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1
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4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
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7 %
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9 % Licensed under the Open Software License version 3.0
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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 Expanded data is created when specifying \code{expanded=True} in the \Data
290 object constructor, while tagged data requires calling the \member{insertTaggedValue}
291 method as shown above.
292
293 Values are accessed through a sample reference number.
294 Operations on expanded \Data objects have to be performed for each sample
295 point individually.
296 When tagged values are used, the values are held in a dictionary.
297 Operations on tagged data require processing the set of tagged values only,
298 rather than processing the value for each individual sample point.
299 \escript allows any mixture of constant, tagged and expanded data in a single expression.
300
301 \subsection{Saving and Restoring Simulation Data}
302 \Data objects can be written to disk files with the \member{dump} method and
303 read back using the \member{load} method, both of which use the
304 \netCDF\cite{NETCDF} file format.
305 Use these to save data for checkpoint/restart or simply to save and reuse data
306 that was expensive to compute.
307 For instance, to save the coordinates of the data points of a
308 \ContinuousFunction to the file \file{x.nc} use
309 \begin{python}
310 x=ContinuousFunction(mydomain).getX()
311 x.dump("x.nc")
312 mydomain.dump("dom.nc")
313 \end{python}
314 To recover the object \var{x}, and you know that \var{mydomain} was an \finley
315 mesh, use
316 \begin{python}
317 from esys.finley import LoadMesh
318 mydomain=LoadMesh("dom.nc")
319 x=load("x.nc", mydomain)
320 \end{python}
321 Obviously, it is possible to execute the same steps that were originally used
322 to generate \var{mydomain} to recreate it. However, in most cases using
323 \member{dump} and \member{load} is faster, particularly if optimization has
324 been applied.
325 If \escript is running on more than one \MPI process \member{dump} will create
326 an individual file for each process containing the local data.
327 In order to avoid conflicts the file names are extended by the \MPI processor
328 rank, that is instead of one file \file{dom.nc} you would get
329 \file{dom.nc.0000}, \file{dom.nc.0001}, etc. You still call
330 \code{LoadMesh("dom.nc")} to load the domain but you have to make sure that
331 the appropriate file is accessible from the corresponding rank, and loading
332 will only succeed if you run with as many processes as were used when calling
333 \member{dump}.
334
335 The function space of the \Data is stored in \file{x.nc}.
336 If the \Data object is expanded, the number of data points in the file and of
337 the \Domain for the particular \FunctionSpace must match.
338 Moreover, the ordering of the values is checked using the reference
339 identifiers provided by \FunctionSpace on the \Domain.
340 In some cases, data points will be reordered so be aware and confirm that you
341 get what you wanted.
342
343 A newer, more flexible way of saving and restoring \escript simulation data
344 is through a \class{DataManager} class object.
345 It has the advantage of allowing to save and load not only a \Domain and
346 \Data objects but also other values\footnote{The \PYTHON \emph{pickle} module
347 is used for other types.} you compute in your simulation script.
348 Further, \class{DataManager} objects can simultaneously create files for
349 visualization so no extra calls to \code{saveVTK} etc. are needed.
350
351 The following example shows how the \class{DataManager} class can be used.
352 For an explanation of all member functions and options see the class reference
353 section \ref{sec:datamanager}.
354 \begin{python}
355 from esys.escript import DataManager, Scalar, Function
356 from esys.finley import Rectangle
357
358 dm = DataManager(formats=[DataManager.RESTART, DataManager.VTK])
359 if dm.hasData():
360 mydomain=dm.getDomain()
361 val=dm.getValue("val")
362 t=dm.getValue("t")
363 t_max=dm.getValue("t_max")
364 else:
365 mydomain=Rectangle()
366 val=Function(mydomain).getX()
367 t=0.
368 t_max=2.5
369
370 while t<t_max:
371 t+=.01
372 val=val+t/2
373 dm.addData(val=val, t=t, t_max=t_max)
374 dm.export()
375 \end{python}
376 In the constructor we specify that we want \code{RESTART} (i.e. dump) files
377 and \code{VTK} files to be saved.
378 By default, the constructor will look for previously saved \code{RESTART}
379 files under the current directory and load them.
380 We can then enquire if such files were found by calling the \member{hasData}
381 method. If it returns \True we retrieve the domain and values into local
382 variables. Otherwise the same variables are initialized with appropriate
383 values to start a new simulation.
384 Note, that \var{t} and \var{t_max} are regular floating point values and not
385 \Data objects. Yet they are treated the same way by the \class{DataManager}.
386
387 After this initialization step the script enters the main simulation loop
388 where calculations are performed.
389 When these are finalized for a time step we call the \member{addData} method
390 to let the manager know which variables to store on disk.
391 This does not actually save the data yet and it is allowed to call
392 \member{addData} more than once to add information incrementally, e.g. from
393 separate functions that have access to the \class{DataManager} instance.
394 Once all variables have been added the \member{export} method has to be called
395 to flush all data to disk and clear the manager.
396 In this example, this call dumps \var{mydomain} and \var{val} to files
397 in a restart directory and also stores \var{t} and \var{t_max} on disk.
398 Additionally, it generates a \VTK file for visualization of the data.
399 If the script would stop running before its completion for some reason (e.g.
400 because its runtime limit was exceeded in a multiuser environment), you could
401 simply run it again and it would resume at the point it stopped before.
402
403 \section{\escript Classes}
404
405 \subsection{The \Domain class}
406 \begin{classdesc}{Domain}{}
407 A \Domain object is used to describe a geometric region together with
408 a way of representing functions over this region.
409 The \Domain class provides an abstract interface to the domain of \FunctionSpace and \Data objects.
410 \Domain needs to be subclassed in order to provide a complete implementation.
411 \end{classdesc}
412
413 \noindent The following methods are available:
414 \begin{methoddesc}[Domain]{getDim}{}
415 returns the spatial dimension of the \Domain.
416 \end{methoddesc}
417 %
418 \begin{methoddesc}[Domain]{dump}{filename}
419 writes the \Domain to the file \var{filename} using the \netCDF file format.
420 \end{methoddesc}
421 %
422 \begin{methoddesc}[Domain]{getX}{}
423 returns the locations in the \Domain. The \FunctionSpace of the returned
424 \Data object is chosen by the \Domain implementation. Typically it will be
425 in the \Function.
426 \end{methoddesc}
427 %
428 \begin{methoddesc}[Domain]{setX}{newX}
429 assigns new locations to the \Domain. \var{newX} has to have \Shape $(d,)$
430 where $d$ is the spatial dimension of the domain. Typically \var{newX}
431 must be in the \ContinuousFunction but the space actually to be used
432 depends on the \Domain implementation.
433 \end{methoddesc}
434 %
435 \begin{methoddesc}[Domain]{getNormal}{}
436 returns the surface normals on the boundary of the \Domain as a \Data object.
437 \end{methoddesc}
438 %
439 \begin{methoddesc}[Domain]{getSize}{}
440 returns the local sample size, i.e. the element diameter, as a \Data object.
441 \end{methoddesc}
442 %
443 \begin{methoddesc}[Domain]{setTagMap}{tag_name, tag}
444 defines a mapping of the tag name \var{tag_name} to the \var{tag}.
445 \end{methoddesc}
446 %
447 \begin{methoddesc}[Domain]{getTag}{tag_name}
448 returns the tag associated with the tag name \var{tag_name}.
449 \end{methoddesc}
450 %
451 \begin{methoddesc}[Domain]{isValidTagName}{tag_name}
452 returns \True if \var{tag_name} is a valid tag name.
453 \end{methoddesc}
454 %
455 \begin{methoddesc}[Domain]{__eq__}{arg}
456 (\PYTHON \var{==} operator) returns \True if the \Domain \var{arg}
457 describes the same domain, \False otherwise.
458 \end{methoddesc}
459 %
460 \begin{methoddesc}[Domain]{__ne__}{arg}
461 (\PYTHON \var{!=} operator) returns \True if the \Domain \var{arg} does
462 not describe the same domain, \False otherwise.
463 \end{methoddesc}
464 %
465 \begin{methoddesc}[Domain]{__str__}{}
466 (\PYTHON \var{str()} function) returns a string representation of the
467 \Domain.
468 \end{methoddesc}
469 %
470 \begin{methoddesc}[Domain]{onMasterProcessor)}{}
471 returns \True if the processor is the master processor within the \MPI
472 processor group used by the \Domain. This is the processor with rank 0.
473 If \MPI support is not enabled the return value is always \True.
474 \end{methoddesc}
475 %
476 \begin{methoddesc}[Domain]{getMPISize}{}
477 returns the number of \MPI processors used for this \Domain. If \MPI
478 support is not enabled 1 is returned.
479 \end{methoddesc}
480 %
481 \begin{methoddesc}[Domain]{getMPIRank}{}
482 returns the rank of the processor executing the statement within the
483 \MPI processor group used by the \Domain. If \MPI support is not enabled
484 0 is returned.
485 \end{methoddesc}
486 %
487 \begin{methoddesc}[Domain]{MPIBarrier}{}
488 executes barrier synchronization within the \MPI processor group used by
489 the \Domain. If \MPI support is not enabled, this command does nothing.
490 \end{methoddesc}
491
492 \subsection{The \FunctionSpace class}
493 \begin{classdesc}{FunctionSpace}{}
494 \FunctionSpace objects, which are instantiated by generator functions, are
495 used to define properties of \Data objects such as continuity.
496 A \Data object in a particular \FunctionSpace is represented by its values at
497 \DataSamplePoints which are defined by the type and the \Domain of the \FunctionSpace.
498 \end{classdesc}
499
500 \noindent The following methods are available:
501 %
502 \begin{methoddesc}[FunctionSpace]{getDim}{}
503 returns the spatial dimension of the \Domain of the \FunctionSpace.
504 \end{methoddesc}
505 %
506 \begin{methoddesc}[FunctionSpace]{getX}{}
507 returns the location of the \DataSamplePoints.
508 \end{methoddesc}
509 %
510 \begin{methoddesc}[FunctionSpace]{getNormal}{}
511 If the domain of functions in the \FunctionSpace is a hyper-manifold (e.g.
512 the boundary of a domain) the method returns the outer normal at each of
513 the \DataSamplePoints. Otherwise an exception is raised.
514 \end{methoddesc}
515 %
516 \begin{methoddesc}[FunctionSpace]{getSize}{}
517 returns a \Data object measuring the spacing of the \DataSamplePoints.
518 The size may be zero.
519 \end{methoddesc}
520 %
521 \begin{methoddesc}[FunctionSpace]{getDomain}{}
522 returns the \Domain of the \FunctionSpace.
523 \end{methoddesc}
524 %
525 \begin{methoddesc}[FunctionSpace]{setTags}{new_tag, mask}
526 assigns a new tag \var{new_tag} to all data samples where \var{mask} is
527 positive for a least one data point.
528 \var{mask} must be defined on this \FunctionSpace.
529 Use the \var{setTagMap} to assign a tag name to \var{new_tag}.
530 \end{methoddesc}
531 %
532 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
533 (\PYTHON \var{==} operator) returns \True if the \FunctionSpace \var{arg}
534 describes the same function space, \False otherwise.
535 \end{methoddesc}
536 %
537 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
538 (\PYTHON \var{!=} operator) returns \True if the \FunctionSpace \var{arg}
539 does not describe the same function space, \False otherwise.
540 \end{methoddesc}
541
542 \begin{methoddesc}[Domain]{__str__}{}
543 (\PYTHON \var{str()} function) returns a string representation of the
544 \FunctionSpace.
545 \end{methoddesc}
546
547 \noindent The following functions provide generators for \FunctionSpace objects:
548
549 \begin{funcdesc}{Function}{domain}
550 returns the \Function on the \Domain \var{domain}. \Data objects in this
551 type of \Function are defined over the whole geometric region defined by
552 \var{domain}.
553 \end{funcdesc}
554 %
555 \begin{funcdesc}{ContinuousFunction}{domain}
556 returns the \ContinuousFunction on the \Domain domain. \Data objects in
557 this type of \Function are defined over the whole geometric region defined
558 by \var{domain} and assumed to represent a continuous function.
559 \end{funcdesc}
560 %
561 \begin{funcdesc}{FunctionOnBoundary}{domain}
562 returns the \FunctionOnBoundary on the \Domain domain. \Data objects in
563 this type of \Function are defined on the boundary of the geometric region
564 defined by \var{domain}.
565 \end{funcdesc}
566 %
567 \begin{funcdesc}{FunctionOnContactZero}{domain}
568 returns the \FunctionOnContactZero the \Domain domain. \Data objects in
569 this type of \Function are defined on side 0 of a discontinuity within
570 the geometric region defined by \var{domain}.
571 The discontinuity is defined when \var{domain} is instantiated.
572 \end{funcdesc}
573 %
574 \begin{funcdesc}{FunctionOnContactOne}{domain}
575 returns the \FunctionOnContactOne on the \Domain domain. \Data objects in
576 this type of \Function are defined on side 1 of a discontinuity within
577 the geometric region defined by \var{domain}.
578 The discontinuity is defined when \var{domain} is instantiated.
579 \end{funcdesc}
580 %
581 \begin{funcdesc}{Solution}{domain}
582 returns the \SolutionFS on the \Domain domain. \Data objects in this type
583 of \Function are defined on the geometric region defined by \var{domain}
584 and are solutions of partial differential equations\index{partial differential equation}.
585 \end{funcdesc}
586 %
587 \begin{funcdesc}{ReducedSolution}{domain}
588 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in
589 this type of \Function are defined on the geometric region defined by
590 \var{domain} and are solutions of partial differential
591 equations\index{partial differential equation} with a reduced smoothness
592 for the solution approximation.
593 \end{funcdesc}
594
595 \subsection{The \Data Class}
596 \label{SEC ESCRIPT DATA}
597
598 The following table shows arithmetic operations that can be performed
599 point-wise on \Data objects:
600 \begin{center}
601 \begin{tabular}{l|l}
602 \textbf{Expression} & \textbf{Description}\\
603 \hline
604 \code{+arg} & identical to \var{arg}\index{+}\\
605 \code{-arg} & negation of \var{arg}\index{-}\\
606 \code{arg0+arg1} & adds \var{arg0} and \var{arg1}\index{+}\\
607 \code{arg0*arg1} & multiplies \var{arg0} and \var{arg1}\index{*}\\
608 \code{arg0-arg1} & subtracts \var{arg1} from \var{arg0}\index{-}\\
609 \code{arg0/arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
610 \code{arg0**arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
611 \end{tabular}
612 \end{center}
613 At least one of the arguments \var{arg0} or \var{arg1} must be a \Data object.
614 Either of the arguments may be a \Data object, a \PYTHON number or a \numpy
615 object.
616 If \var{arg0} or \var{arg1} are not defined on the same \FunctionSpace, then
617 an attempt is made to convert \var{arg0} to the \FunctionSpace of \var{arg1}
618 or to convert \var{arg1} to the \FunctionSpace of \var{arg0}.
619 Both arguments must have the same \Shape or one of the arguments may be of
620 rank 0 (a constant).
621 The returned \Data object has the same \Shape and is defined on
622 the \DataSamplePoints as \var{arg0} or \var{arg1}.
623
624 The following table shows the update operations that can be applied to
625 \Data objects:
626 \begin{center}
627 \begin{tabular}{l|l}
628 \textbf{Expression} & \textbf{Description}\\
629 \hline
630 \code{arg0+=arg1} & adds \var{arg1} to \var{arg0}\index{+}\\
631 \code{arg0*=arg1} & multiplies \var{arg0} by \var{arg1}\index{*}\\
632 \code{arg0-=arg1} & subtracts \var{arg1} from\var{arg0}\index{-}\\
633 \code{arg0/=arg1} & divides \var{arg0} by \var{arg1}\index{/}\\
634 \code{arg0**=arg1} & raises \var{arg0} to the power of \var{arg1}\index{**}\\
635 \end{tabular}
636 \end{center}
637 \var{arg0} must be a \Data object. \var{arg1} must be a \Data object or an
638 object that can be converted into a \Data object.
639 \var{arg1} must have the same \Shape as \var{arg0} or have rank 0.
640 In the latter case it is assumed that the values of \var{arg1} are constant
641 for all components. \var{arg1} must be defined in the same \FunctionSpace as
642 \var{arg0} or it must be possible to interpolate \var{arg1} onto the
643 \FunctionSpace of \var{arg0}.
644
645 The \Data class supports taking slices as well as assigning new values to a
646 slice of an existing \Data object\index{slicing}.
647 The following expressions for taking and setting slices are valid:
648 \begin{center}
649 \begin{tabular}{l|ll}
650 \textbf{Rank of \var{arg}} & \textbf{Slicing expression} & \textbf{\Shape of returned and assigned object}\\
651 \hline
652 0 & no slicing & N/A\\
653 1 & \var{arg[l0:u0]} & (\var{u0}-\var{l0},)\\
654 2 & \var{arg[l0:u0,l1:u1]} & (\var{u0}-\var{l0},\var{u1}-\var{l1})\\
655 3 & \var{arg[l0:u0,l1:u1,l2:u2]} & (\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})\\
656 4 & \var{arg[l0:u0,l1:u1,l2:u2,l3:u3]} & (\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2},\var{u3}-\var{l3})\\
657 \end{tabular}
658 \end{center}
659 Let \var{s} be the \Shape of \var{arg}, then
660 \begin{align*}
661 0 \le \var{l0} \le \var{u0} \le \var{s[0]},\\
662 0 \le \var{l1} \le \var{u1} \le \var{s[1]},\\
663 0 \le \var{l2} \le \var{u2} \le \var{s[2]},\\
664 0 \le \var{l3} \le \var{u3} \le \var{s[3]}.
665 \end{align*}
666 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be
667 present in which case $0$ is assumed.
668 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may be
669 omitted, in which case the upper limit for that dimension is assumed.
670 The lower and upper index may be identical in which case the column and the
671 lower or upper index may be dropped.
672 In the returned or in the object assigned to a slice, the corresponding
673 component is dropped, i.e. the rank is reduced by one in comparison to \var{arg}.
674 The following examples show slicing in action:
675 \begin{python}
676 t=Data(1., (4,4,6,6), Function(mydomain))
677 t[1,1,1,0]=9.
678 s=t[:2,:,2:6,5] # s has rank 3
679 s[:,:,1]=1.
680 t[:2,:2,5,5]=s[2:4,1,:2]
681 \end{python}
682
683 \subsection{Generation of \Data objects}
684 \begin{classdesc}{Data}{value=0, shape=(,), what=FunctionSpace(), expanded=\False}
685 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
686 The values at all \DataSamplePoints are set to the double value \var{value}.
687 If \var{expanded} is \True the \Data object is represented in expanded form.
688 \end{classdesc}
689
690 \begin{classdesc}{Data}{value, what=FunctionSpace(), expanded=\False}
691 creates a \Data object in the \FunctionSpace \var{what}.
692 The value for each data sample point is set to \var{value}, which could be a
693 \numpy object, \Data object or a dictionary of \numpy or floating point
694 numbers. In the latter case the keys must be integers and are used as tags.
695 The \Shape of the returned object is equal to the \Shape of \var{value}.
696 If \var{expanded} is \True the \Data object is represented in expanded form.
697 \end{classdesc}
698
699 \begin{classdesc}{Data}{}
700 creates an \EmptyData object. The \EmptyData object is used to indicate that
701 an argument is not present where a \Data object is required.
702 \end{classdesc}
703
704 \begin{funcdesc}{Scalar}{value=0., what=FunctionSpace(), expanded=\False}
705 returns a \Data object of rank 0 (a constant) in the \FunctionSpace \var{what}.
706 Values are initialized with \var{value}, a double precision quantity.
707 If \var{expanded} is \True the \Data object is represented in expanded form.
708 \end{funcdesc}
709
710 \begin{funcdesc}{Vector}{value=0., what=FunctionSpace(), expanded=\False}
711 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what},
712 where \var{d} is the spatial dimension of the \Domain of \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}{Tensor}{value=0., what=FunctionSpace(), expanded=\False}
718 returns a \Data object of \Shape \var{(d,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}{Tensor3}{value=0., what=FunctionSpace(), expanded=\False}
725 returns a \Data object of \Shape \var{(d,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}{Tensor4}{value=0., what=FunctionSpace(), expanded=\False}
732 returns a \Data object of \Shape \var{(d,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}{load}{filename, domain}
739 recovers a \Data object on \Domain \var{domain} from the file \var{filename},
740 which was created by \function{dump}.
741 \end{funcdesc}
742
743 \subsection{\Data methods}
744 These are the most frequently used methods of the \Data class.
745 A complete list of methods can be found on \ReferenceGuide.
746
747 \begin{methoddesc}[Data]{getFunctionSpace}{}
748 returns the \FunctionSpace of the object.
749 \end{methoddesc}
750
751 \begin{methoddesc}[Data]{getDomain}{}
752 returns the \Domain of the object.
753 \end{methoddesc}
754
755 \begin{methoddesc}[Data]{getShape}{}
756 returns the \Shape of the object as a \class{tuple} of integers.
757 \end{methoddesc}
758
759 \begin{methoddesc}[Data]{getRank}{}
760 returns the rank of the data on each data point\index{rank}.
761 \end{methoddesc}
762
763 \begin{methoddesc}[Data]{isEmpty}{}
764 returns \True if the \Data object is the \EmptyData object, \False otherwise.
765 Note that this is not the same as asking if the object contains no \DataSamplePoints.
766 \end{methoddesc}
767
768 \begin{methoddesc}[Data]{setTaggedValue}{tag_name, value}
769 assigns the \var{value} to all \DataSamplePoints which have the tag
770 assigned to \var{tag_name}. \var{value} must be an object of class
771 \class{numpy.ndarray} or must be convertible into a \class{numpy.ndarray} object.
772 \var{value} (or the corresponding \class{numpy.ndarray} object) must be of
773 rank $0$ or must have the same rank as the object.
774 If a value has already been defined for tag \var{tag_name} within the object
775 it is overwritten by the new \var{value}. If the object is expanded,
776 the value assigned to \DataSamplePoints with tag \var{tag_name} is replaced by
777 \var{value}. If no value is assigned the tag name \var{tag_name}, no value is set.
778 \end{methoddesc}
779
780 \begin{methoddesc}[Data]{dump}{filename}
781 dumps the \Data object to the file \var{filename}. The file stores the
782 function space but not the \Domain. It is the responsibility of the user to
783 save the \Domain in order to be able to recover the \Data object.
784 \end{methoddesc}
785
786 \begin{methoddesc}[Data]{__str__}{}
787 returns a string representation of the object.
788 \end{methoddesc}
789
790 \subsection{Functions of \Data objects}
791 This section lists the most important functions for \Data class objects.
792 A complete list and a more detailed description of the functionality can be
793 found on \ReferenceGuide.
794
795 \begin{funcdesc}{saveVTK}{filename, **kwdata}
796 writes \Data defined by keywords to the file \var{filename} using the \VTK
797 file format. The keyword is used as an identifier. The statement
798 \begin{python}
799 saveVTK("out.vtu", temperature=T, velocity=v)
800 \end{python}
801 writes the scalar \var{T} as \var{temperature} and the vector \var{v} as
802 \var{velocity} into the file \file{out.vtu}.
803 Restrictions on the allowed combinations of \FunctionSpace apply.
804 \end{funcdesc}
805
806 \begin{funcdesc}{saveDX}{filename, **kwdata}
807 writes \Data defined by keywords to the file \var{filename} using the \OpenDX
808 file format. The keyword is used as an identifier. The statement
809 \begin{python}
810 saveDX("out.dx", temperature=T, velocity=v)
811 \end{python}
812 writes the scalar \var{T} as \var{temperature} and the vector \var{v} as
813 \var{velocity} into the file \file{out.dx}.
814 Restrictions on the allowed combinations of \FunctionSpace apply.
815 \end{funcdesc}
816
817 \begin{funcdesc}{kronecker}{d}
818 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
819 \begin{equation}
820 \code{kronecker(d)}\left[ i,j\right] = \left\{
821 \begin{array}{l l}
822 1 & \quad \text{if $i=j$}\\
823 0 & \quad \text{otherwise}
824 \end{array}
825 \right.
826 \end{equation}
827 If \var{d} is an integer a $(d,d)$ \numpy array is returned.
828 \end{funcdesc}
829
830 \begin{funcdesc}{identityTensor}{d}
831 is a synonym for \code{kronecker} (see above).
832 % returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
833 % \begin{equation}
834 % \code{identityTensor(d)}\left[ i,j\right] = \left\{
835 % \begin{array}{cc}
836 % 1 & \mbox{ if } i=j \\
837 % 0 & \mbox{ otherwise }
838 % \end{array}
839 % \right.
840 % \end{equation}
841 % If \var{d} is an integer a $(d,d)$ \numpy array is returned.
842 \end{funcdesc}
843
844 \begin{funcdesc}{identityTensor4}{d}
845 returns a \RankFour \Data object in \FunctionSpace \var{d} such that
846 \begin{equation}
847 \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
848 \begin{array}{l l}
849 1 & \quad \text{if $i=k$ and $j=l$}\\
850 0 & \quad \text{otherwise}
851 \end{array}
852 \right.
853 \end{equation}
854 If \var{d} is an integer a $(d,d,d,d)$ \numpy array is returned.
855 \end{funcdesc}
856
857 \begin{funcdesc}{unitVector}{i,d}
858 returns a \RankOne \Data object in \FunctionSpace \var{d} such that
859 \begin{equation}
860 \code{identityTensor(d)}\left[ j \right] = \left\{
861 \begin{array}{l l}
862 1 & \quad \text{if $j=i$}\\
863 0 & \quad \text{otherwise}
864 \end{array}
865 \right.
866 \end{equation}
867 If \var{d} is an integer a $(d,)$ \numpy array is returned.
868 \end{funcdesc}
869
870 \begin{funcdesc}{Lsup}{a}
871 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute
872 values over all components and all \DataSamplePoints of \var{a}.
873 \end{funcdesc}
874
875 \begin{funcdesc}{sup}{a}
876 returns the maximum value over all components and all \DataSamplePoints of \var{a}.
877 \end{funcdesc}
878
879 \begin{funcdesc}{inf}{a}
880 returns the minimum value over all components and all \DataSamplePoints of \var{a}
881 \end{funcdesc}
882
883 \begin{funcdesc}{minval}{a}
884 returns at each data sample point the minimum value over all components.
885 \end{funcdesc}
886
887 \begin{funcdesc}{maxval}{a}
888 returns at each data sample point the maximum value over all components.
889 \end{funcdesc}
890
891 \begin{funcdesc}{length}{a}
892 returns the Euclidean norm at each data sample point.
893 For a \RankFour \var{a} this is
894 \begin{equation}
895 \code{length(a)}=\sqrt{\sum_{ijkl} \var{a} \left[i,j,k,l\right]^2}
896 \end{equation}
897 \end{funcdesc}
898
899 \begin{funcdesc}{trace}{a\optional{,axis_offset=0}}
900 returns the trace of \var{a}. This is the sum over components \var{axis_offset}
901 and \var{axis_offset+1} with the same index.
902 For instance, in the case of a \RankTwo this is
903 \begin{equation}
904 \code{trace(a)}=\sum_{i} \var{a} \left[i,i\right]
905 \end{equation}
906 and for a \RankFour and \code{axis_offset=1} this is
907 \begin{equation}
908 \code{trace(a,1)}\left[i,j\right]=\sum_{k} \var{a} \left[i,k,k,j\right]
909 \end{equation}
910 \end{funcdesc}
911
912 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
913 returns the transpose of \var{a}. This swaps the first \var{axis_offset}
914 components of \var{a} with the rest. If \var{axis_offset} is not
915 present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
916 For instance, in the case of a \RankTwo this is
917 \begin{equation}
918 \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
919 \end{equation}
920 and for a \RankFour and \code{axis_offset=1} this is
921 \begin{equation}
922 \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
923 \end{equation}
924 \end{funcdesc}
925
926 \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
927 returns \var{a} but with swapped components \var{axis0} and \var{axis1}.
928 The argument \var{a} must be at least of rank 2. For instance, if \var{a}
929 is a \RankFour, \code{axis0=1} and \code{axis1=2}, the result is
930 \begin{equation}
931 \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
932 \end{equation}
933 \end{funcdesc}
934
935 \begin{funcdesc}{symmetric}{a}
936 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
937 \end{funcdesc}
938
939 \begin{funcdesc}{nonsymmetric}{a}
940 returns the non-symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
941 \end{funcdesc}
942
943 \begin{funcdesc}{inverse}{a}
944 return the inverse of \var{a} so that
945 \begin{equation}
946 \code{matrix_mult(inverse(a),a)=kronecker(d)}
947 \end{equation}
948 if \var{a} has shape \code{(d,d)}. The current implementation is restricted to
949 arguments of shape \code{(2,2)} and \code{(3,3)}.
950 \end{funcdesc}
951
952 \begin{funcdesc}{eigenvalues}{a}
953 returns the eigenvalues of \var{a} so that
954 \begin{equation}
955 \code{matrix_mult(a,V)=e[i]*V}
956 \end{equation}
957 where \code{e=eigenvalues(a)} and \var{V} is a suitable non-zero vector.
958 The eigenvalues are ordered in increasing size.
959 The argument \var{a} has to be symmetric, i.e. \code{a=symmetric(a)}.
960 The current implementation is restricted to arguments of shape \code{(2,2)}
961 and \code{(3,3)}.
962 \end{funcdesc}
963
964 \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
965 returns the eigenvalues and eigenvectors of \var{a}.
966 \begin{equation}
967 \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
968 \end{equation}
969 where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are
970 orthogonal and normalized, i.e.
971 \begin{equation}
972 \code{matrix_mult(transpose(V),V)=kronecker(d)}
973 \end{equation}
974 if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing
975 size. The argument \var{a} has to be the symmetric, i.e. \code{a=symmetric(a)}.
976 The current implementation is restricted to arguments of shape \code{(2,2)}
977 and \code{(3,3)}.
978 \end{funcdesc}
979
980 \begin{funcdesc}{maximum}{*a}
981 returns the maximum value over all arguments at all \DataSamplePoints and for each component.
982 \begin{equation}
983 \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
984 \end{equation}
985 at all \DataSamplePoints.
986 \end{funcdesc}
987
988 \begin{funcdesc}{minimum}{*a}
989 returns the minimum value over all arguments at all \DataSamplePoints and for each component.
990 \begin{equation}
991 \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
992 \end{equation}
993 at all \DataSamplePoints.
994 \end{funcdesc}
995
996 \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
997 cuts back \var{a} into the range between \var{minval} and \var{maxval}.
998 A value in the returned object equals \var{minval} if the corresponding value
999 of \var{a} is less than \var{minval}, equals \var{maxval} if the corresponding
1000 value of \var{a} is greater than \var{maxval}, or corresponding value of
1001 \var{a} otherwise.
1002 \end{funcdesc}
1003
1004 \begin{funcdesc}{inner}{a0, a1}
1005 returns the inner product of \var{a0} and \var{a1}. For instance in the
1006 case of a \RankTwo:
1007 \begin{equation}
1008 \code{inner(a)}=\sum_{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
1009 \end{equation}
1010 and for a \RankFour:
1011 \begin{equation}
1012 \code{inner(a)}=\sum_{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
1013 \end{equation}
1014 \end{funcdesc}
1015
1016 \begin{funcdesc}{matrix_mult}{a0, a1}
1017 returns the matrix product of \var{a0} and \var{a1}.
1018 If \var{a1} is a \RankOne this is
1019 \begin{equation}
1020 \code{matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
1021 \end{equation}
1022 and if \var{a1} is a \RankTwo this is
1023 \begin{equation}
1024 \code{matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
1025 \end{equation}
1026 \end{funcdesc}
1027
1028 \begin{funcdesc}{transposed_matrix_mult}{a0,a1}
1029 returns the matrix product of the transposed of \var{a0} and \var{a1}.
1030 The function is equivalent to \code{matrix_mult(transpose(a0),a1)}.
1031 If \var{a1} is a \RankOne this is
1032 \begin{equation}
1033 \code{transposed_matrix_mult(a)}\left[i\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k\right]
1034 \end{equation}
1035 and if \var{a1} is a \RankTwo this is
1036 \begin{equation}
1037 \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[k,i\right]\var{a1} \left[k,j\right]
1038 \end{equation}
1039 \end{funcdesc}
1040
1041 \begin{funcdesc}{matrix_transposed_mult}{a0,a1}
1042 returns the matrix product of \var{a0} and the transposed of \var{a1}.
1043 The function is equivalent to \code{matrix_mult(a0,transpose(a1))}.
1044 If \var{a1} is a \RankTwo this is
1045 \begin{equation}
1046 \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum_{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[j,k\right]
1047 \end{equation}
1048 \end{funcdesc}
1049
1050 \begin{funcdesc}{outer}{a0,a1}
1051 returns the outer product of \var{a0} and \var{a1}.
1052 For instance, if both, \var{a0} and \var{a1} is a \RankOne then
1053 \begin{equation}
1054 \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
1055 \end{equation}
1056 and if \var{a0} is a \RankOne and \var{a1} is a \RankThree:
1057 \begin{equation}
1058 \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
1059 \end{equation}
1060 \end{funcdesc}
1061
1062 \begin{funcdesc}{tensor_mult}{a0,a1}
1063 returns the tensor product of \var{a0} and \var{a1}.
1064 If \var{a1} is a \RankTwo this is
1065 \begin{equation}
1066 \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]
1067 \end{equation}
1068 and if \var{a1} is a \RankFour this is
1069 \begin{equation}
1070 \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]
1071 \end{equation}
1072 \end{funcdesc}
1073
1074 \begin{funcdesc}{transposed_tensor_mult}{a0,a1}
1075 returns the tensor product of the transposed of \var{a0} and \var{a1}.
1076 The function is equivalent to \code{tensor_mult(transpose(a0),a1)}.
1077 If \var{a1} is a \RankTwo this is
1078 \begin{equation}
1079 \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]
1080 \end{equation}
1081 and if \var{a1} is a \RankFour this is
1082 \begin{equation}
1083 \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]
1084 \end{equation}
1085 \end{funcdesc}
1086
1087 \begin{funcdesc}{tensor_transposed_mult}{a0,a1}
1088 returns the tensor product of \var{a0} and the transposed of \var{a1}.
1089 The function is equivalent to \code{tensor_mult(a0,transpose(a1))}.
1090 If \var{a1} is a \RankTwo this is
1091 \begin{equation}
1092 \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]
1093 \end{equation}
1094 and if \var{a1} is a \RankFour this is
1095 \begin{equation}
1096 \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]
1097 \end{equation}
1098 \end{funcdesc}
1099
1100 \begin{funcdesc}{grad}{a\optional{, where=None}}
1101 returns the gradient of \var{a}. If \var{where} is present the gradient will
1102 be calculated in the \FunctionSpace \var{where}, otherwise a default
1103 \FunctionSpace is used. In case that \var{a} is a \RankTwo one has
1104 \begin{equation}
1105 \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x_{k}}
1106 \end{equation}
1107 \end{funcdesc}
1108
1109 \begin{funcdesc}{integrate}{a\optional{ ,where=None}}
1110 returns the integral of \var{a} where the domain of integration is defined by
1111 the \FunctionSpace of \var{a}. If \var{where} is present the argument is
1112 interpolated into \FunctionSpace \var{where} before integration.
1113 For instance in the case of a \RankTwo in \ContinuousFunction it is
1114 \begin{equation}
1115 \code{integrate(a)}\left[i,j\right]=\int_{\Omega}\var{a} \left[i,j\right] \; d\Omega
1116 \end{equation}
1117 where $\Omega$ is the spatial domain and $d\Omega$ volume integration.
1118 To integrate over the boundary of the domain one uses
1119 \begin{equation}
1120 \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int_{\partial \Omega} a\left[i,j\right] \; ds
1121 \end{equation}
1122 where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or
1123 line integration.
1124 \end{funcdesc}
1125
1126 \begin{funcdesc}{interpolate}{a,where}
1127 interpolates argument \var{a} into the \FunctionSpace \var{where}.
1128 \end{funcdesc}
1129
1130 \begin{funcdesc}{div}{a\optional{ ,where=None}}
1131 returns the divergence of \var{a}:
1132 \begin{equation}
1133 \code{div(a)=trace(grad(a),where)}
1134 \end{equation}
1135 \end{funcdesc}
1136
1137 \begin{funcdesc}{jump}{a\optional{ ,domain=None}}
1138 returns the jump of \var{a} over the discontinuity in its domain or if
1139 \Domain \var{domain} is present in \var{domain}.
1140 \begin{equation}
1141 \begin{array}{rcl}
1142 \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
1143 & & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
1144 \end{array}
1145 \end{equation}
1146 \end{funcdesc}
1147
1148 \begin{funcdesc}{L2}{a}
1149 returns the $L^2$-norm of \var{a} in its \FunctionSpace. This is
1150 \begin{equation}
1151 \code{L2(a)=integrate(length(a)}^2\code{)} \; .
1152 \end{equation}
1153 \end{funcdesc}
1154
1155 \noindent The following functions operate ``point-wise''.
1156 That is, the operation is applied to each component of each point individually.
1157
1158 \begin{funcdesc}{sin}{a}
1159 applies the sine function to \var{a}.
1160 \end{funcdesc}
1161
1162 \begin{funcdesc}{cos}{a}
1163 applies the cosine function to \var{a}.
1164 \end{funcdesc}
1165
1166 \begin{funcdesc}{tan}{a}
1167 applies the tangent function to \var{a}.
1168 \end{funcdesc}
1169
1170 \begin{funcdesc}{asin}{a}
1171 applies the arc (inverse) sine function to \var{a}.
1172 \end{funcdesc}
1173
1174 \begin{funcdesc}{acos}{a}
1175 applies the arc (inverse) cosine function to \var{a}.
1176 \end{funcdesc}
1177
1178 \begin{funcdesc}{atan}{a}
1179 applies the arc (inverse) tangent function to \var{a}.
1180 \end{funcdesc}
1181
1182 \begin{funcdesc}{sinh}{a}
1183 applies the hyperbolic sine function to \var{a}.
1184 \end{funcdesc}
1185
1186 \begin{funcdesc}{cosh}{a}
1187 applies the hyperbolic cosine function to \var{a}.
1188 \end{funcdesc}
1189
1190 \begin{funcdesc}{tanh}{a}
1191 applies the hyperbolic tangent function to \var{a}.
1192 \end{funcdesc}
1193
1194 \begin{funcdesc}{asinh}{a}
1195 applies the arc (inverse) hyperbolic sine function to \var{a}.
1196 \end{funcdesc}
1197
1198 \begin{funcdesc}{acosh}{a}
1199 applies the arc (inverse) hyperbolic cosine function to \var{a}.
1200 \end{funcdesc}
1201
1202 \begin{funcdesc}{atanh}{a}
1203 applies the arc (inverse) hyperbolic tangent function to \var{a}.
1204 \end{funcdesc}
1205
1206 \begin{funcdesc}{exp}{a}
1207 applies the exponential function to \var{a}.
1208 \end{funcdesc}
1209
1210 \begin{funcdesc}{sqrt}{a}
1211 applies the square root function to \var{a}.
1212 \end{funcdesc}
1213
1214 \begin{funcdesc}{log}{a}
1215 takes the natural logarithm of \var{a}.
1216 \end{funcdesc}
1217
1218 \begin{funcdesc}{log10}{a}
1219 takes the base-$10$ logarithm of \var{a}.
1220 \end{funcdesc}
1221
1222 \begin{funcdesc}{sign}{a}
1223 applies the sign function to \var{a}. The result is $1$ where \var{a} is
1224 positive, $-1$ where \var{a} is negative, and $0$ otherwise.
1225 \end{funcdesc}
1226
1227 \begin{funcdesc}{wherePositive}{a}
1228 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
1229 \end{funcdesc}
1230
1231 \begin{funcdesc}{whereNegative}{a}
1232 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
1233 \end{funcdesc}
1234
1235 \begin{funcdesc}{whereNonNegative}{a}
1236 returns a function which is $1$ where \var{a} is non-negative and $0$ otherwise.
1237 \end{funcdesc}
1238
1239 \begin{funcdesc}{whereNonPositive}{a}
1240 returns a function which is $1$ where \var{a} is non-positive and $0$ otherwise.
1241 \end{funcdesc}
1242
1243 \begin{funcdesc}{whereZero}{a\optional{, tol=None, \optional{, rtol=1.e-8}}}
1244 returns a function which is $1$ where \var{a} equals zero with tolerance
1245 \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum
1246 value of \var{a} times \var{rtol} is used.
1247 \end{funcdesc}
1248
1249 \begin{funcdesc}{whereNonZero}{a, \optional{, tol=None, \optional{, rtol=1.e-8}}}
1250 returns a function which is $1$ where \var{a} is non-zero with tolerance
1251 \var{tol} and $0$ otherwise. If \var{tol} is not present, the absolute maximum
1252 value of \var{a} times \var{rtol} is used.
1253 \end{funcdesc}
1254
1255 \subsection{Interpolating Data}
1256 \index{interpolateTable}
1257 In some cases, it may be useful to produce Data objects which fit some user
1258 defined function.
1259 Manually modifying each value in the Data object is not a good idea since it
1260 depends on knowing the location and order of each data point in the domain.
1261 Instead, \escript can use an interpolation table to produce a \Data object.
1262
1263 The following example is available as \file{int_save.py} in the \ExampleDirectory.
1264 We will produce a \Data object which approximates a sine curve.
1265
1266 \begin{python}
1267 from esys.escript import saveDataCSV, sup
1268 import numpy
1269 from esys.finley import Rectangle
1270
1271 n=4
1272 r=Rectangle(n,n)
1273 x=r.getX()
1274 x0=x[0]
1275 x1=x[1] #we will use this later
1276 toobig=100
1277 \end{python}
1278
1279 \noindent First we produce an interpolation table:
1280 \begin{python}
1281 sine_table=[0, 0.70710678118654746, 1, 0.70710678118654746, 0,
1282 -0.70710678118654746, -1, -0.70710678118654746, 0]
1283 \end{python}
1284 %
1285 We wish to identify $0$ and $1$ with the ends of the curve, that is
1286 with the first and eighth value in the table.
1287
1288 \begin{python}
1289 numslices=len(sine_table)-1
1290 minval=0
1291 maxval=1
1292 step=sup(maxval-minval)/numslices
1293 \end{python}
1294 %
1295 So the values $v$ from the input lie in the interval
1296 \var{minval} $\leq v <$ \var{maxval}.
1297 \var{step} represents the gap (in the input range) between entries in the table.
1298 By default, values of $v$ outside the table argument range (minval, maxval)
1299 will be pushed back into the range, i.e. if $v <$ \var{minval} the value
1300 \var{minval} will be used to evaluate the table.
1301 Similarly, for values $v>$ \var{maxval} the value \var{maxval} is used.
1302
1303 Now we produce our new \Data object:
1304
1305 \begin{python}
1306 result=x0.interpolateTable(sine_table, minval, step, toobig)
1307 \end{python}
1308 Any values which interpolate to larger than \var{toobig} will raise an
1309 exception. You can switch on boundary checking by adding
1310 \code{check_boundaries=True} to the argument list.
1311
1312 Now consider a 2D example. We will interpolate a surface such that the bottom
1313 edge is the sine curve described above.
1314 The amplitude of the curve decreases as we move towards the top edge.
1315 Our interpolation table will have three rows:
1316
1317 \begin{python}
1318 st=numpy.array(sine_table)
1319 table=[st, 0.5*st, 0*st]
1320 \end{python}
1321 %
1322 The use of \numpy and multiplication here is just to save typing.
1323
1324 \begin{python}
1325 result2=x1.interpolateTable(table, 0, 0.55, x0, minval, step, toobig)
1326 \end{python}
1327
1328 In the 2D case, the parameters for the \var{x1} direction (min=0, step=0.55)
1329 come first followed by the \var{x0} data object and its parameters.
1330 By default, if a point is specified which is outside the boundary, then
1331 \var{interpolateTable} will operate as if the point was on the boundary.
1332 Passing \code{check_boundaries=True} will cause \var{interpolateTable} to
1333 reject any points outside the boundaries.
1334
1335 \subsection{The \var{DataManager} Class}
1336 \label{sec:datamanager}
1337
1338 \begin{classdesc}{DataManager}{formats=[RESTART], work_dir=".", restart_prefix="restart", do_restart=\True}
1339 initializes a new \var{DataManager} object which can be used to save,
1340 restore and export simulation data in a number of formats.
1341 All files and directories saved or restored by this object are located
1342 under the directory specified by \var{work_dir}.
1343 If \var{RESTART} is specified in \var{formats}, the \var{DataManager} will
1344 look for directories whose name starts with \var{restart_prefix}.
1345 In case \var{do_restart} is \True, the last of these directories is used
1346 to restore simulation data while all others are deleted.
1347 If \var{do_restart} is \False, then all of those directories are deleted.
1348 The \var{restart_prefix} and \var{do_restart} parameters are ignored if
1349 \var{RESTART} is not specified in \var{formats}.
1350 \end{classdesc}
1351
1352 \noindent Valid values for the \var{formats} parameter are:
1353 \begin{memberdesc}[DataManager]{RESTART}
1354 enables writing of checkpoint files to be able to continue simulations
1355 as explained in the class description.
1356 \end{memberdesc}
1357 \begin{memberdesc}[DataManager]{SILO}
1358 exports simulation data in the \SILO file format. \escript must have
1359 been compiled with \SILO support for this to work.
1360 \end{memberdesc}
1361 \begin{memberdesc}[DataManager]{VISIT}
1362 enables the \VisIt simulation interface which allows connecting to and
1363 interacting with the running simulation from a compatible \VisIt client.
1364 \escript must have been compiled with \VisIt (version 2) support and the
1365 version of the client has to match the version used at compile time.
1366 In order to connect to the simulation the client needs to have access and
1367 load the file \file{escriptsim.sim2} located under the work directory.
1368 \end{memberdesc}
1369 \begin{memberdesc}[DataManager]{VTK}
1370 exports simulation data in the \VTK file format.
1371 \end{memberdesc}
1372
1373 \noindent The \var{DataManager} class has the following methods:
1374 \begin{methoddesc}[DataManager]{addData}{**data}
1375 adds \Data objects and other data to the manager. Calling this method does
1376 not save or export the data yet so it is allowed to incrementally add data
1377 at various points in the simulation script if required.
1378 Note, that only a single domain is supported so all \Data objects have to
1379 be defined on the same one or an exception is raised.
1380 \end{methoddesc}
1381
1382 \begin{methoddesc}[DataManager]{setDomain}{domain}
1383 explicitly sets the domain for this manager.
1384 It is generally not required to call this method directly.
1385 Instead, the \var{addData} method will set the domain used by the \Data
1386 objects.
1387 An exception is raised if the domain was set to a different domain before
1388 (explicitly or implicitly).
1389 \end{methoddesc}
1390
1391 \begin{methoddesc}[DataManager]{hasData}{}
1392 returns \True if the manager has loaded simulation data for a restart.
1393 \end{methoddesc}
1394
1395 \begin{methoddesc}[DataManager]{getDomain}{}
1396 returns the domain as recovered from a restart.
1397 \end{methoddesc}
1398
1399 \begin{methoddesc}[DataManager]{getValue}{value_name}
1400 returns a \Data object or other value with the name \var{value_name} that
1401 has been recovered after a restart.
1402 \end{methoddesc}
1403
1404 \begin{methoddesc}[DataManager]{getCycle}{}
1405 returns the export cycle, i.e. the number of time \var{export()} has been
1406 called.
1407 \end{methoddesc}
1408
1409 \begin{methoddesc}[DataManager]{setCheckpointFrequency}{freq}
1410 sets the frequency with which checkpoint files are created. This is only
1411 useful if the \var{DataManager} object was created with at least one other
1412 format next to \var{RESTART}. The frequency is 1 by default which means
1413 that checkpoint files are created every time \var{export()} is called.
1414 Unlike visualization output, a simulation checkpoint is usually not
1415 required at every time step. Thus, the frequency can be decreased by
1416 calling this method with $\var{freq}>1$ which would then create restart
1417 files every \var{freq} times \var{export()} is called.
1418 \end{methoddesc}
1419
1420 \begin{methoddesc}[DataManager]{setTime}{time}
1421 sets the simulation time stamp. This floating point number is stored in
1422 the metadata of exported data but not used by \var{RESTART}.
1423 \end{methoddesc}
1424
1425 \begin{methoddesc}[DataManager]{setMeshLabels}{x, y, z=""}
1426 sets labels for the mesh axes. These are currently only used by the \SILO
1427 exporter.
1428 \end{methoddesc}
1429
1430 \begin{methoddesc}[DataManager]{setMeshUnits}{x, y, z=""}
1431 sets units for the mesh axes. These are currently only used by the \SILO
1432 exporter.
1433 \end{methoddesc}
1434
1435 \begin{methoddesc}[DataManager]{setMetadataSchemaString}{schema, metadata=""}
1436 sets metadata namespaces and the corresponding metadata. These are
1437 currently only used by the \VTK exporter.
1438 \var{schema} is a dictionary that maps prefixes to namespace names, e.g.\\
1439 \code{\{"gml": "http://www.opengis.net/gml"\}} and \var{metadata} is a
1440 string with the actual content which will be enclosed in \var{<MetaData>}
1441 tags.
1442 \end{methoddesc}
1443
1444 \begin{methoddesc}[DataManager]{export}{}
1445 executes the actual data export. Depending on the \var{formats} parameter
1446 used in the constructor all data added by \var{addData()} is written to
1447 disk (\var{RESTART,SILO,VTK}) or made available through the \VisIt
1448 simulation interface (\var{VISIT}).
1449 At least the domain must be set for something to be exported.
1450 \end{methoddesc}
1451
1452 \subsection{Saving Data as CSV}
1453 \index{saveDataCSV}\index{CSV}
1454 For simple post-processing, \Data objects can be saved in comma separated
1455 value (\emph{CSV}) format.
1456 If \var{mydata1} and \var{mydata2} are scalar data, the command
1457 \begin{python}
1458 saveDataCSV('output.csv', U=mydata1, V=mydata2)
1459 \end{python}
1460 will record the values in \file{output.csv} in the following format:
1461 \begin{verbatim}
1462 U, V
1463 1.0000000e+0, 2.0000000e-1
1464 5.0000000e-0, 1.0000000e+1
1465 ...
1466 \end{verbatim}
1467
1468 The names of the keyword parameters form the names of columns in the output.
1469 If the data objects are over different function spaces, then \var{saveDataCSV}
1470 will attempt to interpolate to a common function space.
1471 If this is not possible, then an exception is raised.
1472
1473 Output can be restricted using a scalar mask as follows:
1474 \begin{python}
1475 saveDataCSV('outfile.csv', U=mydata1, V=mydata2, mask=myscalar)
1476 \end{python}
1477 This command will only output those rows which correspond to to positive
1478 values of \var{myscalar}.
1479 Some aspects of the output can be tuned using additional parameters:
1480 \begin{python}
1481 saveDataCSV('data.csv', append=True, sep=' ', csep='/', mask=mymask, e=mat1)
1482 \end{python}
1483
1484 \begin{itemize}
1485 \item \var{append} -- specifies that the output should be written to the end of an existing file
1486 \item \var{sep} -- defines the separator between fields
1487 \item \var{csep} -- defines the separator between components in the header
1488 line. For example between the components of a matrix.
1489 \end{itemize}
1490 %
1491 The above command would produce output like this:
1492 \begin{verbatim}
1493 e/0/0 e/1/0 e/0/1 e/1/1
1494 1.0000000000e+00 2.0000000000e+00 3.0000000000e+00 4.0000000000e+00
1495 ...
1496 \end{verbatim}
1497
1498 Note that while the order in which rows are output can vary, all the elements
1499 in a given row always correspond to the same input.
1500
1501 \subsection{The \Operator Class}
1502 The \Operator class provides an abstract access to operators built
1503 within the \LinearPDE class. \Operator objects are created
1504 when a PDE is handed over to a PDE solver library and handled
1505 by the \LinearPDE object defining the PDE. The user can gain access
1506 to the \Operator of a \LinearPDE object through the \var{getOperator}
1507 method.
1508
1509 \begin{classdesc}{Operator}{}
1510 creates an empty \Operator object.
1511 \end{classdesc}
1512
1513 \begin{methoddesc}[Operator]{isEmpty}{fileName}
1514 returns \True is the object is empty, \False otherwise.
1515 \end{methoddesc}
1516
1517 \begin{methoddesc}[Operator]{setValue}{value}
1518 resets all entries in the object representation to \var{value}.
1519 \end{methoddesc}
1520
1521 \begin{methoddesc}[Operator]{solves}{rhs}
1522 solves the operator equation with right hand side \var{rhs}.
1523 \end{methoddesc}
1524
1525 \begin{methoddesc}[Operator]{of}{u}
1526 applies the operator to the \Data object \var{u}.
1527 \end{methoddesc}
1528
1529 \begin{methoddesc}[Operator]{saveMM}{fileName}\index{Matrix Market}
1530 saves the object to a Matrix Market format file with name \var{fileName}, see
1531 \url{http://maths.nist.gov/MatrixMarket}
1532 \end{methoddesc}
1533
1534 \section{Physical Units}
1535 \escript provides support for physical units in the SI system\index{SI units}
1536 including unit conversion. So the user can define variables in the form
1537 \begin{python}
1538 from esys.escript.unitsSI import *
1539 l=20*m
1540 w=30*kg
1541 w2=40*lb
1542 T=100*Celsius
1543 \end{python}
1544 In the two latter cases a conversion from pounds\index{pounds} and degrees
1545 Celsius\index{Celsius} is performed into the appropriate SI units \emph{kg}
1546 and \emph{Kelvin}.
1547 In addition, composed units can be used, for instance
1548 \begin{python}
1549 from esys.escript.unitsSI import *
1550 rho=40*lb/cm**3
1551 \end{python}
1552 defines the density in the units of pounds per cubic centimeter.
1553 The value $40$ will be converted into SI units, in this case kg per cubic
1554 meter. Moreover unit prefixes are supported:
1555 \begin{python}
1556 from esys.escript.unitsSI import *
1557 p=40*Mega*Pa
1558 \end{python}
1559 The pressure \var{p} is set to 40 Mega Pascal. Units can also be converted
1560 back from the SI system into a desired unit, e.g.
1561 \begin{python}
1562 from esys.escript.unitsSI import *
1563 print p/atm
1564 \end{python}
1565 can be used print the pressure in units of atmosphere\index{atmosphere}.
1566
1567 The following is an incomplete list of supported physical units:
1568
1569 \begin{datadesc}{km}
1570 unit of kilometer
1571 \end{datadesc}
1572
1573 \begin{datadesc}{m}
1574 unit of meter
1575 \end{datadesc}
1576
1577 \begin{datadesc}{cm}
1578 unit of centimeter
1579 \end{datadesc}
1580
1581 \begin{datadesc}{mm}
1582 unit of millimeter
1583 \end{datadesc}
1584
1585 \begin{datadesc}{sec}
1586 unit of second
1587 \end{datadesc}
1588
1589 \begin{datadesc}{minute}
1590 unit of minute
1591 \end{datadesc}
1592
1593 \begin{datadesc}{h}
1594 unit of hour
1595 \end{datadesc}
1596
1597 \begin{datadesc}{day}
1598 unit of day
1599 \end{datadesc}
1600
1601 \begin{datadesc}{yr}
1602 unit of year
1603 \end{datadesc}
1604
1605 \begin{datadesc}{gram}
1606 unit of gram
1607 \end{datadesc}
1608
1609 \begin{datadesc}{kg}
1610 unit of kilogram
1611 \end{datadesc}
1612
1613 \begin{datadesc}{lb}
1614 unit of pound
1615 \end{datadesc}
1616
1617 \begin{datadesc}{ton}
1618 metric ton
1619 \end{datadesc}
1620
1621 \begin{datadesc}{A}
1622 unit of Ampere
1623 \end{datadesc}
1624
1625 \begin{datadesc}{Hz}
1626 unit of Hertz
1627 \end{datadesc}
1628
1629 \begin{datadesc}{N}
1630 unit of Newton
1631 \end{datadesc}
1632
1633 \begin{datadesc}{Pa}
1634 unit of Pascal
1635 \end{datadesc}
1636
1637 \begin{datadesc}{atm}
1638 unit of atmosphere
1639 \end{datadesc}
1640
1641 \begin{datadesc}{J}
1642 unit of Joule
1643 \end{datadesc}
1644
1645 \begin{datadesc}{W}
1646 unit of Watt
1647 \end{datadesc}
1648
1649 \begin{datadesc}{C}
1650 unit of Coulomb
1651 \end{datadesc}
1652
1653 \begin{datadesc}{V}
1654 unit of Volt
1655 \end{datadesc}
1656
1657 \begin{datadesc}{F}
1658 unit of Farad
1659 \end{datadesc}
1660
1661 \begin{datadesc}{Ohm}
1662 unit of Ohm
1663 \end{datadesc}
1664
1665 \begin{datadesc}{K}
1666 unit of degrees Kelvin
1667 \end{datadesc}
1668
1669 \begin{datadesc}{Celsius}
1670 unit of degrees Celsius
1671 \end{datadesc}
1672
1673 \begin{datadesc}{Fahrenheit}
1674 unit of degrees Fahrenheit
1675 \end{datadesc}
1676
1677 \noindent Supported unit prefixes:
1678
1679 \begin{datadesc}{Yotta}
1680 prefix yotta = $10^{24}$
1681 \end{datadesc}
1682
1683 \begin{datadesc}{Zetta}
1684 prefix zetta = $10^{21}$
1685 \end{datadesc}
1686
1687 \begin{datadesc}{Exa}
1688 prefix exa = $10^{18}$
1689 \end{datadesc}
1690
1691 \begin{datadesc}{Peta}
1692 prefix peta = $10^{15}$
1693 \end{datadesc}
1694
1695 \begin{datadesc}{Tera}
1696 prefix tera = $10^{12}$
1697 \end{datadesc}
1698
1699 \begin{datadesc}{Giga}
1700 prefix giga = $10^9$
1701 \end{datadesc}
1702
1703 \begin{datadesc}{Mega}
1704 prefix mega = $10^6$
1705 \end{datadesc}
1706
1707 \begin{datadesc}{Kilo}
1708 prefix kilo = $10^3$
1709 \end{datadesc}
1710
1711 \begin{datadesc}{Hecto}
1712 prefix hecto = $10^2$
1713 \end{datadesc}
1714
1715 \begin{datadesc}{Deca}
1716 prefix deca = $10^1$
1717 \end{datadesc}
1718
1719 \begin{datadesc}{Deci}
1720 prefix deci = $10^{-1}$
1721 \end{datadesc}
1722
1723 \begin{datadesc}{Centi}
1724 prefix centi = $10^{-2}$
1725 \end{datadesc}
1726
1727 \begin{datadesc}{Milli}
1728 prefix milli = $10^{-3}$
1729 \end{datadesc}
1730
1731 \begin{datadesc}{Micro}
1732 prefix micro = $10^{-6}$
1733 \end{datadesc}
1734
1735 \begin{datadesc}{Nano}
1736 prefix nano = $10^{-9}$
1737 \end{datadesc}
1738
1739 \begin{datadesc}{Pico}
1740 prefix pico = $10^{-12}$
1741 \end{datadesc}
1742
1743 \begin{datadesc}{Femto}
1744 prefix femto = $10^{-15}$
1745 \end{datadesc}
1746
1747 \begin{datadesc}{Atto}
1748 prefix atto = $10^{-18}$
1749 \end{datadesc}
1750
1751 \begin{datadesc}{Zepto}
1752 prefix zepto = $10^{-21}$
1753 \end{datadesc}
1754
1755 \begin{datadesc}{Yocto}
1756 prefix yocto = $10^{-24}$
1757 \end{datadesc}
1758
1759 \section{Utilities}
1760 The \class{FileWriter} class provides a mechanism to write data to a file.
1761 In essence, this class wraps the standard \PYTHON \class{file} class to write
1762 data that are global in \MPI to a file. In fact, data are written on the
1763 processor with \MPI rank 0 only. It is recommended to use \class{FileWriter}
1764 rather than \class{open} in order to write code that will run with and without
1765 \MPI. It is safe to use \class{open} under \MPI to \emph{read} data which are
1766 global under \MPI.
1767
1768 \begin{classdesc}{FileWriter}{fn\optional{,append=\False, \optional{createLocalFiles=\False}})}
1769 Opens a file with name \var{fn} for writing. If \var{append} is set to \True
1770 data are appended at the end of the file.
1771 If running under \MPI, only the first processor (rank==0) will open the file
1772 and write to it.
1773 If \var{createLocalFiles} is set each individual processor will create a file
1774 where for any processor with rank>0 the file name is extended by its rank.
1775 This option is normally used for debugging purposes only.
1776 \end{classdesc}
1777
1778 \noindent The following methods are available:
1779 \begin{methoddesc}[FileWriter]{close}{}
1780 closes the file.
1781 \end{methoddesc}
1782 \begin{methoddesc}[FileWriter]{flush}{}
1783 flushes the internal buffer to disk.
1784 \end{methoddesc}
1785 \begin{methoddesc}[FileWriter]{write}{txt}
1786 writes string \var{txt} to the file. Note that a newline is not added.
1787 \end{methoddesc}
1788 \begin{methoddesc}[FileWriter]{writelines}{txts}
1789 writes the list \var{txts} of strings to the file.
1790 Note that newlines are not added.
1791 This method is equivalent to calling \var{write()} for each string.
1792 \end{methoddesc}
1793 \begin{memberdesc}[FileWriter]{closed}
1794 this member is \True if the file is closed.
1795 \end{memberdesc}
1796 \begin{memberdesc}[FileWriter]{mode}
1797 holds the access mode.
1798 \end{memberdesc}
1799 \begin{memberdesc}[FileWriter]{name}
1800 holds the file name.
1801 \end{memberdesc}
1802 \begin{memberdesc}[FileWriter]{newlines}
1803 holds the line separator.
1804 \end{memberdesc}
1805
1806 \begin{funcdesc}{setEscriptParamInt}{name,value}
1807 assigns the integer value \var{value} to the parameter \var{name}.
1808 If \var{name}="TOO_MANY_LINES" conversion of any \Data object to a string
1809 switches to a condensed format if more than \var{value} lines would be created.
1810 \end{funcdesc}
1811
1812 \begin{funcdesc}{getEscriptParamInt}{name}
1813 returns the current value of integer parameter \var{name}.
1814 \end{funcdesc}
1815
1816 \begin{funcdesc}{listEscriptParams}{a}
1817 returns a list of valid parameters and their description.
1818 \end{funcdesc}
1819
1820 \begin{funcdesc}{getMPISizeWorld}{}
1821 returns the number of \MPI processors in use in the \env{MPI_COMM_WORLD} processor group.
1822 If \MPI is not used 1 is returned.
1823 \end{funcdesc}
1824
1825 \begin{funcdesc}{getMPIRankWorld}{}
1826 returns the rank of the current process within the \env{MPI_COMM_WORLD}
1827 processor group. If \MPI is not used 0 is returned.
1828 \end{funcdesc}
1829
1830 \begin{funcdesc}{MPIBarrierWorld}{}
1831 performs a barrier synchronization across all processors within the
1832 \env{MPI_COMM_WORLD} processor group.
1833 \end{funcdesc}
1834
1835 \begin{funcdesc}{getMPIWorldMax}{a}
1836 returns the maximum value of the integer \var{a} across all processors within
1837 \env{MPI_COMM_WORLD}.
1838 \end{funcdesc}
1839

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