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 1 jgs 107 2 jgs 82 % $Id$ 3 4 \chapter{The module \escript} 5 6 jgs 102 \declaremodule{extension}{escript} 7 \modulesynopsis{Data manipulation} 8 jgs 82 9 jgs 102 \begin{figure} 10 \includegraphics[width=\textwidth]{EscriptDiagram1.eps} 11 \caption{\label{ESCRIPT DEP}Dependency of Function Spaces. An arrow indicates that a function in the 12 function space at the starting point can be interpreted as a function in the function space of the arrow target.} 13 \end{figure} 14 jgs 82 15 jgs 102 \escript is an extension of Python to handle functions represented by their values on 16 jgs 107 \DataSamplePoints for the geometrical region on which 17 jgs 102 the function is defined. The region as well as the method which is used 18 to interpolate value on the \DataSamplePoints is defined by 19 \Domain class objects. For instance when using 20 the finite element method (FEM) \index{finite element method} 21 \Domain object holds the information about the FEM mesh, eg. 22 a table of nodes and a table of elements. Although \Domain contains 23 the discretization method to be used \escript does not use this information directly. 24 \Domain objects are created from a module which want to make use 25 \escript, e.g. \finley. 26 jgs 82 27 jgs 102 The solution of a PDE is a function of its location in the domain of interest $\Omega$. 28 When solving a partial differential equation \index{partial differential equation} (PDE) using FEM 29 the solution is (piecewise) differentiable but, in general, its gradient 30 is discontinuous. To reflect these different degrees of smoothness different 31 representations of the functions are used. For instance; in FEM 32 the displacement field is represented by its values at the nodes of the mesh, while the 33 strain, which is the symmetric part of the gradient of the displacement field, is stored on the 34 element centers. To be able to classify functions with respect to their smoothness, \escript has the 35 concept of the "function space". A function space is described by a \FunctionSpace object. 36 The following statement generates the object \var{solution_space} which is 37 a \FunctionSpace object and provides access to the function space of 38 PDE solutions on the \Domain \var{mydomain}: 39 \begin{python} 40 solution_space=Solution(mydomain) 41 \end{python} 42 jgs 107 The following generators for function spaces on a \Domain \var{mydomain} are available: 43 jgs 102 \begin{itemize} 44 \item \var{Solution(mydomain)}: solutions of a PDE. 45 \item \var{ContinuousFunction(mydomain)}: continuous functions, eg. a temperature distribution. 46 \item \var{Function(mydomain)}: general functions which are not necessarily continuous, eg. a stress field. 47 \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, eg. a surface pressure. 48 \item \var{FunctionOnContact0(mydomain)}: functions on side $0$ of the discontinuity. 49 \item \var{FunctionOnContact1(mydomain)}: functions on side $1$ of the discontinuity. 50 \end{itemize} 51 A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous. 52 The location of discontinuity is defined in the \Domain object. 53 \fig{ESCRIPT DEP} shows the dependency between the types of function spaces. 54 The solution of a PDE is a continuous function. Any continuous function can be seen as a general function 55 on the domain and can be restricted to the boundary as well as to any side of the 56 discontinuity (the result will be different depending on 57 which side is chosen). Functions on any side of the 58 discontinuity can be seen as a function on the corresponding other side. 59 A function on the boundary or on one side of 60 the discontinuity cannot be seen as a general function on the domain as there are no values 61 defined for the interior. For most PDE solver libraries 62 the space of the solution and continuous functions is identical, however in some cases, eg. 63 when periodic boundary conditions are used in \finley, a solution 64 fulfils periodic boundary conditions while a continuous function does not have to be periodic. 65 66 The concept of function spaces describes the properties of 67 functions and allows abstraction from the actual representation 68 of the function in the context of a particular application. For instance, 69 in the FEM context a 70 function in the \Function function space 71 is typically represented by its values at the element center, 72 but in a finite difference scheme the edge midpoint of cells is preferred. 73 Using the concept of function spaces 74 allows the user to run the same script on different 75 PDE solver libraries by just changing the creator of the \Domain object. 76 Changing the function space of a particular function 77 will typically lead to a change of its representation. 78 So, when seen as a general function, 79 a continuous function which is typically represented by its values 80 on the node of the FEM mesh or finite difference grid 81 jgs 107 must be interpolated to the element centers or the cell edges, 82 jgs 102 respectively. 83 jgs 82 84 jgs 102 \Data class objects store functions of the location in a domain. 85 The function is represented through its values on \DataSamplePoints where 86 the \DataSamplePoints are chosen according to the function space 87 of the function. 88 \Data class objects are used to define the coefficients 89 of the PDEs to be solved by a PDE solver library 90 and to store the returned solutions. 91 jgs 82 92 jgs 102 The values of the function have a rank which gives the 93 jgs 107 number of indices, and a \Shape defining the range of each index. 94 jgs 102 The rank in \escript is limited to the range $0$ through $4$ and 95 it is assumed that the rank and \Shape is the same for all \DataSamplePoints. 96 The \Shape of a \Data object is a tuple \var{s} of integers. The length 97 of \var{s} is the rank of the \Data object and \var{s[i]} is the maximum 98 value for the \var{i}-th index. 99 For instance, a stress field has rank $2$ and 100 \Shape $(d,d)$ where $d$ is the spatial dimension. 101 The following statement creates the \Data object 102 \var{mydat} representing a 103 continuous function with values 104 of \Shape $(2,3)$ and rank $2$: 105 \begin{python} 106 mydat=Data(value=1,what=ContinuousFunction(myDomain),shape=(2,3)) 107 \end{python} 108 The initial value is the constant $1$ for all \DataSamplePoints and 109 all components. 110 jgs 82 111 jgs 102 \Data objects can also be created from any \numarray 112 array or any object, such as a list of floating point numbers, 113 that can be converted into a \numarray array \Ref{NUMARRAY}. 114 The following two statements 115 create objects which are equivalent to \var{mydat}: 116 \begin{python} 117 mydat1=Data(value=numarray.ones((2,3)),what=ContinuousFunction(myDomain)) 118 mydat2=Data(value=[[1,1],[1,1],[1,1]],what=ContinuousFunction(myDomain)) 119 \end{python} 120 In the first case the initial value is \var{numarray.ones((2,3))} 121 which generates a $2 \times 3$ matrix as a \numarray array 122 filled with ones. The \Shape of the created \Data object 123 it taken from the \Shape of the array. In the second 124 case, the creator converts the initial value, which is a list of lists, 125 jgs 107 and converts it into a \numarray array before creating the actual 126 jgs 102 \Data object. 127 jgs 82 128 jgs 102 For convenience \escript provides creators for the most common types 129 of \Data objects in the following forms (\var{d} defines the 130 spatial dimension): 131 \begin{itemize} 132 \item \var{Scalar(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(,))}, 133 e.g a temperature field. 134 \item \var{Vector(0,Function(mydomain))}is the same as \var{Data(0,Function(myDomain),(d))}, e.g 135 a velocity field. 136 \item \var{Tensor(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d))}, 137 eg. a stress field. 138 \item \var{Tensor4(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d,d,d))} 139 eg. a Hook tensor field. 140 \end{itemize} 141 Here the initial value is $0$ but any object that can be converted into a \numarray array and whose \Shape 142 is consistent with \Shape of the \Data object to be created can be used as the initial value. 143 jgs 82 144 jgs 102 \Data objects can be manipulated by applying unitary operations (eg. cos, sin, log) 145 jgs 107 and can be combined by applying binary operations (eg. +, - ,* , /). 146 It is to be emphasized that \escript itself does not handle any spatial dependencies as 147 jgs 102 it does not know how values are interpreted by the processing PDE solver library. 148 However \escript invokes interpolation if this is needed during data manipulations. 149 Typically, this occurs in binary operation when both arguments belong to different 150 function spaces or when data are handed over to a PDE solver library 151 which requires functions to be represented in a particular way. 152 jgs 82 153 jgs 102 The following example shows the usage of {\tt Data} objects: Assume we have a 154 displacement field $u$ and we want to calculate the corresponding stress field 155 $\sigma$ using the linear--elastic isotropic material model 156 \begin{eqnarray}\label{eq: linear elastic stress} 157 \sigma\hackscore {ij}=\lambda u\hackscore {k,k} \delta\hackscore {ij} + \mu ( u\hackscore {i,j} + u\hackscore {j,i}) 158 \end{eqnarray} 159 where $\delta\hackscore {ij}$ is the Kronecker symbol and 160 $\lambda$ and $\mu$ are the Lame coefficients. The following function 161 takes the displacement {\tt u} and the Lame coefficients 162 \var{lam} and \var{mu} as arguments and returns the corresponding stress: 163 \begin{python} 164 import numarray 165 def getStress(u,lam,mu): 166 d=u.getDomain().getDim() 167 g=grad(u) 168 stress=lam*trace(g)*numarray.identity(d)+ \ 169 mu*(g+transpose(g)) 170 return stress 171 \end{python} 172 The variable 173 \var{d} gives the spatial dimension of the 174 domain on which the displacements are defined. 175 \var{identity} is a \numarray function which returns the Kronecker symbol with indexes 176 $i$ and $j$ running from $0$ to \var{d}-1. The call \var{grad(u)} requires 177 the displacement field \var{u} to be in the \var{Solution} or \ContinuousFunction 178 function space. The result \var{g} as well as the returned stress will be in the \Function function space. 179 If \var{u} is available, eg. by solving a PDE, \var{getStress} might be called 180 in the following way: 181 \begin{python} 182 s=getStress(u,1.,2.) 183 \end{python} 184 However \var{getStress} can also be called with \Data objects as values for 185 \var{lam} and \var{mu} which, 186 for instance in the case of a temperature dependency, are calculated by an expression. 187 The following call is equivalent to the previous example: 188 \begin{python} 189 lam=Scalar(1.,ContinuousFunction(mydomain)) 190 mu=Scalar(2.,Function(mydomain)) 191 s=getStress(u,lam,mu) 192 \end{python} 193 The function \var{lam} belongs to the \ContinuousFunction function space 194 but with \var{g} the function \var{trace(g)} is in the \Function function space. 195 Therefore the evaluation of the product \var{lam*trace(g)} in the stress calculation 196 produces a problem, as both functions are represented differently, eg. in FEM 197 \var{lam} by its values on the node, and in \var{trace(g)} by its values at the element centers. 198 In the case of inconsistent function spaces of arguments in a binary operation, \escript 199 interprets the arguments in the appropriate function space according to the inclusion 200 defined in Table~\ref{ESCRIPT DEP}. In this example that means 201 \escript sees \var{lam} as a function of the \Function function space. 202 In the context of FEM this means the nodal values of 203 \var{lam} are interpolated to the element centers. Behind the scenes 204 \escript calls the appropriate function from the PDE solver library. 205 jgs 82 206 jgs 102 \begin{figure} 207 \includegraphics[width=\textwidth]{EscriptDiagram2.eps} 208 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region with two rock types {\it white} and {\it gray}. 209 The number in each cell refers to the major rock type present in the cell ($1$ for {\it white} and $2$ for {\it gray}). 210 } 211 \end{figure} 212 jgs 82 213 jgs 102 Material parameters such as the Lame coefficients are typically dependent on rock types present in the 214 area of interest. A common technique to handle these kinds of material parameters is "tagging". \fig{Figure: tag} 215 shows an example. In this case two rock types {\it white} and {\it gray} can be found in the domain. The domain 216 is subdivided into rectangular shaped cells (which is not necessarily the best subdivision for this case). Each 217 cell has a tag indicating the rock type predominately found in this cell. Here $1$ is used to indicate 218 rock type {\it white} and $2$ for rock type {\it gray}. The tags are assigned at the time when the cells are generated 219 (\escript provides tools to manipulate tags at a later stage) 220 and stored in the \Domain class object. The following statements show how for the 221 example of \fig{Figure: tag} and the stress calculation discussed before tagged values are used for 222 \var{lam}: 223 \begin{python} 224 lam=Scalar(value=2.,what=Function(mydomain)) 225 lam.setTaggedValue(1,30.) 226 lam.setTaggedValue(2,5000.) 227 s=getStress(u,lam,2.) 228 \end{python} 229 In this example \var{lam} is set to $30$ for those cells with tag $1$ and to $5000.$ for those cells 230 with tag $2$. The initial value $2$ of \var{lam} is used as a default value for the case when a tag 231 is encountered which has not been linked with a value. Note that the \var{getStress} method 232 is called without modification. \escript resolves the tags when \var{lam*trace(g)} is calculated. 233 jgs 82 234 jgs 102 The \Data class provides a transparent interface to various data representations and the 235 translations between them. As shown in the example of stress calculation, this allows the user to 236 develop and test algorithms for a simple case (for instance with the Lame coefficients as constants) 237 and then without further modifications of the program code to apply the algorithm in a 238 more complex application (for instance a definition of the Lame coefficients using tags). 239 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): 240 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. 241 Any operation on this constant data will only be performed on the single value. 242 In the expanded case, each sample point has an individual value, eg. the solution of a PDE, 243 and the values are stored as a complete array. The tagged case has already been discussed above. 244 245 Values are accessed through a sample reference number. Operations on expanded \Data 246 objects have to be performed for each sample point individually. If tagged values are used values are 247 held in a dictionary. Operations on tagged data require processing the set of tagged values only, rather than 248 processing the value for each individual sample point. 249 \escript allows use of constant, tagged and expanded data in a single expression. 250 jgs 82 251 252 jgs 102 253 \section{\Domain class} 254 255 \begin{classdesc}{Domain}{} 256 jgs 107 A \Domain object is used to describe a geometrical region together with 257 jgs 102 a way of representing functions over this region. 258 The \Domain class provides an abstract access to the domain of \FunctionSpace and \Data objects. 259 \Domain itself has no initialization but implementations of \Domain are 260 instantiated by numerical libraries making use of \Data objects. 261 jgs 82 \end{classdesc} 262 263 jgs 102 \begin{methoddesc}[Domain]{getDim}{} 264 returns the spatial dimension of the \Domain. 265 \end{methoddesc} 266 267 \begin{methoddesc}[Domain]{getX}{} 268 returns the locations in the \Domain. The \FunctionSpace of the returned 269 \Data object is chosen by the \Domain implementation. Typically it will be 270 in the \Function. 271 \end{methoddesc} 272 273 \begin{methoddesc}[Domain]{setX}{newX} 274 assigns a new location to the \Domain. \var{newX} has to have \Shape $(d,)$ 275 where $d$ is the spatial dimension of the domain. Typically \var{newX} must be 276 in the \ContinuousFunction but the space actually to be used depends on the \Domain implementation. 277 \end{methoddesc} 278 279 \begin{methoddesc}[Domain]{getNormal}{} 280 returns the surface normals on the boundary of the \Domain as \Data object. 281 \end{methoddesc} 282 283 \begin{methoddesc}[Domain]{getSize}{} 284 returns the local sample size, e.g. the element diameter, as \Data object. 285 \end{methoddesc} 286 287 \begin{methoddesc}[Domain]{__eq__}{arg} 288 returns \True of the \Domain \var{arg} describes the same domain. Otherwise 289 \False is returned. 290 \end{methoddesc} 291 292 \begin{methoddesc}[Domain]{__ne__}{arg} 293 returns \True of the \Domain \var{arg} does not describe the same domain. 294 Otherwise \False is returned. 295 \end{methoddesc} 296 297 \section{\Domain class} 298 \begin{classdesc}{FunctionSpace}{} 299 \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects 300 are instantiated by generator functions. \Data objects in particular \FunctionSpace are 301 represented by their values at \DataSamplePoints which are defined by the type and the \Domain of the 302 \FunctionSpace. 303 jgs 82 \end{classdesc} 304 305 jgs 102 \begin{methoddesc}[FunctionSpace]{getDim}{} 306 returns the spatial dimension of the \Domain of the \FunctionSpace. 307 \end{methoddesc} 308 jgs 82 309 jgs 102 \begin{methoddesc}[FunctionSpace]{getX}{} 310 returns the location of the \DataSamplePoints. 311 \end{methoddesc} 312 jgs 82 313 jgs 102 \begin{methoddesc}[FunctionSpace]{getNormal}{} 314 If the domain of functions in the \FunctionSpace 315 is a hypermanifold (e.g. the boundary of a domain) 316 the method returns the outer normal at each of the 317 \DataSamplePoints. Otherwise an exception is raised. 318 \end{methoddesc} 319 jgs 82 320 jgs 102 \begin{methoddesc}[FunctionSpace]{getSize}{} 321 returns a \Data objects measuring the spacing of the \DataSamplePoints. 322 The size may be zero. 323 \end{methoddesc} 324 jgs 82 325 jgs 102 \begin{methoddesc}[FunctionSpace]{getDomain}{} 326 returns the \Domain of the \FunctionSpace. 327 \end{methoddesc} 328 jgs 82 329 jgs 102 \begin{methoddesc}[FunctionSpace]{__eq__}{arg} 330 returns \True of the \Domain \var{arg} describes the same domain. Otherwise 331 \False is returned. 332 \end{methoddesc} 333 jgs 82 334 jgs 102 \begin{methoddesc}[FunctionSpace]{__ne__}{arg} 335 returns \True of the \Domain \var{arg} describes the note same domain. 336 Otherwise \False is returned. 337 \end{methoddesc} 338 jgs 82 339 jgs 102 The following function provide generators for \FunctionSpace objects: 340 \begin{funcdesc}{Function}{domain} 341 returns the \Function on the \Domain \var{domain}. \Data objects in this type of \Function 342 are defined over the whole geometrical region defined by \var{domain}. 343 jgs 82 \end{funcdesc} 344 345 jgs 102 \begin{funcdesc}{ContinuousFunction}{domain} 346 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function 347 are defined over the whole geometrical region defined by \var{domain} and assumed to represent 348 a continuous function. 349 jgs 82 \end{funcdesc} 350 351 jgs 102 \begin{funcdesc}{FunctionOnBoundary}{domain} 352 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function 353 are defined on the boundary of the geometrical region defined by \var{domain}. 354 jgs 82 \end{funcdesc} 355 356 jgs 102 \begin{funcdesc}{FunctionOnContactZero}{domain} 357 returns the \FunctionOnContactZero the \Domain domain. \Data objects in this type of \Function 358 are defined on side 0 of a discontinutiy within the geometrical region defined by \var{domain}. 359 The discontinutiy is defined when \var{domain} is instantiated. 360 jgs 82 \end{funcdesc} 361 362 jgs 102 \begin{funcdesc}{FunctionOnContactOne}{domain} 363 returns the \FunctionOnContactOne on the \Domain domain. 364 \Data objects in this type of \Function 365 are defined on side 1 of a discontinutiy within the geometrical region defined by \var{domain}. 366 The discontinutiy is defined when \var{domain} is instantiated. 367 jgs 82 \end{funcdesc} 368 369 jgs 102 \begin{funcdesc}{Solution}{domain} 370 returns the \SolutionFS on the \Domain domain. \Data objects in this type of \Function 371 are defined on geometrical region defined by \var{domain} and are solutions of 372 partial differential equations \index{partial differential equation}. 373 jgs 82 \end{funcdesc} 374 375 jgs 102 \begin{funcdesc}{ReducedSolution}{domain} 376 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in this type of \Function 377 are defined on geometrical region defined by \var{domain} and are solutions of 378 partial differential equations \index{partial differential equation} with a reduced smoothness 379 for the solution approximation. 380 jgs 82 \end{funcdesc} 381 382 jgs 102 \section{\Data Class} 383 jgs 107 \label{SEC ESCRIPT DATA} 384 jgs 82 385 jgs 102 The following table shows binary and unitary operations that can be applied to 386 \Data objects: 387 \begin{tableii}{l|l}{textrm}{expression}{Description} 388 \lineii{+\var{arg1}} {just \var{arg} \index{+}} 389 \lineii{-\var{arg1}} {swapping the sign\index{-}} 390 \lineii{\var{arg1}+\var{arg2}} {adds \var{arg1} and \var{arg2} \index{+}} 391 \lineii{\var{arg1}*\var{arg2}} {multiplies \var{arg1} and \var{arg2} \index{*}} 392 \lineii{\var{arg1}-\var{arg2}} {difference \var{arg2} from\var{arg2} \index{-}} 393 \lineii{\var{arg1}/\var{arg2}} {ratio \var{arg1} by \var{arg2} \index{/}} 394 \lineii{\var{arg1}**\var{arg2}} {raises \var{arg1} to the power of \var{arg2} \index{**}} 395 \end{tableii} 396 jgs 107 At least one of the arguments \var{arg1} or \var{arg2} must be a 397 jgs 102 \Data object. One of the arguments may be an object that can be 398 converted into a \Data object. If \var{arg1} or \var{arg2} are 399 jgs 107 defined on different \FunctionSpace an attempt is made to embed \var{arg1} 400 jgs 102 into the \FunctionSpace of \var{arg2} or to embed \var{arg2} into 401 the \FunctionSpace of \var{arg1}. Boths arguments must have the same 402 \Shape or one of the arguments my be of rank 0 or \Shape (1,). In the 403 latter case it is assumed that the particular argument is of the same 404 jgs 107 \Shape as the other argument but constant over all components. 405 jgs 82 406 jgs 102 The returned \Data object has the same \Shape and is defined on 407 jgs 107 the \DataSamplePoints as \var{arg1} or \var{arg2}. 408 jgs 82 409 jgs 102 The following table shows the update operations that can be applied to 410 \Data objects: 411 \begin{tableii}{l|l}{textrm}{expression}{Description} 412 \lineii{\var{arg1}+=\var{arg2}} {adds \var{arg1} to \var{arg2} \index{+}} 413 \lineii{\var{arg1}*=\var{arg2}} {multiplies \var{arg1} with \var{arg2} \index{*}} 414 \lineii{\var{arg1}-=\var{arg2}} {subtracts \var{arg2} from\var{arg2} \index{-}} 415 \lineii{\var{arg1}/=\var{arg2}} {divides \var{arg1} by \var{arg2} \index{/}} 416 \end{tableii} 417 \var{arg1} must be a \Data object. \var{arg1} must be a 418 \Data object or an object that can be converted into a 419 \Data object. \var{arg1} must have the same \Shape like 420 \var{arg1} or has rank 0 or \Shape (1,). In the latter case it is 421 assumed that the values of \var{arg1} are constant for all 422 jgs 107 components. \var{arg2} must be defined on the same \DataSamplePoints as 423 jgs 102 \var{arg1} or it must be possible to interpolate \var{arg2} onto the 424 jgs 107 \DataSamplePoints where \var{arg1} is held. 425 jgs 82 426 jgs 102 The \Data class supports getting slices as well as assigning new values to components in an existing 427 jgs 107 \Data object. \index{slicing} 428 jgs 102 The following expression for getting (expression on the right hand side of the 429 equal sign) and setting slices (expression on the left hand side of the 430 equal sign) are valid: 431 \begin{tableiii}{l|ll}{textrm}{rank of \var{arg}}{slicing expression}{\Shape of returned and assigned object} 432 \lineiii{0}{ no slicing } {-} 433 \lineiii{1}{\var{arg[l0:u0]}} {(\var{u0}-\var{l0},)} 434 \lineiii{2}{\var{arg[l0:u0,l1:u1]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1})} 435 \lineiii{3}{\var{arg[l0:u0,l1:u1,l2:u2]} } {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})} 436 \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})} 437 \end{tableiii} 438 where 439 $0 \le \var{l0} \le \var{u0} \le \var{s[0]}$, 440 $0 \le \var{l1} \le \var{u1} \le \var{s[1]}$, 441 $0 \le \var{l2} \le \var{u2} \le \var{s[2]}$, 442 $0 \le \var{l3} \le \var{u3} \le \var{s[3]}$ and \var{s} the \Shape if \var{arg}. 443 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case 444 $0$ is assumed. 445 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may not be present in which case 446 \var{s} is assumed. The lower and upper index may be identical, in which case the column and the lower or upper 447 index may be dropped. In the returned or in the object assigned to a slice the corresponding component is dropped, 448 i.e. the rank is reduced by one in comparison to \var{arg}. 449 jgs 107 The following examples show slicing usage: 450 jgs 102 \begin{python} 451 t[1,1,1,0]=9. 452 s=t[:2,:,2:6,5] # s has rank 3 453 s[:,:,1]=1. 454 t[:2,:2,5,5]=s[2:4,1,:2] 455 \end{python} 456 457 458 459 \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expanded=\False} 460 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}. 461 The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True 462 the \Data object is represented in expanded from. 463 jgs 82 \end{classdesc} 464 465 jgs 102 \begin{classdesc}{Data}{value,what=FunctionSpace(),expanded=\False} 466 creates a \Data object in the \FunctionSpace \var{what}. 467 The value for each \DataSamplePoints is set to \numarray object \var{value}. 468 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True 469 the \Data object is represented in expanded from. 470 \end{classdesc} 471 472 \begin{classdesc}{Data}{value,what=FunctionSpace()} 473 creates a \Data object in the \FunctionSpace \var{what} from the \Data object \var{value}. 474 The \Shape of the created \Data object is equal to the \Shape of \var{value}. 475 If the \FunctionSpace of \var{value} is equal to the \var{what}, a shallow copy, i.e. 476 a reference to the representation of \var{value} is greated. Otherwise, 477 \var{value} is interpolated into the \var{what}. 478 \end{classdesc} 479 480 \begin{classdesc}{Data}{} 481 creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present 482 where a \Data object is required. 483 \end{classdesc} 484 485 \begin{methoddesc}[Data]{getFunctionSpace}{} 486 returns the \FunctionSpace of the object. 487 jgs 82 \end{methoddesc} 488 489 jgs 102 \begin{methoddesc}[Data]{getFunctionSpace}{} 490 returns the \Domain of the object. 491 \end{methoddesc} 492 493 jgs 82 \begin{methoddesc}[Data]{getShape}{} 494 jgs 102 returns the \Shape of the object as a \class{tuple} of 495 integers. 496 jgs 82 \end{methoddesc} 497 498 \begin{methoddesc}[Data]{getRank}{} 499 returns the rank of the data on each data point. \index{rank} 500 \end{methoddesc} 501 502 jgs 102 \begin{methoddesc}[Data]{isEmpty}{} 503 returns \True id the \Data object is the \EmptyData object. 504 Otherwise \False is returned. 505 jgs 82 \end{methoddesc} 506 507 jgs 102 \begin{methoddesc}[Data]{setTaggedValue}{tag,value} 508 assigns the \var{value} to all \DataSamplePoints which have the tag 509 \var{tag}. \var{value} must be an object of class 510 \class{numarray.array} or must be convertable into a 511 \class{numarray.array} object. \var{value} (or the cooresponding 512 \class{numarray.array} object) must be of rank $0$ or must have the 513 same rank like the object. 514 If a value has already be defined for tag \var{tag} within the object 515 it is overwritten by the new \var{value}. If the object is expanded, 516 the value assigned to \DataSamplePoints with tag \var{tag} is replaced by 517 \var{value}. 518 jgs 82 \end{methoddesc} 519 520 jgs 102 \begin{methoddesc}[Data]{wherePositive}{} 521 returns \Data object which has the same \Shape and is defined on 522 the same \FunctionSpace like the object. The returned values are $1$ 523 where the object is positive and $0$ elsewhere. 524 jgs 82 \end{methoddesc} 525 526 jgs 102 \begin{methoddesc}[Data]{wherePositive}{} 527 returns \Data object which has the same \Shape and is defined on 528 the same \FunctionSpace like the object. The returned values are $1$ 529 where the object is non-positive and $0$ elsewhere. 530 jgs 82 \end{methoddesc} 531 532 jgs 102 \begin{methoddesc}[Data]{whereNonnegative}{} 533 returns \Data object which has the same \Shape and is defined on 534 the same \FunctionSpace like the object. The returned values are $1$ 535 where the object is non-negative and $0$ elsewhere. 536 jgs 82 \end{methoddesc} 537 538 jgs 102 \begin{methoddesc}[Data]{whereNegative}{} 539 returns \Data object which has the same \Shape and is defined on 540 the same \FunctionSpace like the object. The returned values are $1$ 541 where the object is negative and $0$ elsewhere. 542 jgs 82 \end{methoddesc} 543 544 jgs 102 \begin{methoddesc}[Data]{whereZero}{tolerance=1.e-8} 545 returns \Data object which has the same \Shape and is defined on 546 the same \FunctionSpace like the object. The returned values are $1$ 547 where the object is nearly zero, i.e. where the absolute value is less 548 than \var{tolerance}, and $0$ elsewhere. 549 \end{methoddesc} 550 jgs 82 551 jgs 102 \begin{methoddesc}[Data]{whereNonzero}{tolerance=1.e-8} 552 returns \Data object which has the same \Shape and is defined on 553 the same \FunctionSpace like the object. The returned values are $1$ 554 where the object is nearly non-zero, i.e. where the absolute value is 555 greater or equal than \var{tolerance}, and $0$ elsewhere. 556 jgs 82 \end{methoddesc} 557 558 jgs 102 \begin{methoddesc}[Data]{sign}{} 559 returns \Data object which has the same \Shape and is defined on 560 the same \FunctionSpace like the object. The returned values are $1$ 561 where the object is positive, $-1$ where the value is negative and $0$ elsewhere. 562 jgs 82 \end{methoddesc} 563 564 jgs 102 \begin{methoddesc}[Data]{copyWithMask}{arg,mask} 565 copies the \Data object \var{arg} into the object 566 where the \Data object \var{mask} is positive. \var{arg} 567 and \var{mask} must have the same \Shape 568 and must belong to same \FunctionSpace as 569 the object. 570 jgs 82 \end{methoddesc} 571 572 jgs 102 \begin{methoddesc}[Data]{Lsup}{} 573 returns the $L^{sup}$-norm of the object. This is maximum absolute values over all components and all \DataSamplePoints. \index{$L^{sup}$-norm}. 574 \end{methoddesc} 575 jgs 117 \begin{methoddesc}[Data]{Linf}{} 576 returns the minimum absolute value over all components and all \DataSamplePoints. \index{$L^{inf}$-norm}. 577 \end{methoddesc} 578 jgs 102 \begin{methoddesc}[Data]{inf}{} 579 returns the minimum value (infimum) of the object. The minimum is 580 taken over all components and all \DataSamplePoints . \index{infimum} 581 \end{methoddesc} 582 583 \begin{methoddesc}[Data]{sup}{} 584 returns the maximum value (supremum) of the object. The maximum is 585 taken over all components and all \DataSamplePoints . \index{supremum} 586 \end{methoddesc} 587 588 \begin{methoddesc}[Data]{grad}{\optional{on}} 589 returns the gradient of the function represented by the object. 590 \Data object is in \FunctionSpace \var{on} and has rank r+1 where r is the rank of the object. 591 If \var{on} is not present, a suitbale \FunctionSpace is used. 592 jgs 82 \index{gradient} 593 \end{methoddesc} 594 595 \begin{methoddesc}[Data]{integrate}{} 596 jgs 102 returns the integral of the function represented by the object. The method returns 597 a \class{numarray.array} object of the same \Shape like the object. A 598 jgs 82 component of the returned object is the integral of the corresponding 599 jgs 102 component of the object. \index{integral} 600 jgs 82 \end{methoddesc} 601 602 jgs 102 \begin{methoddesc}[Data]{interpolate}{on} 603 interpolates 604 the function represented by the object 605 into the \FunctionSpace\var{on}. 606 jgs 82 \index{interpolation} 607 \end{methoddesc} 608 609 jgs 102 \begin{methoddesc}[Data]{abs}{} 610 applies the absolute value function to the object. The 611 return \Data object has the same \Shape and is in the same 612 \FunctionSpace like the object. For all \DataSamplePoints and all 613 components the value is calculated by applying the exponential 614 function. \index{function!absolute value} 615 jgs 82 \end{methoddesc} 616 617 jgs 102 \begin{methoddesc}[Data]{exp}{} 618 applies the exponential function to the object. The 619 return \Data object has the same \Shape and is in the same 620 \FunctionSpace like the object. For all \DataSamplePoints and all 621 components the value is calculated by applying the exponential 622 function. \index{function!exponential} 623 jgs 82 \end{methoddesc} 624 625 \begin{methoddesc}[Data]{sqrt}{} 626 jgs 102 applies the square root function to the object. The 627 return \Data object has the same \Shape and is in the same 628 \FunctionSpace like the object. For all \DataSamplePoints and all 629 components the value is calculated by applying the square root function. 630 An exception is 631 raised if the value is negative. \index{function!square root} 632 jgs 82 \end{methoddesc} 633 634 \begin{methoddesc}[Data]{sin}{} 635 jgs 102 applies the sine function to the object. The 636 return \Data object has the same \Shape and is in the same 637 \FunctionSpace like the object. For all \DataSamplePoints and all 638 components the value is calculated by applying the sine function. \index{function!sine} 639 jgs 82 \end{methoddesc} 640 641 \begin{methoddesc}[Data]{cos}{} 642 jgs 102 applies the cosine function to the object. The 643 return \Data object has the same \Shape and is in the same 644 \FunctionSpace like the object. For all \DataSamplePoints and all 645 components the value is calculated by applying the cosine function. \index{function!cosine} 646 jgs 82 \end{methoddesc} 647 648 jgs 102 \begin{methoddesc}[Data]{tan}{} 649 applies the tangent function to the object. The 650 return \Data object has the same \Shape and is in the same 651 \FunctionSpace like the object. For all \DataSamplePoints and all 652 components the value is calculated by applying the tangent function. \index{function!tangent} 653 jgs 82 \end{methoddesc} 654 655 jgs 102 \begin{methoddesc}[Data]{log}{} 656 applies the logarithmic function to the object. The 657 return \Data object has the same \Shape and is in the same 658 \FunctionSpace like the object. For all \DataSamplePoints and all 659 components the value is calculated by applying the logarithmic function. An exception is 660 raised if the value is negative.\index{function!logarithmic} 661 jgs 82 \end{methoddesc} 662 663 jgs 102 \begin{methoddesc}[Data]{maxval}{} 664 returns the maximum value over all components. The 665 return value is a \Data object of rank 0 666 and is in the same 667 \FunctionSpace like the object. For all \DataSamplePoints 668 the value is calculated as the maximum value over all components. \index{function!maximum} 669 jgs 82 \end{methoddesc} 670 671 jgs 102 \begin{methoddesc}[Data]{minval}{} 672 returns the minimum value over all components. The 673 return value is a \Data object of rank 0 674 and is in the same 675 \FunctionSpace like the object. For all \DataSamplePoints 676 the value is calculated as the minimum value over all components. \index{function!minimum} 677 jgs 82 \end{methoddesc} 678 679 jgs 102 \begin{methoddesc}[Data]{length}{} 680 returns the Euclidean length at all \DataSamplePoints. The 681 return value is a \Data object of rank 0 682 and is in the same 683 \FunctionSpace like the object. For all \DataSamplePoints 684 the value is calculated as the square root of the sum of the square over all over all components. \index{function!length} 685 jgs 82 \end{methoddesc} 686 jgs 102 \begin{methoddesc}[Data]{transpose}{axis} 687 returns the transpose of the object around \var{axis}. \var{axis} is a non-negative integer 688 which is less the rank $r$ of the object. Transpose swaps the indexes $0$ to \var{axis} 689 with the indexes \var{axis}+1 to $r$. If the \var{d} is \RankFour one has 690 \begin{python} 691 d[i,j,k,l]=d.transpose(0)[i,j,k,l] 692 d[i,j,k,l]=d.transpose(1)[j,k,l,i] 693 d[i,j,k,l]=d.transpose(2)[k,l,i,j] 694 d[i,j,k,l]=d.transpose(3)[l,i,j,k] 695 \end{python} 696 \index{function!transpose} 697 jgs 82 \end{methoddesc} 698 699 jgs 102 \begin{methoddesc}[Data]{trace}{} 700 returns sum of the components with identical indexes. 701 The 702 return value is a \Data object of rank 0 703 and is in the same 704 \FunctionSpace like the object. 705 \index{function!trace} 706 jgs 82 \end{methoddesc} 707 jgs 102 \begin{methoddesc}[Data]{saveDX}{fileName} 708 saves the object to an openDX format file of name \var{fileName}, see 709 \ulink{www.opendx.org}{\url{www.opendx.org}}. \index{openDX} 710 \end{methoddesc} 711 jgs 82 712 jgs 102 713 For convenience the following factories are provided to created \Data object: 714 715 \begin{funcdesc}{Scalar}{value=0.,what=escript::FunctionSpace(),expand=\False} 716 returns a \Data object of rank 0 in the \FunctionSpace \var{what}. 717 Values are initialed with the double \var{value}. If \var{expanded} is \True 718 the \Data object is represented in expanded from. 719 \end{funcdesc} 720 721 \begin{funcdesc}{Vector}{value=0.,what=escript::FunctionSpace(),expand=\False} 722 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what} 723 where \var{d} is the spatial dimension of the \Domain of \var{what}. 724 Values are initialed with the double \var{value}. If \var{expanded} is \True 725 the \Data object is represented in expanded from. 726 \end{funcdesc} 727 728 \begin{funcdesc}{Tensor}{value=0.,what=escript::FunctionSpace(),expand=\False} 729 returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what} 730 where \var{d} is the spatial dimension of the \Domain of \var{what}. 731 Values are initialed with the double \var{value}. If \var{expanded} is \True 732 the \Data object is represented in expanded from. 733 \end{funcdesc} 734 735 \begin{funcdesc}{Tensor3}{value=0.,what=escript::FunctionSpace(),expand=\False} 736 returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what} 737 where \var{d} is the spatial dimension of the \Domain of \var{what}. 738 Values are initialed with the double \var{value}. If \var{expanded} is \True 739 the \Data object is re\var{arg}presented in expanded from. 740 \end{funcdesc} 741 742 \begin{funcdesc}{Tensor4}{value=0.,what=escript::FunctionSpace(),expand=\False} 743 returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what} 744 where \var{d} is the spatial dimension of the \Domain of \var{what}. 745 Values are initialed with the double \var{value}. If \var{expanded} is \True 746 the \Data object is represented in expanded from. 747 \end{funcdesc} 748 749 \begin{funcdesc}{abs}{arg} 750 returns the absolute value of \var{arg} where \var{arg} 751 can be double, a \Data object or an \numarray object. 752 \end{funcdesc} 753 754 \begin{funcdesc}{sin}{arg} 755 returns the sine of \var{arg} where \var{arg} 756 can be double, a \Data object or an \numarray object. 757 \end{funcdesc} 758 759 \begin{funcdesc}{cos}{arg} 760 returns the cosine of \var{arg} where \var{arg} 761 can be double, a \Data object or an \numarray object. 762 \end{funcdesc} 763 764 \begin{funcdesc}{exp}{arg} 765 returns the value of the exponential function for \var{arg} where \var{arg} 766 can be double, a \Data object or an \numarray object. 767 \end{funcdesc} 768 769 \begin{funcdesc}{sqrt}{arg} 770 returns the square root of \var{arg} where \var{arg} 771 can be double, a \Data object or an \numarray object. 772 \end{funcdesc} 773 774 \begin{funcdesc}{maxval}{arg} 775 returns the maximum value over all component of \var{arg} where \var{arg} 776 can be double, a \Data object or an \numarray object. 777 \end{funcdesc} 778 779 \begin{funcdesc}{minval}{arg} 780 returns the minumum value over all component of \var{arg} where \var{arg} 781 can be double, a \Data object or an \numarray object. 782 \end{funcdesc} 783 784 \begin{funcdesc}{length}{arg} 785 returns the length of \var{arg} which is the 786 square root of the sum of the squares of all component of \var{arg}. \var{arg} 787 can be double, a \Data object or an \numarray object. 788 \end{funcdesc} 789 790 \begin{funcdesc}{sign}{arg} 791 return the sign of \var{arg} where \var{arg} 792 can be double, a \Data object or an \numarray object. 793 \end{funcdesc} 794 795 \begin{funcdesc}{transpose}{arg,\optional{axis}} 796 returns the transpose of \var{arg} around \var{axis}. \var{axis} is a non-negative integer 797 which is less the rank $r$ of the object. Transpose swaps the indexes $0$ to \var{axis} 798 with the indexes \var{axis}+1 to $r$. If \var{axis} is not present, \var{axis}=$r/2$ is assumed. 799 \var{arg} 800 may be a \Data object or an \numarray object. 801 \end{funcdesc} 802 803 \begin{funcdesc}{transpose}{arg,\optional{axis}} 804 returns the trace the object of \var{arg}. The trace is the sum over those components 805 with identical indexed. 806 \var{arg} 807 may be a \Data object or a \numarray object. 808 \end{funcdesc} 809 810 \begin{funcdesc}{sum}{arg} 811 returns the sum over all components and all 812 \DataSamplePoints of \var{arg}, where \var{arg} 813 is a \Data object. 814 \end{funcdesc} 815 816 \begin{funcdesc}{sup}{arg} 817 returns the maximum over all components and all 818 \DataSamplePoints of \var{arg}, where \var{arg} 819 is a \Data object. 820 \end{funcdesc} 821 822 \begin{funcdesc}{inf}{arg} 823 returns the mimumum over all components and all 824 \DataSamplePoints of \var{arg}, where \var{arg} 825 is a \Data object. 826 \end{funcdesc} 827 828 829 \begin{funcdesc}{L2}{arg} 830 returns the $L^2$ norm of \var{arg}. This is the square root 831 of the sum of the squared value over all components and all 832 \DataSamplePoints of \var{arg}, where \var{arg} 833 is a \Data object. 834 \end{funcdesc} 835 836 \begin{funcdesc}{Lsup}{arg} 837 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values 838 over all components and all 839 \DataSamplePoints of \var{arg}, where \var{arg} 840 is a \Data object. 841 \end{funcdesc} 842 843 \begin{funcdesc}{dot}{arg1,arg2} 844 returns the dot product of of \var{arg1} and \var{arg2}. This is sum 845 of the product of corresponding entries in \var{arg1} and \var{arg2} over all 846 components and and all 847 \DataSamplePoints. \var{arg1} and \var{arg2} are \Data objects of the 848 same \Shape and in the same \FunctionSpace. 849 \end{funcdesc} 850 851 \begin{funcdesc}{grad}{arg,\optional{where}} 852 returns the gradient of \var{arg} as a function in the \FunctionSpace \var{where}. 853 If \var{where} is not present a reasonable \FunctionSpace is chosen. 854 \var{arg} 855 is a \Data object. 856 \end{funcdesc} 857 858 \begin{funcdesc}{integrate}{arg} 859 returns the integral of \var{arg} as a \numarray object. 860 If \var{where} is not present a reasonable \FunctionSpace is chosen. 861 \var{arg} 862 is a \Data object. 863 \end{funcdesc} 864 865 \begin{funcdesc}{interpolate}{arg,where} 866 interpolate \Data object \var{arg} into the \FunctionSpace \var{where} 867 \end{funcdesc} 868 869 870 \section{\Operator Class} 871 872 The \Operator class provides an abstract access to operators build 873 within the \LinearPDE class. \Operator objects are created 874 when a PDE is handed over to a PDE solver library and handled 875 by the \LinearPDE class defining the PDE. The user can gain access 876 to the \Operator of a \LinearPDE object through the \var{getOperator} 877 method. 878 879 \begin{classdesc}{Operator}{} 880 creates an empty \Operator object. 881 \end{classdesc} 882 883 \begin{methoddesc}[Operator]{isEmpty}{fileName} 884 returns \True is the object is empty. Otherwise \True is returned. 885 jgs 82 \end{methoddesc} 886 887 jgs 102 \begin{methoddesc}[Operator]{setValue}{value} 888 resets all entires in the obeject representation to \var{value} 889 jgs 82 \end{methoddesc} 890 891 jgs 102 \begin{methoddesc}[Operator]{solves}{rhs} 892 solves the operator equation with right hand side \var{rhs} 893 jgs 82 \end{methoddesc} 894 895 jgs 102 \begin{methoddesc}[Operator]{of}{u} 896 applies the operator to the \Data object \var{u} 897 jgs 82 \end{methoddesc} 898 899 jgs 102 \begin{methoddesc}[Operator]{saveMM}{fileName} 900 jgs 82 saves the object to a matrix market format file of name 901 \var{fileName}, see 902 \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}. 903 \index{Matrix Market} 904 \end{methoddesc} 905

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