 # Contents of /trunk/doc/user/escript.tex

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


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

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