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

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