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 1 % $Id$ 2 3 \chapter{The module \escript} 4 5 \declaremodule{extension}{escript} 6 \modulesynopsis{Data manipulation} 7 8 \begin{figure} 9 \includegraphics[width=\textwidth]{EscriptDiagram1.eps} 10 \caption{\label{ESCRIPT DEP}Dependency of Function Spaces. An arrow indicates that a function in the 11 function space at the starting point can be interpreted as a function in the function space of the arrow target.} 12 \end{figure} 13 14 \escript is an extension of Python to handle functions represented by their values on 15 \DataSamplePoints. The \DataSamplePoints for the geometrical region on which 16 the function is defined. The region as well as the method which is used 17 to interpolate value on the \DataSamplePoints is defined by 18 \Domain class objects. For instance when using 19 the finite element method (FEM) \index{finite element method} 20 \Domain object holds the information about the FEM mesh, eg. 21 a table of nodes and a table of elements. Although \Domain contains 22 the discretization method to be used \escript does not use this information directly. 23 \Domain objects are created from a module which want to make use 24 \escript, e.g. \finley. 25 26 The solution of a PDE is a function of its location in the domain of interest $\Omega$. 27 When solving a partial differential equation \index{partial differential equation} (PDE) using FEM 28 the solution is (piecewise) differentiable but, in general, its gradient 29 is discontinuous. To reflect these different degrees of smoothness different 30 representations of the functions are used. For instance; in FEM 31 the displacement field is represented by its values at the nodes of the mesh, while the 32 strain, which is the symmetric part of the gradient of the displacement field, is stored on the 33 element centers. To be able to classify functions with respect to their smoothness, \escript has the 34 concept of the "function space". A function space is described by a \FunctionSpace object. 35 The following statement generates the object \var{solution_space} which is 36 a \FunctionSpace object and provides access to the function space of 37 PDE solutions on the \Domain \var{mydomain}: 38 \begin{python} 39 solution_space=Solution(mydomain) 40 \end{python} 41 The following generators for function spaces on a \Domain \var{mydomin} are available: 42 \begin{itemize} 43 \item \var{Solution(mydomain)}: solutions of a PDE. 44 \item \var{ContinuousFunction(mydomain)}: continuous functions, eg. a temperature distribution. 45 \item \var{Function(mydomain)}: general functions which are not necessarily continuous, eg. a stress field. 46 \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, eg. a surface pressure. 47 \item \var{FunctionOnContact0(mydomain)}: functions on side $0$ of the discontinuity. 48 \item \var{FunctionOnContact1(mydomain)}: functions on side $1$ of the discontinuity. 49 \end{itemize} 50 A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous. 51 The location of discontinuity is defined in the \Domain object. 52 \fig{ESCRIPT DEP} shows the dependency between the types of function spaces. 53 The solution of a PDE is a continuous function. Any continuous function can be seen as a general function 54 on the domain and can be restricted to the boundary as well as to any side of the 55 discontinuity (the result will be different depending on 56 which side is chosen). Functions on any side of the 57 discontinuity can be seen as a function on the corresponding other side. 58 A function on the boundary or on one side of 59 the discontinuity cannot be seen as a general function on the domain as there are no values 60 defined for the interior. For most PDE solver libraries 61 the space of the solution and continuous functions is identical, however in some cases, eg. 62 when periodic boundary conditions are used in \finley, a solution 63 fulfils periodic boundary conditions while a continuous function does not have to be periodic. 64 65 The concept of function spaces describes the properties of 66 functions and allows abstraction from the actual representation 67 of the function in the context of a particular application. For instance, 68 in the FEM context a 69 function in the \Function function space 70 is typically represented by its values at the element center, 71 but in a finite difference scheme the edge midpoint of cells is preferred. 72 Using the concept of function spaces 73 allows the user to run the same script on different 74 PDE solver libraries by just changing the creator of the \Domain object. 75 Changing the function space of a particular function 76 will typically lead to a change of its representation. 77 So, when seen as a general function, 78 a continuous function which is typically represented by its values 79 on the node of the FEM mesh or finite difference grid 80 has to be interpolated to the element centers or the cell edges, 81 respectively. 82 83 \Data class objects store functions of the location in a domain. 84 The function is represented through its values on \DataSamplePoints where 85 the \DataSamplePoints are chosen according to the function space 86 of the function. 87 \Data class objects are used to define the coefficients 88 of the PDEs to be solved by a PDE solver library 89 and to store the returned solutions. 90 91 The values of the function have a rank which gives the 92 number of indexes, and a \Shape defining the range of each index. 93 The rank in \escript is limited to the range $0$ through $4$ and 94 it is assumed that the rank and \Shape is the same for all \DataSamplePoints. 95 The \Shape of a \Data object is a tuple \var{s} of integers. The length 96 of \var{s} is the rank of the \Data object and \var{s[i]} is the maximum 97 value for the \var{i}-th index. 98 For instance, a stress field has rank $2$ and 99 \Shape $(d,d)$ where $d$ is the spatial dimension. 100 The following statement creates the \Data object 101 \var{mydat} representing a 102 continuous function with values 103 of \Shape $(2,3)$ and rank $2$: 104 \begin{python} 105 mydat=Data(value=1,what=ContinuousFunction(myDomain),shape=(2,3)) 106 \end{python} 107 The initial value is the constant $1$ for all \DataSamplePoints and 108 all components. 109 110 \Data objects can also be created from any \numarray 111 array or any object, such as a list of floating point numbers, 112 that can be converted into a \numarray array \Ref{NUMARRAY}. 113 The following two statements 114 create objects which are equivalent to \var{mydat}: 115 \begin{python} 116 mydat1=Data(value=numarray.ones((2,3)),what=ContinuousFunction(myDomain)) 117 mydat2=Data(value=[[1,1],[1,1],[1,1]],what=ContinuousFunction(myDomain)) 118 \end{python} 119 In the first case the initial value is \var{numarray.ones((2,3))} 120 which generates a $2 \times 3$ matrix as a \numarray array 121 filled with ones. The \Shape of the created \Data object 122 it taken from the \Shape of the array. In the second 123 case, the creator converts the initial value, which is a list of lists, 124 and converts it into a \numarray array before creating the actuall 125 \Data object. 126 127 For convenience \escript provides creators for the most common types 128 of \Data objects in the following forms (\var{d} defines the 129 spatial dimension): 130 \begin{itemize} 131 \item \var{Scalar(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(,))}, 132 e.g a temperature field. 133 \item \var{Vector(0,Function(mydomain))}is the same as \var{Data(0,Function(myDomain),(d))}, e.g 134 a velocity field. 135 \item \var{Tensor(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d))}, 136 eg. a stress field. 137 \item \var{Tensor4(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d,d,d))} 138 eg. a Hook tensor field. 139 \end{itemize} 140 Here the initial value is $0$ but any object that can be converted into a \numarray array and whose \Shape 141 is consistent with \Shape of the \Data object to be created can be used as the initial value. 142 143 \Data objects can be manipulated by applying unitary operations (eg. cos, sin, log) 144 and be combined by applying binary operations (eg. +, - ,* , /). 145 It is to be emphasized that \escript itself does not handle any spatial dependencies itself as 146 it does not know how values are interpreted by the processing PDE solver library. 147 However \escript invokes interpolation if this is needed during data manipulations. 148 Typically, this occurs in binary operation when both arguments belong to different 149 function spaces or when data are handed over to a PDE solver library 150 which requires functions to be represented in a particular way. 151 152 The following example shows the usage of {\tt Data} objects: Assume we have a 153 displacement field $u$ and we want to calculate the corresponding stress field 154 $\sigma$ using the linear--elastic isotropic material model 155 \begin{eqnarray}\label{eq: linear elastic stress} 156 \sigma\hackscore {ij}=\lambda u\hackscore {k,k} \delta\hackscore {ij} + \mu ( u\hackscore {i,j} + u\hackscore {j,i}) 157 \end{eqnarray} 158 where $\delta\hackscore {ij}$ is the Kronecker symbol and 159 $\lambda$ and $\mu$ are the Lame coefficients. The following function 160 takes the displacement {\tt u} and the Lame coefficients 161 \var{lam} and \var{mu} as arguments and returns the corresponding stress: 162 \begin{python} 163 import numarray 164 def getStress(u,lam,mu): 165 d=u.getDomain().getDim() 166 g=grad(u) 167 stress=lam*trace(g)*numarray.identity(d)+ \ 168 mu*(g+transpose(g)) 169 return stress 170 \end{python} 171 The variable 172 \var{d} gives the spatial dimension of the 173 domain on which the displacements are defined. 174 \var{identity} is a \numarray function which returns the Kronecker symbol with indexes 175 $i$ and $j$ running from $0$ to \var{d}-1. The call \var{grad(u)} requires 176 the displacement field \var{u} to be in the \var{Solution} or \ContinuousFunction 177 function space. The result \var{g} as well as the returned stress will be in the \Function function space. 178 If \var{u} is available, eg. by solving a PDE, \var{getStress} might be called 179 in the following way: 180 \begin{python} 181 s=getStress(u,1.,2.) 182 \end{python} 183 However \var{getStress} can also be called with \Data objects as values for 184 \var{lam} and \var{mu} which, 185 for instance in the case of a temperature dependency, are calculated by an expression. 186 The following call is equivalent to the previous example: 187 \begin{python} 188 lam=Scalar(1.,ContinuousFunction(mydomain)) 189 mu=Scalar(2.,Function(mydomain)) 190 s=getStress(u,lam,mu) 191 \end{python} 192 The function \var{lam} belongs to the \ContinuousFunction function space 193 but with \var{g} the function \var{trace(g)} is in the \Function function space. 194 Therefore the evaluation of the product \var{lam*trace(g)} in the stress calculation 195 produces a problem, as both functions are represented differently, eg. in FEM 196 \var{lam} by its values on the node, and in \var{trace(g)} by its values at the element centers. 197 In the case of inconsistent function spaces of arguments in a binary operation, \escript 198 interprets the arguments in the appropriate function space according to the inclusion 199 defined in Table~\ref{ESCRIPT DEP}. In this example that means 200 \escript sees \var{lam} as a function of the \Function function space. 201 In the context of FEM this means the nodal values of 202 \var{lam} are interpolated to the element centers. Behind the scenes 203 \escript calls the appropriate function from the PDE solver library. 204 205 \begin{figure} 206 \includegraphics[width=\textwidth]{EscriptDiagram2.eps} 207 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region with two rock types {\it white} and {\it gray}. 208 The number in each cell refers to the major rock type present in the cell ($1$ for {\it white} and $2$ for {\it gray}). 209 } 210 \end{figure} 211 212 Material parameters such as the Lame coefficients are typically dependent on rock types present in the 213 area of interest. A common technique to handle these kinds of material parameters is "tagging". \fig{Figure: tag} 214 shows an example. In this case two rock types {\it white} and {\it gray} can be found in the domain. The domain 215 is subdivided into rectangular shaped cells (which is not necessarily the best subdivision for this case). Each 216 cell has a tag indicating the rock type predominately found in this cell. Here $1$ is used to indicate 217 rock type {\it white} and $2$ for rock type {\it gray}. The tags are assigned at the time when the cells are generated 218 (\escript provides tools to manipulate tags at a later stage) 219 and stored in the \Domain class object. The following statements show how for the 220 example of \fig{Figure: tag} and the stress calculation discussed before tagged values are used for 221 \var{lam}: 222 \begin{python} 223 lam=Scalar(value=2.,what=Function(mydomain)) 224 lam.setTaggedValue(1,30.) 225 lam.setTaggedValue(2,5000.) 226 s=getStress(u,lam,2.) 227 \end{python} 228 In this example \var{lam} is set to $30$ for those cells with tag $1$ and to $5000.$ for those cells 229 with tag $2$. The initial value $2$ of \var{lam} is used as a default value for the case when a tag 230 is encountered which has not been linked with a value. Note that the \var{getStress} method 231 is called without modification. \escript resolves the tags when \var{lam*trace(g)} is calculated. 232 233 The \Data class provides a transparent interface to various data representations and the 234 translations between them. As shown in the example of stress calculation, this allows the user to 235 develop and test algorithms for a simple case (for instance with the Lame coefficients as constants) 236 and then without further modifications of the program code to apply the algorithm in a 237 more complex application (for instance a definition of the Lame coefficients using tags). 238 As described here, there are three ways in which \Data objects are represented internally, constant, tagged, and expanded (other representions will become available in later versions of \escript): 239 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. 240 Any operation on this constant data will only be performed on the single value. 241 In the expanded case, each sample point has an individual value, eg. the solution of a PDE, 242 and the values are stored as a complete array. The tagged case has already been discussed above. 243 244 Values are accessed through a sample reference number. Operations on expanded \Data 245 objects have to be performed for each sample point individually. If tagged values are used values are 246 held in a dictionary. Operations on tagged data require processing the set of tagged values only, rather than 247 processing the value for each individual sample point. 248 \escript allows use of constant, tagged and expanded data in a single expression. 249 250 251 252 \section{\Domain class} 253 254 \begin{classdesc}{Domain}{} 255 A \Domain objects is used to describe a geometrical region together with 256 a way of representing functions over this region. 257 The \Domain class provides an abstract access to the domain of \FunctionSpace and \Data objects. 258 \Domain itself has no initialization but implementations of \Domain are 259 instantiated by numerical libraries making use of \Data objects. 260 \end{classdesc} 261 262 \begin{methoddesc}[Domain]{getDim}{} 263 returns the spatial dimension of the \Domain. 264 \end{methoddesc} 265 266 \begin{methoddesc}[Domain]{getX}{} 267 returns the locations in the \Domain. The \FunctionSpace of the returned 268 \Data object is chosen by the \Domain implementation. Typically it will be 269 in the \Function. 270 \end{methoddesc} 271 272 \begin{methoddesc}[Domain]{setX}{newX} 273 assigns a new location to the \Domain. \var{newX} has to have \Shape $(d,)$ 274 where $d$ is the spatial dimension of the domain. Typically \var{newX} must be 275 in the \ContinuousFunction but the space actually to be used depends on the \Domain implementation. 276 \end{methoddesc} 277 278 \begin{methoddesc}[Domain]{getNormal}{} 279 returns the surface normals on the boundary of the \Domain as \Data object. 280 \end{methoddesc} 281 282 \begin{methoddesc}[Domain]{getSize}{} 283 returns the local sample size, e.g. the element diameter, as \Data object. 284 \end{methoddesc} 285 286 \begin{methoddesc}[Domain]{__eq__}{arg} 287 returns \True of the \Domain \var{arg} describes the same domain. Otherwise 288 \False is returned. 289 \end{methoddesc} 290 291 \begin{methoddesc}[Domain]{__ne__}{arg} 292 returns \True of the \Domain \var{arg} does not describe the same domain. 293 Otherwise \False is returned. 294 \end{methoddesc} 295 296 \section{\Domain class} 297 \begin{classdesc}{FunctionSpace}{} 298 \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects 299 are instantiated by generator functions. \Data objects in particular \FunctionSpace are 300 represented by their values at \DataSamplePoints which are defined by the type and the \Domain of the 301 \FunctionSpace. 302 \end{classdesc} 303 304 \begin{methoddesc}[FunctionSpace]{getDim}{} 305 returns the spatial dimension of the \Domain of the \FunctionSpace. 306 \end{methoddesc} 307 308 \begin{methoddesc}[FunctionSpace]{getX}{} 309 returns the location of the \DataSamplePoints. 310 \end{methoddesc} 311 312 \begin{methoddesc}[FunctionSpace]{getNormal}{} 313 If the domain of functions in the \FunctionSpace 314 is a hypermanifold (e.g. the boundary of a domain) 315 the method returns the outer normal at each of the 316 \DataSamplePoints. Otherwise an exception is raised. 317 \end{methoddesc} 318 319 \begin{methoddesc}[FunctionSpace]{getSize}{} 320 returns a \Data objects measuring the spacing of the \DataSamplePoints. 321 The size may be zero. 322 \end{methoddesc} 323 324 \begin{methoddesc}[FunctionSpace]{getDomain}{} 325 returns the \Domain of the \FunctionSpace. 326 \end{methoddesc} 327 328 \begin{methoddesc}[FunctionSpace]{__eq__}{arg} 329 returns \True of the \Domain \var{arg} describes the same domain. Otherwise 330 \False is returned. 331 \end{methoddesc} 332 333 \begin{methoddesc}[FunctionSpace]{__ne__}{arg} 334 returns \True of the \Domain \var{arg} describes the note same domain. 335 Otherwise \False is returned. 336 \end{methoddesc} 337 338 The following function provide generators for \FunctionSpace objects: 339 \begin{funcdesc}{Function}{domain} 340 returns the \Function on the \Domain \var{domain}. \Data objects in this type of \Function 341 are defined over the whole geometrical region defined by \var{domain}. 342 \end{funcdesc} 343 344 \begin{funcdesc}{ContinuousFunction}{domain} 345 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function 346 are defined over the whole geometrical region defined by \var{domain} and assumed to represent 347 a continuous function. 348 \end{funcdesc} 349 350 \begin{funcdesc}{FunctionOnBoundary}{domain} 351 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function 352 are defined on the boundary of the geometrical region defined by \var{domain}. 353 \end{funcdesc} 354 355 \begin{funcdesc}{FunctionOnContactZero}{domain} 356 returns the \FunctionOnContactZero the \Domain domain. \Data objects in this type of \Function 357 are defined on side 0 of a discontinutiy within the geometrical region defined by \var{domain}. 358 The discontinutiy is defined when \var{domain} is instantiated. 359 \end{funcdesc} 360 361 \begin{funcdesc}{FunctionOnContactOne}{domain} 362 returns the \FunctionOnContactOne on the \Domain domain. 363 \Data objects in this type of \Function 364 are defined on side 1 of a discontinutiy within the geometrical region defined by \var{domain}. 365 The discontinutiy is defined when \var{domain} is instantiated. 366 \end{funcdesc} 367 368 \begin{funcdesc}{Solution}{domain} 369 returns the \SolutionFS on the \Domain domain. \Data objects in this type of \Function 370 are defined on geometrical region defined by \var{domain} and are solutions of 371 partial differential equations \index{partial differential equation}. 372 \end{funcdesc} 373 374 \begin{funcdesc}{ReducedSolution}{domain} 375 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in this type of \Function 376 are defined on geometrical region defined by \var{domain} and are solutions of 377 partial differential equations \index{partial differential equation} with a reduced smoothness 378 for the solution approximation. 379 \end{funcdesc} 380 381 \section{\Data Class} 382 383 384 The following table shows binary and unitary operations that can be applied to 385 \Data objects: 386 \begin{tableii}{l|l}{textrm}{expression}{Description} 387 \lineii{+\var{arg1}} {just \var{arg} \index{+}} 388 \lineii{-\var{arg1}} {swapping the sign\index{-}} 389 \lineii{\var{arg1}+\var{arg2}} {adds \var{arg1} and \var{arg2} \index{+}} 390 \lineii{\var{arg1}*\var{arg2}} {multiplies \var{arg1} and \var{arg2} \index{*}} 391 \lineii{\var{arg1}-\var{arg2}} {difference \var{arg2} from\var{arg2} \index{-}} 392 \lineii{\var{arg1}/\var{arg2}} {ratio \var{arg1} by \var{arg2} \index{/}} 393 \lineii{\var{arg1}**\var{arg2}} {raises \var{arg1} to the power of \var{arg2} \index{**}} 394 \end{tableii} 395 At least on of the arguments \var{arg1} or \var{arg2} must be a 396 \Data object. One of the arguments may be an object that can be 397 converted into a \Data object. If \var{arg1} or \var{arg2} are 398 defined on different \FunctionSpace it is tried to embed \var{arg1} 399 into the \FunctionSpace of \var{arg2} or to embed \var{arg2} into 400 the \FunctionSpace of \var{arg1}. Boths arguments must have the same 401 \Shape or one of the arguments my be of rank 0 or \Shape (1,). In the 402 latter case it is assumed that the particular argument is of the same 403 \Shape like the other argument but constant over all components. 404 405 The returned \Data object has the same \Shape and is defined on 406 the \DataSamplePoints like \var{arg1} or \var{arg2}. 407 408 The following table shows the update operations that can be applied to 409 \Data objects: 410 \begin{tableii}{l|l}{textrm}{expression}{Description} 411 \lineii{\var{arg1}+=\var{arg2}} {adds \var{arg1} to \var{arg2} \index{+}} 412 \lineii{\var{arg1}*=\var{arg2}} {multiplies \var{arg1} with \var{arg2} \index{*}} 413 \lineii{\var{arg1}-=\var{arg2}} {subtracts \var{arg2} from\var{arg2} \index{-}} 414 \lineii{\var{arg1}/=\var{arg2}} {divides \var{arg1} by \var{arg2} \index{/}} 415 \end{tableii} 416 \var{arg1} must be a \Data object. \var{arg1} must be a 417 \Data object or an object that can be converted into a 418 \Data object. \var{arg1} must have the same \Shape like 419 \var{arg1} or has rank 0 or \Shape (1,). In the latter case it is 420 assumed that the values of \var{arg1} are constant for all 421 components. \var{arg2} must be defined on the same \DataSamplePoints like 422 \var{arg1} or it must be possible to interpolate \var{arg2} onto the 423 \DataSamplePoints where \var{arg1} is hold. 424 425 The \Data class supports getting slices as well as assigning new values to components in an existing 426 \Data object \index{slicing}. 427 The following expression for getting (expression on the right hand side of the 428 equal sign) and setting slices (expression on the left hand side of the 429 equal sign) are valid: 430 \begin{tableiii}{l|ll}{textrm}{rank of \var{arg}}{slicing expression}{\Shape of returned and assigned object} 431 \lineiii{0}{ no slicing } {-} 432 \lineiii{1}{\var{arg[l0:u0]}} {(\var{u0}-\var{l0},)} 433 \lineiii{2}{\var{arg[l0:u0,l1:u1]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1})} 434 \lineiii{3}{\var{arg[l0:u0,l1:u1,l2:u2]} } {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})} 435 \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})} 436 \end{tableiii} 437 where 438 $0 \le \var{l0} \le \var{u0} \le \var{s}$, 439 $0 \le \var{l1} \le \var{u1} \le \var{s}$, 440 $0 \le \var{l2} \le \var{u2} \le \var{s}$, 441 $0 \le \var{l3} \le \var{u3} \le \var{s}$ and \var{s} the \Shape if \var{arg}. 442 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case 443 $0$ is assumed. 444 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may not be present in which case 445 \var{s} is assumed. The lower and upper index may be identical, in which case the column and the lower or upper 446 index may be dropped. In the returned or in the object assigned to a slice the corresponding component is dropped, 447 i.e. the rank is reduced by one in comparison to \var{arg}. 448 The follwing examples showing the usage of slicing: 449 \begin{python} 450 t[1,1,1,0]=9. 451 s=t[:2,:,2:6,5] # s has rank 3 452 s[:,:,1]=1. 453 t[:2,:2,5,5]=s[2:4,1,:2] 454 \end{python} 455 456 457 458 \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expanded=\False} 459 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}. 460 The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True 461 the \Data object is represented in expanded from. 462 \end{classdesc} 463 464 \begin{classdesc}{Data}{value,what=FunctionSpace(),expanded=\False} 465 creates a \Data object in the \FunctionSpace \var{what}. 466 The value for each \DataSamplePoints is set to \numarray object \var{value}. 467 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True 468 the \Data object is represented in expanded from. 469 \end{classdesc} 470 471 \begin{classdesc}{Data}{value,what=FunctionSpace()} 472 creates a \Data object in the \FunctionSpace \var{what} from the \Data object \var{value}. 473 The \Shape of the created \Data object is equal to the \Shape of \var{value}. 474 If the \FunctionSpace of \var{value} is equal to the \var{what}, a shallow copy, i.e. 475 a reference to the representation of \var{value} is greated. Otherwise, 476 \var{value} is interpolated into the \var{what}. 477 \end{classdesc} 478 479 \begin{classdesc}{Data}{} 480 creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present 481 where a \Data object is required. 482 \end{classdesc} 483 484 \begin{methoddesc}[Data]{getFunctionSpace}{} 485 returns the \FunctionSpace of the object. 486 \end{methoddesc} 487 488 \begin{methoddesc}[Data]{getFunctionSpace}{} 489 returns the \Domain of the object. 490 \end{methoddesc} 491 492 \begin{methoddesc}[Data]{getShape}{} 493 returns the \Shape of the object as a \class{tuple} of 494 integers. 495 \end{methoddesc} 496 497 \begin{methoddesc}[Data]{getRank}{} 498 returns the rank of the data on each data point. \index{rank} 499 \end{methoddesc} 500 501 \begin{methoddesc}[Data]{isEmpty}{} 502 returns \True id the \Data object is the \EmptyData object. 503 Otherwise \False is returned. 504 \end{methoddesc} 505 506 \begin{methoddesc}[Data]{setTaggedValue}{tag,value} 507 assigns the \var{value} to all \DataSamplePoints which have the tag 508 \var{tag}. \var{value} must be an object of class 509 \class{numarray.array} or must be convertable into a 510 \class{numarray.array} object. \var{value} (or the cooresponding 511 \class{numarray.array} object) must be of rank $0$ or must have the 512 same rank like the object. 513 If a value has already be defined for tag \var{tag} within the object 514 it is overwritten by the new \var{value}. If the object is expanded, 515 the value assigned to \DataSamplePoints with tag \var{tag} is replaced by 516 \var{value}. 517 \end{methoddesc} 518 519 \begin{methoddesc}[Data]{wherePositive}{} 520 returns \Data object which has the same \Shape and is defined on 521 the same \FunctionSpace like the object. The returned values are $1$ 522 where the object is positive and $0$ elsewhere. 523 \end{methoddesc} 524 525 \begin{methoddesc}[Data]{wherePositive}{} 526 returns \Data object which has the same \Shape and is defined on 527 the same \FunctionSpace like the object. The returned values are $1$ 528 where the object is non-positive and $0$ elsewhere. 529 \end{methoddesc} 530 531 \begin{methoddesc}[Data]{whereNonnegative}{} 532 returns \Data object which has the same \Shape and is defined on 533 the same \FunctionSpace like the object. The returned values are $1$ 534 where the object is non-negative and $0$ elsewhere. 535 \end{methoddesc} 536 537 \begin{methoddesc}[Data]{whereNegative}{} 538 returns \Data object which has the same \Shape and is defined on 539 the same \FunctionSpace like the object. The returned values are $1$ 540 where the object is negative and $0$ elsewhere. 541 \end{methoddesc} 542 543 \begin{methoddesc}[Data]{whereZero}{tolerance=1.e-8} 544 returns \Data object which has the same \Shape and is defined on 545 the same \FunctionSpace like the object. The returned values are $1$ 546 where the object is nearly zero, i.e. where the absolute value is less 547 than \var{tolerance}, and $0$ elsewhere. 548 \end{methoddesc} 549 550 \begin{methoddesc}[Data]{whereNonzero}{tolerance=1.e-8} 551 returns \Data object which has the same \Shape and is defined on 552 the same \FunctionSpace like the object. The returned values are $1$ 553 where the object is nearly non-zero, i.e. where the absolute value is 554 greater or equal than \var{tolerance}, and $0$ elsewhere. 555 \end{methoddesc} 556 557 \begin{methoddesc}[Data]{sign}{} 558 returns \Data object which has the same \Shape and is defined on 559 the same \FunctionSpace like the object. The returned values are $1$ 560 where the object is positive, $-1$ where the value is negative and $0$ elsewhere. 561 \end{methoddesc} 562 563 \begin{methoddesc}[Data]{copyWithMask}{arg,mask} 564 copies the \Data object \var{arg} into the object 565 where the \Data object \var{mask} is positive. \var{arg} 566 and \var{mask} must have the same \Shape 567 and must belong to same \FunctionSpace as 568 the object. 569 \end{methoddesc} 570 571 \begin{methoddesc}[Data]{Lsup}{} 572 returns the $L^{sup}$-norm of the object. This is maximum absolute values over all components and all \DataSamplePoints. \index{$L^{sup}$-norm}. 573 \end{methoddesc} 574 \begin{methoddesc}[Data]{inf}{} 575 returns the minimum value (infimum) of the object. The minimum is 576 taken over all components and all \DataSamplePoints . \index{infimum} 577 \end{methoddesc} 578 579 \begin{methoddesc}[Data]{sup}{} 580 returns the maximum value (supremum) of the object. The maximum is 581 taken over all components and all \DataSamplePoints . \index{supremum} 582 \end{methoddesc} 583 584 \begin{methoddesc}[Data]{grad}{\optional{on}} 585 returns the gradient of the function represented by the object. 586 \Data object is in \FunctionSpace \var{on} and has rank r+1 where r is the rank of the object. 587 If \var{on} is not present, a suitbale \FunctionSpace is used. 588 \index{gradient} 589 \end{methoddesc} 590 591 \begin{methoddesc}[Data]{integrate}{} 592 returns the integral of the function represented by the object. The method returns 593 a \class{numarray.array} object of the same \Shape like the object. A 594 component of the returned object is the integral of the corresponding 595 component of the object. \index{integral} 596 \end{methoddesc} 597 598 \begin{methoddesc}[Data]{interpolate}{on} 599 interpolates 600 the function represented by the object 601 into the \FunctionSpace\var{on}. 602 \index{interpolation} 603 \end{methoddesc} 604 605 \begin{methoddesc}[Data]{abs}{} 606 applies the absolute value function to the object. The 607 return \Data object has the same \Shape and is in the same 608 \FunctionSpace like the object. For all \DataSamplePoints and all 609 components the value is calculated by applying the exponential 610 function. \index{function!absolute value} 611 \end{methoddesc} 612 613 \begin{methoddesc}[Data]{exp}{} 614 applies the exponential function to the object. The 615 return \Data object has the same \Shape and is in the same 616 \FunctionSpace like the object. For all \DataSamplePoints and all 617 components the value is calculated by applying the exponential 618 function. \index{function!exponential} 619 \end{methoddesc} 620 621 \begin{methoddesc}[Data]{sqrt}{} 622 applies the square root function to the object. The 623 return \Data object has the same \Shape and is in the same 624 \FunctionSpace like the object. For all \DataSamplePoints and all 625 components the value is calculated by applying the square root function. 626 An exception is 627 raised if the value is negative. \index{function!square root} 628 \end{methoddesc} 629 630 \begin{methoddesc}[Data]{sin}{} 631 applies the sine function to the object. The 632 return \Data object has the same \Shape and is in the same 633 \FunctionSpace like the object. For all \DataSamplePoints and all 634 components the value is calculated by applying the sine function. \index{function!sine} 635 \end{methoddesc} 636 637 \begin{methoddesc}[Data]{cos}{} 638 applies the cosine function to the object. The 639 return \Data object has the same \Shape and is in the same 640 \FunctionSpace like the object. For all \DataSamplePoints and all 641 components the value is calculated by applying the cosine function. \index{function!cosine} 642 \end{methoddesc} 643 644 \begin{methoddesc}[Data]{tan}{} 645 applies the tangent function to the object. The 646 return \Data object has the same \Shape and is in the same 647 \FunctionSpace like the object. For all \DataSamplePoints and all 648 components the value is calculated by applying the tangent function. \index{function!tangent} 649 \end{methoddesc} 650 651 \begin{methoddesc}[Data]{log}{} 652 applies the logarithmic function to the object. The 653 return \Data object has the same \Shape and is in the same 654 \FunctionSpace like the object. For all \DataSamplePoints and all 655 components the value is calculated by applying the logarithmic function. An exception is 656 raised if the value is negative.\index{function!logarithmic} 657 \end{methoddesc} 658 659 \begin{methoddesc}[Data]{maxval}{} 660 returns the maximum value over all components. The 661 return value is a \Data object of rank 0 662 and is in the same 663 \FunctionSpace like the object. For all \DataSamplePoints 664 the value is calculated as the maximum value over all components. \index{function!maximum} 665 \end{methoddesc} 666 667 \begin{methoddesc}[Data]{minval}{} 668 returns the minimum value over all components. The 669 return value is a \Data object of rank 0 670 and is in the same 671 \FunctionSpace like the object. For all \DataSamplePoints 672 the value is calculated as the minimum value over all components. \index{function!minimum} 673 \end{methoddesc} 674 675 \begin{methoddesc}[Data]{length}{} 676 returns the Euclidean length at all \DataSamplePoints. The 677 return value is a \Data object of rank 0 678 and is in the same 679 \FunctionSpace like the object. For all \DataSamplePoints 680 the value is calculated as the square root of the sum of the square over all over all components. \index{function!length} 681 \end{methoddesc} 682 \begin{methoddesc}[Data]{transpose}{axis} 683 returns the transpose of the object around \var{axis}. \var{axis} is a non-negative integer 684 which is less the rank $r$ of the object. Transpose swaps the indexes $0$ to \var{axis} 685 with the indexes \var{axis}+1 to $r$. If the \var{d} is \RankFour one has 686 \begin{python} 687 d[i,j,k,l]=d.transpose(0)[i,j,k,l] 688 d[i,j,k,l]=d.transpose(1)[j,k,l,i] 689 d[i,j,k,l]=d.transpose(2)[k,l,i,j] 690 d[i,j,k,l]=d.transpose(3)[l,i,j,k] 691 \end{python} 692 \index{function!transpose} 693 \end{methoddesc} 694 695 \begin{methoddesc}[Data]{trace}{} 696 returns sum of the components with identical indexes. 697 The 698 return value is a \Data object of rank 0 699 and is in the same 700 \FunctionSpace like the object. 701 \index{function!trace} 702 \end{methoddesc} 703 \begin{methoddesc}[Data]{saveDX}{fileName} 704 saves the object to an openDX format file of name \var{fileName}, see 705 \ulink{www.opendx.org}{\url{www.opendx.org}}. \index{openDX} 706 \end{methoddesc} 707 708 709 For convenience the following factories are provided to created \Data object: 710 711 \begin{funcdesc}{Scalar}{value=0.,what=escript::FunctionSpace(),expand=\False} 712 returns a \Data object of rank 0 in the \FunctionSpace \var{what}. 713 Values are initialed with the double \var{value}. If \var{expanded} is \True 714 the \Data object is represented in expanded from. 715 \end{funcdesc} 716 717 \begin{funcdesc}{Vector}{value=0.,what=escript::FunctionSpace(),expand=\False} 718 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what} 719 where \var{d} is the spatial dimension of the \Domain of \var{what}. 720 Values are initialed with the double \var{value}. If \var{expanded} is \True 721 the \Data object is represented in expanded from. 722 \end{funcdesc} 723 724 \begin{funcdesc}{Tensor}{value=0.,what=escript::FunctionSpace(),expand=\False} 725 returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what} 726 where \var{d} is the spatial dimension of the \Domain of \var{what}. 727 Values are initialed with the double \var{value}. If \var{expanded} is \True 728 the \Data object is represented in expanded from. 729 \end{funcdesc} 730 731 \begin{funcdesc}{Tensor3}{value=0.,what=escript::FunctionSpace(),expand=\False} 732 returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what} 733 where \var{d} is the spatial dimension of the \Domain of \var{what}. 734 Values are initialed with the double \var{value}. If \var{expanded} is \True 735 the \Data object is re\var{arg}presented in expanded from. 736 \end{funcdesc} 737 738 \begin{funcdesc}{Tensor4}{value=0.,what=escript::FunctionSpace(),expand=\False} 739 returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what} 740 where \var{d} is the spatial dimension of the \Domain of \var{what}. 741 Values are initialed with the double \var{value}. If \var{expanded} is \True 742 the \Data object is represented in expanded from. 743 \end{funcdesc} 744 745 \begin{funcdesc}{abs}{arg} 746 returns the absolute value of \var{arg} where \var{arg} 747 can be double, a \Data object or an \numarray object. 748 \end{funcdesc} 749 750 \begin{funcdesc}{sin}{arg} 751 returns the sine of \var{arg} where \var{arg} 752 can be double, a \Data object or an \numarray object. 753 \end{funcdesc} 754 755 \begin{funcdesc}{cos}{arg} 756 returns the cosine of \var{arg} where \var{arg} 757 can be double, a \Data object or an \numarray object. 758 \end{funcdesc} 759 760 \begin{funcdesc}{exp}{arg} 761 returns the value of the exponential function for \var{arg} where \var{arg} 762 can be double, a \Data object or an \numarray object. 763 \end{funcdesc} 764 765 \begin{funcdesc}{sqrt}{arg} 766 returns the square root of \var{arg} where \var{arg} 767 can be double, a \Data object or an \numarray object. 768 \end{funcdesc} 769 770 \begin{funcdesc}{maxval}{arg} 771 returns the maximum value over all component of \var{arg} where \var{arg} 772 can be double, a \Data object or an \numarray object. 773 \end{funcdesc} 774 775 \begin{funcdesc}{minval}{arg} 776 returns the minumum value over all component of \var{arg} where \var{arg} 777 can be double, a \Data object or an \numarray object. 778 \end{funcdesc} 779 780 \begin{funcdesc}{length}{arg} 781 returns the length of \var{arg} which is the 782 square root of the sum of the squares of all component of \var{arg}. \var{arg} 783 can be double, a \Data object or an \numarray object. 784 \end{funcdesc} 785 786 \begin{funcdesc}{sign}{arg} 787 return the sign of \var{arg} where \var{arg} 788 can be double, a \Data object or an \numarray object. 789 \end{funcdesc} 790 791 \begin{funcdesc}{transpose}{arg,\optional{axis}} 792 returns the transpose of \var{arg} around \var{axis}. \var{axis} is a non-negative integer 793 which is less the rank $r$ of the object. Transpose swaps the indexes $0$ to \var{axis} 794 with the indexes \var{axis}+1 to $r$. If \var{axis} is not present, \var{axis}=$r/2$ is assumed. 795 \var{arg} 796 may be a \Data object or an \numarray object. 797 \end{funcdesc} 798 799 \begin{funcdesc}{transpose}{arg,\optional{axis}} 800 returns the trace the object of \var{arg}. The trace is the sum over those components 801 with identical indexed. 802 \var{arg} 803 may be a \Data object or a \numarray object. 804 \end{funcdesc} 805 806 \begin{funcdesc}{sum}{arg} 807 returns the sum over all components and all 808 \DataSamplePoints of \var{arg}, where \var{arg} 809 is a \Data object. 810 \end{funcdesc} 811 812 \begin{funcdesc}{sup}{arg} 813 returns the maximum over all components and all 814 \DataSamplePoints of \var{arg}, where \var{arg} 815 is a \Data object. 816 \end{funcdesc} 817 818 \begin{funcdesc}{inf}{arg} 819 returns the mimumum over all components and all 820 \DataSamplePoints of \var{arg}, where \var{arg} 821 is a \Data object. 822 \end{funcdesc} 823 824 825 \begin{funcdesc}{L2}{arg} 826 returns the $L^2$ norm of \var{arg}. This is the square root 827 of the sum of the squared value over all components and all 828 \DataSamplePoints of \var{arg}, where \var{arg} 829 is a \Data object. 830 \end{funcdesc} 831 832 \begin{funcdesc}{Lsup}{arg} 833 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values 834 over all components and all 835 \DataSamplePoints of \var{arg}, where \var{arg} 836 is a \Data object. 837 \end{funcdesc} 838 839 \begin{funcdesc}{dot}{arg1,arg2} 840 returns the dot product of of \var{arg1} and \var{arg2}. This is sum 841 of the product of corresponding entries in \var{arg1} and \var{arg2} over all 842 components and and all 843 \DataSamplePoints. \var{arg1} and \var{arg2} are \Data objects of the 844 same \Shape and in the same \FunctionSpace. 845 \end{funcdesc} 846 847 \begin{funcdesc}{grad}{arg,\optional{where}} 848 returns the gradient of \var{arg} as a function in the \FunctionSpace \var{where}. 849 If \var{where} is not present a reasonable \FunctionSpace is chosen. 850 \var{arg} 851 is a \Data object. 852 \end{funcdesc} 853 854 \begin{funcdesc}{integrate}{arg} 855 returns the integral of \var{arg} as a \numarray object. 856 If \var{where} is not present a reasonable \FunctionSpace is chosen. 857 \var{arg} 858 is a \Data object. 859 \end{funcdesc} 860 861 \begin{funcdesc}{interpolate}{arg,where} 862 interpolate \Data object \var{arg} into the \FunctionSpace \var{where} 863 \end{funcdesc} 864 865 866 \section{\Operator Class} 867 868 The \Operator class provides an abstract access to operators build 869 within the \LinearPDE class. \Operator objects are created 870 when a PDE is handed over to a PDE solver library and handled 871 by the \LinearPDE class defining the PDE. The user can gain access 872 to the \Operator of a \LinearPDE object through the \var{getOperator} 873 method. 874 875 \begin{classdesc}{Operator}{} 876 creates an empty \Operator object. 877 \end{classdesc} 878 879 \begin{methoddesc}[Operator]{isEmpty}{fileName} 880 returns \True is the object is empty. Otherwise \True is returned. 881 \end{methoddesc} 882 883 \begin{methoddesc}[Operator]{setValue}{value} 884 resets all entires in the obeject representation to \var{value} 885 \end{methoddesc} 886 887 \begin{methoddesc}[Operator]{solves}{rhs} 888 solves the operator equation with right hand side \var{rhs} 889 \end{methoddesc} 890 891 \begin{methoddesc}[Operator]{of}{u} 892 applies the operator to the \Data object \var{u} 893 \end{methoddesc} 894 895 \begin{methoddesc}[Operator]{saveMM}{fileName} 896 saves the object to a matrix market format file of name 897 \var{fileName}, see 898 \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}. 899 \index{Matrix Market} 900 \end{methoddesc} 901

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