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

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