FFC User Manual
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1. entire whole and thus to each and every part regardless of who wrote it Thus it is not the intent of this section to claim rights or contest your rights to work written entirely by you rather the intent is to exercise the right to control the distribution of derivative or collective works based on the Program In addition mere aggregation of another work not based on the Program with the Program or with a work based on the Program on a volume of a storage or distribution medium does not bring the other work under the scope of this License 3 You may copy and distribute the Program or a work based on it under Section 2 in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following a Accompany it with the complete corresponding machine readable source code which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or b Accompany it with a written offer valid for at least three years to give any third party for a charge no more than your cost of physically performing source distribution a complete machine readable copy of the corresponding source code to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or c Accompany it with the information you received as to the offer to distribute corresponding source code This alternative is allow2. 4 9 User defined operators A user may define new operators using standard Python syntax As an example consider the strain operator e of linear elasticity defined by dis Vo Vo 4 17 This operator can be implemented as a function using the Python def key word def epsilon v return 0 5 grad v transp grad v 38 FFC User Manual Anders Logg Alternatively using the shorthand lambda notation the strain operator may be defined as follows epsilon lambda v 0 5 grad v transp grad v 39 Chapter 5 Examples The following examples illustrate basic usage of the form language for the definition of a collection of standard multilinear forms We assume that dx has been declared as an integral over the interior of 2 and that both i and j have been declared as a free Index which is always the case if the command line interface is used The examples presented below can all be found in the subdirectory src demo of the FFC source tree 5 1 The mass matrix As a first example consider the bilinear form corresponding to a mass matrix atu U fouas 5 1 which can be implemented in FFC as follows element FiniteElement Lagrange triangle 1 v BasisFunction element 41 FFC User Manual Anders Logg BasisFunction element a Il v Uxdx w ll This example is implemented in the file Mass form in the collection of demon stration forms incl
3. Anders Logg language Hereinafter translation is included without limitation in the term modification Each licensee is addressed as you Activities other than copying distribution and modification are not covered by this License they are outside its scope The act of running the Program is not restricted and the output from the Program is covered only if its contents constitute a work based on the Program independent of having been made by running the Program Whether that is true depends on what the Program does 1 You may copy and distribute verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this License and to the absence of any warranty and give any other recipients of the Program a copy of this License along with the Program You may charge a fee for the physical act of transferring a copy and you may at your option offer warranty protection in exchange for a fee 2 You may modify your copy or copies of the Program or any portion of it thus forming a work based on the Program and copy and distribute such modifications or work under the terms of Section 1 above provided that you also meet all of these conditions a You must cause the modified files to carry prominent notices stating that you changed the files and
4. may also be less stable and even not work at all 61 FFC User Manual Anders Logg B 1 3 Installing the compiler FFC follows the standard for Python packages Enter the source directory of FFC and issue the following command python setup py install This will install the FFC Python package in a subdirectory called ffc in the default location for user installed Python packages usually in the di rectory usr lib python2 4 site packages In addition the compiler executable a Python script will be installed in the default directory for user installed Python scripts usually in usr bin To see a list of optional parameters to the installation script type python setup py install help If you don t have root access to the system you are using you can pass the home option to the installation script to install FFC in your home directory mkdir local python setup py install home local This installs the FFC package in the directory local lib python and the FFC executable in local bin If you use this option make sure to set the environment variable PYTHONPATH to local lib python and to add local bin to the PATH environment variable B 1 4 Compiling the demos To test your installation of FFC enter the subdirectory src demo and com pile some of the demonstration forms With FFC installed on your system just type 62 FFC User Manual Anders Logg ffc
5. on the Python package Numeric which is used by FFC to process multidimensionall arrays tensors Python Numeric can be downloaded from http www scipy org For Debian users the package to install is python2 4 numeric Installing FIAT FFC depends on the latest version of FIAT which can be downloaded from 60 FFC User Manual Anders Logg http www fenics org FIAT is used by FFC to create and evaluate finite element basis functions and quadrature rules The installation instructions for FIAT are similar to those for FFC given in detail below In addition you will need to install the Python package LinearAlgebra which may already be included in your installation of Python Numeric For Debian users the package to install is python2 4 numeric ext B 1 2 Downloading the source code The latest release of FFC can be obtained as a tar gz archive in the down load section at http www fenics org Download the latest release of FFC for example ffc 0 1 0 tar gz and unpack using the command tar zxfv ffc 0 1 0 tar gz This creates a directory ffc 0 1 0 containing the FFC source code If you want the very latest version of FFC there is also a version named ffc cvs current tar gz which provides a snapshot of the current CVS version of FFC updated automatically from the CVS repository each hour This version may contain features not yet present in the latest release but
6. s built in list type This means that all list operations such as slicing list comprehension etc are supported There is one exception to this rule namely vector valued BasisFunctions and Functions which are not lists but can be made into lists using the operator vec discussed below The operators listed below support all objects which are logically vectors thus including both Python lists and vector valued expressions 4 5 1 Component access v i Brackets 1 are used to pick a given component of a logically vector valued expression Thus if v is a vector valued expression then v 0 represents a function corresponding to the first component of the values of v Similarly if i is an Index free or fixed then v i represents a function corresponding to component i of the values of v 32 FFC User Manual Anders Logg 4 5 2 Scalar product dot v w The operator dot accepts as arguments two logically vector valued expres sions and returns the scalar product dot product of the two vectors n 1 dot v w gt v w X vii 4 4 i 0 Note that this operator is only defined for vectors of equal length 4 5 3 Vector product cross v w The operator cross accepts as arguments two logically vector valued expres sions and returns a vector which is the cross product vector product of the two vectors cross v w DUXW viw V2W1 V2W9 UYWa VoW1 vwo 4 5 Note that this operator is only defined for vect
7. the date of any change b You must cause any work that you distribute or publish that in whole or in part contains or is derived from the Program or any part thereof to be licensed as a whole at no charge to all third parties under the terms of this License c If the modified program normally reads commands interactively when run you must cause it when started running for such interactive use in the most ordinary way to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty or else saying that you provide a warranty and that users may redistribute the program under these conditions and telling the user how to view a copy of this License Exception if the Program itself is interactive but does not normally print such an announcement your work based on the Program is not required to print an announcement 71 FFC User Manual Anders Logg These requirements apply to the modified work as a whole If identifiable sections of that work are not derived from the Program and can be reasonably considered independent and separate works in themselves then this License and its terms do not apply to those sections when you distribute them as separate works But when you distribute the same sections as part of a whole which is a work based on the Program the distribution of the whole must be on the terms of this License whose permissions for other licensees extend to the
8. the same FiniteElement v BasisFunction element U BasisFunction element f Function element Function is used to represent user defined functions including right hand sides variable coefficients and stabilization terms FFC treats each Function as a linear combination of basis functions with unknown coefficients It is the responsibility of the user or the system for which the form is compiled to 28 FFC User Manual Anders Logg supply the values of the coefficients at run time In the case of DOLFIN the coefficients are automatically computed from a given user defined function during the assembly of a form Note that the order in which Functions are declared is important The code generated by FFC accepts as arguments a list of functions that should correspond to the Functions appearing in the form in the order they have been declared For a MixedElement the function Functions can be used to construct tuples of Functions as illustrated here for a mixed Taylor Hood element f g Functions TH 4 3 5 Constant The data type Constant represents a constant scalar value that is unknown at compile time A Constant is declared without any arguments c Constant Just as with Functions it is the responsibility of the user or the system for which the form is compiled to supply the value of the constant at run time In the case of DOLFIN a constant is automatically assigned a value
9. 0 tar gz cd ffc 0 1 0 python setup py install 11 FFC User Manual Anders Logg Make sure that you download the latest release which is not 0 1 0 Note that you may need to be root on your system to do the last step You may also need to install the Python packages FIAT and Numeric See Appendix B for detailed instructions 1 2 Compiling Poisson s equation with FFC The discrete variational finite element formulation of Poisson s equation Au f reads Find U V such that a v U L v Ww Va 1 1 with Vh Vp a pair of suitable function spaces the test and trial spaces The bilinear form a Vp x Vp R is given by a v U ve VU dz 1 2 and the linear form L V gt R is given by L v fota 1 3 To compile the pair of forms a L into code that can called to assemble the linear system Ar b corresponding to the variational problem 1 1 for a pair of discrete function spaces specify the forms in a text file with extension form e g Poisson form as follows element FiniteElement Lagrange triangle 1 v BasisFunction element U BasisFunction element f Function element a dot grad v grad U x dx L fx vx xdx 12 FFC User Manual Anders Logg The example is given for piecewise linear finite elements in two dimensions but other choices are available including arbitrary order Lagrange elements in two
10. 2 FiniteElement Vector Lagrange triangle 2 P1 FiniteElement Lagrange triangle 1 TH P2 P1 4 3 3 BasisFunction The data type BasisFunction represents a basis function on a given finite element A BasisFunction must be declared from a previously declared FiniteElement v BasisFunction element Note that more than one BasisFunction can be declared from the same FiniteElement Note that the order in which BasisFunctions are declared is important The order determines the order of arguments to the multilinear form Thus for a bilinear form a v U the test function v should be declared before the trial function U 27 FFC User Manual Anders Logg BasisFunction element BasisFunction element lt I on Il For a MixedElement the function BasisFunctions can be used to construct tuples of BasisFunctions as illustrated here for a mixed Taylor Hood ele ment BasisFunctions TH BasisFunctions TH v q U P 4 3 4 Function The data type Function represents a function belonging to a given finite element space that is a linear combination of basis functions of the finite element space A Function must be declared from a previously declared FiniteElement f Function element Note that more than one BasisFunction can be declared from the same FiniteElement The following example declares two BasisFunctions and one Function from
11. D 35 FiniteElement 25 Functions 28 Function 28 Identity 30 Index 30 MixedElement 27 cross 33 curl 36 div 36 dot 33 grad 35 len 34 rank 34 rot 36 trace 34 transp 33 vec 34 addition 31 backward Euler 45 basis functions 27 boundary measure 37 component access 32 constants 29 contributing 65 TT cross product 33 Debian package 63 def 39 dependencies 59 diff 65 differential operators 35 discontinuous Lagrange element 25 divergence 36 division 32 downloading 11 61 elasticity 43 examples 41 ffc 13 FIAT 12 finite elements 25 fixed point iteration 44 form language 23 functions 28 GNU General Public License 69 GPL 69 gradient 35 heat equation 45 Hessian 35 identity matrix 30 index notation 37 indices 30 FFC User Manual installation 11 59 integrals 36 interior measure 36 Jacobian 35 Lagrange element 25 lambda 39 license 68 69 linear elasticity 43 man page 13 mass matrix 41 matrix product 33 mixed finite elements 27 mixed formulation 46 multiplication 31 Navier Stokes 44 Numeric 12 partial derivative 35 patch 65 67 Poisson s equation 12 42 Python 25 quickstart 11 reference tetrahedron 51 reference triangle 49 rotation 36 scalar operators 31 scalar product 33 source code 61 Stokes equations 46 strain 43 subscripting 32 subtraction 31 Taylor Hood e
12. FFC User Manual February 24 2006 Anders Logg www fenics org Visit http www fenics org for the latest version of this manual Send comments and suggestions to ffc dev fenics org Contents About this manual 1 Quickstart 1 1 Downloading and installing FFC 1 2 Compiling Poisson s equation with FFC 2 Command line interface ZA OVDODSS ee 64 Aa SEA A AA De Te a he BARA OEREE SERA R ARS 23 SOON ec de hae he Ee Ee Ee hw ee he Bx Lal Ey SARRIA ARA RA A E Up PSYOPS ION o ce ated aie aod ae ae Goa me Ge Sa 2 3 3 l language language language 2 3 4 d debuglevel debug debuglevel Bae WE OPIO o e a AY a 11 11 12 15 FFC User Manual Anders Logg LoG E MOB era A A RA A 17 Bee TEA aia AE AA be 17 3 Python interface 19 dl gonpile Torme sre ir e ATA RA 20 Go DUI CIORUTE spede 6 och ea we OS slds 20 33 write Orp ae 245 24 atada Bee 20 3 4 writeFiniteElement element 21 4 Form language 23 AL AIR er e Ge e ee oe 23 4 2 The form language as a Python extension 25 43 DESC OMA DOS er AAA 25 dol e A ces sos raen pak ER ERR OR EO 25 Hoe WISIN e o oy ae katy e ba g e boa p dioe Be 27 doo Basis Function e oo ka g oe kaop ioe baot iiol ket 27 A FINGEION 2 ct ke he EEE RR EA maei 28 ADO CONSAN rs ee Se eS OR ee Oe eR 29 A o sorio R44 65442444 R44 pipia pa 30 A 1 2664 kee aK REA ERA EG EH REE EA 30
13. Poisson form to compile the bilinear and linear forms for Poisson s equation This will generate a C header file called Poisson h that can be used with DOLFIN to implement a solver for Poisson s equation Adding the flag 1 latex generates output in BTEX format ffc l latex Poisson form latex Poisson tex xdvi Poisson dvi It is also possible to compile the forms in src demo without needing to install FFC on your system In that case you need to supply the path to the FFC executable bin ffc Poisson form B 1 5 Verifying the generated code To verify the output generated by the compiler run the script verify from within the FFC source tree scripts verify This script compiles all forms found in src demo and compares the output with previously compiled forms in src reference B 2 Debian package In preparation 63 Appendix C Contributing code If you have created a new module fixed a bug somewhere or have made a small change which you want to contribute to FFC then the best way to do so is to send us your contribution in the form of a patch A patch is a file which describes how to transform a file or directory structure into another The patch is built by comparing a version which both parties have against the modified version which only you have C 1 Creating a patch The tool used to create a patch is called diff and the tool used to apply the patch
14. and three dimensions To compile the pair of forms implemented in the file Poisson form call the compiler on the command line as follows ffc Poisson form This generates the file Poisson h which implements the forms in C for inclusion in DOLFIN For help on the ffc command including compilation for other systems than DOLFIN type ffc h or man ffc 13 Chapter 2 Command line interface The command line interface of FFC is documented by the man page for FFC which can be read by the command man ffc on any system where FFC has been installed A copy of this documentation is included below for convenience 2 1 Synopsis ffc h 1 language d debuglevel f option input form 2 2 Description The FEniCS Form Compiler FFC accepts as input one or more files each specifying one or more multilinear forms and compiles the given forms into 15 FFC User Manual Anders Logg efficent low level code for automatic assembly of the tensors representing the multilinear forms In particular FFC compiles a pair of bilinear and linear forms defining a variational problem into code that can be used to efficiently assemble the corresponding linear system By default FFC generates C code for DOLFIN but this can be changed by specifying a different output language option 1 It is also possible to add new output languages to FFC 2 3 Options 2 3 1 h help Display help text and ex
15. dd Scalar Operators e sosa stt See ROS a REE ee i 31 AA Sealar addition A 31 FFC User Manual Anders Logg 4 5 4 6 4 7 AAD Scalar subtraction e sacca naera wie eee nG 31 4 4 3 Scalar multiplication 2 2 26 cs eee oe we eG 31 AAA Heeler division css oboe RR e E 32 Vector Operators cie bk Gk bE 6 Le P Epa a 32 45 1 Component access vlil s s s eresse sesi 32 452 Scalar product Getty Wis 2 4554048542 aa 33 Aj Vector product cross v W o so so possa fo 33 4 5 4 Matrix product mult A B aaau aaa aaa 33 4 5 5 Transpose transp A 2 cen tee ea ee Rea ss 33 Z2oG Trace trace h 2444444 4464 66484 26 34 4 5 7 Vector length len v 34 458 Rank ene so cc gp eee pee BY RE Be 34 4 5 9 Vectorization vec v 34 Differential operators s secs es eres aona ER EEE 35 4 6 1 Scalar partial derivative D v i 35 A02 Gradient prad V i sa ace eee he aep ee EES 35 46 3 Divergence AVIV lt gt e o cen bb ge ee dup erh 36 AGA Rotation TOELT lt lt s sorito s ee dw E i 36 Pe e TR A 36 4 7 1 Integration over the interior dx 36 4 7 2 Integration over the boundary ds 37 5 FFC User Manual Anders Logg 4 8 Index notation 0 0 0 000 ce ee eee 37 4 9 User defined operators 1 0 0 0 0 eee eee ne 38 5 Examples 41 51 Themass Mee oda cintas A ee 41 Ge Poison cane sesos cra ra a e d 42 63 Ventorv alued Poison co
16. e details of this form language and present a number of examples to illustrate the use of the form language in applications 4 1 Overview A form is expressed using a combination of basic data types and operators FFC compiles a given multilinear form a Vix Vx xV OR 4 1 into code that can be used to compute the corresponding tensor A alo Piz T Qin 4 2 In the form language a multilinear form is defined by first specifying the set of function spaces V V Vf and then expressing the multilinear form in terms of the basis functions of these functions spaces A function space is defined in the form language through a FiniteElement and a corresponding basis function is represented as a BasisFunction The 23 FFC User Manual Anders Logg following code defines a pair of basis functions v and U for a first order La grange finite element on triangles element FiniteElement Lagrange triangle 1 v BasisFunction element U BasisFunction element The two basis functions can now be used to define a bilinear form a v D U 0 dx corresponding to the mathematical notation OU a v U fo an 4 3 w for 4 3 Note the order of the argument list of the multilinear form is determined by the order in which basis functions are declared not by the order in which they appear in the form Thus both a v D U 0 dx and a D U 0 v dx define the same multilinear
17. ed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer in accord with Subsection b above The source code for a work means the preferred form of the work for making modifications to it For an executable work complete source code means all the source code for all modules it contains plus any associated interface definition files plus the scripts used to 72 FFC User Manual Anders Logg control compilation and installation of the executable However as a special exception the source code distributed need not include anything that is normally distributed in either source or binary form with the major components compiler kernel and so on of the operating system on which the executable runs unless that component itself accompanies the executable If distribution of executable or object code is made by offering access to copy from a designated place then offering equivalent access to copy the source code from the same place counts as distribution of the source code even though third parties are not compelled to copy the source along with the object code 4 You may not copy modify sublicense or distribute the Program except as expressly provided under this License Any attempt otherwise to copy modify sublicense or distribute the Program is void and will automatically terminate your rights under this License However parties who have received cop
18. ense and any other pertinent obligations then as a consequence you 73 FFC User Manual Anders Logg may not distribute the Program at all For example if a patent license would not permit royalty free redistribution of the Program by all those who receive copies directly or indirectly through you then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program If any portion of this section is held invalid or unenforceable under any particular circumstance the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system it is up to the author donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License 8 If the distribution and or use of the Program is restricted in certain count
19. er 5 Example 4 The alignment of the first face of a tetrahedron is 5 if the first edge of the face is edge number 4 and the second edge is not edge number 5 It must then be edge number 0 58 Appendix B Installation The source code of FFC is portable and should work on any system with a standard Python installation Questions bug reports and patches concerning the installation should be directed to the FFC mailing list at the address ffc devOfenics org FFC must currently be installed directly from source but effort is underway to provide precompiled Debian packages of FFC and other FENICS compo nents B 1 Installing from source B 1 1 Dependencies and requirements FFC depends on a number of libraries that need to be installed on your system These libraries include FIAT and the Python Numeric module In addition you need to have a working Python installation on your system 59 FFC User Manual Anders Logg Installing Python FFC is developed for Python 2 4 but might also work with Python 2 3 To check which version of Python you have installed issue the command python V python V Python 2 4 1 If Python is not installed on your system it can be downloaded from http www python org Follow the installation instructions for Python given on the Python web page For Debian users the package to install is python2 4 Installing Numeric In addition to Python itself FFC depends
20. essing it with for example gzip is advisable gzip f c lt identifier gt lt date gt patch 66 FFC User Manual Anders Logg C 2 Sending patches Patch files should be sent to the FFC mailing list at the address ffc devOfenics org Include a short description of what your patch accomplishes Small patches have a better chance of being accepted so if you are making a major con tribution please consider breaking your changes up into several small self contained patches if possible C 3 Applying a patch maintainers Let s say that a patch has been built relative to FFC release 0 1 0 The following description then shows how to apply the patch to a clean version of release 0 1 0 1 Unpack the version of FFC which the patch is built relative to tar zxfv ffc 0 1 0 tar gz 2 Check that you have the patch ffc lt identifier gt lt date gt patch and the FFC directory structure in the current directory 1s ffc 0 1 0 ffc lt identifier gt lt date gt patch Unpack the patch file using gunzip if necessary 3 Enter the FFC directory structure cd ffc 0 1 0 67 FFC User Manual Anders Logg 4 Apply the patch patch p1 lt ffc lt identifier gt lt date gt patch The option p1 strips the leading directory from the filename references in the patch to match the fact that we are applying the patch from inside the directory Ano
21. form The arity of a multilinear form is determined by the number of basis functions appearing in the definition of the form Thus a v U dx defines a bilinear form namely a v U Jo vudx whereas L v f dx defines a linear form namely L v fav f dz In the case of a bilinear form the first of the two basis functions is referred to as the test function and the second is referred to as the trial function Not every expression is a valid multilinear form The following list explains some of the basic rules that must be obeyed in the definition of a form e A form must be linear in each of its arguments otherwise it is not a multilinear form Thus a v v U dx is not a valid form since it is quadratic in v 24 FFC User Manual Anders Logg e The value of a form must be a scalar Thus if v is a vector valued basis function see below then L v dx is not a valid form since the value of the form is not a scalar e The integrand of a form must be integrated exactly once Thus neither a v u nor a v u dx dx are valid forms 4 2 The form language as a Python extension The FFC form language is built on top of Python This is true both when calling FFC as a compiler from the command line or when calling the FFC compiler from within a Python program Through the addition of a collection of basic data types and operators FFC allows a form to be specified in a language that is close to the mathematical notation Since the f
22. from a given user defined variable Note that the order in which Constants are declared is important The code generated by FFC accepts as arguments a list of constants that should correspond to the Constants appearing in the form in the order they have been declared 29 FFC User Manual Anders Logg 4 3 6 Index The data type Index represents an index used for subscripting derivatives or taking components of vector valued functions If an Index is declared without any arguments i Index a free Index is created representing an index range determined by the con text if used to subscript a vector valued BasisFunction or a Function the range is given by the number of vector dimensions n and if used to subscript a derivative the range is given by the dimension d of the underlying shape of the finite element space As we shall see below indices can be a powerful tool when used to define forms in tensor notation An Index can also be fixed meaning that the value of the index remains constant i Index 0 When using the command line interface to FFC a sequence of free indices are automatically declared for convenience i j k 1 m n Note however that a user is free to declare new indices with other names or even reuse these variables for other things than indices 4 3 7 Identity The data type Identity represents an n x n unit matrix of given size n An Identity is declared by specifying the dimensi
23. ftware to make sure the software is free for all its users This General Public License applies to most of the Free Software Foundation s software and to any other program whose authors commit to using it Some other Free Software Foundation software is covered by the GNU Library General Public License instead You can apply it to your programs too When we speak of free software we are referring to freedom not price Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software and charge for 69 FFC User Manual Anders Logg this service if you wish that you receive source code or can get it if you want it that you can change the software or use pieces of it in new free programs and that you know you can do these things To protect your rights we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights These restrictions translate to certain responsibilities for you if you distribute copies of the software or if you modify it For example if you distribute copies of such a program whether gratis or for a fee you must give the recipients all the rights that you have You must make sure that they too receive or can get the source code And you must show them these terms so they know their rights We protect your rights with two steps 1 copyright the software and 2 offer you this license which gives you le
24. gal permission to copy distribute and or modify the software Also for each author s protection and ours we want to make certain that everyone understands that there is no warranty for this free software If the software is modified by someone else and passed on we want its recipients to know that what they have is not the original so that any problems introduced by others will not reflect on the original authors reputations Finally any free program is threatened constantly by software patents We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses in effect making the program proprietary To prevent this we have made it clear that any patent must be licensed for everyone s free use or not licensed at all The precise terms and conditions for copying distribution and modification follow GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING DISTRIBUTION AND MODIFICATION O This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License The Program below refers to any such program or work and a work based on the Program means either the Program or any derivative work under copyright law that is to say a work containing the Program or a portion of it either verbatim or with modifications and or translated into another 70 FFC User Manual
25. ies or rights from you under this License will not have their licenses terminated so long as such parties remain in full compliance 5 You are not required to accept this License since you have not signed it However nothing else grants you permission to modify or distribute the Program or its derivative works These actions are prohibited by law if you do not accept this License Therefore by modifying or distributing the Program or any work based on the Program you indicate your acceptance of this License to do so and all its terms and conditions for copying distributing or modifying the Program or works based on it 6 Each time you redistribute the Program or any work based on the Program the recipient automatically receives a license from the original licensor to copy distribute or modify the Program subject to these terms and conditions You may not impose any further restrictions on the recipients exercise of the rights granted herein You are not responsible for enforcing compliance by third parties to this License 7 If as a consequence of a court judgment or allegation of patent infringement or for any other reason not limited to patent issues conditions are imposed on you whether by court order agreement or otherwise that contradict the conditions of this License they do not excuse you from the conditions of this License If you cannot distribute so as to satisfy simultaneously your obligations under this Lic
26. inear elasticity alu U I e v U dz 5 6 Q 43 FFC User Manual Anders Logg where a Vo ia 5 7 can be implemented as follows element FiniteElement Vector Lagrange tetrahedron 1 v BasisFunction element U BasisFunction element def epsilon v return 0 5 grad v transp grad v a dot epsilon v epsilon U dx Alternatively index notation can be used to define the form a 0 25 D v i l j D v jl i D ULi j DC U jl i dx This example is implemented in the file Elasticity form in the collection of demonstration forms included with the FFC source distribution 5 5 The nonlinear term of Navier Stokes The bilinear form for fixed point iteration on the nonlinear term of the in compressible Navier Stokes equations au U vv dz 5 8 with w the frozen velocity from a previous iteration can be conveniently implemented using index notation as follows 44 FFC User Manual Anders Logg element FiniteElement Vector Lagrange tetrahedron 1 v BasisFunction element U BasisFunction element w Function element a vlil w j D U il j dx This example is implemented in the file NavierStokes form in the collection of demonstration forms included with the FFC source distribution 5 6 The heat equation Discretizing the heat equation u V cVu f 5 9 in time using the dG 0 method backward Euler we ob
27. ions vertices topological dimension 0 edges topological dimension 1 faces topological dimension 2 cells topological dimension 2 or 3 A cell can be either a triangle or a tetrahedron depending on the type of mesh For a mesh consisting of triangles the mesh entities involved are vertices edges and cells and for a mesh consisting of tetrahedrons the mesh entities involved are vertices edges faces and cells 52 FFC User Manual Anders Logg Figure A 3 Physical coordinates of the reference tetrahedron 93 FFC User Manual Anders Logg Figure A 4 Ordering of mesh entities vertices edges faces for the reference tetrahedron 54 FFC User Manual Anders Logg A 3 2 Ordering among mesh entities With each mesh entity there can be associated zero or more nodes and the nodes are ordered locally and globally based on the topological dimension of the mesh entity with which they are associated Thus any nodes associated with vertices are ordered first and nodes associated with cells last If more than one node is associated with a single mesh entity the internal ordering of the nodes associated with the mesh entity becomes important in particular for edges and faces where the nodes of two adjacent cells sharing a common edge or face must lign up A 3 3 Internal ordering on edges For edges containing more than one node the nodes are ordered in the di rection from the first vertex v of the edge to the second ve
28. is called patch These tools are free software and are standard on most Unix systems Here s an example of how it works Start from the latest release of FFC which we here assume is release 0 1 0 You then have a directory structure under ffc 0 1 0 where you have made modifications to some files which you think could be useful to other users 1 Clean up your modified directory structure to remove temporary and binary files which will be rebuilt anyway 65 FFC User Manual Anders Logg make clean 2 From the parent directory rename the FFC directory to something else mv ffc 0 1 0 ffc 0 1 0 mod 3 Unpack the version of FFC that you started from tar zxfv f c 0 1 0 tar gz 4 You should now have two FFC directory structures in your current directory 1s ffc 0 1 0 ffc 0 1 0 mod 5 Now use the diff tool to create the patch diff u new file recursive ffc 0 1 0 ffc 0 1 0 mod gt ffc lt identifier gt lt date gt patch written as one line where lt identifier gt is a keyword that can be used to identify the patch as coming from you your username last name first name a nickname etc and lt date gt is today s date in the format yyyy mm dd 6 The patch now exists as ffc lt identifier gt lt date gt patch and can be distributed to other people who already have ffc 0 1 0 to easily create your modified version If the patch is large compr
29. it 2 3 2 v version Display version number and exit 2 3 3 1 language language language Specify output language one of dolfin default latex raw ase or xml 2 3 4 d debuglevel debug debuglevel Specify debug level default is 0 16 FFC User Manual Anders Logg 2 3 5 f option Specify code generation options The list of options available depends on the specified language format Current options include f no gpl and f blas described in detail below 2 3 6 f no gpl Don t add GPL license to generated code This option has only effect when compiling with ldolfin 2 3 7 f blas Generate code that uses BLAS to compute tensor products This option has only effect when compiling with 1do1fin 17 Chapter 3 Python interface FFC provides a Python interface in the form of a standard Python module The following example demonstrates how to define and compile the varia tional problem for Poisson s equation in a Python script from ffc import element FiniteElement Lagrange triangle 1 v BasisFunction element U BasisFunction element f Function element a dot grad v grad U x dx L v dx compile a L At the basic level the only difference between the command line interface and the Python interface is that the function compile must be called when using the Python interface 19 FFC User Manual Anders Logg In addition
30. lement 46 Anders Logg time stepping 45 trace 34 transpose 33 user defined operators 38 vector length 34 vector operators 32 vector product 33 vector rank 34 vector valued Poisson 43 vectorization 34
31. on n I Identity n 30 FFC User Manual Anders Logg 4 4 Scalar operators The basic operators used to define a form are scalar addition subtraction and multiplication Note the absence of division which is intentionally left out but see the comment below 4 4 1 Scalar addition Scalar addition is supported for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types In addition unary plus is supported for all basic data types 4 4 2 Scalar subtraction Scalar subtraction is supported for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types In addition unary minus is supported for all basic data types 4 4 3 Scalar multiplication Scalar multiplication is supported for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types 31 FFC User Manual Anders Logg 4 4 4 Scalar division Division is not allowed in the definition of a form This is because division by a BasisFunction in the definition of a form does not result in a valid multilinear form since a multilinear form must be linear in each of its argu ments Division by Functions and Constants may be implemented in future versions of FFC 4 5 Vector operators Vectors are defined in the form language using Python
32. opers More advanced topics are treated at the end of the manual or in the appendix Typographic conventions e Code is written in monospace typewriter like this e Commands that should be entered in a Unix shell are displayed as follows configure make Commands are written in the dialect of the bash shell For other shells such as tcsh appropriate translations may be needed FFC User Manual Anders Logg Enumeration and list indices Throughout this manual elements x of sets x of size n are enumarated from i 0 toi n 1 Derivatives in R are enumerated similarly 0 0xo 0 0x hens O0 OXn 1 Contact Comments corrections and contributions to this manual are most welcome and should be sent to ffc dev fenics org 10 Chapter 1 Quickstart This chapter demonstrates how to get started with FFC including down loading and installing the latest version of FFC and compiling Poisson s equation These topics are discussed in more detail elsewhere in this man ual In particular see Appendix B for detailed installation instructions and Chapter 4 for a detailed discussion of the form language 1 1 Downloading and installing FFC The latest version of FFC can be found on the FENICS web page http www fenics org The following commands illustrate the installation process assuming that you have downloaded release 0 1 0 of FFC tar zxfv ffc 0 1
33. orm language is built on top of Python any Python code is valid in the definition of a form but not all Python code defines a multilinear form In particular comments lines starting with and functions keyword def see Section 4 9 below are allowed in the definition of a form 4 3 Basic data types 4 3 1 FiniteElement The data type FiniteElement represents a finite element on a triangle or tetrahedron A FiniteElement is declared by specifying the type of element the underlying shape the polynomial order and optionally the number of vector components element FiniteElement type shape lt degree gt lt num_components gt The argument type is a string and possible values include 25 FFC User Manual Anders Logg Lagrange representing a standard Lagrange finite element for con tinuous piecewise polynomial functions Discontinuous Lagrange representing a discontinuous Lagrange finite element for discontinuous piecewise polynomial functions e Vector Lagrange representing a standard vector Lagrange finite element for continuous piecewise polynomial vector valued functions Discontinuous vector Lagrange representing a discontinuous Lagrange finite element for discontinuous piecewise polynomial vector valued functions The argument shape is a string and possible values include e triangle representing a triangle in R e tet
34. ors of length three 4 5 4 Matrix product mult A B The operator mult accepts as arguments two matrices or more generally tensors and returns the matrix tensor product 4 5 5 Transpose transp A The operator transp accept as argument a matrix and return the transpose of the given matrix transp A i j gt 47 Aji 4 6 33 FFC User Manual Anders Logg 4 5 6 Trace trace A The operator trace accepts as argument a square matrix A and returns its trace that is the sum of its diagonal elements trace A gt trace A l gt 4 7 4 5 7 Vector length len v The operator len accepts as argument a logically vector valued expression and returns its length the number of vector components 4 5 8 Rank rank v The operator rank returns the rank of the given argument The rank of an expression is defined as the number of times the operator can be applied to the expression before a scalar is obtained Thus the rank of a scalar is zero the rank of a vector is one and the rank of a matrix is two 4 5 9 Vectorization vec v The operator vec is used to create a Python list object from a logically vector valued expression This operator has no effect on expressions which are already lists Thus if v is a vector valued BasisFunction then vec v returns a list of the components of v This can be used to define forms in terms of standard Python list operators or Python Numeric array opera tors The o
35. perator div accepts as argument a logically vector valued expression and returns its divergence div v gt div v V v 4 11 Note that the length n of the vector v must be equal to the dimension d of the underlying shape of the FiniteElement defining the function space for v 4 6 4 Rotation rot v The operator rot accepts as argument a logically vector valued expression and returns its rotation Ove v Ovo Ove Ov1 Owi Ox Ox Ox Oxo Oxo 0x1 rot v gt rot v V xv 4 12 Note that this operator is only defined for vectors of length three Alternatively the name curl can be used for this operator 4 7 Integrals Each term of a valid form expression must be a scalar valued expression integrated exactly once Integrals are expressed through multiplication with a measure representing either an integral over the interior of the domain Q or the boundary 02 of Q 4 7 1 Integration over the interior dx A measure for integration over the interior of Q is created as follows 36 FFC User Manual Anders Logg dx Integral interior If v is a scalar valued expression then the integral of v over the interior of Q is written as v dx When using the command line interface to FFC the measure dx is automat ically declared as an integral over the interior of 2 Note however that a user is free to declare measures with other names or even reuse the variable dx for something el
36. perator vec does not have to be used if the form is defined only in terms of the basic operators of the form language 34 FFC User Manual Anders Logg 4 6 Differential operators 4 6 1 Scalar partial derivative D v i The basic differential operator is the scalar partial derivative D This dif ferential operator accepts as arguments a scalar or logically vector valued expression v together with a coordinate direction i and returns the partial derivative of the expression in the given coordinate direction Ov Ox i 4 8 Div i e Alternatively the member function dx can be used For v an expression the two expressions D v i and v dx i are equivalent but note that only the operator D works on vector valued expressions that are defined in terms of Python lists 4 6 2 Gradient grad v The operator grad accepts as argument an expression v and returns its gra dient If v is scalar the result is a vector containing the partial derivatives in the coordinate directions v Ov Ov Oxy Ox Ota1 i grad v gt grad v Vv 4 9 If v is logically vector valued the result is a matrix with rows given by the gradients of each component Ou grad v i 5 gt grad v Vv Ox 4 10 Thus if v is scalar valued then grad grad v returns the Hessian of v and if v is vector valued then grad v is the Jacobian of v 39 FFC User Manual Anders Logg 4 6 3 Divergence div v The o
37. rahedron representing a tetrahedron in R The argument order is an integer specifying the polynomial order of the finite element Note that the minimal order for Lagrange finite elements is one whereas the minimal order for discontinuous Lagrange finite elements is Zero The argument num_components is optional and specifies the number of vector components for a vector valued element If not specified the number of vec tor components is assumed to be the same the dimension d of the underlying shape Note that more than one FiniteElement can be declared and used in the definition of a form The following example declares two elements one linear and one quadratic Lagrange finite element P1 P2 FiniteElement Lagrange tetrahedron 1 FiniteElement Lagrange tetrahedron 2 26 FFC User Manual Anders Logg 4 3 2 MixedElement The data type MixedElement represents a mixed finite element on a triangle or tetrahedron The function space of a mixed finite element is defined as the direct sum of the function spaces of a given list of elements A MixedElement is declared by specifying a list of FiniteElements mixed_element FiniteElement e0 el Alternatively a MixedElement can be created as the sum of a sequence of FiniteElements The following example illustrates how to create a Taylor Hood element quadratic velocity and linear pressure P
38. rence tetrahedron are ordered following the convention that face f should be opposite to vertex v for i 0 1 2 3 with the vertices of each face ordered to give a counter clockwise orientation of each face as seen from the outside of the tetrahedron and the first vertex of face f given by vertex vit mod 4 i v v v f v u v A 4 T v9 vt wv l Pos a The edges of the reference tetrahedron are ordered following the convention that edges e e e should correspond to the edges of the reference triangle Edges e e e all ending up at vertex uv are ordered based on their first vertex m N i A 5 00000 v w Nm mS o e w The ordering of vertices on faces implicitly defines an ordering of edges on 51 FFC User Manual Anders Logg faces by identifying an edge on a face with the opposite vertex on the face flee a 1 Bel 0 r or A nat et ee e el e Note that the ordering of edges on f is the same as the ordering of edges on the reference triangle Also note that the internal ordering of vertices on edges does not always follow the orientation of the face which is not possible A 3 Ordering of degrees of freedom The local and global orderings of degrees of freedom or nodes are obtained by associating each node with a mesh entity locally and globally A 3 1 Mesh entities We distinguish between mesh entities of different topological dimens
39. ries either by patents or by copyrighted interfaces the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries so that distribution is permitted only in or among countries not thus excluded In such case this License incorporates the limitation as if written in the body of this License 9 The Free Software Foundation may publish revised and or new versions of the General Public License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns Each version is given a distinguishing version number If the Program specifies a version number of this License which applies to it and any later version you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation If the Program does not specify a version number of this License you may choose any version ever published by the Free Software Foundation 74 FFC User Manual Anders Logg 10 If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different write to the author to ask for permission For software which is copyrighted by the Free Software Foundation write to the Free Software Foundation we sometimes make exceptions for this Our decision will be guided by the
40. rst edge of the face and once the first edge is picked there are two ways to pick the second edge To define an alignment of faces as an integer between 0 and 5 we compare the ordering of edges on a face with the ordering of edges on the corresponding face on the reference tetrahedron If the first edge of the face matches the first edge on the corresponding face on the reference tetrahedron and also the second edge matches the second edge on the reference tetrahedron then the alignment is 0 If only the first 56 FFC User Manual Anders Logg Figure A 6 Internal ordering of nodes on faces 57 FFC User Manual Anders Logg edge matches then the alignment is 1 We similarly define alignments 2 3 by matching the first and second edges with the second and third edges on the corresponding face on the reference tetrahedron and alignments 4 5 by matching the first and second edges with the third and first edges on the corresponding face on the reference tetrahedron Example 1 The alignment of the first face of a tetrahedron is 0 if the first edge of the face is edge number 5 and the second edge is edge number 0 Example 2 The alignment of the first face of a tetrahedron is 1 if the first edge of the face is edge number 5 and the second edge is not edge number 0 It must then be edge number 4 Example 3 The alignment of the first face of a tetrahedron is 4 if the first edge of the face is edge number 4 and the second edge is edge numb
41. rtex vu of the edge as in Figure A 5 ve Figure A 5 Internal ordering of nodes on edges 59 FFC User Manual Anders Logg A 3 4 Alignment of edges Depending on the orientation of any given cell an edge on the cell may be aligned or not aligned with the corresponding edge on the reference cell if the vertices of the cell are mapped to the reference cell We define the alignment of an edge with respect to a cell to be 0 if the edge is aligned with the orientation of the reference cell and 1 otherwise Example 1 The alignment of the first edge e on a triangle is 0 if the first vertex of the edge is the second vertex v of the triangle Example 2 The alignment of the second edge et on a tetrahedron is 0 if the first vertex of the edge is the third vertex v of the tetrahedron If two cells share a common edge and the edge is aligned with one of the cells and not the other we must reverse the order in which the local nodes are mapped to global nodes on one of the two cells As a convention the order is kept if the alignment is 0 and reversed if the alignment is 1 A 3 5 Internal ordering on faces For faces containing more than one node the ordering of nodes is nested going from the first to the third vertex and in each step going from the first to the second vertex as in Figure A 6 A 3 6 Alignment of faces There are six different ways for a face to be aligned on a tetrahedron there are three ways to pick the fi
42. s oa saosa Da a ed eo we HS 43 5 4 The strain strain term of linear elasticity 43 5 5 The nonlinear term of Navier Stokes 44 oo The heat EQUATION 424 4 srate g baia EE BORG GRY 45 5 7 Mixed formulation of Stokes 46 A Reference elements 49 A 1 The reference triangle o 49 A 2 The reference tetrahedron 51 A 3 Ordering of degrees of freedom 52 MAL MIEL OI ecs Seep kh ee eG Re Ge 52 A 3 2 Ordering among mesh entities 55 A 3 3 Internal ordering on edges 55 A 3 4 Alignment of edges 56 A 3 5 Internal ordering on faces 2 56 A 3 6 Alignment of faces 2 220000 56 FFC User Manual B Installation B 1 Installing from source B 1 1 Dependencies and requirements EN Be B13 B 1 4 B 1 5 B 2 Debian package Downloading the source code Installing the compiler Compiling the demos Verifying the generated code C Contributing code C 1 Creating a patch C 2 Sending patches C 3 Applying a patch maintainers C 4 License agreement D License Anders Logg About this manual This manual is currently being written As a consequence some sections may be incomplete or inaccurate Intended audience This manual is written both for the beginning and the advanced user There is also some useful information for devel
43. se 4 7 2 Integration over the boundary ds A measure for integration over the boundary of 2 is created as follows ds Integral boundary If v is a scalar valued expression then the integral of v over the boundary of Q is written as v ds When using the command line interface to FFC the measure ds is automat ically declared as an integral over the boundary of Q Note however that a user is free to declare measures with other names or even reuse the variable ds for something else At this point complete support has not been added to FFC for boundary integrals which means that all boundary integrals are currently evaluated to zero 4 8 Index notation FFC supports index notation which is often a convenient way to express forms The basic principle of index notation is that summation is implicit 37 FFC User Manual Anders Logg over indices repeated twice in each term of an expression The following examples illustrate the index notation assuming that each of the variables i and j have been declared as a free Index vlilewli o Y viw 4 13 D v i D w i gt Yo en 4 14 D vlil D e X V v 4 15 D v i j D wlil j Index notation is used internally by FFC to represent multilinear forms and in most cases FFC is capable of generating an efficient tensor representation of any given expression However in some cases index notation might generate more efficient code
44. tain the following variational problem for the discrete solution U U z t Find U U tn with U U tn_1 given such that 1 E U u vds f ever Wear prods 5 10 kn Q Q Q for all test functions v where k tn tn_ denotes the time step In the example below we implement this variational problem with piecewise linear test and trial functions but other choices are possible just choose another finite element Rewriting the variational problem in the standard form a v U L v for all v we obtain the following pair of bilinear and linear forms au U ourar tin f cv VU de 5 11 o o Lw foU tans ky fv fra 5 12 Q Q which can be implemented as follows 45 FFC User Manual Anders Logg element FiniteElement Lagrange triangle 1 v BasisFunction element Test function U1 BasisFunction element Value at t_n UO Function element Value at t_n 1 c Function element Heat conductivity f Function element Heat source k Constant Time step a v Ulx dx k c dot grad v grad Ul x dx L v U0 xdx k xv f dx 5 7 Mixed formulation of Stokes To solve Stokes equations Au Vp f 5 13 V u 0 5 14 we write the variational problem in standard form a v U L v for all v to obtain the following pair of bilinear and linear forms alv U P ve VU V v P q V U dz 5 15 L v fo Fda 5 16 Using a mixed form
45. ther useful option to patch is dry run which can be used to test the patch without actually applying it 5 The modified version now exists as ffc 0 1 0 C 4 License agreement By contributing a patch to FFC you agree to license your contributed code under the GNU General Public License a condition also built into the GPL license of the code you have modified Before creating the patch please update the author and date information of the file s you have modified according to the following example __author__ Anders Logg logg tti c org __date__ 2004 11 17 2005 09 09 __copyright__ Copyright c 2004 2005 Anders Logg __license__ GNU GPL Version 2 Modified by Johan Jansson 2005 As a rule of thumb the original author of a file holds the copyright 68 Appendix D License FFC is licensed under the GNU General Public License GPL version 2 included verbatim below GNU GENERAL PUBLIC LICENSE Version 2 June 1991 Copyright C 1989 1991 Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed Preamble The licenses for most software are designed to take away your freedom to share and change it By contrast the GNU General Public License is intended to guarantee your freedom to share and change free so
46. to the function compile the Python interface provides the func tions build write and writeFiniteElement These functions are docu mented below Documentation can also be accessed from within Python To read the docu mentation for the function compile run the following commands in a Python shell from ffc import help compile 3 1 compile forms This function takes as argument a form or list of forms and compiles it into low level code for assembly Calling this function is equivalent to first calling build followed by write 3 2 build forms This function takes as argument a form or list of forms and does preprocessing of the forms including computation of the reference tensor but does not generate any code 3 3 write forms This function takes a preprocessed form or list of forms and generates code Note that build must be called before write forms build a L write forms 20 FFC User Manual Anders Logg 3 4 writeFiniteFlement element This function generates code for a given FiniteElement Use this function if you just want to generate code for a finite element including mapping of nodes and nodal points element FiniteElement Lagrange triangle 1 writeFiniteElement element 21 Chapter 4 Form language FFC uses a flexible and extensible language to define and process multilinear forms In this chapter we give th
47. two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally NO WARRANTY 11 BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE THERE IS NO WARRANTY FOR THE PROGRAM TO THE EXTENT PERMITTED BY APPLICABLE LAW EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND OR OTHER PARTIES PROVIDE THE PROGRAM AS IS WITHOUT WARRANTY OF ANY KIND EITHER EXPRESSED OR IMPLIED INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU SHOULD THE PROGRAM PROVE DEFECTIVE YOU ASSUME THE COST OF ALL NECESSARY SERVICING REPAIR OR CORRECTION 12 IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER OR ANY OTHER PARTY WHO MAY MODIFY AND OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE BE LIABLE TO YOU FOR DAMAGES INCLUDING ANY GENERAL SPECIAL INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES END OF TERMS AND CONDITIONS 75 Index BasisFunctions 27 BasisFunction 27 Constant 29
48. uded with the FFC source distribution 5 2 Poisson s equation The bilinear and linear forms form for Poisson s equation alu U vo VU dz 5 2 Le vtec 5 3 can be implemented as follows element FiniteElement Lagrange triangle 1 v BasisFunction element U BasisFunction element f Function element a dot grad v grad U x dx L v f dx Alternatively index notation can be used to express the scalar product a D v i D U i xdx This example is implemented in the file Poisson form in the collection of demonstration forms included with the FFC source distribution 42 FFC User Manual Anders Logg 5 3 Vector valued Poisson The bilinear and linear forms for a system of independent Poisson equa tions a v U ve VU de 5 4 Lw fv t s 5 5 with v U and f vector valued can be implemented as follows element FiniteElement Vector Lagrange triangle 1 v BasisFunction element U BasisFunction element f Function element a dot grad v grad U dx L dot v f dx Alternatively index notation may be used D v i j D U il j dx v i f i dx m w toll This example is implemented in the file PoissonSystem form in the collec tion of demonstration forms included with the FFC source distribution 5 4 The strain strain term of linear elasticity The strain strain term of l
49. ulation with Taylor Hood elements this can be imple mented as follows P2 FiniteEFlement Vector Lagrange triangle 2 Pi FiniteElement Lagrange triangle 1 TH P2 Pl v q BasisFunctions TH 46 FFC User Manual Anders Logg U P BasisFunctions TH f Function P2 a dot grad v grad U div v P q div U dx L dot v f x dx 47 Appendix A Reference elements A 1 The reference triangle The reference triangle Figure A 1 is defined by the following three vertices v 0 0 v 1 0 A 1 v 0 1 Note that this corresponds to a counter clockwise orientation of the vertices in the plane The edges of the reference triangle are ordered following the convention that edge e should be opposite to vertex v for i 0 1 2 with the vertices of each edge ordered to give a counter clockwise orientation of the triangle in the plane e uv v e v 49 A 2 e v v 49 FFC User Manual Anders Logg Lo vu yi Figure A 1 Physical coordinates of the reference triangle v e e yo e v Figure A 2 Ordering of mesh entities vertices and edges for the reference triangle 50 FFC User Manual Anders Logg A 2 The reference tetrahedron The reference tetrahedron Figure A 3 is defined by the following four ver tices v 0 0 0 v 1 0 0 Dar A 3 v 0 1 0 Aa v 0 0 1 The faces of the refe
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