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

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

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