226 mu=Scalar(2., Function(mydomain))
227 s=getStress(u, lam, mu)
228 \end{python}
229 %
230 The function \var{lam} belongs to the \ContinuousFunction but with \var{g} the
231 function \var{trace(g)} is in the \Function.
232 In the evaluation of the product \var{lam*trace(g)} we have different function
233 spaces (on the nodes versus in the centers) and at first glance we have incompatible data.
234 \escript converts the arguments into an appropriate function space according
235 to \fig{ESCRIPT DEP}.
236 In this example that means \escript sees \var{lam} as a function of the \Function.
237 In the context of FEM this means the nodal values of \var{lam} are
238 interpolated to the element centers.
239 The interpolation is automatic and requires no special handling.
240
241 \begin{figure}
242 \centering
243 \includegraphics{EscriptDiagram2}
244 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region
245 with two rock types {\it white} and {\it gray} is shown.
246 The number in each cell refers to the major rock type present in the cell
247 ($1$ for {\it white} and $2$ for {\it gray}).}
248 \end{figure}
249
250 \subsection{Tagged, Expanded and Constant Data}
251 Material parameters such as the Lam\'e coefficients are typically dependent on
252 rock types present in the area of interest.
253 A common technique to handle these kinds of material parameters is
254 \emph{tagging}\index{tagging}, which uses storage efficiently.
255 \fig{Figure: tag} shows an example. In this case two rock types {\it white}
256 and {\it gray} can be found in the domain.
257 The domain is subdivided into triangular shaped cells.
258 Each cell has a tag indicating the rock type predominantly found in this cell.
259 Here $1$ is used to indicate rock type {\it white} and $2$ for rock type {\it gray}.
260 The tags are assigned at the time when the cells are generated and stored in
261 the \Domain class object. To allow easier usage of tags, names can be used
262 instead of numbers. These names are typically defined at the time when the
263 geometry is generated.
264
265 The following statements show how to use tagged values for \var{lam} as shown
266 in \fig{Figure: tag} for the stress calculation discussed above:
267 \begin{python}
268 lam=Scalar(value=2., what=Function(mydomain))
269 insertTaggedValue(lam, white=30., gray=5000.)
270 s=getStress(u, lam, 2.)
271 \end{python}
272 In this example \var{lam} is set to $30$ for those cells with tag {\it white}
273 (=$1$) and to $5000$ for cells with tag {\it gray} (=$2$).
274 The initial value $2$ of \var{lam} is used as a default value for the case
275 when a tag is encountered which has not been linked with a value.
276 The \code{getStress} method does not need to be changed now that we are using tags.
277 \escript resolves the tags when \var{lam*trace(g)} is calculated.
278
279 This brings us to a very important point about \escript.
280 You can develop a simulation with constant Lam\'e coefficients, and then later
281 switch to tagged Lam\'e coefficients without otherwise changing your \PYTHON script.
282 In short, you can use the same script for models with different domains and
283 different types of input data.
284
285 There are three main ways in which \Data objects are represented internally --
286 constant, tagged, and expanded.
287 In the constant case, the same value is used at each sample point while only a
288 single value is stored to save memory.
289 In the expanded case, each sample point has an individual value (such as for the solution of a PDE).
290 This is where your largest data sets will be created because the values are
291 stored as a complete array.
292 The tagged case has already been discussed above.
293 Expanded data is created when specifying \code{expanded=True} in the \Data
294 object constructor, while tagged data requires calling the \member{insertTaggedValue}
295 method as shown above.
296
297 Values are accessed through a sample reference number.
298 Operations on expanded \Data objects have to be performed for each sample
299 point individually.
300 When tagged values are used, the values are held in a dictionary.
301 Operations on tagged data require processing the set of tagged values only,
302 rather than processing the value for each individual sample point.
303 \escript allows any mixture of constant, tagged and expanded data in a single expression.
304
305 \subsection{Saving and Restoring Simulation Data}
306 \Data objects can be written to disk files with the \member{dump} method and
307 read back using the \member{load} method, both of which use the
308 \netCDF\cite{NETCDF} file format.
309 Use these to save data for checkpoint/restart or simply to save and reuse data
310 that was expensive to compute.
311 For instance, to save the coordinates of the data points of a
312 \ContinuousFunction to the file \file{x.nc} use
313 \begin{python}
314 x=ContinuousFunction(mydomain).getX()
315 x.dump("x.nc")
316 mydomain.dump("dom.nc")
317 \end{python}
318 To recover the object \var{x}, and you know that \var{mydomain} was an \finley
319 mesh, use
320 \begin{python}
321 from esys.finley import LoadMesh
322 mydomain=LoadMesh("dom.nc")
323 x=load("x.nc", mydomain)
324 \end{python}
325 Obviously, it is possible to execute the same steps that were originally used
326 to generate \var{mydomain} to recreate it. However, in most cases using
327 \member{dump} and \member{load} is faster, particularly if optimization has
328 been applied.
329 If \escript is running on more than one \MPI process \member{dump} will create
330 an individual file for each process containing the local data.
331 In order to avoid conflicts the \MPI processor
332 rank is appended to the file names.
333 That is instead of one file \file{dom.nc} you would get
334 \file{dom.nc.0000}, \file{dom.nc.0001}, etc.
335 You still call \code{LoadMesh("dom.nc")} to load the domain but you have to
336 make sure that the appropriate file is accessible from the corresponding rank,
337 and loading will only succeed if you run with as many processes as were used
338 when calling \member{dump}.
339
340 The function space of the \Data is stored in \file{x.nc}.
341 If the \Data object is expanded, the number of data points in the file and of
342 the \Domain for the particular \FunctionSpace must match.
343 Moreover, the ordering of the values is checked using the reference
344 identifiers provided by the \FunctionSpace on the \Domain.
345 In some cases, data points will be reordered so be aware and confirm that you
346 get what you wanted.
347
348 A more flexible way of saving and restoring \escript simulation data
349 is through an instance of the \class{DataManager} class.
350 It has the advantage of allowing to save and load not only a \Domain and
351 \Data objects but also other values\footnote{The \PYTHON \emph{pickle} module
352 is used for other types.} you compute in your simulation script.
353 Further, \class{DataManager} objects can simultaneously create files for
354 visualization so no extra calls to \code{saveVTK} etc. are needed.
355
356 The following example shows how the \class{DataManager} class can be used.
357 For an explanation of all member functions and options see the class reference
358 Section \ref{sec:datamanager}.
359 \begin{python}
360 from esys.escript import DataManager, Scalar, Function
361 from esys.finley import Rectangle
362
363 dm = DataManager(formats=[DataManager.RESTART, DataManager.VTK])
364 if dm.hasData():
365 mydomain=dm.getDomain()
366 val=dm.getValue("val")
367 t=dm.getValue("t")
368 t_max=dm.getValue("t_max")
369 else:
370 mydomain=Rectangle()
371 val=Function(mydomain).getX()
372 t=0.
373 t_max=2.5
374
375 while t<t_max:
376 t+=.01
377 val=val+t/2
378 dm.addData(val=val, t=t, t_max=t_max)
379 dm.export()
380 \end{python}
381 In the constructor we specify that we want \code{RESTART} (i.e. dump) files
382 and \code{VTK} files to be saved.
383 By default, the constructor will look for previously saved \code{RESTART}
384 files under the current directory and load them.
385 We can then enquire if such files were found by calling the \member{hasData}
386 method. If it returns \True we retrieve the domain and values into local
387 variables. Otherwise the same variables are initialized with appropriate
388 values to start a new simulation.
389 Note, that \var{t} and \var{t_max} are regular floating point values and not
390 \Data objects. Yet they are treated the same way by the \class{DataManager}.
391
392 After this initialization step the script enters the main simulation loop
393 where calculations are performed.
394 When these are finalized for a time step we call the \member{addData} method
395 to let the manager know which variables to store on disk.
396 This does not actually save the data yet and it is allowed to call
397 \member{addData} more than once to add information incrementally, e.g. from
398 separate functions that have access to the \class{DataManager} instance.
399 Once all variables have been added the \member{export} method has to be called
400 to flush all data to disk and clear the manager.
401 In this example, this call dumps \var{mydomain} and \var{val} to files
402 in a restart directory and also stores \var{t} and \var{t_max} on disk.
403 Additionally, it generates a \VTK file for visualization of the data.
404 If the script would stop running before its completion for some reason (e.g.
405 because its runtime limit was exceeded in a batch job environment), you could
406 simply run it again and it would resume at the point it stopped before.
407
408 \section{\escript Classes}
409
410 \subsection{The \Domain class}
411 \begin{classdesc}{Domain}{}
412 A \Domain object is used to describe a geometric region together with
413 a way of representing functions over this region.
414 The \Domain class provides an abstract interface to the domain of \FunctionSpace and \Data objects.
415 \Domain needs to be subclassed in order to provide a complete implementation.
416 \end{classdesc}
417
418 \vspace{1em}\noindent The following methods are available:
419 \begin{methoddesc}[Domain]{getDim}{}
420 returns the number of spatial dimensions of the \Domain.
421 \end{methoddesc}
422 %
423 \begin{methoddesc}[Domain]{dump}{filename}
424 writes the \Domain to the file \var{filename} using the \netCDF file format.
425 \end{methoddesc}
426 %
427 \begin{methoddesc}[Domain]{getX}{}
428 returns the locations in the \Domain. The \FunctionSpace of the returned
429 \Data object is chosen by the \Domain implementation. Typically it will be
430 in the \ContinuousFunction.
431 \end{methoddesc}
432 %
433 \begin{methoddesc}[Domain]{setX}{newX}
434 assigns new locations to the \Domain. \var{newX} has to have \Shape $(d,)$
435 where $d$ is the spatial dimensionality of the domain. Typically \var{newX}
436 must be in the \ContinuousFunction but the space actually to be used
437 depends on the \Domain implementation. Not all domain families support
438 setting locations.
439 \end{methoddesc}
440 %
441 \begin{methoddesc}[Domain]{getNormal}{}
442 returns the surface normals on the boundary of the \Domain as a \Data object.
443 \end{methoddesc}
444 %
445 \begin{methoddesc}[Domain]{getSize}{}
446 returns the local sample size, i.e. the element diameter, as a \Data object.
447 \end{methoddesc}
448 %
449 \begin{methoddesc}[Domain]{setTagMap}{tag_name, tag}
450 defines a mapping of the tag name \var{tag_name} to the \var{tag}.
451 \end{methoddesc}
452 %
453 \begin{methoddesc}[Domain]{getTag}{tag_name}
454 returns the tag associated with the tag name \var{tag_name}.
455 \end{methoddesc}
456 %
457 \begin{methoddesc}[Domain]{isValidTagName}{tag_name}
458 returns \True if \var{tag_name} is a valid tag name.
459 \end{methoddesc}
460 %
461 \begin{methoddesc}[Domain]{__eq__}{arg}
462 (\PYTHON \var{==} operator) returns \True if the \Domain \var{arg}
463 describes the same domain, \False otherwise.
464 \end{methoddesc}
465 %
466 \begin{methoddesc}[Domain]{__ne__}{arg}
467 (\PYTHON \var{!=} operator) returns \True if the \Domain \var{arg} does
468 not describe the same domain, \False otherwise.
469 \end{methoddesc}
470 %
471 \begin{methoddesc}[Domain]{__str__}{}
472 (\PYTHON \var{str()} function) returns a string representation of the
473 \Domain.
474 \end{methoddesc}
475 %
476 \begin{methoddesc}[Domain]{onMasterProcessor}{}
477 returns \True if the process is the master process within the \MPI
478 process group used by the \Domain. This is the process with rank 0.
479 If \MPI support is not enabled the return value is always \True.
480 \end{methoddesc}
481 %
482 \begin{methoddesc}[Domain]{getMPISize}{}
483 returns the number of \MPI processes used for this \Domain. If \MPI
484 support is not enabled 1 is returned.
485 \end{methoddesc}
486 %
487 \begin{methoddesc}[Domain]{getMPIRank}{}
488 returns the rank of the process executing the statement within the
489 \MPI process group used by the \Domain. If \MPI support is not enabled
490 0 is returned.
491 \end{methoddesc}
492 %
493 \begin{methoddesc}[Domain]{MPIBarrier}{}
494 executes barrier synchronization within the \MPI process group used by
495 the \Domain. If \MPI support is not enabled, this command does nothing.
496 \end{methoddesc}
497
498 \subsection{The \FunctionSpace class}
499 \begin{classdesc}{FunctionSpace}{}
500 \FunctionSpace objects, which are instantiated by generator functions, are
501 used to define properties of \Data objects such as continuity.
502 A \Data object in a particular \FunctionSpace is represented by its values at
503 \DataSamplePoints which are defined by the type and the \Domain of the \FunctionSpace.
504 \end{classdesc}
505
506 \vspace{1em}\noindent The following methods are available:
507 %
508 \begin{methoddesc}[FunctionSpace]{getDim}{}
509 returns the spatial dimensionality of the \Domain of the \FunctionSpace.
510 \end{methoddesc}
511 %
512 \begin{methoddesc}[FunctionSpace]{getX}{}
513 returns the location of the \DataSamplePoints.
514 \end{methoddesc}
515 %
516 \begin{methoddesc}[FunctionSpace]{getNormal}{}
517 If the domain of functions in the \FunctionSpace is a hyper-manifold (e.g.
518 the boundary of a domain) the method returns the outer normal at each of
519 the \DataSamplePoints. Otherwise an exception is raised.
520 \end{methoddesc}
521 %
522 \begin{methoddesc}[FunctionSpace]{getSize}{}
523 returns a \Data object measuring the spacing of the \DataSamplePoints.
524 The size may be zero.
525 \end{methoddesc}
526 %
527 \begin{methoddesc}[FunctionSpace]{getDomain}{}
528 returns the \Domain of the \FunctionSpace.
529 \end{methoddesc}
530 %
531 \begin{methoddesc}[FunctionSpace]{setTags}{new_tag, mask}
532 assigns a new tag \var{new_tag} to all data samples where \var{mask} is
533 positive for a least one data point.
534 \var{mask} must be defined on this \FunctionSpace.
535 Use the \var{setTagMap} to assign a tag name to \var{new_tag}.
536 \end{methoddesc}
537 %
538 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
539 (\PYTHON \var{==} operator) returns \True if the \FunctionSpace \var{arg}
540 describes the same function space, \False otherwise.
541 \end{methoddesc}
542 %
543 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
544 (\PYTHON \var{!=} operator) returns \True if the \FunctionSpace \var{arg}
545 does not describe the same function space, \False otherwise.
546 \end{methoddesc}
547
548 \begin{methoddesc}[Domain]{__str__}{}
549 (\PYTHON \var{str()} function) returns a string representation of the
550 \FunctionSpace.
551 \end{methoddesc}
552
553 \noindent The following functions provide generators for \FunctionSpace objects:
554
555 \begin{funcdesc}{Function}{domain}
556 returns the \Function on the \Domain \var{domain}. \Data objects in this
557 type of \Function are defined over the whole geometric region defined by
558 \var{domain}.
559 \end{funcdesc}
560 %
561 \begin{funcdesc}{ContinuousFunction}{domain}
562 returns the \ContinuousFunction on the \Domain domain. \Data objects in
563 this type of \Function are defined over the whole geometric region defined
564 by \var{domain} and assumed to represent a continuous function.
565 \end{funcdesc}
566 %
567 \begin{funcdesc}{FunctionOnBoundary}{domain}
568 returns the \FunctionOnBoundary on the \Domain domain. \Data objects in
569 this type of \Function are defined on the boundary of the geometric region
570 defined by \var{domain}.
571 \end{funcdesc}
572 %
573 \begin{funcdesc}{FunctionOnContactZero}{domain}
574 returns the \FunctionOnContactZero the \Domain domain. \Data objects in
575 this type of \Function are defined on side 0 of a discontinuity within
576 the geometric region defined by \var{domain}.
577 The discontinuity is defined when \var{domain} is instantiated.
578 \end{funcdesc}
579 %
580 \begin{funcdesc}{FunctionOnContactOne}{domain}
581 returns the \FunctionOnContactOne on the \Domain domain. \Data objects in
582 this type of \Function are defined on side 1 of a discontinuity within
583 the geometric region defined by \var{domain}.
584 The discontinuity is defined when \var{domain} is instantiated.
585 \end{funcdesc}
586 %
587 \begin{funcdesc}{Solution}{domain}
588 returns the \SolutionFS on the \Domain domain. \Data objects in this type
589 of \Function are defined on the geometric region defined by \var{domain}
590 and are solutions of partial differential equations\index{partial differential equation}.
591 \end{funcdesc}
592 %
593 \begin{funcdesc}{ReducedSolution}{domain}
594 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in
595 this type of \Function are defined on the geometric region defined by
596 \var{domain} and are solutions of partial differential
597 equations\index{partial differential equation} with a reduced smoothness
598 for the solution approximation.
599 \end{funcdesc}
600
601 \subsection{The \Data Class}
602 \label{SEC ESCRIPT DATA}
603
604 The following table shows arithmetic operations that can be performed
605 point-wise on \Data objects:
606 \begin{center}
607 \begin{tabular}{l|l}
608 \textbf{Expression} & \textbf{Description}\\
609 \hline
610 \code{+arg} & identical to \var{arg}\index{+}\\
611 \code{-arg} & negation of \var{arg}\index{-}\\
612 \code{arg0+arg1} & adds \var{arg0} and \var{arg1}\index{+}\\
613 \code{arg0*arg1} & multiplies \var{arg0} and \var{arg1}\index{*}\\
614 \code{arg0-arg1} & subtracts \var{arg1} from \var{arg0}\index{-}\\
615 \code{arg0/arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
616 \code{arg0**arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
617 \end{tabular}
618 \end{center}
619 At least one of the arguments \var{arg0} or \var{arg1} must be a \Data object.
620 Either of the arguments may be a \Data object, a \PYTHON number or a \numpy
621 object.
622 If \var{arg0} or \var{arg1} are not defined on the same \FunctionSpace, then
623 an attempt is made to convert \var{arg0} to the \FunctionSpace of \var{arg1}
624 or to convert \var{arg1} to the \FunctionSpace of \var{arg0}.
625 Both arguments must have the same \Shape or one of the arguments may be of
626 rank 0 (a constant).
627 The returned \Data object has the same \Shape and is defined on
628 the \DataSamplePoints as \var{arg0} or \var{arg1}.
629
630 The following table shows the update operations that can be applied to
631 \Data objects:
632 \begin{center}
633 \begin{tabular}{l|l}
634 \textbf{Expression} & \textbf{Description}\\
635 \hline
636 \code{arg0+=arg1} & adds \var{arg1} to \var{arg0}\index{+}\\
637 \code{arg0*=arg1} & multiplies \var{arg0} by \var{arg1}\index{*}\\
638 \code{arg0-=arg1} & subtracts \var{arg1} from\var{arg0}\index{-}\\
639 \code{arg0/=arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
640 \code{arg0**=arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
641 \end{tabular}
642 \end{center}
643 \var{arg0} must be a \Data object. \var{arg1} must be a \Data object or an
644 object that can be converted into a \Data object.
645 \var{arg1} must have the same \Shape as \var{arg0} or have rank 0.
646 In the latter case it is assumed that the values of \var{arg1} are constant
647 for all components. \var{arg1} must be defined in the same \FunctionSpace as
648 \var{arg0} or it must be possible to interpolate \var{arg1} onto the
649 \FunctionSpace of \var{arg0}.
650
651 The \Data class supports taking slices as well as assigning new values to a
652 slice of an existing \Data object\index{slicing}.
653 The following expressions for taking and setting slices are valid:
654 \begin{center}
655 \begin{tabular}{l|ll}
656 \textbf{Rank of \var{arg}} & \textbf{Slicing expression} & \textbf{\Shape of returned and assigned object}\\
657 \hline
658 0 & no slicing & N/A\\
659 1 & \var{arg[l0:u0]} & (\var{u0}-\var{l0},)\\
660 2 & \var{arg[l0:u0,l1:u1]} & (\var{u0}-\var{l0},\var{u1}-\var{l1})\\
661 3 & \var{arg[l0:u0,l1:u1,l2:u2]} & (\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})\\
662 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})\\
663 \end{tabular}
664 \end{center}
665 Let \var{s} be the \Shape of \var{arg}, then
666 \begin{align*}
667 0 \le \var{l0} \le \var{u0} \le \var{s[0]},\\
668 0 \le \var{l1} \le \var{u1} \le \var{s[1]},\\
669 0 \le \var{l2} \le \var{u2} \le \var{s[2]},\\
670 0 \le \var{l3} \le \var{u3} \le \var{s[3]}.
671 \end{align*}
672 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be
673 present in which case $0$ is assumed.
674 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may be
675 omitted, in which case the upper limit for that dimension is assumed.
676 The lower and upper index may be identical in which case the column and the
677 lower or upper index may be dropped.
678 In the returned or in the object assigned to a slice, the corresponding
679 component is dropped, i.e. the rank is reduced by one in comparison to \var{arg}.
680 The following examples show slicing in action:
681 \begin{python}
682 t=Data(1., (4,4,6,6), Function(mydomain))
683 t[1,1,1,0]=9.
684 s=t[:2,:,2:6,5] # s has rank 3
685 s[:,:,1]=1.
686 t[:2,:2,5,5]=s[2:4,1,:2]
687 \end{python}
688
689
690 \subsection{Generation of \Data objects}
691 \begin{classdesc}{Data}{value=0, shape=(,), what=FunctionSpace(), expanded=\False}
692 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
693 The values at all \DataSamplePoints are set to the double value \var{value}.
694 If \var{expanded} is \True the \Data object is represented in expanded form.
695 \end{classdesc}
696
697 \begin{classdesc}{Data}{value, what=FunctionSpace(), expanded=\False}
698 creates a \Data object in the \FunctionSpace \var{what}.
699 The value for each data sample point is set to \var{value}, which could be a
700 \numpy object, \Data object or a dictionary of \numpy or floating point
701 numbers. In the latter case the keys must be integers and are used as tags.
702 The \Shape of the returned object is equal to the \Shape of \var{value}.
703 If \var{expanded} is \True the \Data object is represented in expanded form.
704 \end{classdesc}
705
706 \begin{classdesc}{Data}{}
707 creates an \EmptyData object. The \EmptyData object is used to indicate that
708 an argument is not present where a \Data object is required.
709 \end{classdesc}
710
711 \begin{funcdesc}{Scalar}{value=0., what=FunctionSpace(), expanded=\False}
712 returns a \Data object of rank 0 (a constant) in the \FunctionSpace \var{what}.
713 Values are initialized with \var{value}, a double precision quantity.
714 If \var{expanded} is \True the \Data object is represented in expanded form.
715 \end{funcdesc}
716
717 \begin{funcdesc}{Vector}{value=0., what=FunctionSpace(), expanded=\False}
718 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what},
719 where \var{d} is the spatial dimension of the \Domain of \var{what}.
720 Values are initialized with \var{value}, a double precision quantity.
721 If \var{expanded} is \True the \Data object is represented in expanded form.
722 \end{funcdesc}
723
724 \begin{funcdesc}{Tensor}{value=0., what=FunctionSpace(), expanded=\False}
725 returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what},
726 where \var{d} is the spatial dimension of the \Domain of \var{what}.
727 Values are initialized with \var{value}, a double precision quantity.
728 If \var{expanded} is \True the \Data object is represented in expanded form.
729 \end{funcdesc}
730
731 \begin{funcdesc}{Tensor3}{value=0., what=FunctionSpace(), expanded=\False}
732 returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what},
733 where \var{d} is the spatial dimension of the \Domain of \var{what}.
734 Values are initialized with \var{value}, a double precision quantity.
735 If \var{expanded} is \True the \Data object is represented in expanded form.
736 \end{funcdesc}
737
738 \begin{funcdesc}{Tensor4}{value=0., what=FunctionSpace(), expanded=\False}
739 returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what},
740 where \var{d} is the spatial dimension of the \Domain of \var{what}.
741 Values are initialized with \var{value}, a double precision quantity.
742 If \var{expanded} is \True the \Data object is represented in expanded form.
743 \end{funcdesc}
744
745 \begin{funcdesc}{load}{filename, domain}
746 recovers a \Data object on \Domain \var{domain} from the file \var{filename},
747 which was created by \function{dump}.
748 \end{funcdesc}
749
750 \subsection{Generating random \Data objects}
751 A \Data object filled with random values can be produced using the
752 \function{RandomData} function.
753 By default values are drawn uniformly at random from the interval $[0,1]$ (i.e.
754 including end points).
755 The function takes a shape for the data points and a \FunctionSpace for the new
756 \Data as arguments.
757 For example:
758 \begin{python}
759 from esys.finley import *
760 from esys.escript import *
761
762 domain=Rectangle(11,11)
763 fs=ContinuousFunction(domain)
764 d=RandomData((), fs)
765 \end{python}
766 would result in \var{d} being filled with scalar random data since \texttt{()}
767 is an empty tuple.
768
769 \begin{python}
770 from esys.finley import *
771 from esys.escript import *
772
773 domain=Rectangle(11,11)
774 fs=ContinuousFunction(domain)
775 d=RandomData((2,2), fs)
776 \end{python}
777 would give \var{d} the same number of data points, but each point would be a
778 $2\times 2$ matrix instead of a scalar.
779
780 By default, the seed used to generate the random values will be different each
781 time.
782 If required, you can specify a seed to ensure the same sequence is produced.
783 \begin{python}
784 from esys.dudley import *
785 from esys.escript import *
786
787 seed=-17171717
788 domain=Brick(10,10,10)
789 fs=Function(domain)
790 d=RandomData((2,2), fs, seed)
791 \end{python}
792
793 The \var{seed} can be any integer value\footnote{which can be converted to a
794 C++ long} but 0 is special.
795 A seed of zero will cause \escript to use a different seed each time.
796 Also, note that the mechanism used to produce the random values could be
797 different in different releases.
798
799 \noindent\textbf{Note for MPI users:}
800 \textsl{
801 Even if you specify a seed, you will only get the same results if you are running with the same
802 number of ranks.
803 If you change the number of ranks, you will get different values for the same seed.
804 }
805
806 \subsubsection{Smoothed randoms}
807 The \ripley domains (see Chapter \ref{chap:ripley}) support generating random
808 scalars which are smoothed using Gaussian blur.
809 To use this, you need to supply the radius of the filter kernel (in elements)
810 and the \var{sigma} value used in the filter.
811 For example:
812 \begin{python}
813 from esys.ripley import *
814 from esys.escript import *
815
816 fs=ContinuousFunction(Rectangle(11,11, d1=2,d0=2))
817 d=RandomData((), fs, 0, ('gaussian', 1, 0.5))
818 \end{python}
819 will use a filter that uses the immediate neighbours of each point with a sigma
820 value of $0.5$.
821 The random values will be different each time this code is executed due to the
822 seed of $0$.
823
824 Ripley's Gaussian smoothing has the following requirements:
825 \begin{enumerate}
826 \item If \MPI is in use, then each rank must have at least $5$ elements in
827 it \emph{in each dimension}. This value increases as the radius of
828 the blur increases.
829 \item The data being generated must be scalar. (You can generate random
830 data objects for \ripley domains with whatever shape you require, you
831 just can't smooth them unless that shape is scalar).
832 \end{enumerate}
833 An exception will be raised if either of these requirements is not met.
834
835 The components of the matrix used in the kernal for the 2D case are
836 defined\cite{gaussfilter} by:
837
838 \[ G(x,y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}} \]
839
840 \noindent For the 3D case, we use:
841
842 \[ G(x,y) = \frac{1}{(\sqrt{2\pi\sigma^2})^3} e^{-\frac{x^2+y^2+z^2}{2\sigma^2}} \]
843
844 All distances ($x$,$y$,$z$) refer to the number of points from the centre point.
845 That is, the closest neighbours have at least one distance of $1$, the next
846 ``ring'' of neighbours have at least one $2$ and so on.
847 The matrix is normalised before use.
848
849 \subsection{\Data methods}
850 These are the most frequently used methods of the \Data class.
851 A complete list of methods can be found in the reference guide,
852 see \ReferenceGuide.
853
854 \begin{methoddesc}[Data]{getFunctionSpace}{}
855 returns the \FunctionSpace of the object.
856 \end{methoddesc}
857
858 \begin{methoddesc}[Data]{getDomain}{}
859 returns the \Domain of the object.
860 \end{methoddesc}
861
862 \begin{methoddesc}[Data]{getShape}{}
863 returns the \Shape of the object as a \class{tuple} of integers.
864 \end{methoddesc}
865
866 \begin{methoddesc}[Data]{getRank}{}
867 returns the rank of the data on each data point\index{rank}.
868 \end{methoddesc}
869
870 \begin{methoddesc}[Data]{isEmpty}{}
871 returns \True if the \Data object is the \EmptyData object, \False otherwise.
872 Note that this is not the same as asking if the object contains no \DataSamplePoints.
873 \end{methoddesc}
874
875 \begin{methoddesc}[Data]{setTaggedValue}{tag_name, value}
876 assigns the \var{value} to all \DataSamplePoints which have the tag
877 assigned to \var{tag_name}. \var{value} must be an object of class
878 \class{numpy.ndarray} or must be convertible into a \class{numpy.ndarray} object.
879 \var{value} (or the corresponding \class{numpy.ndarray} object) must be of
880 rank $0$ or must have the same rank as the object.
881 If a value has already been defined for tag \var{tag_name} within the object
882 it is overwritten by the new \var{value}. If the object is expanded,
883 the value assigned to \DataSamplePoints with tag \var{tag_name} is replaced by
884 \var{value}. If no value is assigned the tag name \var{tag_name}, no value is set.
885 \end{methoddesc}
886
887 \begin{methoddesc}[Data]{dump}{filename}
888 dumps the \Data object to the file \var{filename}. The file stores the
889 function space but not the \Domain. It is the responsibility of the user to
890 save the \Domain in order to be able to recover the \Data object.
891 \end{methoddesc}
892
893 \begin{methoddesc}[Data]{__str__}{}
894 returns a string representation of the object.
895 \end{methoddesc}
896
897 \subsection{Functions of \Data objects}
898 This section lists the most important functions for \Data class objects.
899 A complete list and a more detailed description of the functionality can be
900 found on \ReferenceGuide.
901
902 \begin{funcdesc}{kronecker}{d}
903 returns a \RankTwo in \FunctionSpace \var{d} such that
904 \begin{equation}
905 \code{kronecker(d)}\left[ i,j\right] = \left\{
906 \begin{array}{l l}
907 1 & \quad \text{if $i=j$}\\
908 0 & \quad \text{otherwise}
909 \end{array}
910 \right.
911 \end{equation}
912 If \var{d} is an integer a $(d,d)$ \numpy array is returned.
913 \end{funcdesc}
914
915 \begin{funcdesc}{identityTensor}{d}
916 is a synonym for \code{kronecker} (see above).
917 \end{funcdesc}
918
919 \begin{funcdesc}{identityTensor4}{d}
920 returns a \RankFour in \FunctionSpace \var{d} such that
921 \begin{equation}
922 \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
923 \begin{array}{l l}
924 1 & \quad \text{if $i=k$ and $j=l$}\\
925 0 & \quad \text{otherwise}
926 \end{array}
927 \right.
928 \end{equation}
929 If \var{d} is an integer a $(d,d,d,d)$ \numpy array is returned.
930 \end{funcdesc}
931
932 \begin{funcdesc}{unitVector}{i,d}
933 returns a \RankOne in \FunctionSpace \var{d} such that
934 \begin{equation}
935 \code{identityTensor(d)}\left[ j \right] = \left\{
936 \begin{array}{l l}
937 1 & \quad \text{if $j=i$}\\
938 0 & \quad \text{otherwise}
939 \end{array}
940 \right.
941 \end{equation}
942 If \var{d} is an integer a $(d,)$ \numpy array is returned.
943 \end{funcdesc}
944
945 \begin{funcdesc}{Lsup}{a}
946 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute
947 values over all components and all \DataSamplePoints of \var{a}.
948 \end{funcdesc}
949
950 \begin{funcdesc}{sup}{a}
951 returns the maximum value over all components and all \DataSamplePoints of \var{a}.
952 \end{funcdesc}
953
954 \begin{funcdesc}{inf}{a}
955 returns the minimum value over all components and all \DataSamplePoints of \var{a}
956 \end{funcdesc}
957
958 \begin{funcdesc}{minval}{a}
959 returns at each data sample point the minimum value over all components.
960 \end{funcdesc}
961
962 \begin{funcdesc}{maxval}{a}
963 returns at each data sample point the maximum value over all components.
964 \end{funcdesc}
965
966 \begin{funcdesc}{length}{a}
967 returns the Euclidean norm at each data sample point.
968 For a \RankFour \var{a} this is
969 \begin{equation}
970 \code{length(a)}=\sqrt{\sum_{ijkl} \var{a} \left[i,j,k,l\right]^2}
971 \end{equation}
972 \end{funcdesc}
973
974 \begin{funcdesc}{trace}{a\optional{, axis_offset=0}}
975 returns the trace of \var{a}. This is the sum over components \var{axis_offset}
976 and \var{axis_offset+1} with the same index.
977 For instance, in the case of a \RankTwo this is
978 \begin{equation}
979 \code{trace(a)}=\sum_{i} \var{a} \left[i,i\right]
980 \end{equation}
981 and for a \RankFour and \code{axis_offset=1} this is
982 \begin{equation}
983 \code{trace(a,1)}\left[i,j\right]=\sum_{k} \var{a} \left[i,k,k,j\right]
984 \end{equation}
985 \end{funcdesc}
986
987 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
988 returns the transpose of \var{a}. This swaps the first \var{axis_offset}
989 components of \var{a} with the rest. If \var{axis_offset} is not
990 present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
991 For instance, in the case of a \RankTwo this is
992 \begin{equation}
993 \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
994 \end{equation}
995 and for a \RankFour and \code{axis_offset=1} this is
996 \begin{equation}
997 \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
998 \end{equation}
999 \end{funcdesc}
1000
1001 \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
1002 returns \var{a} but with swapped components \var{axis0} and \var{axis1}.
1003 The argument \var{a} must be at least of rank 2. For instance, if \var{a}
1004 is a \RankFour, \code{axis0=1} and \code{axis1=2}, the result is
1005 \begin{equation}
1006 \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
1007 \end{equation}
1008 \end{funcdesc}
1009
1010 \begin{funcdesc}{symmetric}{a}
1011 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
1012 \end{funcdesc}
1013
1014 \begin{funcdesc}{nonsymmetric}{a}
1015 returns the non-symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
1016 \end{funcdesc}
1017
1018 \begin{funcdesc}{inverse}{a}
1019 return the inverse of \var{a} so that
1020 \begin{equation}
1021 \code{matrix_mult(inverse(a),a)=kronecker(d)}
1022 \end{equation}
1023 if \var{a} has shape \code{(d,d)}. The current implementation is restricted to
1024 arguments of shape \code{(2,2)} and \code{(3,3)}.
1025 \end{funcdesc}
1026
1027 \begin{funcdesc}{eigenvalues}{a}
1028 returns the eigenvalues of \var{a} so that
1029 \begin{equation}
1030 \code{matrix_mult(a,V)=e[i]*V}
1031 \end{equation}
1032 where \code{e=eigenvalues(a)} and \var{V} is a suitable non-zero vector.
1033 The eigenvalues are ordered in increasing size.
1034 The argument \var{a} has to be symmetric, i.e. \code{a=symmetric(a)}.
1035 The current implementation is restricted to arguments of shape \code{(2,2)}
1036 and \code{(3,3)}.
1037 \end{funcdesc}
1038
1039 \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
1040 returns the eigenvalues and eigenvectors of \var{a}.
1041 \begin{equation}
1042 \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
1043 \end{equation}
1044 where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are
1045 orthogonal and normalized, i.e.
1046 \begin{equation}
1047 \code{matrix_mult(transpose(V),V)=kronecker(d)}
1048 \end{equation}
1049 if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing
1050 size. The argument \var{a} has to be the symmetric, i.e. \code{a=symmetric(a)}.
1051 The current implementation is restricted to arguments of shape \code{(2,2)}
1052 and \code{(3,3)}.
1053 \end{funcdesc}
1054
1055 \begin{funcdesc}{maximum}{*a}
1056 returns the maximum value over all arguments at all \DataSamplePoints and for each component.
1057 \begin{equation}
1058 \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
1059 \end{equation}
1060 at all \DataSamplePoints.
1061 \end{funcdesc}
1062
1063 \begin{funcdesc}{minimum}{*a}
1064 returns the minimum value over all arguments at all \DataSamplePoints and for each component.
1065 \begin{equation}
1066 \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
1067 \end{equation}
1068 at all \DataSamplePoints.
1069 \end{funcdesc}
1070
1071 \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
1072 cuts back \var{a} into the range between \var{minval} and \var{maxval}.
1073 A value in the returned object equals \var{minval} if the corresponding value
1074 of \var{a} is less than \var{minval}, equals \var{maxval} if the corresponding
1075 value of \var{a} is greater than \var{maxval}, or corresponding value of
1076 \var{a} otherwise.
1077 \end{funcdesc}
1078
1079 \begin{funcdesc}{inner}{a0, a1}
1080 returns the inner product of \var{a0} and \var{a1}. For instance in the
1081 case of a \RankTwo:
1082 \begin{equation}
1083 \code{inner(a)}=\sum_{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
1084 \end{equation}
1085 and for a \RankFour:
1086 \begin{equation}
1087 \code{inner(a)}=\sum_{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
1088 \end{equation}
1089 \end{funcdesc}
1090
1091 \begin{funcdesc}{matrix_mult}{a0, a1}
1092 returns the matrix product of \var{a0} and \var{a1}.
1093 If \var{a1} is a \RankOne this is
1094 \begin{equation}
1095 \code{matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
1096 \end{equation}
1097 and if \var{a1} is a \RankTwo this is
1098 \begin{equation}
1099 \code{matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
1100 \end{equation}
1101 \end{funcdesc}
1102
1103 \begin{funcdesc}{transposed_matrix_mult}{a0, a1}
1104 returns the matrix product of the transposed of \var{a0} and \var{a1}.
1105 The function is equivalent to \code{matrix_mult(transpose(a0),a1)}.
1106 If \var{a1} is a \RankOne this is
1107 \begin{equation}
1108 \code{transposed_matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k\right]
1109 \end{equation}
1110 and if \var{a1} is a \RankTwo this is
1111 \begin{equation}
1112 \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k,j\right]
1113 \end{equation}
1114 \end{funcdesc}
1115
1116 \begin{funcdesc}{matrix_transposed_mult}{a0, a1}
1117 returns the matrix product of \var{a0} and the transposed of \var{a1}.
1118 The function is equivalent to \code{matrix_mult(a0,transpose(a1))}.
1119 If \var{a1} is a \RankTwo this is
1120 \begin{equation}
1121 \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[j,k\right]
1122 \end{equation}
1123 \end{funcdesc}
1124
1125 \begin{funcdesc}{outer}{a0, a1}
1126 returns the outer product of \var{a0} and \var{a1}.
1127 For instance, if both, \var{a0} and \var{a1} is a \RankOne then
1128 \begin{equation}
1129 \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
1130 \end{equation}
1131 and if \var{a0} is a \RankOne and \var{a1} is a \RankThree:
1132 \begin{equation}
1133 \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
1134 \end{equation}
1135 \end{funcdesc}
1136
1137 \begin{funcdesc}{tensor_mult}{a0, a1}
1138 returns the tensor product of \var{a0} and \var{a1}.
1139 If \var{a1} is a \RankTwo this is
1140 \begin{equation}
1141 \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]
1142 \end{equation}
1143 and if \var{a1} is a \RankFour this is
1144 \begin{equation}
1145 \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]
1146 \end{equation}
1147 \end{funcdesc}
1148
1149 \begin{funcdesc}{transposed_tensor_mult}{a0, a1}
1150 returns the tensor product of the transposed of \var{a0} and \var{a1}.
1151 The function is equivalent to \code{tensor_mult(transpose(a0),a1)}.
1152 If \var{a1} is a \RankTwo this is
1153 \begin{equation}
1154 \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]
1155 \end{equation}
1156 and if \var{a1} is a \RankFour this is
1157 \begin{equation}
1158 \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]
1159 \end{equation}
1160 \end{funcdesc}
1161
1162 \begin{funcdesc}{tensor_transposed_mult}{a0, a1}
1163 returns the tensor product of \var{a0} and the transposed of \var{a1}.
1164 The function is equivalent to \code{tensor_mult(a0,transpose(a1))}.
1165 If \var{a1} is a \RankTwo this is
1166 \begin{equation}
1167 \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]
1168 \end{equation}
1169 and if \var{a1} is a \RankFour this is
1170 \begin{equation}
1171 \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]
1172 \end{equation}
1173 \end{funcdesc}
1174
1175 \begin{funcdesc}{grad}{a\optional{, where=None}}
1176 returns the gradient of \var{a}. If \var{where} is present the gradient will
1177 be calculated in the \FunctionSpace \var{where}, otherwise a default
1178 \FunctionSpace is used. In case that \var{a} is a \RankTwo one has
1179 \begin{equation}
1180 \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x_{k}}
1181 \end{equation}
1182 \end{funcdesc}
1183
1184 \begin{funcdesc}{integrate}{a\optional{, where=None}}
1185 returns the integral of \var{a} where the domain of integration is defined by
1186 the \FunctionSpace of \var{a}. If \var{where} is present the argument is
1187 interpolated into \FunctionSpace \var{where} before integration.
1188 For instance in the case of a \RankTwo in \ContinuousFunction it is
1189 \begin{equation}
1190 \code{integrate(a)}\left[i,j\right]=\int_{\Omega}\var{a} \left[i,j\right] \; d\Omega
1191 \end{equation}
1192 where $\Omega$ is the spatial domain and $d\Omega$ volume integration.
1193 To integrate over the boundary of the domain one uses
1194 \begin{equation}
1195 \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int_{\partial \Omega} a\left[i,j\right] \; ds
1196 \end{equation}
1197 where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or
1198 line integration.
1199 \end{funcdesc}
1200
1201 \begin{funcdesc}{interpolate}{a, where}
1202 interpolates argument \var{a} into the \FunctionSpace \var{where}.
1203 \end{funcdesc}
1204
1205 \begin{funcdesc}{div}{a\optional{, where=None}}
1206 returns the divergence of \var{a}:
1207 \begin{equation}
1208 \code{div(a)=trace(grad(a),where)}
1209 \end{equation}
1210 \end{funcdesc}
1211
1212 \begin{funcdesc}{jump}{a\optional{, domain=None}}
1213 returns the jump of \var{a} over the discontinuity in its domain or if
1214 \Domain \var{domain} is present in \var{domain}.
1215 \begin{equation}
1216 \begin{array}{rcl}
1217 \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
1218 & & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
1219 \end{array}
1220 \end{equation}
1221 \end{funcdesc}
1222
1223 \begin{funcdesc}{L2}{a}
1224 returns the $L^2$-norm of \var{a} in its \FunctionSpace. This is
1225 \begin{equation}
1226 \code{L2(a)=integrate(length(a)}^2\code{)} \; .
1227 \end{equation}
1228 \end{funcdesc}
1229
1230 \noindent The following functions operate ``point-wise''.
1231 That is, the operation is applied to each component of each point individually.
1232
1233 \begin{funcdesc}{sin}{a}
1234 applies the sine function to \var{a}.
1235 \end{funcdesc}
1236
1237 \begin{funcdesc}{cos}{a}
1238 applies the cosine function to \var{a}.
1239 \end{funcdesc}
1240
1241 \begin{funcdesc}{tan}{a}
1242 applies the tangent function to \var{a}.
1243 \end{funcdesc}
1244
1245 \begin{funcdesc}{asin}{a}
1246 applies the arc (inverse) sine function to \var{a}.
1247 \end{funcdesc}
1248
1249 \begin{funcdesc}{acos}{a}
1250 applies the arc (inverse) cosine function to \var{a}.
1251 \end{funcdesc}
1252
1253 \begin{funcdesc}{atan}{a}
1254 applies the arc (inverse) tangent function to \var{a}.
1255 \end{funcdesc}
1256
1257 \begin{funcdesc}{sinh}{a}
1258 applies the hyperbolic sine function to \var{a}.
1259 \end{funcdesc}
1260
1261 \begin{funcdesc}{cosh}{a}
1262 applies the hyperbolic cosine function to \var{a}.
1263 \end{funcdesc}
1264
1265 \begin{funcdesc}{tanh}{a}
1266 applies the hyperbolic tangent function to \var{a}.
1267 \end{funcdesc}
1268
1269 \begin{funcdesc}{asinh}{a}
1270 applies the arc (inverse) hyperbolic sine function to \var{a}.
1271 \end{funcdesc}
1272
1273 \begin{funcdesc}{acosh}{a}
1274 applies the arc (inverse) hyperbolic cosine function to \var{a}.
1275 \end{funcdesc}
1276
1277 \begin{funcdesc}{atanh}{a}
1278 applies the arc (inverse) hyperbolic tangent function to \var{a}.
1279 \end{funcdesc}
1280
1281 \begin{funcdesc}{exp}{a}
1282 applies the exponential function to \var{a}.
1283 \end{funcdesc}
1284
1285 \begin{funcdesc}{sqrt}{a}
1286 applies the square root function to \var{a}.
1287 \end{funcdesc}
1288
1289 \begin{funcdesc}{log}{a}
1290 takes the natural logarithm of \var{a}.
1291 \end{funcdesc}
1292
1293 \begin{funcdesc}{log10}{a}
1294 takes the base-$10$ logarithm of \var{a}.
1295 \end{funcdesc}
1296
1297 \begin{funcdesc}{sign}{a}
1298 applies the sign function to \var{a}. The result is $1$ where \var{a} is
1299 positive, $-1$ where \var{a} is negative, and $0$ otherwise.
1300 \end{funcdesc}
1301
1302 \begin{funcdesc}{wherePositive}{a}
1303 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
1304 \end{funcdesc}
1305
1306 \begin{funcdesc}{whereNegative}{a}
1307 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
1308 \end{funcdesc}
1309
1310 \begin{funcdesc}{whereNonNegative}{a}
1311 returns a function which is $1$ where \var{a} is non-negative and $0$ otherwise.
1312 \end{funcdesc}
1313
1314 \begin{funcdesc}{whereNonPositive}{a}
1315 returns a function which is $1$ where \var{a} is non-positive and $0$ otherwise.
1316 \end{funcdesc}
1317
1318 \begin{funcdesc}{whereZero}{a\optional{, tol=None\optional{, rtol=1.e-8}}}
1319 returns a function which is $1$ where \var{a} equals zero with tolerance
1320 \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum
1321 value of \var{a} times \var{rtol} is used.
1322 \end{funcdesc}
1323
1324 \begin{funcdesc}{whereNonZero}{a\optional{, tol=None\optional{, rtol=1.e-8}}}
1325 returns a function which is $1$ where \var{a} is non-zero with tolerance
1326 \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum
1327 value of \var{a} times \var{rtol} is used.
1328 \end{funcdesc}
1329
1330 \subsection{Interpolating Data}
1331 \index{interpolateTable}
1332 \label{sec:interpolation}
1333 In some cases, it may be useful to produce Data objects which fit some user
1334 defined function.
1335 Manually modifying each value in the Data object is not a good idea since it
1336 depends on knowing the location and order of each data point in the domain.
1337 Instead, \escript can use an interpolation table to produce a \Data object.
1338
1339 The following example is available as \file{int_save.py} in the \ExampleDirectory.
1340 We will produce a \Data object which approximates a sine curve.
1341
1342 \begin{python}
1343 from esys.escript import saveDataCSV, sup, interpolateTable
1344 import numpy
1345 from esys.finley import Rectangle
1346
1347 n=4
1348 r=Rectangle(n,n)
1349 x=r.getX()
1350 toobig=100
1351 \end{python}
1352
1353 \noindent First we produce an interpolation table:
1354 \begin{python}
1355 sine_table=[0, 0.70710678118654746, 1, 0.70710678118654746, 0,
1356 -0.70710678118654746, -1, -0.70710678118654746, 0]
1357 \end{python}
1358 %
1359 We wish to identify $0$ and $1$ with the ends of the curve, that is
1360 with the first and eighth value in the table.
1361
1362 \begin{python}
1363 numslices=len(sine_table)-1
1364 minval=0.
1365 maxval=1.
1366 step=sup(maxval-minval)/numslices
1367 \end{python}
1368 %
1369 So the values $v$ from the input lie in the interval
1370 \var{minval} $\leq v <$ \var{maxval}.
1371 \var{step} represents the gap (in the input range) between entries in the table.
1372 By default, values of $v$ outside the table argument range (minval, maxval)
1373 will be pushed back into the range, i.e. if $v <$ \var{minval} the value
1374 \var{minval} will be used to evaluate the table.
1375 Similarly, for values $v>$ \var{maxval} the value \var{maxval} is used.
1376
1377 Now we produce our new \Data object:
1378
1379 \begin{python}
1380 result=interpolateTable(sine_table, x[0], minval, step, toobig)
1381 \end{python}
1382 Any values which interpolate to larger than \var{toobig} will raise an
1383 exception. You can switch on boundary checking by adding
1384 \code{check_boundaries=True} to the argument list.
1385
1386 Now consider a 2D example. We will interpolate from a plane where $\forall x,y\in[0,9]:(x,y)=x+y\cdot10$.
1387
1388 \begin{python}
1389 from esys.escript import whereZero
1390 table2=[]
1391 for y in range(0,10):
1392 r=[]
1393 for x in range(0,10):
1394 r.append(x+y*10)
1395 table2.append(r)
1396 xstep=(maxval-minval)/(10-1)
1397 ystep=(maxval-minval)/(10-1)
1398
1399 xmin=minval
1400 ymin=minval
1401
1402 result2=interpolateTable(table2, x2, (xmin, ymin), (xstep, ystep), toobig)
1403 \end{python}
1404
1405 We can check the values using \function{whereZero}.
1406 For example, for $x=0$:
1407 \begin{python}
1408 print(result2*whereZero(x[0]))
1409 \end{python}
1410
1411 Finally let us look at a 3D example. Note that the parameter tuples should be
1412 $(x,y,z)$ but that in the interpolation table, $x$ is the innermost dimension.
1413 \begin{python}
1414 b=Brick(n,n,n)
1415 x3=b.getX()
1416 toobig=1000000
1417
1418 table3=[]
1419 for z in range(0,10):
1420 face=[]
1421 for y in range(0,10):
1422 r=[]
1423 for x in range(0,10):
1424 r.append(x+y*10+z*100)
1425 face.append(r)
1426 table3.append(face);
1427
1428 zstep=(maxval-minval)/(10-1)
1429
1430 zmin=minval
1431
1432 result3=interpolateTable(table3, x3, (xmin, ymin, zmin), (xstep, ystep, zstep), toobig)
1433 \end{python}
1434
1435
1436 \subsubsection{Non-uniform Interpolation}
1437 Non-uniform interpolation is also supported for the one dimensional case.
1438 \begin{python}
1439 Data.nonuniformInterpolate(in, out, check_boundaries)
1440 Data.nonuniformSlope(in, out, check_boundaries)
1441 \end{python}
1442
1443 Will produce a new \Data object by mapping the given \Data object through the user-defined function
1444 specified by \texttt{in} and \texttt{out}.
1445 The \ldots Interpolate version gives the value of the function at the specified point and the
1446 \ldots Slope version gives the slope at those points.
1447 The check_boundaries boolean argument specifies what the function should do if the \Data object contains
1448 values outside the range specified by the \texttt{in} parameter.
1449 If the argument is \texttt{False}, then those datapoints will be interpolated to the value of the edge
1450 they are closest to (or assigned a slope of zero).
1451 If the argument is \texttt{True}, then an exception will be thrown if out of bounds values are detected.
1452 Note that the values given by the \texttt{in} parameter must be monotonically increasing.
1453
1454 \noindent For example:\\
1455 If \texttt{d} contains the values \texttt{\{1,2,3,4,5\}}, then
1456 \begin{python}
1457 d.nonuniformInterpolate([1.5, 2, 2.8, 4.6], [4, 5, -1, 1], False)
1458 \end{python}
1459 would produce a \Data object containing \texttt{\{4, 5, -0.7777, 0.3333, 1\}}.\\
1460 A similar call to \texttt{nonuniformSlope} would produce a \Data object containing \texttt{\{0, 2, 1.1111, 1.1111, 0\}}.
1461 %
1462 %
1463 % We will interpolate a surface such that the bottom
1464 % edge is the sine curve described above.
1465 % The amplitude of the curve decreases as we move towards the top edge.
1466 % Our interpolation table will have three rows:
1467 %
1468 % \begin{python}
1469 % st=numpy.array(sine_table)
1470 % table=[st, 0.5*st, 0*st]
1471 % \end{python}
1472 % %
1473 % The use of \numpy and multiplication here is just to save typing.
1474 %
1475 % % result2=x1.interpolateTable(table, 0, 0.55, x0, minval, step, toobig)
1476 % \begin{python}
1477 % result=interpolateTable(table, x (minval,0), (0.55, step), toobig)
1478 % \end{python}
1479 %
1480 % In the 2D case the start and step parameters are tuples $(x,y)$.
1481 % By default, if a point is specified which is outside the boundary, then
1482 % \var{interpolateTable} will operate as if the point was on the boundary.
1483 % Passing \code{check_boundaries=True} will lead to the rejection of any points
1484 % outside the boundaries by \var{interpolateTable}.
1485 %
1486 % This method can also be called with three dimensional tables and \Data objects.
1487 % Tuples should be ordered $(x,y,z)$.
1488
1489 \subsection{The \var{DataManager} Class}
1490 \label{sec:datamanager}
1491
1492 The \var{DataManager} class can be used to conveniently add checkpoint/restart
1493 functionality to \escript simulations.
1494 Once an instance is created \Data objects and other values can be added and
1495 dumped to disk by a single method call.
1496 If required the object can be set up to also save the data in a format suitable
1497 for visualization.
1498 Internally the \var{DataManager} interfaces with \weipa for this.
1499
1500 \begin{classdesc}{DataManager}{formats=[RESTART], work_dir=".", restart_prefix="restart", do_restart=\True}
1501 initializes a new \var{DataManager} object which can be used to save,
1502 restore and export simulation data in a number of formats.
1503 All files and directories saved or restored by this object are located
1504 under the directory specified by \var{work_dir}.
1505 If \var{RESTART} is specified in \var{formats}, the \var{DataManager} will
1506 look for directories whose name starts with \var{restart_prefix}.
1507 In case \var{do_restart} is \True, the last of these directories is used
1508 to restore simulation data while all others are deleted.
1509 If \var{do_restart} is \False, then all of those directories are deleted.
1510 The \var{restart_prefix} and \var{do_restart} parameters are ignored if
1511 \var{RESTART} is not specified in \var{formats}.
1512 \end{classdesc}
1513
1514 \noindent Valid values for the \var{formats} parameter are:
1515 \begin{memberdesc}[DataManager]{RESTART}
1516 enables writing of checkpoint files to be able to continue simulations
1517 as explained in the class description.
1518 \end{memberdesc}
1519 \begin{memberdesc}[DataManager]{SILO}
1520 exports simulation data in the \SILO file format. \escript must have
1521 been compiled with \SILO support for this to work.
1522 \end{memberdesc}
1523 \begin{memberdesc}[DataManager]{VISIT}
1524 enables the \VisIt simulation interface which allows connecting to and
1525 interacting with the running simulation from a compatible \VisIt client.
1526 \escript must have been compiled with \VisIt (version 2) support and the
1527 version of the client has to match the version used at compile time.
1528 In order to connect to the simulation the client needs to have access and
1529 load the file \file{escriptsim.sim2} located under the work directory.
1530 \end{memberdesc}
1531 \begin{memberdesc}[DataManager]{VTK}
1532 exports simulation data in the \VTK file format.
1533 \end{memberdesc}
1534
1535 \noindent The \var{DataManager} class has the following methods:
1536 \begin{methoddesc}[DataManager]{addData}{**data}
1537 adds \Data objects and other data to the manager. Calling this method does
1538 not save or export the data yet so it is allowed to incrementally add data
1539 at various points in the simulation script if required.
1540 Note, that only a single domain is supported so all \Data objects have to
1541 be defined on the same one or an exception is raised.
1542 \end{methoddesc}
1543
1544 \begin{methoddesc}[DataManager]{setDomain}{domain}
1545 explicitly sets the domain for this manager.
1546 It is generally not required to call this method directly.
1547 Instead, the \var{addData} method will set the domain used by the \Data
1548 objects.
1549 An exception is raised if the domain was set to a different domain before
1550 (explicitly or implicitly).
1551 \end{methoddesc}
1552
1553 \begin{methoddesc}[DataManager]{hasData}{}
1554 returns \True if the manager has loaded simulation data for a restart.
1555 \end{methoddesc}
1556
1557 \begin{methoddesc}[DataManager]{getDomain}{}
1558 returns the domain as recovered from a restart.
1559 \end{methoddesc}
1560
1561 \begin{methoddesc}[DataManager]{getValue}{value_name}
1562 returns a \Data object or other value with the name \var{value_name} that
1563 has been recovered after a restart.
1564 \end{methoddesc}
1565
1566 \begin{methoddesc}[DataManager]{getCycle}{}
1567 returns the export cycle, i.e. the number of times \var{export()} has been
1568 called.
1569 \end{methoddesc}
1570
1571 \begin{methoddesc}[DataManager]{setCheckpointFrequency}{freq}
1572 sets the frequency with which checkpoint files are created. This is only
1573 useful if the \var{DataManager} object was created with at least one other
1574 format next to \var{RESTART}. The frequency is 1 by default which means
1575 that checkpoint files are created every time \var{export()} is called.
1576 Unlike visualization output, a simulation checkpoint is usually not
1577 required at every time step. Thus, the frequency can be decreased by
1578 calling this method with $\var{freq}>1$ which would then create restart
1579 files every \var{freq} times \var{export()} is called.
1580 \end{methoddesc}
1581
1582 \begin{methoddesc}[DataManager]{setTime}{time}
1583 sets the simulation time stamp. This floating point number is stored in
1584 the metadata of exported data but not used by \var{RESTART}.
1585 \end{methoddesc}
1586
1587 \begin{methoddesc}[DataManager]{setMeshLabels}{x, y, z=""}
1588 sets labels for the mesh axes. These are currently only used by the \SILO
1589 exporter.
1590 \end{methoddesc}
1591
1592 \begin{methoddesc}[DataManager]{setMeshUnits}{x, y, z=""}
1593 sets units for the mesh axes. These are currently only used by the \SILO
1594 exporter.
1595 \end{methoddesc}
1596
1597 \begin{methoddesc}[DataManager]{setMetadataSchemaString}{schema, metadata=""}
1598 sets metadata namespaces and the corresponding metadata. These are
1599 currently only used by the \VTK exporter.
1600 \var{schema} is a dictionary that maps prefixes to namespace names, e.g.\\
1601 \code{\{"gml": "http://www.opengis.net/gml"\}} and \var{metadata} is a
1602 string with the actual content which will be enclosed in \var{<MetaData>}
1603 tags.
1604 \end{methoddesc}
1605
1606 \begin{methoddesc}[DataManager]{export}{}
1607 executes the actual data export. Depending on the \var{formats} parameter
1608 used in the constructor all data added by \var{addData()} is written to
1609 disk (\var{RESTART,SILO,VTK}) or made available through the \VisIt
1610 simulation interface (\var{VISIT}).
1611 At least the domain must be set for something to be exported.
1612 \end{methoddesc}
1613
1614 \subsection{Saving Data as CSV}
1615 \label{sec:savedatacsv}
1616 \index{saveDataCSV}\index{CSV}
1617 For simple post-processing, \Data objects can be saved in comma separated
1618 value (\emph{CSV}) format.
1619 If \var{mydata1} and \var{mydata2} are scalar data, the command
1620 \begin{python}
1621 saveDataCSV('output.csv', U=mydata1, V=mydata2)
1622 \end{python}
1623 will record the values in \file{output.csv} in the following format:
1624 \begin{verbatim}
1625 U, V
1626 1.0000000e+0, 2.0000000e-1
1627 5.0000000e-0, 1.0000000e+1
1628 ...
1629 \end{verbatim}
1630
1631 The names of the keyword parameters form the names of columns in the output.
1632 If the data objects are over different function spaces, then \var{saveDataCSV}
1633 will attempt to interpolate to a common function space.
1634 If this is not possible, then an exception is raised.
1635
1636 Output can be restricted using a scalar mask as follows:
1637 \begin{python}
1638 saveDataCSV('outfile.csv', U=mydata1, V=mydata2, mask=myscalar)
1639 \end{python}
1640 This command will only output those rows which correspond to to positive
1641 values of \var{myscalar}.
1642 Some aspects of the output can be tuned using additional parameters:
1643 \begin{python}
1644 saveDataCSV('data.csv', append=True, sep=' ', csep='/', mask=mymask, e=mat1)
1645 \end{python}
1646
1647 \begin{itemize}
1648 \item \var{append} -- specifies that the output should be written to the end of an existing file
1649 \item \var{sep} -- defines the separator between fields
1650 \item \var{csep} -- defines the separator between components in the header
1651 line. For example between the components of a matrix.
1652 \end{itemize}
1653 %
1654 The above command would produce output like this:
1655 \begin{verbatim}
1656 e/0/0 e/1/0 e/0/1 e/1/1
1657 1.0000000000e+00 2.0000000000e+00 3.0000000000e+00 4.0000000000e+00
1658 ...
1659 \end{verbatim}
1660
1661 Note that while the order in which rows are output can vary, all the elements
1662 in a given row always correspond to the same input.
1663 When run on more than one \MPI rank, \function{saveDataCSV} is currently
1664 limited to certain domain and function space combinations throwing an exception
1665 in other cases. Writing data on \ContinuousFunction is always supported.
1666
1667 \subsection{The \Operator Class}
1668 The \Operator class provides an abstract access to operators built
1669 within the \LinearPDE class. \Operator objects are created
1670 when a PDE is handed over to a PDE solver library and handled
1671 by the \LinearPDE object defining the PDE. The user can gain access
1672 to the \Operator of a \LinearPDE object through the \var{getOperator}
1673 method.
1674
1675 \begin{classdesc}{Operator}{}
1676 creates an empty \Operator object.
1677 \end{classdesc}
1678
1679 \begin{methoddesc}[Operator]{isEmpty}{fileName}
1680 returns \True is the object is empty, \False otherwise.
1681 \end{methoddesc}
1682
1683 \begin{methoddesc}[Operator]{resetValues}{}
1684 resets all entries in the operator.
1685 \end{methoddesc}
1686
1687 \begin{methoddesc}[Operator]{solve}{rhs}
1688 returns the solution \var{u} of: operator * \var{u} = \var{rhs}.
1689 \end{methoddesc}
1690
1691 \begin{methoddesc}[Operator]{of}{u}
1692 applies the operator to the \Data object \var{u}, i.e. performs a matrix-vector
1693 multiplication.
1694 \end{methoddesc}
1695
1696 \begin{methoddesc}[Operator]{saveMM}{fileName}\index{Matrix Market}
1697 saves the object to a Matrix Market format file with name \var{fileName}, see
1698 \url{http://math.nist.gov/MatrixMarket}
1699 \end{methoddesc}
1700
1701 \section{Physical Units}
1702 \escript provides support for physical units in the SI system\index{SI units}
1703 including unit conversion. So the user can define variables in the form
1704 \begin{python}
1705 from esys.escript.unitsSI import *
1706 l=20*m
1707 w=30*kg
1708 w2=40*lb
1709 T=100*Celsius
1710 \end{python}
1711 In the two latter cases a conversion from pounds\index{pounds} and degrees
1712 Celsius\index{Celsius} is performed into the appropriate SI units \emph{kg}
1713 and \emph{Kelvin}.
1714 In addition, composed units can be used, for instance
1715 \begin{python}
1716 from esys.escript.unitsSI import *
1717 rho=40*lb/cm**3
1718 \end{python}
1719 defines the density in the units of pounds per cubic centimeter.
1720 The value $40$ will be converted into SI units, in this case kg per cubic
1721 meter. Moreover unit prefixes are supported:
1722 \begin{python}
1723 from esys.escript.unitsSI import *
1724 p=40*Mega*Pa
1725 \end{python}
1726 The pressure \var{p} is set to 40 Mega Pascal. Units can also be converted
1727 back from the SI system into a desired unit, e.g.
1728 \begin{python}
1729 from esys.escript.unitsSI import *
1730 print(p/atm)
1731 \end{python}
1732 can be used print the pressure in units of atmosphere\index{atmosphere}.
1733
1734 The following is an incomplete list of supported physical units:
1735
1736 \begin{datadesc}{km}
1737 unit of kilometer
1738 \end{datadesc}
1739
1740 \begin{datadesc}{m}
1741 unit of meter
1742 \end{datadesc}
1743
1744 \begin{datadesc}{cm}
1745 unit of centimeter
1746 \end{datadesc}
1747
1748 \begin{datadesc}{mm}
1749 unit of millimeter
1750 \end{datadesc}
1751
1752 \begin{datadesc}{sec}
1753 unit of second
1754 \end{datadesc}
1755
1756 \begin{datadesc}{minute}
1757 unit of minute
1758 \end{datadesc}
1759
1760 \begin{datadesc}{h}
1761 unit of hour
1762 \end{datadesc}
1763
1764 \begin{datadesc}{day}
1765 unit of day
1766 \end{datadesc}
1767
1768 \begin{datadesc}{yr}
1769 unit of year
1770 \end{datadesc}
1771
1772 \begin{datadesc}{gram}
1773 unit of gram
1774 \end{datadesc}
1775
1776 \begin{datadesc}{kg}
1777 unit of kilogram
1778 \end{datadesc}
1779
1780 \begin{datadesc}{lb}
1781 unit of pound
1782 \end{datadesc}
1783
1784 \begin{datadesc}{ton}
1785 metric ton
1786 \end{datadesc}
1787
1788 \begin{datadesc}{A}
1789 unit of Ampere
1790 \end{datadesc}
1791
1792 \begin{datadesc}{Hz}
1793 unit of Hertz
1794 \end{datadesc}
1795
1796 \begin{datadesc}{N}
1797 unit of Newton
1798 \end{datadesc}
1799
1800 \begin{datadesc}{Pa}
1801 unit of Pascal
1802 \end{datadesc}
1803
1804 \begin{datadesc}{atm}
1805 unit of atmosphere
1806 \end{datadesc}
1807
1808 \begin{datadesc}{J}
1809 unit of Joule
1810 \end{datadesc}
1811
1812 \begin{datadesc}{W}
1813 unit of Watt
1814 \end{datadesc}
1815
1816 \begin{datadesc}{C}
1817 unit of Coulomb
1818 \end{datadesc}
1819
1820 \begin{datadesc}{V}
1821 unit of Volt
1822 \end{datadesc}
1823
1824 \begin{datadesc}{F}
1825 unit of Farad
1826 \end{datadesc}
1827
1828 \begin{datadesc}{Ohm}
1829 unit of Ohm
1830 \end{datadesc}
1831
1832 \begin{datadesc}{K}
1833 unit of degrees Kelvin
1834 \end{datadesc}
1835
1836 \begin{datadesc}{Celsius}
1837 unit of degrees Celsius
1838 \end{datadesc}
1839
1840 \begin{datadesc}{Fahrenheit}
1841 unit of degrees Fahrenheit
1842 \end{datadesc}
1843
1844 \noindent Supported unit prefixes:
1845
1846 \begin{datadesc}{Yotta}
1847 prefix yotta = $10^{24}$
1848 \end{datadesc}
1849
1850 \begin{datadesc}{Zetta}
1851 prefix zetta = $10^{21}$
1852 \end{datadesc}
1853
1854 \begin{datadesc}{Exa}
1855 prefix exa = $10^{18}$
1856 \end{datadesc}
1857
1858 \begin{datadesc}{Peta}
1859 prefix peta = $10^{15}$
1860 \end{datadesc}
1861
1862 \begin{datadesc}{Tera}
1863 prefix tera = $10^{12}$
1864 \end{datadesc}
1865
1866 \begin{datadesc}{Giga}
1867 prefix giga = $10^9$
1868 \end{datadesc}
1869
1870 \begin{datadesc}{Mega}
1871 prefix mega = $10^6$
1872 \end{datadesc}
1873
1874 \begin{datadesc}{Kilo}
1875 prefix kilo = $10^3$
1876 \end{datadesc}
1877
1878 \begin{datadesc}{Hecto}
1879 prefix hecto = $10^2$
1880 \end{datadesc}
1881
1882 \begin{datadesc}{Deca}
1883 prefix deca = $10^1$
1884 \end{datadesc}
1885
1886 \begin{datadesc}{Deci}
1887 prefix deci = $10^{-1}$
1888 \end{datadesc}
1889
1890 \begin{datadesc}{Centi}
1891 prefix centi = $10^{-2}$
1892 \end{datadesc}
1893
1894 \begin{datadesc}{Milli}
1895 prefix milli = $10^{-3}$
1896 \end{datadesc}
1897
1898 \begin{datadesc}{Micro}
1899 prefix micro = $10^{-6}$
1900 \end{datadesc}
1901
1902 \begin{datadesc}{Nano}
1903 prefix nano = $10^{-9}$
1904 \end{datadesc}
1905
1906 \begin{datadesc}{Pico}
1907 prefix pico = $10^{-12}$
1908 \end{datadesc}
1909
1910 \begin{datadesc}{Femto}
1911 prefix femto = $10^{-15}$
1912 \end{datadesc}
1913
1914 \begin{datadesc}{Atto}
1915 prefix atto = $10^{-18}$
1916 \end{datadesc}
1917
1918 \begin{datadesc}{Zepto}
1919 prefix zepto = $10^{-21}$
1920 \end{datadesc}
1921
1922 \begin{datadesc}{Yocto}
1923 prefix yocto = $10^{-24}$
1924 \end{datadesc}
1925
1926 \section{Utilities}
1927 The \class{FileWriter} class provides a mechanism to write data to a file.
1928 In essence, this class wraps the standard \PYTHON \class{file} class to write
1929 data that are global in \MPI to a file. In fact, data are written on the
1930 processor with \MPI rank 0 only. It is recommended to use \class{FileWriter}
1931 rather than \class{open} in order to write code that will run with and without
1932 \MPI. It is safe to use \class{open} under \MPI to \emph{read} data which are
1933 global under \MPI.
1934
1935 \begin{classdesc}{FileWriter}{fn\optional{,append=\False, \optional{createLocalFiles=\False}})}
1936 Opens a file with name \var{fn} for writing. If \var{append} is set to \True
1937 data are appended at the end of the file.
1938 If running under \MPI, only the first processor (rank==0) will open the file
1939 and write to it.
1940 If \var{createLocalFiles} is set each individual processor will create a file
1941 where for any processor with rank $> 0$ the file name is extended by its rank.
1942 This option is normally used for debugging purposes only.
1943 \end{classdesc}
1944
1945 \vspace{1em}\noindent The following methods are available:
1946 \begin{methoddesc}[FileWriter]{close}{}
1947 closes the file.
1948 \end{methoddesc}
1949 \begin{methoddesc}[FileWriter]{flush}{}
1950 flushes the internal buffer to disk.
1951 \end{methoddesc}
1952 \begin{methoddesc}[FileWriter]{write}{txt}
1953 writes string \var{txt} to the file. Note that a newline is not added.
1954 \end{methoddesc}
1955 \begin{methoddesc}[FileWriter]{writelines}{txts}
1956 writes the list \var{txts} of strings to the file.
1957 Note that newlines are not added.
1958 This method is equivalent to calling \var{write()} for each string.
1959 \end{methoddesc}
1960 \begin{memberdesc}[FileWriter]{closed}
1961 this member is \True if the file is closed.
1962 \end{memberdesc}
1963 \begin{memberdesc}[FileWriter]{mode}
1964 holds the access mode.
1965 \end{memberdesc}
1966 \begin{memberdesc}[FileWriter]{name}
1967 holds the file name.
1968 \end{memberdesc}
1969 \begin{memberdesc}[FileWriter]{newlines}
1970 holds the line separator.
1971 \end{memberdesc}
1972
1973 \noindent The following additional functions are available in the \escript
1974 module:
1975 \begin{funcdesc}{setEscriptParamInt}{name,value}
1976 assigns the integer value \var{value} to the internal Escript parameter
1977 \var{name}. This should be considered an advanced feature and it is generally
1978 not required to call this function. One parameter worth mentioning is
1979 \var{name}="TOO_MANY_LINES" which affects the conversion of \Data objects to a
1980 string. If more than \var{value} lines would be created, a condensed format is
1981 used instead which reports the minimum and maximum values and general
1982 information about the \Data object rather than all values.
1983 \end{funcdesc}
1984
1985 \begin{funcdesc}{getEscriptParamInt}{name}
1986 returns the current value of internal Escript parameter \var{name}.
1987 \end{funcdesc}
1988
1989 \begin{funcdesc}{listEscriptParams}{a}
1990 returns a list of valid Escript parameters and their description.
1991 \end{funcdesc}
1992
1993 \begin{funcdesc}{getMPISizeWorld}{}
1994 returns the number of \MPI processes in use in the \env{MPI_COMM_WORLD}
1995 process group. If \MPI is not used 1 is returned.
1996 \end{funcdesc}
1997
1998 \begin{funcdesc}{getMPIRankWorld}{}
1999 returns the rank of the current process within the \env{MPI_COMM_WORLD}
2000 process group. If \MPI is not used 0 is returned.
2001 \end{funcdesc}
2002
2003 \begin{funcdesc}{MPIBarrierWorld}{}
2004 performs a barrier synchronization across all processes within the
2005 \env{MPI_COMM_WORLD} process group.
2006 \end{funcdesc}
2007
2008 \begin{funcdesc}{getMPIWorldMax}{a}
2009 returns the maximum value of the integer \var{a} across all processes within
2010 \env{MPI_COMM_WORLD}.
2011 \end{funcdesc}
2012

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