This is the `GetDP Reference Manual' for GetDP 2.0 (15 March 2010). Copyright (C) 1997-2010 Patrick Dular, Christophe Geuzaine. Short Contents ************** GetDP Obtaining GetDP Copying conditions 1 Overview 2 How to read this manual 3 Running GetDP 4 Expressions 5 Objects 6 Types for objects 7 Short examples 8 Complete examples Appendix A File formats Appendix B Gmsh examples Appendix C Frequently asked questions Appendix D Tips and tricks Appendix E Version history Appendix F Copyright and credits Appendix G License Concept index Metasyntactic variable index Syntax index Table of Contents ***************** GetDP Obtaining GetDP Copying conditions 1 Overview 1.1 Numerical tools as objects 1.2 Which problems can GetDP actually solve? 1.3 Bug reports 2 How to read this manual 2.1 Syntactic rules used in this document 3 Running GetDP 4 Expressions 4.1 Comments 4.2 Includes 4.3 Expressions definition 4.4 Constants 4.5 Operators 4.5.1 Operator types 4.5.2 Evaluation order 4.6 Functions 4.7 Current values 4.8 Arguments 4.9 Registers 4.10 Fields 4.11 Loops and conditionals 5 Objects 5.1 `Group': defining topological entities 5.2 `Function': defining global and piecewise expressions 5.3 `Constraint': specifying constraints on function spaces and formulations 5.4 `FunctionSpace': building function spaces 5.5 `Jacobian': defining jacobian methods 5.6 `Integration': defining integration methods 5.7 `Formulation': building equations 5.8 `Resolution': solving systems of equations 5.9 `PostProcessing': exploiting computational results 5.10 `PostOperation': exporting results 6 Types for objects 6.1 Types for `Group' 6.2 Types for `Function' 6.2.1 Math functions 6.2.2 Extended math functions 6.2.3 Green functions 6.2.4 Type manipulation functions 6.2.5 Coordinate functions 6.2.6 Miscellaneous functions 6.3 Types for `Constraint' 6.4 Types for `FunctionSpace' 6.5 Types for `Jacobian' 6.6 Types for `Integration' 6.7 Types for `Formulation' 6.8 Types for `Resolution' 6.9 Types for `PostProcessing' 6.10 Types for `PostOperation' 7 Short examples 7.1 Constant expression examples 7.2 `Group' examples 7.3 `Function' examples 7.4 `Constraint' examples 7.5 `FunctionSpace' examples 7.5.1 Nodal finite element spaces 7.5.2 High order nodal finite element space 7.5.3 Nodal finite element space with floating potentials 7.5.4 Edge finite element space 7.5.5 Edge finite element space with gauge condition 7.5.6 Coupled edge and nodal finite element spaces 7.5.7 Coupled edge and nodal finite element spaces for multiply connected domains 7.6 `Jacobian' examples 7.7 `Integration' examples 7.8 `Formulation' examples 7.8.1 Electrostatic scalar potential formulation 7.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges 7.8.3 Magnetostatic 3D vector potential formulation 7.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation 7.8.5 Nonlinearities, Mixed formulations, ... 7.9 `Resolution' examples 7.9.1 Static resolution (electrostatic problem) 7.9.2 Frequency domain resolution (magnetodynamic problem) 7.9.3 Time domain resolution (magnetodynamic problem) 7.9.4 Nonlinear time domain resolution (magnetodynamic problem) 7.9.5 Coupled formulations 7.10 `PostProcessing' examples 7.11 `PostOperation' examples 8 Complete examples 8.1 Electrostatic problem 8.2 Magnetostatic problem 8.3 Magnetodynamic problem Appendix A File formats A.1 Input file format A.2 Output file format A.2.1 File `.pre' A.2.2 File `.res' Appendix B Gmsh examples Appendix C Frequently asked questions C.1 The basics C.2 Installation C.3 Usage Appendix D Tips and tricks Appendix E Version history Appendix F Copyright and credits Appendix G License Concept index Metasyntactic variable index Syntax index GetDP ***** Patrick Dular and Christophe Geuzaine GetDP is a general finite element solver that uses mixed finite elements to discretize de Rham-type complexes in one, two and three dimensions. This is the `GetDP Reference Manual' for GetDP 2.0 (15 March 2010). Obtaining GetDP *************** The source code and various pre-compiled versions of GetDP (for Unix, Windows and Mac OS) can be downloaded from `http://geuz.org/getdp'. If you use GetDP, we would appreciate that you mention it in your work. References and the latest news about GetDP are always available on `http://geuz.org/getdp'. Copying conditions ****************** GetDP is "free software"; this means that everyone is free to use it and to redistribute it on a free basis. GetDP is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of GetDP that they might get from you. Specifically, we want to make sure that you have the right to give away copies of GetDP, that you receive source code or else can get it if you want it, that you can change GetDP or use pieces of GetDP in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of GetDP, 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 tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for GetDP. If GetDP is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions of the license for GetDP are found in the General Public License that accompanies the source code (*note License::). Further information about this license is available from the GNU Project webpage `http://www.gnu.org/copyleft/gpl-faq.html'. Detailed copyright information can be found in *note Copyright and credits::. If you want to integrate parts of GetDP into a closed-source software, or want to sell a modified closed-source version of GetDP, you will need to obtain a different license. Please contact us directly (http://geuz.org) for more information. 1 Overview ********** GetDP (a "General environment for the treatment of Discrete Problems") is a scientific software environment for the numerical solution of integro-differential equations, open to the coupling of physical problems (electromagnetic, thermal, etc.) as well as of numerical methods (finite element method, integral methods, etc.). It can deal with such problems of various dimensions (1D, 2D or 3D) and time states (static, transient or harmonic). The main feature of GetDP is the closeness between its internal structure (written in C), the organization of data defining discrete problems (written by the user in ASCII data files) and the symbolic mathematical expressions of these problems. Its aim is to be welcoming and of easy use for both development and application levels: it consists of a working environment in which the definition of any problem makes use of a limited number of objects, which makes the environment structured and concise. It therefore gives researchers advanced developing tools and a large freedom in adding new functionalities. The modeling tools provided by GetDP can be tackled at various levels of complexity: this opens the software to a wide range of activities, such as research, collaboration, education, training and industrial studies. 1.1 Numerical tools as objects ============================== An assembly of computational tools (or objects) in GetDP leads to a problem definition structure, which is a transcription of the mathematical expression of the problem, and forms a text data file: the equations describing a phenomenon, written in a mathematical form adapted to a chosen numerical method, directly constitute data for GetDP. The resolution of a discrete problem with GetDP requires the definition, in a text data file, of the GetDP objects listed (together with their dependencies) in the following figure and table. ---------------------------------------------------------------------------- | | | -------------- | | -------->| Function |-------------------------------- | | | -------------- \ \ | | | | | --------------------------- \ | | | | \ | \ | | | \|/ ------------ | \ \|/ -------------- -------------- | | \ -------------- | Group |-->| Constraint |-------- | | \ |PostOperation | -------------- -------------- | | | \ -------------- | | | || | | | | \ /||\ *********************************************************************************************** * | | | || \|/ \|/\|/ \|/ _\| || * * | | | || -------------- -------------- -------------- -------------- * * | | | =====>|FunctionSpace |==>| Formulation |==>| Resolution |==>|PostProcessing| * * | | | -------------- -------------- -------------- -------------- * * | | | /|\/|\/|\ | /|\/|\/|\/|\ * * | | -------------------------------- | | | | | | | * * | | ----------------- | --------------------------- | | | * * | | | -------------- | | | * * | | | | Integration |---------------------------- | | * * | | | -------------- | | * * | | | | | * * | | -------------- | | * * | ----------->| Jacobian |-------------------------------------------------- | * * | -------------- | * * | | * * ------------------------------------------------------------------------------------ * * * *********************************************************************************************** Group -- Function Group Constraint Group, Function, (Resolution) FunctionSpace Group, Constraint, (Formulation), (Resolution) Jacobian Group Integration -- Formulation Group, Function, (Constraint), FunctionSpace, Jacobian, Integration Resolution Function, Formulation PostProcessing Group, Function, Jacobian, Integration, Formulation, Resolution PostOperation Group, PostProcessing The gathering of all these objects constitutes the problem definition structure, which is a copy of the formal mathematical formulation of the problem. Reading the first column of the table from top to bottom pictures the working philosophy and the linking of operations peculiar to GetDP, from group definition to results visualization. The decomposition highlighted in the figure points out the separation between the objects defining the method of resolution, which may be isolated in a "black box" (bottom) and those defining the data peculiar to a given problem (top). The computational tools which are in the center of a problem definition structure are formulations (`Formulation') and function spaces (`FunctionSpace'). Formulations define systems of equations that have to be built and solved, while function spaces contain all the quantities, i.e., functions, fields of vectors or covectors, known or not, involved in formulations. Each object of a problem definition structure must be defined before being referred to by others. A linking which always respects this property is the following: it first contains the objects defining particular data of a problem, such as geometry, physical characteristics and boundary conditions (i.e., `Group', `Function' and `Constraint') followed by those defining a resolution method, such as unknowns, equations and related objects (i.e., `Jacobian', `Integration', `FunctionSpace', `Formulation', `Resolution' and `PostProcessing'). The processing cycle ends with the presentation of the results (i.e., lists of numbers in various formats), defined in `PostOperation' fields. This decomposition points out the possibility of building black boxes, containing objects of the second group, adapted to treatment of general classes of problems that share the same resolution methods. 1.2 Which problems can GetDP actually solve? ============================================ The preceding explanations may seem very (too) general. Which are the problems that GetDP can actually solve? To answer this question, here is a list of methods that we have considered and coupled until now: Numerical methods finite element method boundary element method (experimental, undocumented) volume integral methods (experimental, undocumented) Geometrical models one-dimensional models (1D) two-dimensional models (2D), plane and axisymmetric three-dimensional models (3D) Time states static states sinusoidal and harmonic states transient states eigenvalue problems These methods have been successfully applied to build coupled physical models involving electromagnetic phenomena (magnetostatics, magnetodynamics, electrostatics, electrokinetics, electrodynamics, wave propagation, lumped electric circuits), acoustic phenomena, thermal phenomena and mechanical phenomena (elasticity, rigid body movement). As can be guessed from the preceding list, GetDP has been initially developed in the field of computational electromagnetics, which fully uses all the offered coupling features. We believe that this does not interfere with the expected generality of the software because a particular modeling forms a problem definition structure which is totally external to the software: GetDP offers computational tools; the user freely applies them to define and solve his problem. Nevertheless, specific numerical tools will _always_ need to be implemented to solve specific problems in areas other than those mentionned above. If you think the general phisosophy of GetDP is right for you and your problem, but you discover that GetDP lacks the tools necessary to handle it, let us know: we would love to discuss it with you. For example, at the time of this writing, many areas of GetDP would need to be improved to make GetDP as useful for computational mechanics or computational fluid dynamics as it is for computational electromagnetics... So if you have the skills and some free time, feel free to join the project: we gladly accept all code contributions! 1.3 Bug reports =============== If you think you have found a bug in GetDP, you can report it by electronic mail to the GetDP mailing list at , or file it directly into our bug tracking system at `https://geuz.org/trac/getdp/report' (login: getdp, password: getdp). Please send as precise a description of the problem as you can, including sample input files that produce the bug (problem definition and mesh files). Don't forget to mention both the version of GetDP and the version of your operation system (*note Running GetDP:: to see how to get this information). See *note Frequently asked questions::, and the bug tracking system to see which problems we already know about. 2 How to read this manual ************************* After reading *note Overview::, which depicts the general philosophy of GetDP, you might want to skip *note Expressions::, *note Objects:: and *note Types for objects:: and directly run the demo files bundled in the distribution on your computer (*note Running GetDP::). You should then open these examples with a text editor and compare their structure with the examples given in *note Short examples:: and *note Complete examples::. For each new syntax element that you fall onto, you can then go back to *note Expressions::, *note Objects::, and *note Types for objects::, and find in these chapters the detailed description of the syntactic rules as well as all the available options. Indexes for many concepts (*note Concept index::) and for all the syntax elements (*note Syntax index::) are available at the end of this manual. 2.1 Syntactic rules used in this document ========================================= Here are the rules we tried to follow when writing this user's guide. Note that metasyntactic variable definitions stay valid throughout all the manual (and not only in the sections where the definitions appear). See *note Metasyntactic variable index::, for an index of all metasyntactic variables. 1. Keywords and literal symbols are printed like `this'. 2. Metasyntactic variables (i.e., text bits that are not part of the syntax, but stand for other text bits) are printed like THIS. 3. A colon (`:') after a metasyntactic variable separates the variable from its definition. 4. Optional rules are enclosed in `<' `>' pairs. 5. Multiple choices are separated by `|'. 6. Three dots (...) indicate a possible repetition of the preceding rule. 7. For conciseness, the notation `RULE <, RULE > ...' is replaced by `RULE <,...>'. 8. The ETC symbol replaces nonlisted rules. 3 Running GetDP *************** GetDP has no graphical interface(1) (*note Running GetDP-Footnote-1::). It is a command-line driven program that reads a problem definition file once at the beginning of the processing. This problem definition file is a regular ASCII text file (*note Numerical tools as objects::), hence created with whatever text editor you like. If you just type the program name at your shell prompt (without any argument), you will get a short help on how to run GetDP. All GetDP calls look like getdp FILENAME OPTIONS where FILENAME is the ASCII file containing the problem definition, i.e., the structures this user's guide has taught you to create. This file can include other files (*note Includes::), so that only one problem definition file should always be given on the command line. The input files containing the problem definition structure are usually given the `.pro' extension (if so, there is no need to specify the extension on the command line). The name of this file (without the extension) is used as a basis for the creation of intermediate files during the pre-processing and the processing stages. The OPTIONS are a combination of the following commands (in any order): `-pre' RESOLUTION-ID Performs the pre-processing associated with the resolution RESOLUTION-ID. In the pre-processing stage, GetDP creates the geometric database (from the mesh file), identifies the degrees of freedom (the unknowns) of the problem and sets up the constraints on these degrees of freedom. The pre-processing creates a file with a `.pre' extension. If RESOLUTION-ID is omitted, the list of available choices is displayed. `-cal' Performs the processing. This requires that a pre-processing has been performed previously, or that a `-pre' option is given on the same command line. The performed resolution is the one given as an argument to the `-pre' option. In the processing stage, GetDP executes all the commands given in the `Operation' field of the selected `Resolution' object (such as matrix assemblies, system resolutions, ...). `-pos' POST-OPERATION-ID ... Performs the operations in the `PostOperation'(s) selected by the POST-OPERATION-ID(s). This requires that a processing has been performed previously, or that a `-cal' option is given on the same command line. If POST-OPERATION-ID is omitted, the list of available choices is displayed. `-msh' FILENAME Reads the mesh (in `.msh' format) from FILENAME (*note File formats::) rather than from the default problem file name (with the `.msh' extension appended). `-split' Saves processing results in separate files (one for each timestep). `-res' FILENAME ... Loads processing results from file(s). `-name' STRING Uses STRING as the default generic file name for input or output of mesh, pre-processing and processing files. `-restart' Restarts processing of a time stepping resolution interrupted before being complete. `-solve' RESOLUTION-ID Same as `-pre RESOLUTION-ID -cal'. `-adapt' FILE Reads adaptation constraints from file. `-order' REAL Specifies the maximum interpolation order. `-bin' Selects binary format for output files. `-socket' STRING Communicates through socket STRING. `-check' Lets you check the problem structure interactively. `-v' `-verbose' INTEGER Sets the verbosity level. A value of 0 means that no information will be displayed during the processing. `-p' `-progress' INTEGER Sets the progress update rate. This controls the refreshment rate of the counter indicating the progress of the current computation (in %). `-info' Displays the version information. `-version' Displays the version number. `-help' Displays a message listing basic usage and available options. (1) If you are looking for a graphical front-end to GetDP, you may consider using Gmsh (available at `http://geuz.org/gmsh'). Gmsh permits to construct geometries, generate meshes, launch computations and visualize results directly from within a user-friendly graphical interface. The file formats used by Gmsh for mesh generation and post-processing are the default file formats accepted by GetDP (see *note Input file format::, and *note Types for PostOperation::). 4 Expressions ************* This chapter and the next two describe in a rather formal way all the commands that can be used in the ASCII text input files. If you are just beginning to use GetDP, or just want to see what GetDP is all about, you should skip this chapter and the next two for now, have a quick look at *note Running GetDP::, and run the demo problems bundled in the distribution on your computer. You should then open the `.pro' files in a text editor and compare their structure with the examples given in *note Short examples:: and *note Complete examples::. Once you have a general idea of how the files are organized, you might want to come back here to learn more about the specific syntax of all the objects, and all the available options. 4.1 Comments ============ Both C and C++ style comments are supported and can be used in the input data files to comment selected text regions: 1. the text region comprised between `/*' and `*/' pairs is ignored; 2. the rest of a line after a double slash `//' is ignored. Comments cannot be used inside double quotes or inside GetDP keywords. 4.2 Includes ============ An input data file can be included in another input data file by placing one of the following commands (EXPRESSION-CHAR represents a file name) on a separate line, outside the GetDP objects. Any text placed after an include command on the same line is ignored. `Include EXPRESSION-CHAR' `#include EXPRESSION-CHAR' See *note Constants::, for the definition of the character expression EXPRESSION-CHAR. 4.3 Expressions definition ========================== Expressions are the basic tool of GetDP. They cover a wide range of functional expressions, from constants to formal expressions containing functions (built-in or user-defined, depending on space and time, etc.), arguments, discrete quantities and their associated differential operators, etc. Note that `white space' (spaces, tabs, new line characters) is ignored inside expressions (as well as inside all GetDP objects). Expressions are denoted by the metasyntactic variable EXPRESSION (remember the definition of the syntactic rules in *note Syntactic rules::): EXPRESSION: ( EXPRESSION ) | INTEGER | REAL | CONSTANT-ID | QUANTITY | ARGUMENT | CURRENT-VALUE | REGISTER-VALUE-SET | REGISTER-VALUE-GET | OPERATOR-UNARY EXPRESSION | EXPRESSION OPERATOR-BINARY EXPRESSION | EXPRESSION OPERATOR-TERNARY-LEFT EXPRESSION OPERATOR-TERNARY-RIGHT EXPRESSION | BUILT-IN-FUNCTION-ID [ < EXPRESSION-LIST > ] < { EXPRESSION-CST-LIST } > | FUNCTION-ID [ < EXPRESSION-LIST > ] | < Real | Complex > [ EXPRESSION ] | Dt [ EXPRESSION ] | AtAnteriorTimeStep [ EXPRESSION, INTEGER ] | Order [ QUANTITY ] | Trace [ EXPRESSION, GROUP-ID ] | EXPRESSION ##INTEGER The following sections introduce the quantities that can appear in expressions, i.e., constant terminals (INTEGER, REAL) and constant expression identifiers (CONSTANT-ID, EXPRESSION-CST-LIST), discretized fields (QUANTITY), arguments (ARGUMENT), current values (CURRENT-VALUE), register values (REGISTER-VALUE-SET, REGISTER-VALUE-GET), operators (OPERATOR-UNARY, OPERATOR-BINARY, OPERATOR-TERNARY-LEFT, OPERATOR-TERNARY-RIGHT) and built-in or user-defined functions (BUILT-IN-FUNCTION-ID, FUNCTION-ID). The last seven cases in this definition permit to cast an expression as real or complex, get the time derivative or evaluate an expression at an anterior time step, retrieve the interpolation order of a discretized quantity, evaluate the trace of an expression, and print the value of an expression for debugging purposes. List of expressions are defined as: EXPRESSION-LIST: EXPRESSION <,...> 4.4 Constants ============= The three constant types used in GetDP are INTEGER, REAL and STRING. These types have the same meaning and syntax as in the C or C++ programming languages. Besides general expressions (EXPRESSION), purely constant expressions, denoted by the metasyntactic variable EXPRESSION-CST, are also used: EXPRESSION-CST: ( EXPRESSION-CST ) | INTEGER | REAL | CONSTANT-ID | OPERATOR-UNARY EXPRESSION-CST | EXPRESSION-CST OPERATOR-BINARY EXPRESSION-CST | EXPRESSION-CST OPERATOR-TERNARY-LEFT EXPRESSION-CST OPERATOR-TERNARY-RIGHT EXPRESSION-CST | MATH-FUNCTION-ID [ < EXPRESSION-CST-LIST > ] List of constant expressions are defined as: EXPRESSION-CST-LIST: EXPRESSION-CST-LIST-ITEM <,...> with EXPRESSION-CST-LIST-ITEM: EXPRESSION-CST | EXPRESSION-CST : EXPRESSION-CST | EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST | CONSTANT-ID {} | CONSTANT-ID { EXPRESSION-CST-LIST } | List[ CONSTANT-ID ] | ListAlt[ CONSTANT-ID, CONSTANT-ID ] | LinSpace[ EXPRESSION-CST, EXPRESSION-CST, EXPRESSION-CST ] | LogSpace[ EXPRESSION-CST, EXPRESSION-CST, EXPRESSION-CST ] The second case in this last definition permits to create a list containing the range of numbers comprised between the two EXPRESSION-CST, with a unit incrementation step. The third case also permits to create a list containing the range of numbers comprised between the two EXPRESSION-CST, but with a positive or negative incrementation step equal to the third EXPRESSION-CST. The fourth and fifth cases permit to reference constant identifiers (CONSTANT-IDs) of lists of constants and constant identifiers of sublists of constants (see below for the definition of constant identifiers) . The sixth case is a synonym for the fourth. The seventh case permits to create alternate lists: the arguments of `ListAlt' must be CONSTANT-IDs of lists of constants of the same dimension. The result is an alternate list of these constants: first constant of argument 1, first constant of argument 2, second constant of argument 1, etc. These kinds of lists of constants are for example often used for function parameters (*note Functions::). The last two cases permit to create linear and logarithmic lists of numbers, respectively. Contrary to a general EXPRESSION which is evaluated at runtime (thanks to an internal stack mechanism), an EXPRESSION-CST is completely evaluated during the syntactic analysis of the problem (when GetDP reads the `.pro' file). The definition of such constants or lists of constants with identifiers can be made outside or inside any GetDP object. The syntax for the definition of constants is: AFFECTATION: DefineConstant [ CONSTANT-ID < = EXPRESSION-CST > STRING-ID < = "STRING" > <,...> ]; | CONSTANT-ID = CONSTANT-DEF; | STRING-ID = STRING-DEF; | Printf("STRING"); | Printf("STRING", EXPRESSION-CST-LIST); | Read(CONSTANT-ID); | Read(CONSTANT-ID)[EXPRESSION-CST]; with CONSTANT-ID: STRING | STRING ~ { EXPRESSION-CST } CONSTANT-DEF: EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST } | ListFromFile [ EXPRESSION-CHAR ] STRING-ID: STRING | STRING ~ { EXPRESSION-CST } STRING-DEF: "STRING" | Str[ EXPRESSION-CHAR ] | StrCat[ EXPRESSION-CHAR, EXPRESSION-CHAR ] Notes: 1. Five constants are predefined in GetDP: `Pi' (3.1415926535897932), `0D' (0), `1D' (1), `2D' (2) and `3D' (3). 2. When `~{EXPRESSION-CST}' is appended to a string STRING, the result is a new string formed by the concatenation of STRING, `_' (an underscore) and the value of the EXPRESSION-CST. This is most useful in loops (*note Loops and conditionals::), where it permits to define unique strings automatically. For example, For i In {1:3} x~{i} = i; EndFor is the same as x_1 = 1; x_2 = 2; x_3 = 3; 3. The assignment in `DefineConstant' (zero if no EXPRESSION-CST is given) is performed only if CONSTANT-ID has not yet been defined. This kind of explicit default definition mechanism is most useful in general problem definition structures making use of a large number of generic constants, functions or groups. When exploiting only a part of a complex problem definition structure, the default definition mechanism allows to define the quantities of interest only, the others being assigned a default value (that will not be used during the processing but that avoids the error messages produced when references to undefined quantities are made). See *note Constant expression examples::, as well as *note Function examples::, for some examples. Character expressions are defined as follows: EXPRESSION-CHAR: "STRING" | STRING-ID | StrCat[ EXPRESSION-CHAR , EXPRESSION-CHAR ] | Sprintf( EXPRESSION-CHAR ) | Sprintf( EXPRESSION-CHAR, EXPRESSION-CST-LIST ) | Date The third case in this definition permits to concatenate two character expressions; the next two cases permit to print the value of variables using standard C formatting; the last case permits to access the current date. 4.5 Operators ============= 4.5.1 Operator types -------------------- The operators in GetDP are similar to the corresponding operators in the C or C++ programming languages. OPERATOR-UNARY: `-' Unary minus. `!' Logical not. OPERATOR-BINARY: `^' Exponentiation. The evaluation of the both arguments must result in a scalar value. `*' Multiplication or scalar product, depending on the type of the arguments. `/\' Cross product. The evaluation of both arguments must result in vectors. `/' Division. `%' Modulo. The evaluation of the second argument must result in a scalar value. `+' Addition. `-' Subtraction. `==' Equality. `!=' Inequality. `>' Greater. The evaluation of both arguments must result in scalar values. `>=' Greater or equality. The evaluation of both arguments must result in scalar values. `<' Less. The evaluation of both arguments must result in scalar values. `<=' Less or equality. The evaluation of both arguments must result in scalar values. `&&' Logical `and'. The evaluation of both arguments must result in scalar values. `||' Logical `or'. The evaluation of both arguments must result in floating point values. Warning: the logical `or' always (unlike in C or C++) implies the evaluation of both arguments. That is, the second operand of `||' is evaluated even if the first one is true. `&' Binary `and'. `|' Binary `or'. OPERATOR-TERNARY-LEFT: `?' OPERATOR-TERNARY-RIGHT: `:' The only ternary operator, formed by OPERATOR-TERNARY-LEFT and OPERATOR-TERNARY-RIGHT is defined as in the C or C++ programming languages. The ternary operator first evaluates its first argument (the EXPRESSION-CST located before the `?'), which must result in a scalar value. If it is true (non-zero) the second argument (located between `?' and `:') is evaluated and returned; otherwise the third argument (located after `:') is evaluated and returned. 4.5.2 Evaluation order ---------------------- The evaluation priorities are summarized below (from stronger to weaker, i.e., `^' has the highest evaluation priority). Parentheses `()' may be used anywhere to change the order of evaluation. `^' `- (unary), !' `| &' `/\' `*, /, %' `+, -' `<, >, <=, >=' `!=, ==' `&&, ||' `?:' 4.6 Functions ============= Two types of functions coexist in GetDP: user-defined functions (FUNCTION-ID, see *note Function::) and built-in functions (BUILT-IN-FUNCTION-ID, defined in this section). Both types of functions are always followed by a pair of brackets `[]' that can possibly contain arguments (*note Arguments::). This makes it simple to distinguish a FUNCTION-ID or a BUILT-IN-FUNCTION-ID from a CONSTANT-ID. As shown below, built-in functions might also have parameters, given between braces `{}', and which are completely evaluated during the analysis of the syntax (since they are of EXPRESSION-CST-LIST type): BUILT-IN-FUNCTION-ID [ < EXPRESSION-LIST > ] < { EXPRESSION-CST-LIST } > with BUILT-IN-FUNCTION-ID: MATH-FUNCTION-ID | EXTENDED-MATH-FUNCTION-ID | GREEN-FUNCTION-ID | TYPE-FUNCTION-ID | COORD-FUNCTION-ID | MISC-FUNCTION-ID Notes: 1. All possible values for BUILT-IN-FUNCTION-ID are listed in *note Types for Function::. 2. Classical mathematical functions (*note Math functions::) are the only functions allowed in a constant definition (see the definition of EXPRESSION-CST in *note Constants::). 4.7 Current values ================== Current values are a special kind of arguments (*note Arguments::) which return the current integer or floating point value of an internal GetDP variable: `$Time' Value of the current time. This value is set to zero for non time dependent analyses. `$DTime' Value of the current time increment used in a time stepping algorithm. `$Theta' Current theta value in a theta time stepping algorithm. `$TimeStep' Number of the current time step in a time stepping algorithm. `$Iteration' Number of the current iteration in a nonlinear loop. `$EigenvalueReal' Real part of the current eigenvalue. `$EigenvalueImag' Imaginary part of the current eigenvalue. `$X, $XS' Value of the current (destination or source) X-coordinate. `$Y, $YS' Value of the current (destination or source) Y-coordinate. `$Z, $ZS' Value of the current (destination or source) Z-coordinate. `$A, $B, $C' Value of the current parametric coordinates used in the parametric `OnGrid' `PostOperation' (*note Types for PostOperation::). Note: 1. The current X, Y and Z coordinates refer to the `physical world' coordinates, i.e., coordinates in which the mesh is expressed. 4.8 Arguments ============= Function arguments can be used in expressions and have the following syntax (INTEGER indicates the position of the argument in the EXPRESSION-LIST of the function, starting from 1): ARGUMENT: $INTEGER See *note Function::, and *note Function examples::, for more details. 4.9 Registers ============= In many situations, identical parts of expressions are used more than once. If this is not a problem with constant expressions (since EXPRESSION-CSTs are evaluated only once during the analysis of the problem definition structure, cf. *note Constants::), it may introduce some important overhead while evaluating complex EXPRESSIONs (which are evaluated at runtime, thanks to an internal stack mechanism). In order to circumvent this problem, the evaluation result of any part of an EXPRESSION can be saved in a register: a memory location where this partial result will be accessible without any costly reevaluation of the partial expression. Registers have the following syntax: REGISTER-VALUE-SET: EXPRESSION#INTEGER REGISTER-VALUE-GET: #INTEGER Thus, to store any part of an expression in the register 5, one should add `#5' directly after the expression. To reuse the value stored in this register, one simply uses `#5' instead of the expression it should replace. See *note Function examples::, for an example. 4.10 Fields =========== A discretized quantity (defined in a function space, cf. *note FunctionSpace::) is represented between braces `{}', and can only appear in well-defined expressions in `Formulation' (*note Formulation::) and `PostProcessing' (*note PostProcessing::) objects: QUANTITY: < QUANTITY-DOF > { < QUANTITY-OPERATOR > QUANTITY-ID } | { < QUANTITY-OPERATOR > QUANTITY-ID } [ EXPRESSION-CST-LIST ] with QUANTITY-ID: STRING | STRING ~ { EXPRESSION-CST } and QUANTITY-DOF: `Dof' Defines a vector of discrete quantities (vector of `D'egrees `o'f `f'reedom), to be used only in `Equation' terms of formulations to define (elementary) matrices. Roughly said, the `Dof' symbol in front of a discrete quantity indicates that this quantity is an unknown quantity, and should therefore not be considered as already computed. An `Equation' term must be linear with respect to the `Dof'. Thus, for example, a nonlinear term like Galerkin { [ f[] * Dof{T}^4 , {T} ]; ... } must first be linearized; and while Galerkin { [ f[] * Dof{T} , {T} ]; ... } Galerkin { [ -f[] * 12 , {T} ]; ... } is valid, the following, which is affine but not linear, is not: Galerkin { [ f[] * (Dof{T} - 12) , {T} ]; ... } GetDP supports two linearization techniques. The first is functional iteration (or Picard method), where one simply plugs the value obtained at the previous iteration into the nonlinear equation (the previous value is known, and is accessed e.g. with `{T}' instead `Dof{T}'). The second is the Newton-Raphson iteration, where the Jacobian is specified with a `JacNL' equation term (see `https://geuz.org/trac/getdp' for an example). `BF' Indicates that only a basis function will be used (only valid with basis functions associated with regions). QUANTITY-OPERATOR: `d' Exterior derivative (d): applied to a P-form, gives a (P+1)-form. `Grad' Gradient: applied to a scalar field, gives a vector. `Curl' `Rot' Curl: applied to a vector field, gives a vector. `Div' Divergence (div): applied to a vector field, gives a scalar. `dInv' d^(-1): applied to a p-form, gives a (p-1)-form. `GradInv' Inverse grad: applied to a gradient field, gives a scalar. `CurlInv' `RotInv' Inverse curl: applied to a curl field, gives a vector. `DivInv' Inverse div: applied to a divergence field. Notes: 1. While the operators `Grad', `Curl' and `Div' can be applied to 0, 1 and 2-forms respectively, the exterior derivative operator `d' is usually preferred with such fields. 2. The second case permits to evaluate a discretized quantity at a certain position X, Y, Z (when EXPRESSION-CST-LIST contains three items) or at a specific time, N time steps ago (when EXPRESSION-CST-LIST contains a single item). 4.11 Loops and conditionals =========================== Loops and conditionals are defined as follows, and can be imbricated: LOOP: `For ( EXPRESSION-CST : EXPRESSION-CST )' Iterates from the value of the first EXPRESSION-CST to the value of the second EXPRESSION-CST, with a unit incrementation step. At each iteration, the commands comprised between ``For ( EXPRESSION-CST : EXPRESSION-CST )'' and the matching `EndFor' are executed. `For ( EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST )' Iterates from the value of the first EXPRESSION-CST to the value of the second EXPRESSION-CST, with a positive or negative incrementation step equal to the third EXPRESSION-CST. At each iteration, the commands comprised between ``For ( EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST )'' and the matching `EndFor' are executed. `For STRING In { EXPRESSION-CST : EXPRESSION-CST }' Iterates from the value of the first EXPRESSION-CST to the value of the second EXPRESSION-CST, with a unit incrementation step. At each iteration, the value of the iterate is affected to an expression named STRING, and the commands comprised between ``For STRING In { EXPRESSION-CST : EXPRESSION-CST }'' and the matching `EndFor' are executed. `For STRING In { EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST }' Iterates from the value of the first EXPRESSION-CST to the value of the second EXPRESSION-CST, with a positive or negative incrementation step equal to the third EXPRESSION-CST. At each iteration, the value of the iterate is affected to an expression named STRING, and the commands comprised between ``For STRING In { EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST }'' and the matching `EndFor' are executed. `EndFor' Ends a matching `For' command. `If ( EXPRESSION-CST )' The body enclosed between ``If ( EXPRESSION-CST )'' and the matching `Endif' is evaluated if EXPRESSION-CST is non-zero. `EndIf' Ends a matching `If' command. Loops and conditionals can be used in any of the following objects: Group, Function, Constraint (as well as in a contraint-case), FunctionSpace, Formulation (as well as in the quantity and equation defintions), Resolution (as well as resolution-term, system defintion and operations), PostProcessing (in the definition of the PostQuantities) and PostOperation (as well as in the operation list). 5 Objects ********* This chapter presents the formal definition of the ten GetDP objects mentioned in *note Overview::. To be concise, all the possible parameters for these objects are not given here (cf. the ETC syntactic rule defined in *note Syntactic rules::). Please refer to *note Types for objects::, for the list of all available options. 5.1 `Group': defining topological entities ========================================== Meshes (grids) constitute the input data of GetDP. All that is needed by GetDP as a mesh is a file containing a list of nodes (with their coordinates) and a list of geometrical elements with, for each one, a number characterizing its geometrical type (i.e., line, triangle, quadrangle, tetrahedron, hexahedron, prism, etc.), a number characterizing the physical region to which it belongs and the list of its nodes. This minimal input set should be easy to extract from most of the classical mesh file formats (*note Input file format::, for a complete description of the mesh file format read by GetDP). Groups of geometrical entities of various types can be considered and are used in many objects. There are region groups, of which the entities are regions, and function groups, with nodes, edges, facets, volumes, groups of nodes, edges of tree, facets of tree, ... of regions. Amongst region groups, elementary and global groups can be distinguished: elementary groups are relative to single regions (e.g., physical regions in which piecewise defined functions or constraints can be defined) while global groups are relative to sets of regions for which given treatments have to be performed (e.g., domain of integration, support of a function space, etc.). Groups of function type contain lists of entities built on some region groups (e.g., nodes for nodal elements, edges for edge elements, edges of tree for gauge conditions, groups of nodes for floating potentials, elements on one side of a surface for cuts, etc.). A definition of initially empty groups can be obtained thanks to a `DefineGroup' command, so that their identifiers exist and can be referred to in other objects, even if these groups are not explicitly defined. This procedure is similar to the `DefineConstant' procedure introduced for constants in *note Constants::. The syntax for the definition of groups is: Group { < DefineGroup [ GROUP-ID <{INTEGER}> <,...> ]; > ... < GROUP-ID = GROUP-DEF; > ... < GROUP-ID += GROUP-DEF; > ... < AFFECTATION > ... < LOOP > ... } with GROUP-ID: STRING | STRING ~ { EXPRESSION-CST } GROUP-DEF: GROUP-TYPE [ GROUP-LIST <, GROUP-SUB-TYPE GROUP-LIST > ] | GROUP-ID <{}> | #GROUP-LIST GROUP-TYPE: Region | Global | NodesOf | EdgesOf | ETC GROUP-LIST: All | GROUP-LIST-ITEM | { GROUP-LIST-ITEM <,...> } GROUP-LIST-ITEM: INTEGER | INTEGER : INTEGER | INTEGER : INTEGER : INTEGER | GROUP-ID <{}> GROUP-SUB-TYPE: Not | StartingOn | OnOneSideOf | ETC Notes: 1. INTEGER as a GROUP-LIST-ITEM is the only interface with the mesh; with each element is associated a region number, being this INTEGER, and a geometrical type (*note Input file format::). Ranges of integers can be specified in the same way as ranges of constant expressions in an EXPRESSION-CST-LIST-ITEM (*note Constants::). For example, `I:J' replaces the list of consecutive integers I, I+1, ..., J-1, J. 2. Array of groups: `DefineGroup[GROUP-ID{N}]' defines the empty groups `GROUP-ID{I}', I=1, ..., n. Such a definition is optional, i.e., each `GROUP-ID{I}' can be separately defined, in any order. 3. `#GROUP-LIST' is an abbreviation of `Region[GROUP-LIST]'. See *note Types for Group::, for the complete list of options and *note Group examples::, for some examples. 5.2 `Function': defining global and piecewise expressions ========================================================= A user-defined function can be global in space or piecewise defined in region groups. A physical characteristic is an example of a piecewise defined function (e.g., magnetic permeability, electric conductivity, etc.) and can be simply a constant, for linear materials, or a function of one or several arguments for nonlinear materials. Such functions can of course depend on space coordinates or time, which can be needed to express complex constraints. A definition of initially empty functions can be made thanks to the `DefineFunction' command so that their identifiers exist and can be referred to (but cannot be used) in other objects. The syntax for the definition of functions is: Function { < DefineFunction [ FUNCTION-ID <,...> ]; > ... < FUNCTION-ID [ < GROUP-DEF > ] = EXPRESSION; > ... < AFFECTATION > ... < LOOP > ... } with FUNCTION-ID: STRING Note: 1. The optional GROUP-DEF in brackets must be of `Region' type, and indicates on which region the (piecewise) function is defined. Warning: it is incorrect to write `f[reg1]=1; g[reg2]=f[]+1;' since the domains of definition of `f[]' and `g[]' don't match. 2. One can also define initially empty functions inline by replacing the expression with `***'. See *note Types for Function::, for the complete list of built-in functions and *note Function examples::, for some examples. 5.3 `Constraint': specifying constraints on function spaces and formulations ============================================================================ Constraints can be referred to in `FunctionSpace' objects to be used for boundary conditions, to impose global quantities or to initialize quantities. These constraints can be expressed with functions or be imposed by the pre-resolution of another discrete problem. Other constraints can also be defined, e.g., constraints of network type for the definition of circuit connections, to be used in `Formulation' objects. The syntax for the definition of constraints is: Constraint { { Name CONSTRAINT-ID; Type CONSTRAINT-TYPE; Case { { Region GROUP-DEF; < Type CONSTRAINT-TYPE; > < SubRegion GROUP-DEF; > < TimeFunction EXPRESSION; > < RegionRef GROUP-DEF; > < SubRegionRef GROUP-DEF; > < Coefficient EXPRESSION; > < Function EXPRESSION; > < Filter EXPRESSION; > CONSTRAINT-VAL; } ... < LOOP > ... } | Case CONSTRAINT-CASE-ID { { Region GROUP-DEF; < Type CONSTRAINT-TYPE; > CONSTRAINT-CASE-VAL; } ... < LOOP > ... } ... } ... < AFFECTATION > ... < LOOP > ... } with CONSTRAINT-ID: CONSTRAINT-CASE-ID: STRING | STRING ~ { EXPRESSION-CST } CONSTRAINT-TYPE: Assign | Init | Network | Link | ETC CONSTRAINT-VAL: Value EXPRESSION | NameOfResolution RESOLUTION-ID | ETC CONSTRAINT-CASE-VAL: Branch { INTEGER, INTEGER } | ETC Notes: 1. The constraint type CONSTRAINT-TYPE defined outside the `Case' fields is applied to all the cases of the constraint, unless other types are explicitly given in these cases. The default type is `Assign'. 2. The region type `Region GROUP-DEF' will be the main GROUP-LIST argument of the GROUP-DEF to be built for the constraints of `FunctionSpace's. The optional region type `SubRegion GROUP-DEF' will be the argument of the associated GROUP-SUB-TYPE. 3. EXPRESSION in `Value' of CONSTRAINT-VAL cannot be time dependent (`$Time') because it is evaluated only once during the pre-processing (for efficiency reasons). Time dependences must be defined in `TimeFunction EXPRESSION'. See *note Types for Constraint::, for the complete list of options and *note Constraint examples::, for some examples. 5.4 `FunctionSpace': building function spaces ============================================= A `FunctionSpace' is characterized by the type of its interpolated fields, one or several basis functions and optional constraints (in space and time). Subspaces of a function space can be defined (e.g., for the use with hierarchical elements), as well as direct associations of global quantities (e.g., floating potential, electric charge, current, voltage, magnetomotive force, etc.). A key point is that basis functions are defined by any number of subsets of functions, being added. Each subset is characterized by associated built-in functions for evaluation, a support of definition and a set of associated supporting geometrical entities (e.g., nodes, edges, facets, volumes, groups of nodes, edges incident to a node, etc.). The freedom in defining various kinds of basis functions associated with different geometrical entities to interpolate a field permits to build made-to-measure function spaces adapted to a wide variety of field approximations (*note FunctionSpace examples::). The syntax for the definition of function spaces is: FunctionSpace { { Name FUNCTION-SPACE-ID; Type FUNCTION-SPACE-TYPE; BasisFunction { { Name BASIS-FUNCTION-ID; NameOfCoef COEF-ID; Function BASIS-FUNCTION-TYPE < { Quantity QUANTITY-ID; Formulation FORMULATION-ID {#INTEGER}; Group GROUP-DEF; Resolution RESOLUTION-ID {} } >; Support GROUP-DEF; Entity GROUP-DEF; } ... } < SubSpace { { Name SUB-SPACE-ID; NameOfBasisFunction BASIS-FUNCTION-LIST; } ... } > < GlobalQuantity { { Name GLOBAL-QUANTITY-ID; Type GLOBAL-QUANTITY-TYPE; NameOfCoef COEF-ID; } ... } > < Constraint { { NameOfCoef COEF-ID; EntityType GROUP-TYPE; < EntitySubType GROUP-SUB-TYPE; > NameOfConstraint CONSTRAINT-ID <{}>; } ... } > } ... < AFFECTATION > ... < LOOP > ... } with FUNCTION-SPACE-ID: FORMULATION-ID: RESOLUTION-ID: STRING | STRING ~ { EXPRESSION-CST } BASIS-FUNCTION-ID: COEF-ID: SUB-SPACE-ID: GLOBAL-QUANTITY-ID: STRING FUNCTION-SPACE-TYPE: Scalar | Vector | Form0 | Form1 | ETC BASIS-FUNCTION-TYPE: BF_Node | BF_Edge | ETC BASIS-FUNCTION-LIST: BASIS-FUNCTION-ID | { BASIS-FUNCTION-ID <,...> } GLOBAL-QUANTITY-TYPE: AliasOf | AssociatedWith Notes: 1. When the definition region of a function type group used as an `Entity' of a `BasisFunction' is the same as that of the associated `Support', it is replaced by `All' for more efficient treatments during the computation process (this prevents the construction and the analysis of a list of geometrical entities). 2. Piecewise defined basis functions: the same `Name' for several `BasisFunction' fields permits to define piecewise basis functions; separate `NameOfCoef's must be defined for those fields. 3. Constraint: a constraint is associated with geometrical entities defined by an automatically created `Group' of type GROUP-TYPE, using the `Region' defined in a `Constraint' object as its main argument, and the optional `SubRegion' in the same object as a GROUP-SUB-TYPE argument. 4. Function: a global basis function (`BF_Global' or `BF_dGlobal') needs parameters, i.e., it is given by the quantity (QUANTITY-ID) pre-computed from multiresolutions performed on multiformulations. See *note Types for FunctionSpace::, for the complete list of options and *note FunctionSpace examples::, for some examples. 5.5 `Jacobian': defining jacobian methods ========================================= Jacobian methods can be referred to in `Formulation' and `PostProcessing' objects to be used in the computation of integral terms and for changes of coordinates. They are based on `Group' objects and define the geometrical transformations applied to the reference elements (i.e., lines, triangles, quadrangles, tetrahedra, prisms, hexahedra, etc.). Besides the classical lineic, surfacic and volume Jacobians, the `Jacobian' object allows the construction of various transformation methods (e.g., infinite transformations for unbounded domains) thanks to dedicated jacobian methods. The syntax for the definition of Jacobian methods is: Jacobian { { Name JACOBIAN-ID; Case { { Region GROUP-DEF | All; Jacobian JACOBIAN-TYPE < { EXPRESSION-CST-LIST } >; } ... } } ... } with JACOBIAN-ID: STRING JACOBIAN-TYPE: Vol | Sur | VolAxi | ETC Note: 1. The default case of a `Jacobian' object is defined by `Region All' and must follow all the other cases. See *note Types for Jacobian::, for the complete list of options and *note Jacobian examples::, for some examples. 5.6 `Integration': defining integration methods =============================================== Various numerical or analytical integration methods can be referred to in `Formulation' and `PostProcessing' objects to be used in the computation of integral terms, each with a set of particular options (number of integration points for quadrature methods--which can be linked to an error criterion for adaptative methods, definition of transformations for singular integrations, etc.). Moreover, a choice can be made between several integration methods according to a criterion (e.g., on the proximity between the source and computation points in integral formulations). The syntax for the definition of integration methods is: Integration { { Name INTEGRATION-ID; < Criterion EXPRESSION; > Case { < { Type INTEGRATION-TYPE; Case { { GeoElement ELEMENT-TYPE; NumberOfPoints EXPRESSION-CST } ... } } ... > < { Type Analytic; } ... > } } ... } with INTEGRATION-ID: STRING INTEGRATION-TYPE: Gauss | ETC ELEMENT-TYPE: Line | Triangle | Tetrahedron ETC See *note Types for Integration::, for the complete list of options and *note Integration examples::, for some examples. 5.7 `Formulation': building equations ===================================== The `Formulation' tool permits to deal with volume, surface and line integrals with many kinds of densities to integrate, written in a form that is similar to their symbolic expressions (it uses the same EXPRESSION syntax as elsewhere in GetDP), which therefore permits to directly take into account various kinds of elementary matrices (e.g., with scalar or cross products, anisotropies, nonlinearities, time derivatives, various test functions, etc.). In case nonlinear physical characteristics are considered, arguments are used for associated functions. In that way, many formulations can be directly written in the data file, as they are written symbolically. Fields involved in each formulation are declared as belonging to beforehand defined function spaces. The uncoupling between formulations and function spaces allows to maintain a generality in both their definitions. A `Formulation' is characterized by its type, the involved quantities (of local, global or integral type) and a list of equation terms. Global equations can also be considered, e.g., for the coupling with network relations. The syntax for the definition of formulations is: Formulation { { Name FORMULATION-ID; Type FORMULATION-TYPE; Quantity { { Name QUANTITY-ID; Type QUANTITY-TYPE; NameOfSpace FUNCTION-SPACE-ID <{}> < [ SUB-SPACE-ID | GLOBAL-QUANTITY-ID ] >; < Symmetry EXPRESSION-CST; > < [ EXPRESSION ]; In GROUP-DEF; Jacobian JACOBIAN-ID; Integration INTEGRATION-ID; > < IndexOfSystem INTEGER; > } ... } Equation { < LOCAL-TERM-TYPE { < TERM-OP-TYPE > [ EXPRESSION, EXPRESSION ]; In GROUP-DEF; Jacobian JACOBIAN-ID; Integration INTEGRATION-ID; } > ... < GlobalTerm { < TERM-OP-TYPE > [ EXPRESSION, EXPRESSION ]; In GROUP-DEF; } > ... < GlobalEquation { Type Network; NameOfConstraint CONSTRAINT-ID; { Node EXPRESSION; Loop EXPRESSION; Equation EXPRESSION; In GROUP-DEF; } ... } > ... < AFFECTATION > ... < LOOP > ... } } ... < AFFECTATION > ... < LOOP > ... } with FORMULATION-ID: STRING | STRING ~ { EXPRESSION-CST } FORMULATION-TYPE: FemEquation | ETC LOCAL-TERM-TYPE: Galerkin | deRham QUANTITY-TYPE: Local | Global | Integral TERM-OP-TYPE: Dt | DtDt | JacNL | ETC Note: 1. `IndexOfSystem' permits to resolve ambiguous cases when several quantities belong to the same function space, but to different systems of equations. The INTEGER parameter then specifies the index in the list of an `OriginSystem' command (*note Resolution::). 2. A `GlobalTerm' defines a term to be assembled in an equation associated with a global quantity. This equation is a finite element equation if that global quantity is linked with local quantities. 3. A `GlobalEquation' defines a global equation to be assembled in the matrix of the system. See *note Types for Formulation::, for the complete list of options and *note Formulation examples::, for some examples. 5.8 `Resolution': solving systems of equations ============================================== The operations available in a `Resolution' include: the generation of a linear system, its solving with various kinds of linear solvers, the saving of the solution or its transfer to another system, the definition of various time stepping methods, the construction of iterative loops for nonlinear problems (Newton-Raphson and fixed point methods), etc. Multi-harmonic resolutions, coupled problems (e.g., magneto-thermal) or linked problems (e.g., pre-computations of source fields) are thus easily defined in GetDP. The `Resolution' object is characterized by a list of systems to build and their associated formulations, using time or frequency domain, and a list of elementary operations: Resolution { { Name RESOLUTION-ID; System { { Name SYSTEM-ID; NameOfFormulation FORMULATION-LIST; < Type SYSTEM-TYPE; > < Frequency EXPRESSION-CST-LIST-ITEM | Frequency { EXPRESSION-CST-LIST }; > < DestinationSystem SYSTEM-ID; > < OriginSystem SYSTEM-ID; | OriginSystem { SYSTEM-ID <,...> }; > < NameOfMesh EXPRESSION-CHAR > < Solver EXPRESSION-CHAR > < LOOP > } ... < LOOP > ... } Operation { < RESOLUTION-OP; > ... < LOOP > ... } } ... < AFFECTATION > ... < LOOP > ... } with RESOLUTION-ID: SYSTEM-ID: STRING | STRING ~ { EXPRESSION-CST } FORMULATION-LIST: FORMULATION-ID <{}> | { FORMULATION-ID <{}> <,...> } SYSTEM-TYPE: Real | Complex RESOLUTION-OP: Generate[SYSTEM-ID] | Solve[SYSTEM-ID] | ETC Notes: 1. The default type for a system of equations is `Real'. A frequency domain analysis is defined through the definition of one or several frequencies (`Frequency EXPRESSION-CST-LIST-ITEM | Frequency { EXPRESSION-CST-LIST }'). Complex systems of equations with no predefined list of frequencies (e.g., in modal analyses) can be explicitely defined with `Type Complex'. 2. `NameOfMesh' permits to explicitely specify the mesh to be used for the construction of the system of equations. 3. `Solver' permits to explicitely specify the name of the solver parameter file to use for the solving of the system of equations. This is ony valid if GetDP was compiled against the default solver library (it is the case if you downloaded a pre-compiled copy of GetDP from the internet). 4. `DestinationSystem' permits to specify the destination system of a `TransferSolution' operation (*note Types for Resolution::). 5. `OriginSystem' permits to specify the systems from which ambiguous quantity definitions can be solved (*note Formulation::). See *note Types for Resolution::, for the complete list of options and *note Resolution examples::, for some examples. 5.9 `PostProcessing': exploiting computational results ====================================================== The `PostProcessing' object is based on the quantities defined in a `Formulation' and permits the construction (thanks to the EXPRESSION syntax) of any useful piecewise defined quantity of interest: PostProcessing { { Name POST-PROCESSING-ID; NameOfFormulation FORMULATION-ID <{}>; < NameOfSystem SYSTEM-ID; > Quantity { { Name POST-QUANTITY-ID; Value { POST-VALUE ... } } ... < LOOP > ... } } ... < AFFECTATION > ... < LOOP > ... } with POST-PROCESSING-ID: POST-QUANTITY-ID: STRING | STRING ~ { EXPRESSION-CST } POST-VALUE: Local { LOCAL-VALUE } | Integral { INTEGRAL-VALUE } LOCAL-VALUE: [ EXPRESSION ]; In GROUP-DEF; Jacobian JACOBIAN-ID; INTEGRAL-VALUE: [ EXPRESSION ]; In GROUP-DEF; Integration INTEGRATION-ID; Jacobian JACOBIAN-ID; Notes: 1. The quantity defined with INTEGRAL-VALUE is piecewise defined over the elements of the mesh of GROUP-DEF, and takes, in each element, the value of the integration of EXPRESSION over this element. The global integral of EXPRESSION over a whole region (being either GROUP-DEF or a subset of GROUP-DEF) has to be defined in the `PostOperation' with the `POST-QUANTITY-ID[GROUP-DEF]' command (*note PostOperation::). 2. If `NameOfSystem SYSTEM-ID' is not given, the system is automatically selected as the one to which the first quantity listed in the `Quantity' field of FORMULATION-ID is associated. See *note Types for PostProcessing::, for the complete list of options and *note PostProcessing examples::, for some examples. 5.10 `PostOperation': exporting results ======================================= The `PostOperation' is the bridge between results obtained with GetDP and the external world. It defines several elementary operations on `PostProcessing' quantities (e.g., plot on a region, section on a user-defined plane, etc.), and outputs the results in several file formats. PostOperation { { Name POST-OPERATION-ID; NameOfPostProcessing POST-PROCESSING-ID; < Format POST-OPERATION-FMT; > < Append EXPRESSION-CHAR; > Operation { < POST-OPERATION-OP; > ... } } ... < AFFECTATION > ... < LOOP > ... } | PostOperation POST-OPERATION-ID UsingPost POST-PROCESSING-ID { < POST-OPERATION-OP; > ... } ... with POST-OPERATION-ID: STRING | STRING ~ { EXPRESSION-CST } POST-OPERATION-OP: Print[ POST-QUANTITY-TERM, PRINT-SUPPORT <,PRINT-OPTION> ... ] | Print[ "STRING", EXPRESSION <,PRINT-OPTION> ... ] | Print[ "STRING", Str[ EXPRESSION-CHAR ] <,PRINT-OPTION> ... ] | Echo[ "STRING" <,PRINT-OPTION> ... ] | PrintGroup[ GROUP-ID, PRINT-SUPPORT <,PRINT-OPTION> ... ] | < LOOP > ... ETC POST-QUANTITY-TERM: POST-QUANTITY-ID <[GROUP-DEF]> | POST-QUANTITY-ID POST-QUANTITY-OP POST-QUANTITY-ID[GROUP-DEF] | POST-QUANTITY-ID[GROUP-DEF] POST-QUANTITY-OP POST-QUANTITY-ID POST-QUANTITY-OP: + | - | * | / PRINT-SUPPORT: OnElementsOf GROUP-DEF | OnRegion GROUP-DEF | OnGlobal | ETC PRINT-OPTION: File EXPRESSION-CHAR | Format POST-OPERATION-FMT | ETC POST-OPERATION-FMT: Table | TimeTable | ETC Notes: 1. Both `PostOperation' syntaxes are equivalent. The first one conforms to the overall interface, but the second one is more concise. 2. The format POST-OPERATION-FMT defined outside the `Operation' field is applied to all the post-processing operations, unless other formats are explicitly given in these operations with the `Format' option (*note Types for PostOperation::). The default format is `Gmsh'. 3. The optional argument `[GROUP-DEF]' of the POST-QUANTITY-ID can only be used when this quantity has been defined as an INTEGRAL-VALUE (*note PostProcessing::). In this case, the sum of all elementary integrals is performed over the region GROUP-DEF. 4. The POST-QUANTITY-OP allows the simple combination of space-dependent quantities (POST-QUANTITY-ID) with global integral quantities (`POST-QUANTITY-ID[GROUP-DEF]'). See *note Types for PostOperation::, for the complete list of options and *note PostOperation examples::, for some examples. 6 Types for objects ******************* This chapter presents the complete list of choices associated with metasyntactic variables introduced for the ten GetDP objects. 6.1 Types for `Group' ===================== Types in GROUP-TYPE [ R1 <, GROUP-SUB-TYPE R2 > ] `GROUP-TYPE < GROUP-SUB-TYPE >': `Region' Regions in R1. `Global' Regions in R1 (variant of `Region' used with global `BasisFunction's `BF_Global' and `BF_dGlobal'). `NodesOf' Nodes of elements of R1 < `Not': but not those of R2 >. `EdgesOf' Edges of elements of R1 < `Not': but not those of R2 >. `FacetsOf' Facets of elements of R1 < `Not': but not those of R2 >. `VolumesOf' Volumes of elements of R1 < `Not': but not those of R2 >. `ElementsOf' Elements of regions in R1 < `OnOneSideOf': only elements on one side of R2) >. `GroupsOfNodesOf' Groups of nodes of elements of R1 (a group is associated with each region). `GroupsOfEdgesOf' Groups of edges of elements of R1 (a group is associated with each region). < `InSupport': in a support R2 being a group of type `ElementOf', i.e., containing elements >. `GroupsOfEdgesOnNodesOf' Groups of edges incident to nodes of elements of R1 (a group is associated with each node). < `Not': but not those of R2) >. `EdgesOfTreeIn' Edges of a tree of edges of R1 < `StartingOn': a complete tree is first built on R2 >. `FacetsOfTreeIn' Facets of a tree of facets of R1 < `StartingOn': a complete tree is first built on R2 >. `DualNodesOf' Dual nodes of elements of R1. `DualEdgesOf' Dual edges of elements of R1. `DualFacetsOf' Dual facets of elements of R1. `DualVolumesOf' Dual volumes of elements of R1. 6.2 Types for `Function' ======================== 6.2.1 Math functions -------------------- The following functions are the equivalent of the functions of the C math library, and always return real-valued expressions. These are the only functions allowed in constant expressions (EXPRESSION-CST, see *note Constants::). MATH-FUNCTION-ID: `Exp' `[EXPRESSION]' Exponential function: e^EXPRESSION. `Log' `[EXPRESSION]' Natural logarithm: ln(EXPRESSION), EXPRESSION>0. `Log10' `[EXPRESSION]' Base 10 logarithm: log10(EXPRESSION), EXPRESSION>0. `Sqrt' `[EXPRESSION]' Square root, EXPRESSION>=0. `Sin' `[EXPRESSION]' Sine of EXPRESSION. `Asin' `[EXPRESSION]' Arc sine (inverse sine) of EXPRESSION in [-Pi/2,Pi/2], EXPRESSION in [-1,1]. `Cos' `[EXPRESSION]' Cosine of EXPRESSION. `Acos' `[EXPRESSION]' Arc cosine (inverse cosine) of EXPRESSION in [0,Pi], EXPRESSION in [-1,1]. `Tan' `[EXPRESSION]' Tangent of EXPRESSION. `Atan' `[EXPRESSION]' Arc tangent (inverse tangent) of EXPRESSION in [-Pi/2,Pi/2]. `Atan2' `[EXPRESSION,EXPRESSION]' Arc tangent (inverse tangent) of the first EXPRESSION divided by the second, in [-Pi,Pi]. `Sinh' `[EXPRESSION]' Hyperbolic sine of EXPRESSION. `Cosh' `[EXPRESSION]' Hyperbolic cosine of EXPRESSION. `Tanh' `[EXPRESSION]' Hyperbolic tangent of EXPRESSION. `Fabs' `[EXPRESSION]' Absolute value of EXPRESSION. `Fmod' `[EXPRESSION,EXPRESSION]' Remainder of the division of the first EXPRESSION by the second, with the sign of the first. 6.2.2 Extended math functions ----------------------------- EXTENDED-MATH-FUNCTION-ID: `Cross' `[EXPRESSION,EXPRESSION]' Cross product of the two arguments; EXPRESSION must be a vector. `Hypot' `[EXPRESSION,EXPRESSION]' Square root of the sum of the squares of its arguments. `Norm' `[EXPRESSION]' Absolute value if EXPRESSION is a scalar; euclidian norm if EXPRESSION is a vector. `SquNorm' `[EXPRESSION]' Square norm: `Norm[EXPRESSION]^2'. `Unit' `[EXPRESSION]' Normalization: `EXPRESSION/Norm[EXPRESSION]'. Returns 0 if the norm is smaller than 1.e-30. `Transpose' `[EXPRESSION]' Transposition; EXPRESSION must be a tensor. `TTrace' `[EXPRESSION]' Trace; EXPRESSION must be a tensor. `F_Cos_wt_p' `[]{EXPRESSION-CST,EXPRESSION-CST}' The first parameter represents the angular frequency and the second represents the phase. If the type of the current system is `Real', `F_Cos_wt_p[]{w,p}' is identical to `Cos[w*$Time+p]'. If the type of the current system is `Complex', it is identical to `Complex[Cos[w],Sin[w]]'. `F_Sin_wt_p' `[]{EXPRESSION-CST,EXPRESSION-CST}' The first parameter represents the angular frequency and the second represents the phase. If the type of the current system is `Real', `F_Sin_wt_p[]{w,p}' is identical to `Sin[w*$Time+p]'. If the type of the current system is `Complex', it is identical to `Complex[Sin[w],-Cos[w]]'. `F_Period' `[EXPRESSION]{EXPRESSION-CST}' `Fmod[EXPRESSION,EXPRESSION-CST]' `+' `(EXPRESSION<0 ? EXPRESSION-CST : 0)'; the result is always in [0,EXPRESSION-CST[. 6.2.3 Green functions --------------------- The Green functions are only used in integral quantities (*note Formulation::). The first parameter represents the dimension of the problem: * `1D': `r = Fabs[$X-$XS]' * `2D': `r = Sqrt[($X-$XS)^2+($Y-$YS)^2]' * `3D': `r = Sqrt[($X-$XS)^2+($Y-$YS)^2+($Z-$ZS)^2]' The triplets of values given in the definitions below correspond to the `1D', `2D' and `3D' cases. GREEN-FUNCTION-ID: `Laplace' `[]{EXPRESSION-CST}' `r/2', `1/(2*Pi)*ln(1/r)', `1/(4*Pi*r)'. `GradLaplace' `[]{EXPRESSION-CST}' Gradient of `Laplace' relative to the destination point (`$X', `$Y', `$Z'). `Helmholtz' `[]{EXPRESSION-CST, EXPRESSION-CST}' `exp(j*k0*r)/(4*Pi*r)', where `k0' is given by the second parameter. `GradHelmholtz' `[]{EXPRESSION-CST, EXPRESSION-CST}' Gradient of `Helmholtz' relative to the destination point (`$X', `$Y', `$Z'). 6.2.4 Type manipulation functions --------------------------------- TYPE-FUNCTION-ID: `Complex' `[EXPRESSION-LIST]' Creates a (multi-harmonic) complex expression from an number of real-valued expressions. The number of expressions in EXPRESSION-LIST must be even. `Re' `[EXPRESSION]' Takes the real part of a complex-valued expression. `Im' `[EXPRESSION]' Takes the imaginary part of a complex-valued expression. `Vector' `[EXPRESSION,EXPRESSION,EXPRESSION]' Creates a vector from 3 scalars. `Tensor' `[EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION,' `EXPRESSION,EXPRESSION,EXPRESSION]' Creates a second-rank tensor of order 3 from 9 scalars. `TensorV' `[EXPRESSION,EXPRESSION,EXPRESSION]' Creates a second-rank tensor of order 3 from 3 vectors. `TensorSym' `[EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION,EXPRESSION]' Creates a symmetrical second-rank tensor of order 3 from 6 scalars. `TensorDiag' `[EXPRESSION,EXPRESSION,EXPRESSION]' Creates a diagonal second-rank tensor of order 3 from 3 scalars. `CompX' `[EXPRESSION]' Gets the X component of a vector. `CompY' `[EXPRESSION]' Gets the Y component of a vector. `CompZ' `[EXPRESSION]' Gets the Z component of a vector. `CompXX' `[EXPRESSION]' Gets the XX component of a tensor. `CompXY' `[EXPRESSION]' Gets the XY component of a tensor. `CompXZ' `[EXPRESSION]' Gets the XZ component of a tensor. `CompYX' `[EXPRESSION]' Gets the YX component of a tensor. `CompYY' `[EXPRESSION]' Gets the YY component of a tensor. `CompYZ' `[EXPRESSION]' Gets the YZ component of a tensor. `CompZX' `[EXPRESSION]' Gets the ZX component of a tensor. `CompZY' `[EXPRESSION]' Gets the ZY component of a tensor. `CompZZ' `[EXPRESSION]' Gets the ZZ component of a tensor. 6.2.5 Coordinate functions -------------------------- COORD-FUNCTION-ID: `X' `[]' Gets the X coordinate. `Y' `[]' Gets the Y coordinate. `Z' `[]' Gets the Z coordinate. `XYZ' `[]' Gets X, Y and Z in a vector. 6.2.6 Miscellaneous functions ----------------------------- MISC-FUNCTION-ID: `Printf' `[EXPRESSION]' Prints the value of EXPRESSION when evaluated. `Rand' `[EXPRESSION]' Returns a pseudo-random number in [0, EXPRESSION]. `Normal' `[]' Computes the normal to the element. `NormalSource' `[]' Computes the normal to the source element (only valid in a quantity of Integral type). `F_CompElementNum' `[]' Returns 0 if the current element and the current source element are identical. `InterpolationLinear' `[]{EXPRESSION-CST-LIST}' Linear interpolation of points. The number of constant expressions in EXPRESSION-CST-LIST must be even. `dInterpolationLinear' `[]{EXPRESSION-CST-LIST}' Derivative of linear interpolation of points. The number of constant expressions in EXPRESSION-CST-LIST must be even. `InterpolationAkima' `[]{EXPRESSION-CST-LIST}' Akima interpolation of points. The number of constant expressions in EXPRESSION-CST-LIST must be even. `dInterpolationAkima' `[]{EXPRESSION-CST-LIST}' Derivative of Akima interpolation of points. The number of constant expressions in EXPRESSION-CST-LIST must be even. `Order' `[QUANTITY]' Returns the interpolation order of the QUANTITY. 6.3 Types for `Constraint' ========================== CONSTRAINT-TYPE: `Assign' To assign a value (e.g., for boundary condition). `Init' To give an initial value (e.g., initial value in a time analysis). `AssignFromResolution' To assign a value to be computed by a pre-resolution. `InitFromResolution' To give an initial value to be computed by a pre-resolution. `Network' To describe the node connections of branches in a network. `Link' To define links between degrees of freedom in the constrained region with degrees of freedom in a "reference" region, with some coefficient. For example, to link the degrees of freedom in the contrained region `Left' with the degrees of freedom in the reference region `Right', located 1 unit to the right of the region `Left' along the X-axis, with the coeficient `-1', one could write: { Name periodic; Case { { Region Left; Type Link ; RegionRef Right; Coefficient -1; Function Vector[$X+Pi,$Y,$Z] ; } } } In this example, `Function' defines the mapping that translates the geometrical elements in the region `Left' by 1 along the X-axis, so that they correspond with the elements in the region `Right'. For this mapping to work, the meshes of `Left' and `Right' must be identical. `LinkCplx' To define complex-valued links between degrees of freedom. The syntax is the same as for constraints of type `Link', but `Coeficient' can be complex. 6.4 Types for `FunctionSpace' ============================= FUNCTION-SPACE-TYPE: `Form0' 0-form, i.e., scalar field of potential type. `Form1' 1-form, i.e., curl-conform field (associated with a curl). `Form2' 2-form, i.e., div-conform field (associated with a divergence). `Form3' 3-form, i.e., scalar field of density type. `Form1P' 1-form perpendicular to the Z=0 plane, i.e., perpendicular curl-conform field (associated with a curl). `Form2P' 2-form in the Z=0 plane, i.e., parallel div-conform field (associated with a divergence). `Scalar' Scalar field. `Vector' Vector field. BASIS-FUNCTION-TYPE: `BF_Node' Nodal function (on `NodesOf', value `Form0'). `BF_Edge' Edge function (on `EdgesOf', value `Form1'). `BF_Facet' Facet function (on `FacetsOf', value `Form2'). `BF_Volume' Volume function (on `VolumesOf', value `Form3'). `BF_GradNode' Gradient of nodal function (on `NodesOf', value `Form1'). `BF_CurlEdge' Curl of edge function (on `EdgesOf', value `Form2'). `BF_DivFacet' Divergence of facet function (on `FacetsOf', value `Form3'). `BF_GroupOfNodes' Sum of nodal functions (on `GroupsOfNodesOf', value `Form0'). `BF_GradGroupOfNodes' Gradient of sum of nodal functions (on `GroupsOfNodesOf', value `Form1'). `BF_GroupOfEdges' Sum of edge functions (on `GroupsOfEdgesOf', value `Form1'). `BF_CurlGroupOfEdges' Curl of sum of edge functions (on `GroupsOfEdgesOf', value `Form2'). `BF_PerpendicularEdge' 1-form (0, 0, `BF_Node') (on `NodesOf', value `Form1P'). `BF_CurlPerpendicularEdge' Curl of 1-form (0, 0, `BF_Node') (on `NodesOf', value `Form2P'). `BF_GroupOfPerpendicularEdge' Sum of 1-forms (0, 0, `BF_Node') (on `NodesOf', value `Form1P'). `BF_CurlGroupOfPerpendicularEdge' Curl of sum of 1-forms (0, 0, `BF_Node') (on `NodesOf', value `Form2P'). `BF_PerpendicularFacet' 2-form (90 degree rotation of `BF_Edge') (on `EdgesOf', value `Form2P'). `BF_DivPerpendicularFacet' Div of 2-form (90 degree rotation of `BF_Edge') (on `EdgesOf', value `Form3'). `BF_Region' Unit value 1 (on `Region', value `Scalar'). `BF_RegionX' Unit vector (1, 0, 0) (on `Region', value `Vector'). `BF_RegionY' Unit vector (0, 1, 0) (on `Region', value `Vector'). `BF_RegionZ' Unit vector (0, 0, 1) (on `Region', value `Vector'). `BF_Global' Global pre-computed quantity (on `Global', value depends on parameters). `BF_dGlobal' Exterior derivative of global pre-computed quantity (on `Global', value depends on parameters). `BF_NodeX' Vector (`BF_Node', 0, 0) (on `NodesOf', value `Vector'). `BF_NodeY' Vector (0, `BF_Node', 0) (on `NodesOf', value `Vector'). `BF_NodeZ' Vector (0, 0, `BF_Node') (on `NodesOf', value `Vector'). `BF_Zero' Zero value 0 (on all regions, value `Scalar'). `BF_One' Unit value 1 (on all regions, value `Scalar'). GLOBAL-QUANTITY-TYPE: `AliasOf' Another name for a name of coefficient of basis function. `AssociatedWith' A global quantity associated with a name of coefficient of basis function, and therefore with this basis function. 6.5 Types for `Jacobian' ======================== JACOBIAN-TYPE: `Vol' Volume Jacobian, for N-D regions in N-D geometries, N = 1, 2 or 3. `Sur' Surface Jacobian, for (N-1)-D regions in N-D geometries, N = 1, 2 or 3. `Lin' Line Jacobian, for (N-2)-D regions in N-D geometries, N = 2 or 3. `VolAxi' Axisymmetrical volume Jacobian (1st type: r), for 2-D regions in axisymmetrical geometries. `SurAxi' Axisymmetrical surface Jacobian (1st type: r), for 1-D regions in axisymmetrical geometries. `VolAxiSqu' Axisymmetrical volume Jacobian (2nd type: r^2), for 2-D regions in axisymmetrical geometries. `VolSphShell' Volume Jacobian with spherical shell transformation, for N-D regions in N-D geometries, N = 2 or 3. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. `VolAxiSphShell' Same as `VolAxi', but with spherical shell transformation. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. `VolAxiSquSphShell' Same as `VolAxiSqu', but with spherical shell transformation. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. `VolRectShell' Volume Jacobian with rectangular shell transformation, for N-D regions in N-D geometries, N = 2 or 3. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, DIRECTION, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. `VolAxiRectShell' Same as `VolAxi', but with rectangular shell transformation. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, DIRECTION, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. `VolAxiSquRectShell' Same as `VolAxiSqu', but with rectangular shell transformation. Parameters: RADIUS-INTERNAL, RADIUS-EXTERNAL <, DIRECTION, CENTER-X, CENTER-Y, CENTER-Z, POWER, 1/INFINITY >. 6.6 Types for `Integration' =========================== INTEGRATION-TYPE: `Gauss' Numerical Gauss integration. `GaussLegendre' Numerical Gauss integration obtained by application of a multiplicative rule on the one-dimensional Gauss integration. ELEMENT-TYPE: `Line' Line (2 nodes, 1 edge, 1 volume) (#1). `Triangle' Triangle (3 nodes, 3 edges, 1 facet, 1 volume) (#2). `Quadrangle' Quadrangle (4 nodes, 4 edges, 1 facet, 1 volume) (#3). `Tetrahedron' Tetrahedron (4 nodes, 6 edges, 4 facets, 1 volume) (#4). `Hexahedron' Hexahedron (8 nodes, 12 edges, 6 facets, 1 volume) (#5). `Prism' Prism (6 nodes, 9 edges, 5 facets, 1 volume) (#6). `Pyramid' Pyramid (5 nodes, 8 edges, 5 facets, 1 volume) (#7). `Point' Point (1 node) (#15). Note: 1. N in (#N) is the type number of the element (*note Input file format::). 6.7 Types for `Formulation' =========================== FORMULATION-TYPE: `FemEquation' Finite element method formulation (all methods of moments, integral methods). LOCAL-TERM-TYPE: `Galerkin' Integral of Galerkin type. `deRham' deRham projection (collocation). QUANTITY-TYPE: `Local' Local quantity defining a field in a function space. In case a subspace is considered, its identifier has to be given between the brackets following the `NameOfSpace FUNCTION-SPACE-ID'. `Global' Global quantity defining a global quantity from a function space. The identifier of this quantity has to be given between the brackets following the `NameOfSpace FUNCTION-SPACE-ID'. `Integral' Integral quantity obtained by the integration of a `LocalQuantity' before its use in an `Equation' term. TERM-OP-TYPE: `Dt' Time derivative applied to the whole term of the equation. `DtDof' Time derivative applied only to the `Dof{}' term of the equation. `DtDt' Time derivative of 2nd order applied to the whole term of the equation. `DtDtDof' Time derivative of 2nd order applied only to the `Dof{}' term of the equation. `JacNL' Jacobian term to be assembled in the Jacobian matrix for nonlinear analysis. `NeverDt' No time scheme applied to the term (e.g., Theta is always 1 even if a theta scheme is applied). 6.8 Types for `Resolution' ========================== RESOLUTION-OP: `Generate' `[SYSTEM-ID]' Generate the system of equations SYSTEM-ID. `Solve' `[SYSTEM-ID]' Solve the system of equations SYSTEM-ID. `GenerateJac' `[SYSTEM-ID]' Generate the system of equations SYSTEM-ID using a jacobian matrix (of which the unknowns are corrections DX of the current solution X). `SolveJac' `[SYSTEM-ID]' Solve the system of equations SYSTEM-ID using a jacobian matrix (of which the unknowns are corrections DX of the current solution X). Then, Increment the solution (X=X+DX) and compute the relative error DX/X. `GenerateSeparate' `[SYSTEM-ID]' Generate iteration matrices separately for system SYSTEM-ID. It is destined to be used with `Update' in order to create more efficiently the actual system to solve (this is only useful in linear transient problems with one single excitation) or with `EigenSolve' in order to generate the matrices of a (generalized) eigenvalue problem. `GenerateOnly' `[SYSTEM-ID, EXPRESSION-CST-LIST]' Not documented yet. `GenerateOnlyJac' `[SYSTEM-ID, EXPRESSION-CST-LIST]' Not documented yet. `Update' `[SYSTEM-ID, EXPRESSION]' Update the system of equations SYSTEM-ID (built from iteration matrices generated separately with `GenerateSeparate') with EXPRESSION `UpdateConstraint' `[SYSTEM-ID, GROUP-ID, CONSTRAINT-TYPE]' Not documented yet. `InitSolution' `[SYSTEM-ID]' Initialize the solution of SYSTEM-ID to zero (default) or to the values given in a `Constraint' of `Init' type. `SaveSolution' `[SYSTEM-ID]' Save the solution of the system of equations SYSTEM-ID. `SaveSolutions' `[SYSTEM-ID]' Save all the solutions available for the system of equations SYSTEM-ID. This should be used with algorithms that generate more than one solution at once, e.g., `EigenSolve' or `FourierTransform'. `TransferSolution' `[SYSTEM-ID]' Transfer the solution of system SYSTEM-ID, as an `Assign' constraint, to the system of equations defined with a `DestinationSystem' command. This is used with the `AssignFromResolution' constraint type (*note Types for Constraint::). `TransferInitSolution' `[SYSTEM-ID]' Transfer the solution of system SYSTEM-ID, as an `Init' constraint, to the system of equations defined with a `DestinationSystem' command. This is used with the `InitFromResolution' constraint type (*note Types for Constraint::). `Evaluate' `[EXPRESSION]' Evaluate EXPRESSION. `SetTime' `[EXPRESSION]' Change the current time. `SetFrequency' `[SYSTEM-ID, EXPRESSION]' Change the frequency of system SYSTEM-ID. `SystemCommand' `[EXPRESSION-CHAR]' Execute the system command given by EXPRESSION-CHAR. `If' `[EXPRESSION] { RESOLUTION-OP }' If EXPRESSION is true (nonzero), perform the operations in RESOLUTION-OP. `If' `[EXPRESSION] { RESOLUTION-OP }' `Else' `{ RESOLUTION-OP }' If EXPRESSION is true (nonzero), perform the operations in the first RESOLUTION-OP, else perform the operations in the second RESOLUTION-OP. `Break' Not implemented yet. `Print' `[ { EXPRESSION-LIST }, < File EXPRESSION-CHAR > ]' Print the expressions listed in EXPRESSION-LIST. `Print' `[ SYSTEM-ID <, File EXPRESSION-CHAR > <, { EXPRESSION-CST-LIST } >' `<, TimeStep { EXPRESSION-CST-LIST } >]' Print the system SYSTEM-ID. If the EXPRESSION-CST-LIST is given, print only the values of the degrees of freedom given in that list. If the `TimeStep' option is present, limit the printing to the selected time steps. `EigenSolve' `[SYSTEM-ID, EXPRESSION-CST, EXPRESSION-CST, EXPRESSION-CST]' Eigenvalue/eigenvector computation using Arpack. The parameters are: the system (which has to be generated with `GenerateSeparate[]'), the number of eigenvalues/eigenvectors to compute and the real and imaginary spectral shift (around which to look for eigenvalues). `Lanczos' `[SYSTEM-ID, EXPRESSION-CST, { EXPRESSION-CST-LIST } , EXPRESSION-CST]' Eigenvalue/eigenvector computation using the Lanczos algorithm. The parameters are: the system (which has to be generated with `GenerateSeparate[]'), the size of the Lanczos space, the indices of the eigenvalues/eigenvectors to store, the spectral shift. This routine is deprecated: use `EigenSolve' instead. `FourierTransform' `[SYSTEM-ID, SYSTEM-ID, { EXPRESSION-CST-LIST }]' On-the-fly computation of a discrete Fourier transform. The parameters are: the (time domain) system, the destination system in which the result of the Fourier tranform is to be saved (it should be declared with `Type Complex'), the list of frequencies to consider in the discrete Fourier transform. `TimeLoopTheta' `[EXPRESSION-CST,EXPRESSION-CST,EXPRESSION,EXPRESSION-CST]' `{ RESOLUTION-OP }' Time loop of a theta scheme. The parameters are: the initial time, the end time, the time step and the theta parameter (e.g., 1 for implicit Euler, 0.5 for Crank-Nicholson). `TimeLoopNewmark' `[EXPRESSION-CST,EXPRESSION-CST,EXPRESSION,EXPRESSION-CST,EXPRESSION-CST]' { RESOLUTION-OP } Time loop of a Newmark scheme. The parameters are: the initial time, the end time, the time step, the beta and the gamma parameter. `IterativeLoop' `[EXPRESSION-CST,EXPRESSION,EXPRESSION-CST<,EXPRESSION-CST>]' { RESOLUTION-OP } Iterative loop for nonlinear analysis. The parameters are: the maximum number of iterations (if no convergence), the relaxation factor (multiplies the iterative correction DX) and the relative error to achieve. The optional parameter is a flag for testing purposes. `PostOperation' `[POST-OPERATION-ID]' Perform the specified `PostOperation'. 6.9 Types for `PostProcessing' ============================== POST-VALUE: `Local' { LOCAL-VALUE } To compute a local quantity. `Integral' { INTEGRAL-VALUE } To integrate the expression over each element. 6.10 Types for `PostOperation' ============================== PRINT-SUPPORT: `OnElementsOf' GROUP-DEF To compute a quantity on the elements belonging to the region GROUP-DEF, where the solution was computed during the processing stage. `OnRegion' GROUP-DEF To compute a global quantity associated with the region GROUP-DEF. `OnGlobal' To compute a global integral quantity, with no associated region. `OnSection' { { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } } To compute a quantity on a section of the mesh defined by three points (i.e., on the intersection of the mesh with a cutting a plane, specified by three points). Each EXPRESSION-CST-LIST must contain exactly three elements (the coordinates of the points). `OnGrid' GROUP-DEF To compute a quantity in elements of a mesh which differs from the real support of the solution. `OnGrid GROUP-DEF' differs from `OnElementsOf GROUP-DEF' by the reinterpolation that must be performed. `OnGrid' `{ EXPRESSION, EXPRESSION, EXPRESSION }' `{ EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST } ,' ` EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST } ,' ` EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST } }' To compute a quantity on a parametric grid. The three EXPRESSIONs represent the three cartesian coordinates X, Y and Z, and can be functions of the current values `$A', `$B' and `$C'. The values for `$A', `$B' and `$C' are specified by each EXPRESSION-CST-LIST-ITEM or EXPRESSION-CST-LIST. For example, `OnGrid {Cos[$A], Sin[$A], 0} { 0:2*Pi:Pi/180, 0, 0 }' will compute the quantity on 360 points equally distributed on a circle in the z=0 plane, and centered on the origin. `OnPoint' { EXPRESSION-CST-LIST } To compute a quantity at a point. The EXPRESSION-CST-LIST must contain exactly three elements (the coordinates of the point). `OnLine' { { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } } { EXPRESSION-CST } To compute a quantity along a line (given by its two end points), with an associated number of divisions equal to EXPRESSION-CST. The interpolation points on the line are equidistant. Each EXPRESSION-CST-LIST must contain exactly three elements (the coordinates of the points). `OnPlane' { { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } } `{ EXPRESSION-CST, EXPRESSION-CST }' To compute a quantity on a plane (specified by three points), with an associated number of divisions equal to each EXPRESSION-CST along both generating directions. Each EXPRESSION-CST-LIST must contain exactly three elements (the coordinates of the points). `OnBox' { { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } { EXPRESSION-CST-LIST } ` { EXPRESSION-CST-LIST } } { EXPRESSION-CST, EXPRESSION-CST, EXPRESSION-CST }' To compute a quantity in a box (specified by four points), with an associated number of divisions equal to each EXPRESSION-CST along the three generating directions. Each EXPRESSION-CST-LIST must contain exactly three elements (the coordinates of the points). PRINT-OPTION: `File' `EXPRESSION-CHAR' Outputs the result in a file named EXPRESSION-CHAR. `File' `> EXPRESSION-CHAR' Same as `File EXPRESSION-CHAR', except that, if several `File > EXPRESSION-CHAR' options appear in the same `PostOperation', the results are concatenated in the file EXPRESSION-CHAR. `File' `>> EXPRESSION-CHAR' Appends the result to a file named EXPRESSION-CHAR. `Depth' EXPRESSION-CST Recursive division of the elements if EXPRESSION-CST is greater than zero, derefinement if EXPRESSION-CST is smaller than zero. If EXPRESSION-CST is equal to zero, evaluation at the barycenter of the elements. `Skin' Computes the result on the boundary of the region. `Smoothing' Smoothes the solution at the nodes. `HarmonicToTime' EXPRESSION-CST Converts a harmonic solution into a time-dependent one (with EXPRESSION-CST steps). `Dimension' EXPRESSION-CST Forces the dimension of the elements to consider in an element search. Specifies the problem dimension during an adaptation (h- or p-refinement). `TimeStep' `EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST }' Outputs results for the specified time steps only. `LastTimeStepOnly' Outputs results for the last time step only (useful when calling a `PostOperation' directly in a `Resolution', for example). `Frequency' `EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST }' Outputs results for the specified frequencies only. `Format' POST-OPERATION-FMT Outputs results in the specified format. `Adapt' `P1 | H1 | H2' Performs p- or h-refinement on the post-processing result, considered as an error map. `Target' EXPRESSION-CST Specifies the target for the optimizer during adaptation (error for `P1|H1', number of elements for `H2'). `Value' `EXPRESSION-CST-LIST-ITEM | { EXPRESSION-CST-LIST }' Specifies acceptable output values for discrete optimization (e.g. the available interpolation orders with `Adapt P1'). `Sort' `Position | Connection' Sorts the output by position (x, y, z) or by connection (for `LINE' elements only). `Iso' EXPRESSION-CST Outputs directly contour prints (with EXPRESSION-CST values) instead of elementary values. `Iso' `{ EXPRESSION-CST-LIST }' Outputs directly contour prints for the values specified in the EXPRESSION-CST-LIST instead of elementary values. `NoNewLine' Suppresses the new lines in the output when printing global quantities (i.e., with `Print OnRegion' or `Print OnGlobal'). `ChangeOfCoordinates' `{ EXPRESSION, EXPRESSION, EXPRESSION }' Changes the coordinates of the results according to the three expressions given in argument. The three EXPRESSIONs represent the three new cartesian coordinates X, Y and Z, and can be functions of the current values of the cartesian coordinates `$X', `$Y' and `$Z'. `ChangeOfValues' `{ EXPRESSION-LIST }' Changes the values of the results according to the expressions given in argument. The EXPRESSIONs represent the new values (X-compoment, Y-component, etc.), and can be functions of the current values of the solution ($VAL0, $VAL1, etc.). `DecomposeInSimplex' Decomposes all output elements in simplices (points, lines, triangles or tetrahedra). `Store' `EXPRESSION-CST' Stores the result of an `OnRegion' post-processing operation in the register EXPRESSION-CST. `TimeLegend' `< { EXPRESSION, EXPRESSION, EXPRESSION } >' Includes a time legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot. `FrequencyLegend' `< { EXPRESSION, EXPRESSION, EXPRESSION } >' Includes a frequency legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot. `EigenvalueLegend' `< { EXPRESSION, EXPRESSION, EXPRESSION } >' Includes an eigenvalue legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot. POST-OPERATION-FMT: `Gmsh' `GmshParsed' Gmsh output. See *note Input file format:: and the documentation of Gmsh (`http://geuz.org/gmsh') for a description of the file formats. `Table' Space oriented column output, e.g., suitable for Gnuplot, Excel, Caleida Graph, etc. The columns are: ELEMENT-TYPE ELEMENT-INDEX X-COORD Y-COORD Z-COORD ... REAL REAL REAL VALUES. The three REAL numbers preceding the VALUES contain context-dependent information, depending on the type of plot: curvilinear abscissa for `OnLine' plots, normal to the plane for `OnPlane' plots, parametric coordinates for parametric `OnGrid' plots, etc. `SimpleTable' Like `Table', but with only the X-COORD Y-COORD Z-COORD and VALUES columns. `TimeTable' Time oriented column output, e.g., suitable for Gnuplot, Excel, Caleida Graph, etc. The columns are: TIME-STEP TIME X-COORD Y-COORD Z-COORD ... VALUE. `Gnuplot' Space oriented column output similar to the `Table' format, except that a new line is created for each node of each element, with a repetition of the first node if the number of nodes in the element is greater than 2. This permits to draw unstructured meshes and nice three-dimensional elevation plots in Gnuplot. The columns are: ELEMENT-TYPE ELEMENT-INDEX X-COORD Y-COORD Z-COORD REAL REAL REAL VALUES. The three REAL numbers preceding the VALUES contain context-dependent information, depending on the type of plot: curvilinear abscissa for `OnLine' plots, normal to the plane for `OnPlane' plots, parametric coordinates for parametric `OnGrid' plots, etc. `Adaptation' Adaptation map, suitable for the GetDP `-adapt' command line option. 7 Short examples **************** 7.1 Constant expression examples ================================ The simplest constant expression consists of an INTEGER or a REAL number as in 21 -3 or -3.1415 27e3 -290.53e-12 Using operators and the classic math functions, CONSTANT-IDs can be defined: c1 = Sin[2/3*3.1415] * 5000^2; c2 = -1/c1; 7.2 `Group' examples ==================== Let us assume that some elements in the input mesh have the region numbers 1000, 2000 and 3000. In the definitions Group { Air = Region[1000]; Core = Region[2000]; Inductor = Region[3000]; NonConductingDomain = Region[{Air, Core}]; ConductingDomain = Region[{Inductor}]; } `Air', `Core', `Inductor' are identifiers of elementary region groups while `NonConductingDomain' and `ConductingDomain' are global region groups. Groups of function type contain lists of entities built on the region groups appearing in their arguments. For example, NodesOf[NonConductingDomain] represents the group of nodes of geometrical elements belonging to the regions in `NonConductingDomain' and EdgesOf[DomainC, Not SkinDomainC] represents the group of edges of geometrical elements belonging to the regions in `DomainC' but not to those of `SkinDomainC'. 7.3 `Function' examples ======================= A physical characteristic is a piecewise defined function. The magnetic permeability `mu[]' can for example be defined in the considered regions by Function { mu[Air] = 4.e-7*Pi; mu[Core] = 1000.*4.e-7*Pi; } A nonlinear characteristic can be defined through an EXPRESSION with arguments, e.g., Function { mu0 = 4.e-7*Pi; a1 = 1000.; b1 = 100.; // Constants mu[NonlinearCore] = mu0 + 1./(a1+b1*Norm[$1]^6); } where function `mu[]' in region `NonLinearCore' has one argument `$1' which has to be the magnetic flux density. This function is actually called when writing the equations of a formulation, which permits to directly extend it to a nonlinear form by adding only the necessary arguments. For example, in a magnetic vector potential formulation, one may write `mu[{Curl a}]' instead of `mu[]' in `Equation' terms (*note Formulation examples::). Multiple arguments can be specified in a similar way: writing `mu[{Curl a},{T}]' in an `Equation' term will provide the function `mu[]' with two usable arguments, `$1' (the magnetic flux density) and `$2' (the temperature). It is also possible to directly interpolate one-dimensional functions from tabulated data. In the following example, the function F(X) as well as its derivative F'(X) are interpolated from the (X,F(X)) couples (0,0.65), (1,0.72), (2,0.98) and (3,1.12): Function { couples = {0, 0.65 , 1, 0.72 , 2, 0.98 , 3, 1.12}; f[] = InterpolationLinear[$1]{List[couples]}; dfdx[] = dInterpolationLinear[$1]{List[couples]}; } The function `f[]' may then be called in an `Equation' term of a `Formulation' with one argument, X. Notice how the list of constants `List[couples]' is supplied as a list of parameters to the built-in function `InterpolationLinear' (*note Constants::, as well as *note Functions::). In order to facilitate the construction of such interpolations, the couples can also be specified in two separate lists, merged with the alternate list `ListAlt' command (*note Constants::): Function { data_x = {0, 1, 2, 3}; data_f = {0.65, 0.72, 0.98, 1.12}; f[] = InterpolationLinear[$1]{ListAlt[data_x, data_f]}; dfdx[] = dInterpolationLinear[$1]{ListAlt[data_x, data_f]}; } In order to optimize the evaluation time of complex expressions, registers may be used (*note Registers::). For example, the evaluation of `g[] = f[$1]*Sin[f[$1]^2]' would require two (costly) linear interpolations. But the result of the evaluation of `f[]' may be stored in a register (for example the register 0) with g[] = f[$1]#0 * Sin[#0^2]; thus reducing the number of evaluations of `f[]' (and of the argument `$1') to one. A function can also be time dependent, e.g., Function { Freq = 50.; Phase = 30./180.*Pi; // Constants TimeFct_Sin[] = Sin [ 2.*Pi*Freq * $Time + Phase ]; TimeFct_Exp[] = Exp [ - $Time / 0.0119 ]; TimeFct_ExtSin[] = F_Sin_wt_p [] {2.*Pi*Freq, Phase}; } Note that `TimeFct_ExtSin[]' is identical to `TimeFct_Sin[]' in a time domain analysis, but also permits to define phasors implicitely in the case of harmonic analyses. 7.4 `Constraint' examples ========================= Constraints are referred to in `FunctionSpace's and are usually used for boundary conditions (`Assign' type). For example, essential conditions on two surface regions, `Surf0' and `Surf1', will be first defined by Constraint { { Name DirichletBoundaryCondition1; Type Assign; Case { { Region Surf0; Value 0.; } { Region Surf1; Value 1.; } } } } The way the `Value's are associated with `Region's (with their nodes, their edges, their global regions, ...) is defined in the `FunctionSpace's which use the `Constraint'. In other words, a `Constraint' defines data but does not define the method to process them. A time dependent essential boundary condition on `Surf1' would be introduced as (cf. *note Function examples:: for the definition of `TimeFct_Exp[]'): { Region Surf1; Value 1.; TimeFunction 3*TimeFct_Exp[] } It is important to notice that the time dependence cannot be introduced in the `Value' field, since the `Value' is only evaluated once during the pre-processing. Other constraints can be referred to in `Formulation's. It is the case of those defining electrical circuit connections (`Network' type), e.g., Constraint { { Name ElectricalCircuit; Type Network; Case Circuit1 { { Region VoltageSource; Branch {1,2}; } { Region PrimaryCoil; Branch {1,2}; } } Case Circuit2 { { Region SecondaryCoil; Branch {1,2}; } { Region Charge; Branch {1,2}; } } } } which defines two non-connected circuits (`Circuit1' and `Circuit2'), with an independent numbering of nodes: region `VoltageSource' is connected in parallel with region `PrimaryCoil', and region `SecondaryCoil' is connected in parallel with region `Charge'. 7.5 `FunctionSpace' examples ============================ Various discrete function spaces can be defined in the frame of the finite element method. 7.5.1 Nodal finite element spaces --------------------------------- The most elementary function space is the nodal finite element space, defined on a mesh of a domain W and denoted S0(W) (associated finite elements can be of various geometries), and associated with essential boundary conditions (Dirichlet conditions). It contains 0-forms, i.e., scalar fields of potential type: V = Sum [ VN * SN, for all N in N ], V in S0(W) where N is the set of nodes of W, SN is the nodal basis function associated with node N and VN is the value of V at node N. It is defined by FunctionSpace { { Name Hgrad_v; Type Form0; BasisFunction { { Name sn; NameOfCoef vn; Function BF_Node; Support Domain; Entity NodesOf[All]; } } Constraint { { NameOfCoef vn; EntityType NodesOf; NameOfConstraint DirichletBoundaryCondition1; } } } } Function `sn' is the built-in basis function BF_Node associated with all nodes (`NodesOf') in the mesh of W (`Domain'). Previously defined `Constraint DirichletBoundaryCondition1' (*note Constraint examples::) is used as boundary condition. In the example above, `Entity NodesOf[All]' is preferred to `Entity NodesOf[Domain]'. In this way, the list of all the nodes of `Domain' will not have to be generated. All the nodes of each geometrical element in `Support Domain' will be directly taken into account. 7.5.2 High order nodal finite element space ------------------------------------------- Higher order finite elements can be directly taken into account by `BF_Node'. Hierarchical finite elements for 0-forms can be used by simply adding other basis functions (associated with other geometrical entities, e.g., edges and facets) to `BasisFunction', e.g., ... BasisFunction { { Name sn; NameOfCoef vn; Function BF_Node; Support Domain; Entity NodesOf[All]; } { Name s2; NameOfCoef v2; Function BF_Node_2E; Support Domain; Entity EdgesOf[All]; } } ... 7.5.3 Nodal finite element space with floating potentials --------------------------------------------------------- A scalar potential with floating values VF on certain boundaries GF, F in CF, e.g., for electrostatic problems, can be expressed as V = Sum [ VN * SN, for all N in NV ] + Sum [ VF * SF, for all F in CF ], V in S0(W) where NV is the set of inner nodes of W and each function SF is associated with the group of nodes of boundary GF, F in CF (`SkinDomainC'); SF is the sum of the nodal basis functions of all the nodes of CF. Its function space is defined by FunctionSpace { { Name Hgrad_v_floating; Type Form0; BasisFunction { { Name sn; NameOfCoef vn; Function BF_Node; Support Domain; Entity NodesOf[All, Not SkinDomainC]; } { Name sf; NameOfCoef vf; Function BF_GroupOfNodes; Support Domain; Entity GroupsOfNodesOf[SkinDomainC]; } } GlobalQuantity { { Name GlobalElectricPotential; Type AliasOf; NameOfCoef vf; } { Name GlobalElectricCharge; Type AssociatedWith; NameOfCoef vf; } } Constraint { ... } } } Two global quantities have been associated with this space: the electric potential `GlobalElectricPotential', being an alias of coefficient `vf', and the electric charge `GlobalElectricCharge', being associated with coefficient `vf'. 7.5.4 Edge finite element space ------------------------------- Another space is the edge finite element space, denoted S1(W), containing 1-forms, i.e., curl-conform fields: H = Sum [ HE * SE, for all E in E ], H in S1(W) where E is the set of edges of W, SE is the edge basis function for edge E and HE is the circulation of H along edge E. It is defined by FunctionSpace { { Name Hcurl_h; Type Form1; BasisFunction { { Name se; NameOfCoef he; Function BF_Edge; Support Domain; Entity EdgesOf[All]; } } Constraint { ... } } } 7.5.5 Edge finite element space with gauge condition ---------------------------------------------------- A 1-form function space containing vector potentials can be associated with a gauge condition, which can be defined as a constraint, e.g., a zero value is fixed for all circulations along edges of a tree (`EdgesOfTreeIn') built in the mesh (`Domain'), having to be complete on certain boundaries (`StartingOn Surf'): Constraint { { Name GaugeCondition_a_Mag_3D; Type Assign; Case { { Region Domain; SubRegion Surf; Value 0.; } } } } FunctionSpace { { Name Hcurl_a_Gauge; Type Form1; BasisFunction { { Name se; NameOfCoef ae; Function BF_Edge; Support Domain; Entity EdgesOf[All]; } } Constraint { { NameOfCoef ae; EntityType EdgesOfTreeIn; EntitySubType StartingOn; NameOfConstraint GaugeCondition_a_Mag_3D; } ... } } } 7.5.6 Coupled edge and nodal finite element spaces -------------------------------------------------- A 1-form function space, containing curl free fields in certain regions WCC (`DomainCC') of W, which are the complementary part of WC (`DomainC') in W, can be explicitly characterized by H = Sum [ HK * SK, for all E in EC ] + Sum [ PHIN * VN, for all N in NCC ], H in S1(W) where EC is the set of inner edges of W, NCC is the set of nodes inside WCC and on its boundary DWCC, SK is an edge basis function and VN is a vector nodal function. Such a space, coupling a vector field with a scalar potential, can be defined by FunctionSpace { { Name Hcurl_hphi; Type Form1; BasisFunction { { Name sk; NameOfCoef hk; Function BF_Edge; Support DomainC; Entity EdgesOf[All, Not SkinDomainC]; } { Name vn; NameOfCoef phin; Function BF_GradNode; Support DomainCC; Entity NodesOf[All]; } { Name vn; NameOfCoef phic; Function BF_GroupOfEdges; Support DomainC; Entity GroupsOfEdgesOnNodesOf[SkinDomainC];} } Constraint { { NameOfCoef hk; EntityType EdgesOf; NameOfConstraint MagneticField; } { NameOfCoef phin; EntityType NodesOf; NameOfConstraint MagneticScalarPotential; } { NameOfCoef phic; EntityType NodesOf; NameOfConstraint MagneticScalarPotential; } } } } This example points out the definition of a piecewise defined basis function, e.g., function `vn' being defined with `BF_GradNode' in `DomainCC' and `BF_GroupOfEdges' in `DomainC'. This leads to an easy coupling between these regions. 7.5.7 Coupled edge and nodal finite element spaces for multiply connected domains --------------------------------------------------------------------------------- In case a multiply connected domain WCC is considered, basis functions associated with cuts (`SurfaceCut') have to be added to the previous basis functions, which gives the function space below: Group { _TransitionLayer_SkinDomainC_ = ElementsOf[SkinDomainC, OnOneSideOf SurfaceCut]; } FunctionSpace { { Name Hcurl_hphi; Type Form1; BasisFunction { ... SAME AS ABOVE ... { Name sc; NameOfCoef Ic; Function BF_GradGroupOfNodes; Support ElementsOf[DomainCC, OnOneSideOf SurfaceCut]; Entity GroupsOfNodesOf[SurfaceCut]; } { Name sc; NameOfCoef Icc; Function BF_GroupOfEdges; Support DomainC; Entity GroupsOfEdgesOf [SurfaceCut, InSupport _TransitionLayer_SkinDomainC_]; } } GlobalQuantity { { Name I; Type AliasOf ; NameOfCoef Ic; } { Name U; Type AssociatedWith; NameOfCoef Ic; } } Constraint { ... SAME AS ABOVE ... { NameOfCoef Ic; EntityType GroupsOfNodesOf; NameOfConstraint Current; } { NameOfCoef Icc; EntityType GroupsOfNodesOf; NameOfConstraint Current; } { NameOfCoef U; EntityType GroupsOfNodesOf; NameOfConstraint Voltage; } } } } Global quantities associated with the cuts, i.e., currents and voltages if H is the magnetic field, have also been defined. 7.6 `Jacobian' examples ======================= A simple Jacobian method is for volume transformations (of N-D regions in N-D geometries; N = 1, 2 or 3), e.g., in region `Domain', Jacobian { { Name Vol; Case { { Region Domain; Jacobian Vol; } } } } `Jacobian VolAxi' would define a volume Jacobian for axisymmetrical problems. A Jacobian method can also be piecewise defined, in `DomainInf', where an infinite geometrical transformation has to be made using two constant parameters (inner and outer radius of a spherical shell), and in all the other regions (`All', being the default); in each case, a volume Jacobian is used. This method is defined by: Jacobian { { Name Vol; Case { { Region DomainInf; Jacobian VolSphShell {Val_Rint, Val_Rext}; } { Region All; Jacobian Vol; } } } } 7.7 `Integration' examples ========================== A commonly used numerical integration method is the `Gauss' integration, with a number of integration points (`NumberOfPoints') depending on geometrical element types (`GeoElement'), i.e. Integration { { Name Int_1; Case { {Type Gauss; Case { { GeoElement Triangle ; NumberOfPoints 4; } { GeoElement Quadrangle ; NumberOfPoints 4; } { GeoElement Tetrahedron; NumberOfPoints 4; } { GeoElement Hexahedron ; NumberOfPoints 6; } { GeoElement Prism ; NumberOfPoints 9; } } } } } } The method above is valid for both 2D and 3D problems, for different kinds of elements. 7.8 `Formulation' examples ========================== 7.8.1 Electrostatic scalar potential formulation ------------------------------------------------ An electrostatic formulation using an electric scalar potential V, i.e. ( epsr grad V, grad VP ) W = 0, for all VP in S0(W), is expressed by Formulation { { Name Electrostatics_v; Type FemEquation; Quantity { { Name v; Type Local; NameOfSpace Hgrad_v; } } Equation { Galerkin { [ epsr[] * Dof{Grad v} , {Grad v} ]; In Domain; Jacobian Vol; Integration Int_1; } } } } The density of the `Galerkin' term is a copy of the symbolic form of the formulation, i.e., the product of a relative permittivity function `epsr[]' by a vector of degrees of freedom (`Dof{.}'); the scalar product of this with the gradient of test function `v' results in a symmetrical matrix. Note that another `Quantity' could be defined for test functions, e.g., `vp' defined by `{ Name vp; Type Local; NameOfSpace Hgrad_v; }'. However, its use would result in the computation of a full matrix and consequently in a loss of efficiency. 7.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges ---------------------------------------------------------------------------------------------- An extension of the formulation above can be made to take floating potentials and electrical charges into account (the latter being defined in `FunctionSpace Hgrad_v_floating'), i.e. Formulation { { Name Electrostatics_v_floating; Type FemEquation; Quantity { { Name v; Type Local; NameOfSpace Hgrad_v_floating; } { Name V; Type Global; NameOfSpace Hgrad_v_floating [GlobalElectricPotential]; } { Name Q; Type Global; NameOfSpace Hgrad_v_floating [GlobalElectricCharge]; } } Equation { Galerkin { [ epsr[] * Dof{Grad v} , {Grad v} ]; In Domain; Jacobian Vol; Integration Int_1; } GlobalTerm { [ - Dof{Q}/eps0 , {V} ]; In SkinDomainC; } } } } with the predefinition `Function { eps0 = 8.854187818e-12; }'. 7.8.3 Magnetostatic 3D vector potential formulation --------------------------------------------------- A magnetostatic 3D vector potential formulation ( NU curl A , curl AP ) W - ( JS , AP ) WS = 0, for all AP in S1(W) with gauge condition, with a source current density JS in inductors WS, is expressed by Formulation { { Name Magnetostatics_a_3D; Type FemEquation; Quantity { { Name a; Type Local; NameOfSpace Hcurl_a_Gauge; } } Equation { Galerkin { [ nu[] * Dof{Curl a} , {Curl a} ]; In Domain; Jacobian Vol; Integration Int_1; } Galerkin { [ - SourceCurrentDensity[] , {a} ]; In DomainWithSourceCurrentDensity; Jacobian Vol; Integration Int_1; } } } } Note that JS is here given by a function `SourceCurrentDensity[]', but could also be given by data computed from another problem, e.g., from an electrokinetic problem (coupling of formulations) or from a fully fixed function space (constraints fixing the density, which is usually more efficient in time domain analyses). 7.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation -------------------------------------------------------------------------------------- A magnetodynamic 3D or 2D H-PHI formulation, i.e., coupling the magnetic field H with a magnetic scalar potential PHI, Dt ( MU H , HP ) W + ( RO curl H , curl HP ) WC = 0, for all HP in S1(W), can be expressed by Formulation { { Name Magnetodynamics_hphi; Type FemEquation; Quantity { { Name h; Type Local; NameOfSpace Hcurl_hphi; } } Equation { Galerkin { Dt [ mu[] * Dof{h} , {h} ]; In Domain; Jacobian Vol; Integration Int_1; } Galerkin { [ rho[] * Dof{Curl h} , {Curl h} ]; In DomainC; Jacobian Vol; Integration Int_1; } } } } 7.8.5 Nonlinearities, Mixed formulations, ... --------------------------------------------- In case nonlinear physical characteristics are considered, arguments are used for associated functions, e.g., `mu[{h}]'. Several test functions can be considered in an `Equation' field. Consequently, mixed formulations can be defined. 7.9 `Resolution' examples ========================= 7.9.1 Static resolution (electrostatic problem) ----------------------------------------------- A static resolution, e.g., for the electrostatic formulation (*note Formulation examples::), can be defined by Resolution { { Name Electrostatics_v; System { { Name Sys_Ele; NameOfFormulation Electrostatics_v; } } Operation { Generate[Sys_Ele]; Solve[Sys_Ele]; SaveSolution[Sys_Ele]; } } } The generation (`Generate') of the matrix of the system `Sys_Ele' will be made with the formulation `Electrostatics_v', followed by its solving (`Solve') and the saving of the solution (`SaveSolution'). 7.9.2 Frequency domain resolution (magnetodynamic problem) ---------------------------------------------------------- A frequency domain resolution, e.g., for the magnetodynamic H-PHI formulation (*note Formulation examples::), is given by Resolution { { Name Magnetodynamics_hphi; System { { Name Sys_Mag; NameOfFormulation Magnetodynamics_hphi; Frequency Freq; } } Operation { Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag]; } } } preceded by the definition of constant `Freq', e.g., Function { Freq = 50.; } 7.9.3 Time domain resolution (magnetodynamic problem) ----------------------------------------------------- A time domain resolution, e.g., for the same magnetodynamic H-PHI formulation (*note Formulation examples::), is given by Resolution { { Name Magnetodynamics_hphi_Time; System { { Name Sys_Mag; NameOfFormulation Magnetodynamics_hphi; } } Operation { InitSolution[Sys_Mag]; SaveSolution[Sys_Mag]; TimeLoopTheta[Mag_Time0, Mag_TimeMax, Mag_DTime[], Mag_Theta[]] { Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag]; } } } } If, e.g., the `Resolution' above is preceded by the constant and function definitions below Function { Tc = 10.e-3; Mag_Time0 = 0.; Mag_TimeMax = 2.*Tc; Mag_DTime[] = Tc/20.; Mag_Theta[] = 1./2.; } the performed time analysis will be a Crank-Nicolson scheme (theta-scheme with `Theta = 0.5') with initial time 0 ms, end time 20 ms and time step 1 ms. 7.9.4 Nonlinear time domain resolution (magnetodynamic problem) --------------------------------------------------------------- In case a nonlinear problem is solved, an iterative loop has to be defined in an appropriate level of the recursive resolution operations, e.g., for the magnetodynamic problem above, ... Operation { InitSolution[Sys_Mag]; SaveSolution[Sys_Mag]; TimeLoopTheta[Mag_Time0, Mag_TimeMax, Mag_DTime[], Mag_Theta[]] { IterativeLoop[NL_NbrMax, NL_Eps, NL_Relax] { GenerateJac[Sys_Mag]; SolveJac[Sys_Mag]; } SaveSolution[Sys_Mag]; } } ... preceded by constant definitions, e.g., Function { NL_Eps = 1.e-4; NL_Relax = 1.; NL_NbrMax = 80; } 7.9.5 Coupled formulations -------------------------- A coupled problem, e.g., magnetodynamic (in frequency domain; `Frequency Freq') - thermal (in time domain) coupling, with temperature dependent characteristics (e.g., `rho[{T}]', ...), can be defined by: Resolution { { Name MagnetoThermalCoupling_hphi_T; System { { Name Sys_Mag; NameOfFormulation Magnetodynamics_hphi; Frequency Freq; } { Name Sys_The; NameOfFormulation Thermal_T; } } Operation { InitSolution[Sys_Mag]; InitSolution[Sys_The]; IterativeLoop[NL_NbrMax, NL_Eps, NL_Relax] { GenerateJac[Sys_Mag]; SolveJac[Sys_Mag]; GenerateJac[Sys_The]; SolveJac[Sys_The]; } SaveSolution[Sys_Mag]; SaveSolution[Sys_The]; } } } Two systems of equations, `Sys_Mag' and `Sys_The', will be solved iteratively until convergence (`Criterion'), using a relaxation factor (`RelaxationFactor'). It can be seen through these examples that many resolutions can be linked or nested directly by the user, which gives a great freedom for coupled problems. 7.10 `PostProcessing' examples ============================== The quantities to be post-computed based on a solution of a resolution are defined, e.g., for the electrostatic problem (*note Formulation examples::; *note Resolution examples::), for the solution associated with the formulation `Electrostatics_v', by PostProcessing { { Name EleSta_v; NameOfFormulation Electrostatics_v; Quantity { { Name v; Value { Local { [ {v} ]; In Domain; } } } { Name e; Value { Local { [ -{Grad v} ]; In Domain; } } } { Name d; Value { Local { [ -eps0*epsr[] *{Grad v} ]; In Domain; } } } } } } The electric scalar potential V (`v'), the electric field E (`e') and the electric flux density D (`d') can all be computed from the solution. They are all defined in the region `Domain'. The quantities for the solution associated with the formulation `Electrostatics_v_floating' are defined by PostProcessing { { Name EleSta_vf; NameOfFormulation Electrostatics_v_floating; Quantity { ... SAME AS ABOVE ... { Name Q; Value { Local { [ {Q} ]; In SkinDomainC; } } } { Name V; Value { Local { [ {V} ]; In SkinDomainC; } } } } } } which points out the way to define post-quantities based on global quantities. 7.11 `PostOperation' examples ============================= The simplest post-processing operation is the generation of maps of local quantities, i.e., the display of the computed fields on the mesh. For example, using the `PostProcessing' defined in *note PostProcessing examples::, the maps of the electric scalar potential and of the electric field on the elements of the region `Domain' are defined as: PostOperation { { Name Map_v_e; NameOfPostProcessing EleSta_v ; Operation { Print [ v, OnElementsOf Domain, File "map_v.pos" ]; Print [ e, OnElementsOf Domain, File "map_e.pos" ]; } } } It is also possible to display local quantities on sections of the mesh, here for example on the plane containing the points (0,0,1), (1,0,1) and (0,1,1): Print [ v, OnSection { {0,0,1} {1,0,1} {0,1,1} }, File "sec_v.pos" ]; Finally, local quantities can also be interpolated on another mesh than the one on which they have been computed. Six types of grids can be specified for this interpolation: a single point, a set of points evenly distributed on a line, a set of points evenly distributed on a plane, a set of points evenly distributed in a box, a set of points defined by a parametric equation, and a set of elements belonging to a different mesh than the original one: Print [ e, OnPoint {0,0,1} ]; Print [ e, OnLine { {0,0,1} {1,0,1} } {125} ]; Print [ e, OnPlane { {0,0,1} {1,0,1} {0,1,1} } {125, 75} ]; Print [ e, OnBox { {0,0,1} {1,0,1} {0,1,1} {0,0,2} } {125, 75, 85} ]; Print [ e, OnGrid {$A, $B, 1} { 0:1:1/125, 0:1:1/75, 0 } ]; Print [ e, OnGrid Domain2 ]; Many options can be used to modify the aspect of all these maps, as well as the default behaviour of the `Print' commands. See *note Types for PostOperation::, to get the list of all these options. For example, to obtain a map of the scalar potential at the barycenters of the elements on the boundary of the region `Domain', in a table oriented format appended to an already existing file `out.txt', the operation would be: Print [ v, OnElementsOf Domain, Depth 0, Skin, Format Table, File >> "out.txt" ]; Global quantities, which are associated with regions (and not with the elements of the mesh of these regions), are displayed thanks to the `OnRegion' operation. For example, the global potential and charge on the region `SkinDomainC' can be displayed with: PostOperation { { Name Val_V_Q; NameOfPostProcessing EleSta_vf ; Operation { Print [ V, OnRegion SkinDomainC ]; Print [ Q, OnRegion SkinDomainC ]; } } } 8 Complete examples ******************* This chapter presents complete examples that can be run "as is" with GetDP (*note Running GetDP::). Many other ready-to-use examples are available in the GetDP wiki at the following address: `https://geuz.org/trac/getdp' (username=getdp; password=wiki). 8.1 Electrostatic problem ========================= Let us first consider a simple electrostatic problem. The formulation used is an electric scalar potential formulation (file `EleSta_v.pro', including files `Jacobian_Lib.pro' and `Integration_Lib.pro'). It is applied to a microstrip line (file `mStrip.pro'), whose geometry is defined in the file `mStrip.geo' (*note Gmsh examples::). The geometry is two-dimensional and by symmetry only one half of the structure is modeled. SurfInf / / +------------------------------------+ / | | / | Air |/ | | | Line | | / / / | 2D elements in: +-------/---+ / | Air, Diel1 / |- | +-----------+------------------------+ 1D elements in: | Diel1 | Line, Ground, SurfInf | | +------------------------------------+ \ Ground Note that the structure of the following files points out the separation of the data describing the particular problem and the method used to solve it (*note Numerical tools as objects::), and therefore how it is possible to build black boxes adapted to well defined categories of problems. The files are commented (*note Comments::) and can be run without any modification. /* ------------------------------------------------------------------- File "mStrip.pro" This file defines the problem dependent data structures for the microstrip problem. To compute the solution: getdp mStrip -solve EleSta_v To compute post-results: getdp mStrip -pos Map or getdp mStrip -pos Cut ------------------------------------------------------------------- */ Group { /* Let's start by defining the interface (i.e. elementary groups) between the mesh file and GetDP (no mesh object is defined, so the default mesh will be assumed to be in GMSH format and located in "mStrip.msh") */ Air = Region[101]; Diel1 = Region[111]; Ground = Region[120]; Line = Region[121]; SurfInf = Region[130]; /* We can then define a global group (used in "EleSta_v.pro", the file containing the function spaces and formulations) */ DomainCC_Ele = Region[{Air, Diel1}]; } Function { /* The relative permittivity (needed in the formulation) is piecewise defined in elementary groups */ epsr[Air] = 1.; epsr[Diel1] = 9.8; } Constraint { /* Now, some Dirichlet conditions are defined. The name 'ElectricScalarPotential' refers to the constraint name given in the function space */ { Name ElectricScalarPotential; Type Assign; Case { { Region Region[{Ground, SurfInf}]; Value 0.; } { Region Line; Value 1.e-3; } } } } /* The formulation used and its tools, considered as being in a black box, can now be included */ Include "Jacobian_Lib.pro" Include "Integration_Lib.pro" Include "EleSta_v.pro" /* Finally, we can define some operations to output results */ e = 1.e-7; PostOperation { { Name Map; NameOfPostProcessing EleSta_v; Operation { Print [ v, OnElementsOf DomainCC_Ele, File "mStrip_v.pos" ]; Print [ e, OnElementsOf DomainCC_Ele, File "mStrip_e.pos" ]; } } { Name Cut; NameOfPostProcessing EleSta_v; Operation { Print [ e, OnLine {{e,e,0}{10.e-3,e,0}} {500}, File "Cut_e" ]; } } } /* ------------------------------------------------------------------- File "EleSta_v.pro" Electrostatics - Electric scalar potential v formulation ------------------------------------------------------------------- I N P U T --------- Global Groups : (Extension '_Ele' is for Electric problem) ------------- Domain_Ele Whole electric domain (not used) DomainCC_Ele Nonconducting regions DomainC_Ele Conducting regions (not used) Function : -------- epsr[] Relative permittivity Constraint : ---------- ElectricScalarPotential Fixed electric scalar potential (classical boundary condition) Physical constants : ------------------ */ eps0 = 8.854187818e-12; Group { DefineGroup[ Domain_Ele, DomainCC_Ele, DomainC_Ele ]; } Function { DefineFunction[ epsr ]; } FunctionSpace { { Name Hgrad_v_Ele; Type Form0; BasisFunction { // v = v s , for all nodes // n n { Name sn; NameOfCoef vn; Function BF_Node; Support DomainCC_Ele; Entity NodesOf[ All ]; } } Constraint { { NameOfCoef vn; EntityType NodesOf; NameOfConstraint ElectricScalarPotential; } } } } Formulation { { Name Electrostatics_v; Type FemEquation; Quantity { { Name v; Type Local; NameOfSpace Hgrad_v_Ele; } } Equation { Galerkin { [ epsr[] * Dof{d v} , {d v} ]; In DomainCC_Ele; Jacobian Vol; Integration GradGrad; } } } } Resolution { { Name EleSta_v; System { { Name Sys_Ele; NameOfFormulation Electrostatics_v; } } Operation { Generate[Sys_Ele]; Solve[Sys_Ele]; SaveSolution[Sys_Ele]; } } } PostProcessing { { Name EleSta_v; NameOfFormulation Electrostatics_v; Quantity { { Name v; Value { Local { [ {v} ]; In DomainCC_Ele; Jacobian Vol; } } } { Name e; Value { Local { [ -{d v} ]; In DomainCC_Ele; Jacobian Vol; } } } { Name d; Value { Local { [ -eps0*epsr[] * {d v} ]; In DomainCC_Ele; Jacobian Vol; } } } } } } /* ------------------------------------------------------------------- File "Jacobian_Lib.pro" Definition of a jacobian method ------------------------------------------------------------------- I N P U T --------- GlobalGroup : ----------- DomainInf Regions with Spherical Shell Transformation Parameters : ---------- Val_Rint, Val_Rext Inner and outer radius of the Spherical Shell of DomainInf */ Group { DefineGroup[ DomainInf ] ; DefineVariable[ Val_Rint, Val_Rext ] ; } Jacobian { { Name Vol ; Case { { Region DomainInf ; Jacobian VolSphShell {Val_Rint, Val_Rext} ; } { Region All ; Jacobian Vol ; } } } } /* ------------------------------------------------------------------- File "Integration_Lib.pro" Definition of integration methods ------------------------------------------------------------------- */ Integration { { Name GradGrad ; Case { {Type Gauss ; Case { { GeoElement Triangle ; NumberOfPoints 4 ; } { GeoElement Quadrangle ; NumberOfPoints 4 ; } { GeoElement Tetrahedron ; NumberOfPoints 4 ; } { GeoElement Hexahedron ; NumberOfPoints 6 ; } { GeoElement Prism ; NumberOfPoints 9 ; } } } } } { Name CurlCurl ; Case { {Type Gauss ; Case { { GeoElement Triangle ; NumberOfPoints 4 ; } { GeoElement Quadrangle ; NumberOfPoints 4 ; } { GeoElement Tetrahedron ; NumberOfPoints 4 ; } { GeoElement Hexahedron ; NumberOfPoints 6 ; } { GeoElement Prism ; NumberOfPoints 9 ; } } } } } } 8.2 Magnetostatic problem ========================= We now consider a magnetostatic problem. The formulation used is a 2D magnetic vector potential formulation (see file `MagSta_a_2D.pro'). It is applied to a core-inductor system (file `CoreSta.pro'), whose geometry is defined in theh file `Core.geo' (*note Gmsh examples::). The geometry is two-dimensional and, by symmetry, one fourth of the structure is modeled. SurfaceGInf ______ / | -----____/ | \___ |____ AirInf \__ S | ------_ \ u | \ \ r | Air \ \ f | \ \ a | \ | c | | | e +------+ | | G | | | | e | Core | +---+ | | 0 | | |Ind| | | | | | | | | +------+---+---+---------+------+ \ SurfaceGh0 2D elements in: Air, AirInf, Core, Ind 1D elements in: SurfaceGh0, SurfaceGe0, SurfaceGInf (AirInf is a spherical shell corresponding to an infinite region) The jacobian and integration methods used are the same as for the electrostatic problem presented in *note Electrostatic problem::. /* ------------------------------------------------------------------- File "CoreSta.pro" This file defines the problem dependent data structures for the static core-inductor problem. To compute the solution: getdp CoreSta -msh Core.msh -solve MagSta_a_2D To compute post-results: getdp CoreSta -msh Core.msh -pos Map_a ------------------------------------------------------------------- */ Group { Air = Region[ 101 ]; Core = Region[ 102 ]; Ind = Region[ 103 ]; AirInf = Region[ 111 ]; SurfaceGh0 = Region[ 1100 ]; SurfaceGe0 = Region[ 1101 ]; SurfaceGInf = Region[ 1102 ]; Val_Rint = 200.e-3; Val_Rext = 250.e-3; DomainCC_Mag = Region[ {Air, AirInf, Core, Ind} ]; DomainC_Mag = Region[ {} ]; DomainS_Mag = Region[ {Ind} ]; // Stranded inductor DomainInf = Region[ {AirInf} ]; Domain_Mag = Region[ {DomainCC_Mag, DomainC_Mag} ]; } Function { mu0 = 4.e-7 * Pi; murCore = 100.; nu [ Region[{Air, Ind, AirInf}] ] = 1. / mu0; nu [ Core ] = 1. / (murCore * mu0); Sc[ Ind ] = 2.5e-2 * 5.e-2; } Constraint { { Name MagneticVectorPotential_2D; Case { { Region SurfaceGe0 ; Value 0.; } { Region SurfaceGInf; Value 0.; } } } Val_I_1_ = 0.01 * 1000.; { Name SourceCurrentDensityZ; Case { { Region Ind; Value Val_I_1_/Sc[]; } } } } Include "Jacobian_Lib.pro" Include "Integration_Lib.pro" Include "MagSta_a_2D.pro" e = 1.e-5; p1 = {e,e,0}; p2 = {0.12,e,0}; PostOperation { { Name Map_a; NameOfPostProcessing MagSta_a_2D; Operation { Print[ az, OnElementsOf Domain_Mag, File "CoreSta_a.pos" ]; Print[ b, OnLine{{List[p1]}{List[p2]}} {1000}, File "k_a" ]; } } } /* ------------------------------------------------------------------- File "MagSta_a_2D.pro" Magnetostatics - Magnetic vector potential a formulation (2D) ------------------------------------------------------------------- I N P U T --------- GlobalGroup : (Extension '_Mag' is for Magnetic problem) ----------- Domain_Mag Whole magnetic domain DomainS_Mag Inductor regions (Source) Function : -------- nu[] Magnetic reluctivity Constraint : ---------- MagneticVectorPotential_2D Fixed magnetic vector potential (2D) (classical boundary condition) SourceCurrentDensityZ Fixed source current density (in Z direction) */ Group { DefineGroup[ Domain_Mag, DomainS_Mag ]; } Function { DefineFunction[ nu ]; } FunctionSpace { // Magnetic vector potential a (b = curl a) { Name Hcurl_a_Mag_2D; Type Form1P; BasisFunction { // a = a s // e e { Name se; NameOfCoef ae; Function BF_PerpendicularEdge; Support Domain_Mag; Entity NodesOf[ All ]; } } Constraint { { NameOfCoef ae; EntityType NodesOf; NameOfConstraint MagneticVectorPotential_2D; } } } // Source current density js (fully fixed space) { Name Hregion_j_Mag_2D; Type Vector; BasisFunction { { Name sr; NameOfCoef jsr; Function BF_RegionZ; Support DomainS_Mag; Entity DomainS_Mag; } } Constraint { { NameOfCoef jsr; EntityType Region; NameOfConstraint SourceCurrentDensityZ; } } } } Formulation { { Name Magnetostatics_a_2D; Type FemEquation; Quantity { { Name a ; Type Local; NameOfSpace Hcurl_a_Mag_2D; } { Name js; Type Local; NameOfSpace Hregion_j_Mag_2D; } } Equation { Galerkin { [ nu[] * Dof{d a} , {d a} ]; In Domain_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { [ - Dof{js} , {a} ]; In DomainS_Mag; Jacobian Vol; Integration CurlCurl; } } } } Resolution { { Name MagSta_a_2D; System { { Name Sys_Mag; NameOfFormulation Magnetostatics_a_2D; } } Operation { Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag]; } } } PostProcessing { { Name MagSta_a_2D; NameOfFormulation Magnetostatics_a_2D; Quantity { { Name a; Value { Local { [ {a} ]; In Domain_Mag; Jacobian Vol; } } } { Name az; Value { Local { [ CompZ[{a}] ]; In Domain_Mag; Jacobian Vol; } } } { Name b; Value { Local { [ {d a} ]; In Domain_Mag; Jacobian Vol; } } } { Name h; Value { Local { [ nu[] * {d a} ]; In Domain_Mag; Jacobian Vol; } } } } } } 8.3 Magnetodynamic problem ========================== As a third example we consider a magnetodynamic problem. The formulation is a two-dimensional a-v formulation (see file `MagDyn_av_2D.pro', which includes the same jacobian and integration library files as in *note Electrostatic problem::). It is applied to a core-inductor system (defined in file `CoreMassive.pro'), whose geometry has already been defined in file `Core.geo'. /* ------------------------------------------------------------------- File "CoreMassive.pro" This file defines the problem dependent data structures for the dynamic core-inductor problem. To compute the solution: getdp CoreMassive -msh Core.msh -solve MagDyn_av_2D To compute post-results: getdp CoreMassive -msh Core.msh -pos Map_a getdp CoreMassive -msh Core.msh -pos U_av ------------------------------------------------------------------- */ Group { Air = Region[ 101 ]; Core = Region[ 102 ]; Ind = Region[ 103 ]; AirInf = Region[ 111 ]; SurfaceGh0 = Region[ 1100 ]; SurfaceGe0 = Region[ 1101 ]; SurfaceGInf = Region[ 1102 ]; Val_Rint = 200.e-3; Val_Rext = 250.e-3; DomainCC_Mag = Region[ {Air, AirInf} ]; DomainC_Mag = Region[ {Ind, Core} ]; // Massive inductor + conducting core DomainB_Mag = Region[ {} ]; DomainS_Mag = Region[ {} ]; DomainInf = Region[ {AirInf} ]; Domain_Mag = Region[ {DomainCC_Mag, DomainC_Mag} ]; } Function { mu0 = 4.e-7 * Pi; murCore = 100.; nu [ #{Air, Ind, AirInf} ] = 1. / mu0; nu [ Core ] = 1. / (murCore * mu0); sigma [ Ind ] = 5.9e7; sigma [ Core ] = 2.5e7; Freq = 1.; } Constraint { { Name MagneticVectorPotential_2D; Case { { Region SurfaceGe0 ; Value 0.; } { Region SurfaceGInf; Value 0.; } } } { Name SourceCurrentDensityZ; Case { } } Val_I_ = 0.01 * 1000.; { Name Current_2D; Case { { Region Ind; Value Val_I_; } } } { Name Voltage_2D; Case { { Region Core; Value 0.; } } } } Include "Jacobian_Lib.pro" Include "Integration_Lib.pro" Include "MagDyn_av_2D.pro" PostOperation { { Name Map_a; NameOfPostProcessing MagDyn_av_2D; Operation { Print[ az, OnElementsOf Domain_Mag, File "Core_m_a.pos" ]; Print[ j, OnElementsOf Domain_Mag, File "Core_m_j.pos" ]; } } { Name U_av; NameOfPostProcessing MagDyn_av_2D; Operation { Print[ U, OnRegion Ind ]; Print[ I, OnRegion Ind ]; } } } /* ------------------------------------------------------------------- File "MagDyn_av_2D.pro" Magnetodynamics - Magnetic vector potential and electric scalar potential a-v formulation (2D) ------------------------------------------------------------------- I N P U T --------- GlobalGroup : (Extension '_Mag' is for Magnetic problem) ----------- Domain_Mag Whole magnetic domain DomainCC_Mag Nonconducting regions (not used) DomainC_Mag Conducting regions DomainS_Mag Inductor regions (Source) DomainV_Mag All regions in movement (for speed term) Function : -------- nu[] Magnetic reluctivity sigma[] Electric conductivity Velocity[] Velocity of regions Constraint : ---------- MagneticVectorPotential_2D Fixed magnetic vector potential (2D) (classical boundary condition) SourceCurrentDensityZ Fixed source current density (in Z direction) Voltage_2D Fixed voltage Current_2D Fixed Current Parameters : ---------- Freq Frequency (Hz) Parameters for time loop with theta scheme : Mag_Time0, Mag_TimeMax, Mag_DTime Initial time, Maximum time, Time step (s) Mag_Theta Theta (e.g. 1. : Implicit Euler, 0.5 : Cranck Nicholson) */ Group { DefineGroup[ Domain_Mag, DomainCC_Mag, DomainC_Mag, DomainS_Mag, DomainV_Mag ]; } Function { DefineFunction[ nu, sigma ]; DefineFunction[ Velocity ]; DefineVariable[ Freq ]; DefineVariable[ Mag_Time0, Mag_TimeMax, Mag_DTime, Mag_Theta ]; } FunctionSpace { // Magnetic vector potential a (b = curl a) { Name Hcurl_a_Mag_2D; Type Form1P; BasisFunction { // a = a s // e e { Name se; NameOfCoef ae; Function BF_PerpendicularEdge; Support Domain_Mag; Entity NodesOf[ All ]; } } Constraint { { NameOfCoef ae; EntityType NodesOf; NameOfConstraint MagneticVectorPotential_2D; } } } // Gradient of Electric scalar potential (2D) { Name Hregion_u_Mag_2D; Type Form1P; BasisFunction { { Name sr; NameOfCoef ur; Function BF_RegionZ; Support DomainC_Mag; Entity DomainC_Mag; } } GlobalQuantity { { Name U; Type AliasOf ; NameOfCoef ur; } { Name I; Type AssociatedWith; NameOfCoef ur; } } Constraint { { NameOfCoef U; EntityType Region; NameOfConstraint Voltage_2D; } { NameOfCoef I; EntityType Region; NameOfConstraint Current_2D; } } } // Source current density js (fully fixed space) { Name Hregion_j_Mag_2D; Type Vector; BasisFunction { { Name sr; NameOfCoef jsr; Function BF_RegionZ; Support DomainS_Mag; Entity DomainS_Mag; } } Constraint { { NameOfCoef jsr; EntityType Region; NameOfConstraint SourceCurrentDensityZ; } } } } Formulation { { Name Magnetodynamics_av_2D; Type FemEquation; Quantity { { Name a ; Type Local ; NameOfSpace Hcurl_a_Mag_2D; } { Name ur; Type Local ; NameOfSpace Hregion_u_Mag_2D; } { Name I ; Type Global; NameOfSpace Hregion_u_Mag_2D [I]; } { Name U ; Type Global; NameOfSpace Hregion_u_Mag_2D [U]; } { Name js; Type Local ; NameOfSpace Hregion_j_Mag_2D; } } Equation { Galerkin { [ nu[] * Dof{d a} , {d a} ]; In Domain_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { DtDof [ sigma[] * Dof{a} , {a} ]; In DomainC_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { [ sigma[] * Dof{ur} , {a} ]; In DomainC_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { [ - sigma[] * (Velocity[] *^ Dof{d a}) , {a} ]; In DomainV_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { [ - Dof{js} , {a} ]; In DomainS_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { DtDof [ sigma[] * Dof{a} , {ur} ]; In DomainC_Mag; Jacobian Vol; Integration CurlCurl; } Galerkin { [ sigma[] * Dof{ur} , {ur} ]; In DomainC_Mag; Jacobian Vol; Integration CurlCurl; } GlobalTerm { [ Dof{I} , {U} ]; In DomainC_Mag; } } } } Resolution { { Name MagDyn_av_2D; System { { Name Sys_Mag; NameOfFormulation Magnetodynamics_av_2D; Type ComplexValue; Frequency Freq; } } Operation { Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag]; } } { Name MagDyn_t_av_2D; System { { Name Sys_Mag; NameOfFormulation Magnetodynamics_av_2D; } } Operation { InitSolution[Sys_Mag]; SaveSolution[Sys_Mag]; TimeLoopTheta[Mag_Time0, Mag_TimeMax, Mag_DTime, Mag_Theta] { Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag]; } } } } PostProcessing { { Name MagDyn_av_2D; NameOfFormulation Magnetodynamics_av_2D; Quantity { { Name a; Value { Local { [ {a} ]; In Domain_Mag; Jacobian Vol; } } } { Name az; Value { Local { [ CompZ[{a}] ]; In Domain_Mag; Jacobian Vol; } } } { Name b; Value { Local { [ {d a} ]; In Domain_Mag; Jacobian Vol; } } } { Name h; Value { Local { [ nu[] * {d a} ]; In Domain_Mag; Jacobian Vol; } } } { Name j; Value { Local { [ - sigma[]*(Dt[{a}]+{ur}) ]; In DomainC_Mag; Jacobian Vol; } } } { Name jz; Value { Local { [ - sigma[]*CompZ[Dt[{a}]+{ur}] ]; In DomainC_Mag; Jacobian Vol; } } } { Name roj2; Value { Local { [ sigma[]*SquNorm[Dt[{a}]+{ur}] ]; In DomainC_Mag; Jacobian Vol; } } } { Name U; Value { Local { [ {U} ]; In DomainC_Mag; } } } { Name I; Value { Local { [ {I} ]; In DomainC_Mag; } } } } } } Appendix A File formats *********************** This chapter describes the file formats that cannot be modified by the user. The format of the problem definition structure is explained in *note Objects::, and *note Types for objects::. The format of the post-processing files is explained in *note Types for PostOperation::. A.1 Input file format ===================== The native mesh format read by GetDP is the mesh file format produced by Gmsh (`http://geuz.org/gmsh'). In its "version 1" incarnation, an `msh' file is divided into two sections, defining the nodes and the elements in the mesh. $NOD NUMBER-OF-NODES NODE-NUMBER X-COORD Y-COORD Z-COORD ... $ENDNOD $ELM NUMBER-OF-ELEMENTS ELM-NUMBER ELM-TYPE ELM-REGION UNUSED NUMBER-OF-NODES NODE-NUMBERS ... $ENDELM All the syntactic variables stand for integers except X-COORD, Y-COORD and Z-COORD which stand for floating point values. The ELM-TYPE value defines the geometrical type for the element: ELM-TYPE: `1' Line (2 nodes, 1 edge). `2' Triangle (3 nodes, 3 edges). `3' Quadrangle (4 nodes, 4 edges). `4' Tetrahedron (4 nodes, 6 edges, 4 facets). `5' Hexahedron (8 nodes, 12 edges, 6 facets). `6' Prism (6 nodes, 9 edges, 5 facets). `7' Pyramid (5 nodes, 8 edges, 5 facets). `15' Point (1 node). GetDP can also read more recent version of the `msh' format (2.0 and above). See the Gmsh documentation for more information. A.2 Output file format ====================== A.2.1 File `.pre' ----------------- The `.pre' file is generated by the pre-processing stage. It contains all the information about the degrees of freedom to be considered during the processing stage for a given resolution (i.e., unknowns, fixed values, initial values, etc.). $Resolution /* 'RESOLUTION-ID' */ MAIN-RESOLUTION-NUMBER NUMBER-OF-DOFDATA $EndResolution $DofData /* #DOFDATA-NUMBER */ RESOLUTION-NUMBER SYSTEM-NUMBER NUMBER-OF-FUNCTION-SPACES FUNCTION-SPACE-NUMBER ... NUMBER-OF-TIME-FUNCTIONS TIME-FUNCTION-NUMBER ... NUMBER-OF-PARTITIONS PARTITION-INDEX ... NUMBER-OF-ANY-DOF NUMBER-OF-DOF DOF-BASIS-FUNCTION-NUMBER DOF-ENTITY DOF-HARMONIC DOF-TYPE DOF-DATA ... $EndDofData ... with DOF-DATA: EQUATION-NUMBER NNZ (DOF-TYPE: 1; UNKNOWN) | DOF-VALUE DOF-TIME-FUNCTION-NUMBER (DOF-TYPE: 2; FIXED VALUE) | DOF-ASSOCIATE-DOF-NUMBER DOF-VALUE DOF-TIME-FUNCTION-NUMBER (DOF-TYPE: 3; ASSOCIATED DEGREE OF FREEDOM) | EQUATION-NUMBER DOF-VALUE (DOF-TYPE: 5; INITIAL VALUE FOR AN UNKNOWN) Notes: 1. There is one `$DofData' field for each system of equations considered in the resolution (including those considered in pre-resolutions). 2. The DOFDATA-NUMBER of a `$DofData' field is determined by the order of this field in the `.pre' file. 3. NUMBER-OF-DOF is the dimension of the considered system of equations, while NUMBER-OF-ANY-DOF is the total number of degrees of freedom before the application of constraints. 4. Each degree of freedom is coded with three integer values, which are the associated basis function, entity and harmonic numbers, i.e., DOF-BASIS-FUNCTION-NUMBER, DOF-ENTITY and DOF-HARMONIC. 5. NNZ is not used at the moment. A.2.2 File `.res' ----------------- The `.res' file is generated by the processing stage. It contains the solution of the problem (or a part of it in case of program interruption). $ResFormat /* GetDP vGETDP-VERSION-NUMBER, STRING-FOR-FORMAT */ 1.1 FILE-RES-FORMAT $EndResFormat $Solution /* DofData #DOFDATA-NUMBER */ DOFDATA-NUMBER TIME-VALUE TIME-IMAG-VALUE TIME-STEP-NUMBER SOLUTION-VALUE ... $EndSolution ... Notes: 1. A `$Solution' field contains the solution associated with a `$DofData' field. 2. There is one `$Solution' field for each time step, of which the time is TIME-VALUE (0 for non time dependent or non modal analyses) and the imaginary time is TIME-IMAG-VALUE (0 for non time dependent or non modal analyses). 3. The order of the SOLUTION-VALUEs in a `$Solution' field follows the numbering of the equations given in the `.pre' file (one floating point value for each degree of freedom). Appendix B Gmsh examples ************************ Gmsh is a three-dimensional finite element mesh generator with simple CAD and post-processing capabilities that can be used as a graphical front-end for GetDP. Gmsh can be downloaded from `http://geuz.org/gmsh'. This appendix reproduces verbatim the input files needed by Gmsh to produce the mesh files `mStrip.msh' and `Core.msh' used in the examples of *note Complete examples::. /* ------------------------------------------------------------------- File "mStrip.geo" This file is the geometrical description used by GMSH to produce the file "mStrip.msh". ------------------------------------------------------------------- */ /* Definition of some parameters for geometrical dimensions, i.e. h (height of 'Diel1'), w (width of 'Line'), t (thickness of 'Line') xBox (width of the air box) and yBox (height of the air box) */ h = 1.e-3 ; w = 4.72e-3 ; t = 0.035e-3 ; xBox = w/2. * 6. ; yBox = h * 12. ; /* Definition of parameters for local mesh dimensions */ s = 1. ; p0 = h / 10. * s ; pLine0 = w/2. / 10. * s ; pLine1 = w/2. / 50. * s ; pxBox = xBox / 10. * s ; pyBox = yBox / 8. * s ; /* Definition of gemetrical points */ Point(1) = { 0 , 0, 0, p0} ; Point(2) = { xBox, 0, 0, pxBox} ; Point(3) = { xBox, h, 0, pxBox} ; Point(4) = { 0 , h, 0, pLine0} ; Point(5) = { w/2., h, 0, pLine1} ; Point(6) = { 0 , h+t, 0, pLine0} ; Point(7) = { w/2., h+t, 0, pLine1} ; Point(8) = { 0 , yBox, 0, pyBox} ; Point(9) = { xBox, yBox, 0, pyBox} ; /* Definition of gemetrical lines */ Line(1) = {1,2}; Line(2) = {2,3}; Line(3) = {3,9}; Line(4) = {9,8}; Line(5) = {8,6}; Line(7) = {4,1}; Line(8) = {5,3}; Line(9) = {4,5}; Line(10) = {6,7}; Line(11) = {5,7}; /* Definition of geometrical surfaces */ Line Loop(12) = {8,-2,-1,-7,9}; Plane Surface(13) = {12}; Line Loop(14) = {10,-11,8,3,4,5}; Plane Surface(15) = {14}; /* Definition of Physical entities (surfaces, lines). The Physical entities tell GMSH the elements and their associated region numbers to save in the file 'mStrip.msh'. For example, the Region 111 is made of elements of surface 13, while the Region 121 is made of elements of lines 9, 10 and 11 */ Physical Surface (101) = {15} ; /* Air */ Physical Surface (111) = {13} ; /* Diel1 */ Physical Line (120) = {1} ; /* Ground */ Physical Line (121) = {9,10,11} ; /* Line */ Physical Line (130) = {2,3,4} ; /* SurfInf */ /* ------------------------------------------------------------------- File "Core.geo" This file is the geometrical description used by GMSH to produce the file "Core.msh". ------------------------------------------------------------------- */ dxCore = 50.e-3; dyCore = 100.e-3; xInd = 75.e-3; dxInd = 25.e-3; dyInd = 50.e-3; rInt = 200.e-3; rExt = 250.e-3; s = 1.; p0 = 12.e-3 *s; pCorex = 4.e-3 *s; pCorey0 = 8.e-3 *s; pCorey = 4.e-3 *s; pIndx = 5.e-3 *s; pIndy = 5.e-3 *s; pInt = 12.5e-3*s; pExt = 12.5e-3*s; Point(1) = {0,0,0,p0}; Point(2) = {dxCore,0,0,pCorex}; Point(3) = {dxCore,dyCore,0,pCorey}; Point(4) = {0,dyCore,0,pCorey0}; Point(5) = {xInd,0,0,pIndx}; Point(6) = {xInd+dxInd,0,0,pIndx}; Point(7) = {xInd+dxInd,dyInd,0,pIndy}; Point(8) = {xInd,dyInd,0,pIndy}; Point(9) = {rInt,0,0,pInt}; Point(10) = {rExt,0,0,pExt}; Point(11) = {0,rInt,0,pInt}; Point(12) = {0,rExt,0,pExt}; Line(1) = {1,2}; Line(2) = {2,5}; Line(3) = {5,6}; Line(4) = {6,9}; Line(5) = {9,10}; Line(6) = {1,4}; Line(7) = {4,11}; Line(8) = {11,12}; Line(9) = {2,3}; Line(10) = {3,4}; Line(11) = {6,7}; Line(12) = {7,8}; Line(13) = {8,5}; Circle(14) = {9,1,11}; Circle(15) = {10,1,12}; Line Loop(16) = {-6,1,9,10}; Plane Surface(17) = {16}; Line Loop(18) = {11,12,13,3}; Plane Surface(19) = {18}; Line Loop(20) = {7,-14,-4,11,12,13,-2,9,10}; Plane Surface(21) = {20}; Line Loop(22) = {8,-15,-5,14}; Plane Surface(23) = {22}; Physical Surface(101) = {21}; /* Air */ Physical Surface(102) = {17}; /* Core */ Physical Surface(103) = {19}; /* Ind */ Physical Surface(111) = {23}; /* AirInf */ Physical Line(1000) = {1,2}; /* Cut */ Physical Line(1001) = {2}; /* CutAir */ Physical Line(202) = {9,10}; /* SkinCore */ Physical Line(203) = {11,12,13}; /* SkinInd */ Physical Line(1100) = {1,2,3,4,5}; /* SurfaceGh0 */ Physical Line(1101) = {6,7,8}; /* SurfaceGe0 */ Physical Line(1102) = {15}; /* SurfaceGInf */ Appendix C Frequently asked questions ************************************* C.1 The basics ============== 1. What is GetDP? GetDP is a scientific software environment for the numerical solution of integro-differential equations, open to the coupling of physical problems (electromagnetic, thermal, mechanical, etc) as well as of numerical methods (finite element method, integral methods, etc). It can deal with such problems of various dimensions (1D, 2D, 2D axisymmetric or 3D) and time states (static, transient or harmonic). The main feature of GetDP is the closeness between the organization of data defining discrete problems (written by the user in ASCII data files) and the symbolic mathematical expressions of these problems. 2. What are the terms and conditions of use? GetDP is distributed under the terms of the GNU General Public License. See *note License:: for more information. 3. What does `GetDP' mean? It's an acronym for a "General environment for the treatment of Discrete Problems". 4. Where can I find more information? `http://geuz.org/getdp' is the primary site to obtain information about GetDP. You will find a short presentation, a complete reference guide as well as a searchable archive of the GetDP mailing list () on this site. C.2 Installation ================ 1. Which OSes does GetDP run on? Gmsh runs on Windows XP/Vista, Mac OS X, Linux and most Unix variants. 2. What do I need to compile GetDP from the sources? You need a C++ and a Fortran 77 compiler as well as the GSL (version 1.2 or higher; freely available from `http://sources.redhat.com/gsl'). 3. How do I compile GetDP? Just type ./configure; make; make install If you change some configuration options (type `./configure --help' to get the list of all available choices), don't forget to do `make clean' before rebuilding GetDP. 4. GetDP [from a binary distribution] complains about missing libraries. Try `ldd getdp' (or `otool -L getdp' on Mac OS X) to check if all the required shared libraries are installed on your system. If not, install them. If it still doesn't work, recompile GetDP from the sources. C.3 Usage ========= 1. How can I provide a mesh to GetDP? The only meshing format accepted by this version of GetDP is the `msh' format created by Gmsh `http://geuz.org/gmsh'. This format being very simple (see the Gmsh reference manual for more details), it should be straightforward to write a converter from your mesh format to the `msh' format. 2. How can I visualize the results produced by GetDP? You can specify a format in all post-processing operations. Available formats include `Table', `SimpleTable', `TimeTable' and `Gmsh'. `Table', `SimpleTable' and `TimeTable' output lists of numbers easily readable by Excel/gnuplot/Caleida Graph/etc. `Gmsh' outputs post-processing views directly loadable by Gmsh. 3. How do I change the linear solver used by GetDP? It depends on which linear solver toolkit was enabled when GetDP was compiled. With PETSc-based linear solvers you can either specify options on the command line, or in the `.petscrc' file located in your home directly. With Sparskit-based linear solvers you should edit the `solver.par' file in the current working directory. You can also remove the file: GetDP will give you the opportunity to create it dynamically next time you perform a linear system solution. Appendix D Tips and tricks ************************** * Install the 'info' version of this user's guide! On your (Unix) system, this can be achieved by 1) copying all getdp.info* files to the place where your info files live (usually /usr/info), and 2) issuing the command 'install-info /usr/info/getdp.info /usr/info/dir'. You will then be able to access the documentation with the command 'info getdp'. Note that particular sections ("nodes") can be accessed directly. For example, 'info getdp functionspace' will take you directly to the definition of the FunctionSpace object. * Use emacs to edit your files, and load the C++ mode! This permits automatic syntax highlighting and easy indentation. Automatic loading of the C++ mode for `.pro' files can be achieved by adding the following command in your `.emacs' file: `(setq auto-mode-alist (append '(("\\.pro$" . c++-mode)) auto-mode-alist))'. * Define integration and Jacobian method in separate files, reusable in all your problem definition structures (*note Includes::). Define meshes, groups, functions and constraints in one file dependent of the geometrical model, and function spaces, formulations, resolutions and post-processings in files independent of the geometrical model. * Use `All' as soon as possible in the definition of topological entities used as `Entity' of `BasisFunction's. This will prevent GetDP from constructing unnecessary lists of entities. * Intentionally misspelling an object type in the problem definition structure will produce an error message listing all available types in the particular context. * If you don't specify the mandatory arguments on the command line, GetDP will give you the available choices. For example, 'getdp test -pos' (the name of the PostOperation is missing) will produce an error message listing all available PostOperations. Appendix E Version history ************************** New in 2.0.0: general code cleanup (separated interface from legacy code; removed various undocumented, unstable and otherwise experimental features; moved to C++); updated input file formats; default solvers are now based on PETSc; small bug fixes (binary .res read, Newmark -restart). New in 1.2: Windows versions do not depend on Cygwin anymore; major parser cleanup (loops & co). New in 1.1: New eigensolver based on Arpack (EigenSolve); generalized old Lanczos solver to work with GSL+lapack; reworked PETSc interface, which now requires PETSc 2.3; documented many previously undocumented features (loops, conditionals, strings, link constraints, etc.); various improvements and bug fixes. New in 1.0: New license (GNU GPL); added support for latest Gmsh mesh file format; more code cleanups. New in 0.91: Merged moving band and multi-harmonic code; new loops and conditionals in the parser; removed old readline code (just use GNU readline if available); upgraded to latest Gmsh post-processing format; various small enhancements and bug fixes. New in 0.89: Code cleanup. New in 0.88: Integrated FMM code. New in 0.87: Fixed major performance problem on Windows (matrix assembly and post-processing can be up to 3-4 times faster with 0.87 compared to 0.86, bringing performance much closer to Unix versions); fixed stack overflow on Mac OS X; Re-introduced face basis functions mistakenly removed in 0.86; fixed post-processing bug with pyramidal basis functions; new build system based on autoconf. New in 0.86: Updated Gmsh output format; many small bug fixes. New in 0.85: Upgraded communication interface with Gmsh; new ChangeOfValues option in PostOperation; many internal changes. New in 0.84: New ChangeOfCoordinate option in PostOperation; fixed crash in InterpolationAkima; improved interactive postprocessing (-ipos); changed syntax of parametric OnGrid ($S, $T -> $A, $B, $C); corrected Skin for non simplicial meshes; fixed floating point exception in diagonal matrix scaling; many other small fixes and cleanups. New in 0.83: Fixed bugs in SaveSolutions[] and InitSolution[]; fixed corrupted binary post-processing files in the harmonic case for the Gmsh format; output files are now created relatively to the input file directory; made solver options available on the command line; added optional matrix scaling and changed default parameter file name to 'solver.par' (Warning: please check the scaling definition in your old SOLVER.PAR files); generalized syntax for lists (start:[incr]end -> start:end:incr); updated reference guide; added a new short presentation on the web site; OnCut -> OnSection; new functional syntax for resolution operations (e.g. Generate X -> Generate[X]); many other small fixes and cleanups. New in 0.82: Added communication socket for interactive use with Gmsh; corrected (again) memory problem (leak + seg. fault) in time stepping schemes; corrected bug in Update[]. New in 0.81: Generalization of transformation jacobians (spherical and rectangular, with optional parameters); changed handling of missing command line arguments; enhanced Print OnCut; fixed memory leak for time domain analysis of coupled problems; -name option; fixed seg. fault in ILUK. New in 0.80: Fixed computation of time derivatives on first time step (in post-processing); added tolerance in transformation jacobians; fixed parsing of DOS files (carriage return problems); automatic memory reallocation in ILUD/ILUK. New in 0.79: Various bug fixes (mainly for the post-processing of intergal quantities); automatic treatment of degenerated cases in axisymmetrical problems. New in 0.78: Various bug fixes. New in 0.77: Changed syntax for PostOperations (Plot suppressed in favour of Print; Plot OnRegion becomes Print OnElementsOf); changed table oriented post-processing formats; new binary formats; new error diagnostics. New in 0.76: Reorganized high order shape functions; optimization of the post-processing (faster and less bloated); lots of internal cleanups. New in 0.74: High order shape functions; lots of small bug fixes. New in 0.73: Eigen value problems (Lanczos); minor corrections. New in 0.7: constraint syntax; fourier transform; unary minus correction; complex integral quantity correction; separate iteration matrix generation. New in 0.6: Second order time derivatives; Newton nonlinear scheme; Newmark time stepping scheme; global quantity syntax; interactive post-processing; tensors; integral quantities; post-processing facilities. New in 0.3: First distributed version. Appendix F Copyright and credits ******************************** GetDP is copyright (C) 1997-2010 Patrick Dular and Christophe Geuzaine Major code contributions to GetDP have been provided by Johan Gyselinck and Ruth Sabariego. Other code contributors include: David Colignon, Tuan Ledinh, Patrick Lefevre, Andre Nicolet, Jean-Francois Remacle, Timo Tarhasaari, Christophe Trophime and Marc Ume. See the source code for more details. The AVL tree code (DataStr/avl.*) is copyright (C) 1988-1993, 1995 The Regents of the University of California. Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the name of the University of California not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission. The University of California makes no representations about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty. The Arpack code (in the Arpack subdirectory) was written by Danny Sorensen, Richard Lehoucq, Chao Yang and Kristi Maschhoff from the Dept. of Computational & Applied Mathematics at Rice University, Houston, Texas, USA. See http://www.caam.rice.edu/software/ARPACK/ for more info. This version of GetDP may contain code (in the Sparskit subdirectory) copyright (C) 1990 Yousef Saad: check the configuration options. Thanks to the following folks who have contributed by providing fresh ideas on theoretical or programming topics, who have sent patches, requests for changes or improvements, or who gave us access to exotic machines for testing GetDP: Olivier Adam, Alejandro Angulo, Geoffrey Deliege, Mark Evans, Philippe Geuzaine, Eric Godard, Sebastien Guenneau, Francois Henrotte, Daniel Kedzierski, Samuel Kvasnica, Benoit Meys, Uwe Pahner, Georgia Psoni, Robert Struijs, Ahmed Rassili, Thierry Scordilis, Herve Tortel, Jose Geraldo A. Brito Neto, Matthias Fenner, Daryl Van Vorst. Appendix G License ****************** 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 software--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 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 legal 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 0. 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 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. 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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. 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 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. 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It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. Copyright (C) This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Also add information on how to contact you by electronic and paper mail. If the program is interactive, make it output a short notice like this when it starts in an interactive mode: Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program. You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names: Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker. , 1 April 1989 Ty Coon, President of Vice This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License. Concept index ************* .msh file: See A.1. (line 4943) .pre file: See A.2.1. (line 4998) .res file: See A.2.2. (line 5050) Acknowledgments: See Appendix F. (line 5453) Analytical integration: See 5.6. (line 1536) Approximation spaces: See 5.4. (line 1396) Arguments: See 4.6. (line 922) Arguments, definition: See 4.8. (line 1004) Authors, e-mail: See 1.3. (line 365) Axisymmetric, transformation: See 5.5. (line 1497) Basis Functions: See 5.4. (line 1396) Binary operators: See 4.5.1. (line 809) Boundary conditions: See 5.3. (line 1327) Boundary Element Method: See 1.2. (line 317) Bugs, reporting: See 1.3. (line 365) Built-in functions: See 4.6. (line 922) Change of coordinates: See 5.5. (line 1497) Changelog: See Appendix E. (line 5347) Circuit equations: See 5.3. (line 1327) Command line options: See 3. (line 427) Comments: See 4.1. (line 582) Complete examples: See 8. (line 4077) Complex-valued, system: See 5.8. (line 1670) Concepts, index: See ``Concept index''. (line 5850) Conditionals: See 4.11. (line 1133) Constant, definition: See 4.4. (line 664) Constant, evaluation: See 4.4. (line 664) Constraint, definition: See 5.3. (line 1327) Constraint, examples: See 7.4. (line 3362) Constraint, types: See 6.3. (line 2363) Contact information: See 1.3. (line 365) Contributors, list: See Appendix F. (line 5453) Coordinate change: See 5.5. (line 1497) Copyright <1>: See Appendix F. (line 5453) Copyright: See ``Copying conditions''. (line 159) Credits: See Appendix F. (line 5453) Curl: See 4.10. (line 1044) Current values: See 4.7. (line 957) Dependences, objects: See 1.1. (line 223) Derivative, exterior: See 4.10. (line 1044) Derivative, time: See 5.7. (line 1578) Developments, future: See 1.2. (line 317) Differential operators: See 4.10. (line 1044) Discrete function spaces: See 5.4. (line 1396) Discrete quantities: See 4.10. (line 1044) Discretized Geometry: See 5.1. (line 1197) Divergence: See 4.10. (line 1044) Document syntax: See 2.1. (line 398) Download <1>: See ``Copying conditions''. (line 159) Download: See ``Obtaining GetDP''. (line 149) E-mail, authors: See 1.3. (line 365) Edge element space, example: See 7.8.3. (line 3769) Efficiency, tips: See Appendix D. (line 5307) Electromagnetism: See 1.2. (line 317) Electrostatic formulation: See 7.8.1. (line 3711) Elementary matrices: See 5.7. (line 1578) Entities, topological: See 5.1. (line 1197) Equations: See 5.7. (line 1578) Evaluation mechanism: See 4.4. (line 664) Evaluation, order: See 4.5.2. (line 895) Examples, complete: See 8. (line 4077) Examples, short: See 7. (line 3227) Exporting results: See 5.10. (line 1802) Expression, definition: See 4.3. (line 608) Exterior derivative: See 4.10. (line 1044) FAQ: See Appendix C. (line 5206) Fields: See 4.10. (line 1044) File, .msh: See A.1. (line 4943) File, .pre: See A.2.1. (line 4998) File, .res: See A.2.2. (line 5050) File, comment: See 4.1. (line 582) File, include: See 4.2. (line 594) File, mesh: See A.1. (line 4943) File, pre-processing: See A.2.1. (line 4998) File, result: See A.2.2. (line 5050) Finite Difference Method: See 1.2. (line 317) Finite Element Method: See 1.2. (line 317) Finite Volume Method: See 1.2. (line 317) Floating point numbers: See 4.4. (line 664) Floating potential, example: See 7.8.2. (line 3743) Format, output: See 5.10. (line 1802) Formulation, definition: See 5.7. (line 1578) Formulation, electrostatics: See 7.8.1. (line 3711) Formulation, examples: See 7.8. (line 3708) Formulation, types: See 6.7. (line 2657) Frequency: See 5.8. (line 1670) Frequently asked questions: See Appendix C. (line 5206) Function groups: See 5.1. (line 1197) Function space, definition: See 5.4. (line 1396) Function space, examples: See 7.5. (line 3414) Function space, types: See 6.4. (line 2411) Function, definition <1>: See 5.2. (line 1287) Function, definition: See 4.6. (line 922) Function, examples: See 7.3. (line 3280) Future developments: See 1.2. (line 317) Gauss, integration: See 5.6. (line 1536) Geometric transformations: See 5.5. (line 1497) Global quantity: See 5.7. (line 1578) Global quantity, example: See 7.8.2. (line 3743) Gmsh, examples: See Appendix B. (line 5080) Gmsh, file format: See A.1. (line 4943) Gradient: See 4.10. (line 1044) Grid: See 5.1. (line 1197) Group, definition: See 5.1. (line 1197) Group, examples: See 7.2. (line 3251) Group, types: See 6.1. (line 1886) Hierarchical basis functions: See 5.4. (line 1396) History, versions: See Appendix E. (line 5347) Includes: See 4.2. (line 594) Index, concepts: See ``Concept index''. (line 5850) Index, metasyntactic variables: See ``Metasyntactic variable index''. (line 5853) Index, syntax: See ``Syntax index''. (line 5856) Input file format: See A.1. (line 4943) Integer numbers: See 4.4. (line 664) Integral Equation Method: See 1.2. (line 317) Integral quantity: See 5.7. (line 1578) Integration, definition: See 5.6. (line 1536) Integration, examples: See 7.7. (line 3685) Integration, types: See 6.6. (line 2615) Internet address <1>: See ``Copying conditions''. (line 159) Internet address: See ``Obtaining GetDP''. (line 149) Interpolation <1>: See 5.4. (line 1396) Interpolation: See 4.10. (line 1044) Introduction: See 1. (line 198) Iterative loop: See 5.8. (line 1670) Jacobian, definition: See 5.5. (line 1497) Jacobian, examples: See 7.6. (line 3653) Jacobian, types: See 6.5. (line 2544) Keywords, index: See ``Syntax index''. (line 5856) License <1>: See Appendix G. (line 5506) License: See ``Copying conditions''. (line 159) Linear system solving: See 5.8. (line 1670) Linking, objects: See 1.1. (line 223) Local quantity: See 5.7. (line 1578) Loops: See 4.11. (line 1133) Mailing list <1>: See 1.3. (line 365) Mailing list: See ``Copying conditions''. (line 159) Maps: See 5.10. (line 1802) Matrices, elementary: See 5.7. (line 1578) Mechanics: See 1.2. (line 317) Mesh: See 5.1. (line 1197) Mesh, examples: See Appendix B. (line 5080) Mesh, file format: See A.1. (line 4943) Metasyntactic variables, index: See ``Metasyntactic variable index''. (line 5853) Method of Moments: See 1.2. (line 317) Networks: See 5.3. (line 1327) Newmark, time scheme: See 5.8. (line 1670) Newton, nonlinear scheme: See 5.8. (line 1670) Nodal function space, example: See 7.8.1. (line 3711) Nonlinear system solving: See 5.8. (line 1670) Numbers, integer: See 4.4. (line 664) Numbers, real: See 4.4. (line 664) Numerical integration: See 5.6. (line 1536) Objects, definition: See 5. (line 1188) Objects, dependences: See 1.1. (line 223) Objects, types: See 6. (line 1883) Operating system: See 3. (line 427) Operation, priorities: See 4.5.2. (line 895) Operators, definition: See 4.5.1. (line 809) Operators, differential: See 4.10. (line 1044) Options, command line: See 3. (line 427) Order of evaluation: See 4.5.2. (line 895) Output file format: See A.2. (line 4995) Overview: See 1. (line 198) Parameters: See 4.6. (line 922) Philosophy, general: See 1.1. (line 223) Physical problems: See 1.2. (line 317) Picard, nonlinear scheme: See 5.8. (line 1670) Piecewise functions <1>: See 5.2. (line 1287) Piecewise functions: See 4.6. (line 922) Platforms: See 3. (line 427) Post-operation, definition: See 5.10. (line 1802) Post-operation, examples: See 7.11. (line 4013) Post-operation, types: See 6.10. (line 2941) Post-processing, definition: See 5.9. (line 1750) Post-processing, examples: See 7.10. (line 3972) Post-processing, types: See 6.9. (line 2926) Priorities, operations: See 4.5.2. (line 895) Processing cycle: See 1.1. (line 223) Quantities, discrete: See 4.10. (line 1044) Quantity, global: See 5.7. (line 1578) Quantity, integral: See 5.7. (line 1578) Quantity, local: See 5.7. (line 1578) Quantity, post-processing: See 5.9. (line 1750) Questions, frequently asked: See Appendix C. (line 5206) Reading, guidelines: See 2. (line 381) Real numbers: See 4.4. (line 664) Region groups: See 5.1. (line 1197) Registers, definition: See 4.9. (line 1016) Relaxation factor: See 5.8. (line 1670) Reporting bugs: See 1.3. (line 365) Resolution, definition: See 5.8. (line 1670) Resolution, examples: See 7.9. (line 3833) Resolution, types: See 6.8. (line 2714) Results, exploitation: See 5.9. (line 1750) Results, export: See 5.10. (line 1802) Rules, syntactic: See 2.1. (line 398) Running GetDP: See 3. (line 427) Scope of GetDP: See 1.2. (line 317) Sections: See 5.10. (line 1802) Short examples: See 7. (line 3227) Solving, system: See 5.8. (line 1670) Spaces, discrete: See 5.4. (line 1396) String: See 4.4. (line 664) Symmetry, integral kernel: See 5.7. (line 1578) Syntax, index: See ``Syntax index''. (line 5856) Syntax, rules: See 2.1. (line 398) System, definition: See 5.8. (line 1670) Ternary operators: See 4.5.1. (line 809) Thermics: See 1.2. (line 317) Theta, time scheme: See 5.8. (line 1670) Time derivative: See 5.7. (line 1578) Time stepping: See 5.8. (line 1670) Time, discretization: See 5.8. (line 1670) Tips: See Appendix D. (line 5307) Tools, order of definition: See 1.1. (line 223) Topology: See 5.1. (line 1197) Transformations, geometric: See 5.5. (line 1497) Tree: See 5.1. (line 1197) Tricks: See Appendix D. (line 5307) Types, definition: See 6. (line 1883) Unary operators: See 4.5.1. (line 809) User-defined functions: See 5.2. (line 1287) Values, current: See 4.7. (line 957) Variables, index: See ``Metasyntactic variable index''. (line 5853) Versions: See Appendix E. (line 5347) Web site <1>: See ``Copying conditions''. (line 159) Web site: See ``Obtaining GetDP''. (line 149) Wiki: See 8. (line 4077) Metasyntactic variable index **************************** ...: See 2.1. (line 398) :: See 2.1. (line 398) <, >: See 2.1. (line 398) AFFECTATION: See 4.4. (line 664) ARGUMENT: See 4.8. (line 1004) BASIS-FUNCTION-ID: See 5.4. (line 1396) BASIS-FUNCTION-LIST: See 5.4. (line 1396) BASIS-FUNCTION-TYPE <1>: See 6.4. (line 2411) BASIS-FUNCTION-TYPE: See 5.4. (line 1396) BUILT-IN-FUNCTION-ID: See 4.6. (line 922) COEF-ID: See 5.4. (line 1396) CONSTANT-DEF: See 4.4. (line 664) CONSTANT-ID: See 4.4. (line 664) CONSTRAINT-CASE-ID: See 5.3. (line 1327) CONSTRAINT-CASE-VAL: See 5.3. (line 1327) CONSTRAINT-ID: See 5.3. (line 1327) CONSTRAINT-TYPE <1>: See 6.3. (line 2363) CONSTRAINT-TYPE: See 5.3. (line 1327) CONSTRAINT-VAL: See 5.3. (line 1327) COORD-FUNCTION-ID: See 6.2.5. (line 2277) ELEMENT-TYPE <1>: See 6.6. (line 2615) ELEMENT-TYPE: See 5.6. (line 1536) ETC: See 2.1. (line 398) EXPRESSION: See 4.3. (line 608) EXPRESSION-CHAR: See 4.4. (line 664) EXPRESSION-CST: See 4.4. (line 664) EXPRESSION-CST-LIST: See 4.4. (line 664) EXPRESSION-CST-LIST-ITEM: See 4.4. (line 664) EXPRESSION-LIST: See 4.3. (line 608) EXTENDED-MATH-FUNCTION-ID: See 6.2.2. (line 2063) FORMULATION-ID: See 5.7. (line 1578) FORMULATION-LIST: See 5.8. (line 1670) FORMULATION-TYPE <1>: See 6.7. (line 2657) FORMULATION-TYPE: See 5.7. (line 1578) FUNCTION-ID: See 5.2. (line 1287) FUNCTION-SPACE-ID: See 5.4. (line 1396) FUNCTION-SPACE-TYPE <1>: See 6.4. (line 2411) FUNCTION-SPACE-TYPE: See 5.4. (line 1396) GLOBAL-QUANTITY-ID: See 5.4. (line 1396) GLOBAL-QUANTITY-TYPE <1>: See 6.4. (line 2411) GLOBAL-QUANTITY-TYPE: See 5.4. (line 1396) GREEN-FUNCTION-ID: See 6.2.3. (line 2129) GROUP-DEF: See 5.1. (line 1197) GROUP-ID: See 5.1. (line 1197) GROUP-LIST: See 5.1. (line 1197) GROUP-LIST-ITEM: See 5.1. (line 1197) GROUP-SUB-TYPE: See 5.1. (line 1197) GROUP-TYPE <1>: See 6.1. (line 1886) GROUP-TYPE: See 5.1. (line 1197) INTEGER: See 4.4. (line 664) INTEGRAL-VALUE: See 5.9. (line 1750) INTEGRATION-ID: See 5.6. (line 1536) INTEGRATION-TYPE <1>: See 6.6. (line 2615) INTEGRATION-TYPE: See 5.6. (line 1536) JACOBIAN-ID: See 5.5. (line 1497) JACOBIAN-TYPE <1>: See 6.5. (line 2544) JACOBIAN-TYPE: See 5.5. (line 1497) LOCAL-TERM-TYPE <1>: See 6.7. (line 2657) LOCAL-TERM-TYPE: See 5.7. (line 1578) LOCAL-VALUE: See 5.9. (line 1750) LOOP: See 4.11. (line 1133) MATH-FUNCTION-ID: See 6.2.1. (line 1969) MISC-FUNCTION-ID: See 6.2.6. (line 2302) OPERATOR-BINARY: See 4.5.1. (line 809) OPERATOR-TERNARY-LEFT: See 4.5.1. (line 809) OPERATOR-TERNARY-RIGHT: See 4.5.1. (line 809) OPERATOR-UNARY: See 4.5.1. (line 809) POST-OPERATION-FMT <1>: See 6.10. (line 3180) POST-OPERATION-FMT: See 5.10. (line 1802) POST-OPERATION-ID: See 5.10. (line 1802) POST-OPERATION-OP: See 5.10. (line 1802) POST-PROCESSING-ID: See 5.9. (line 1750) POST-QUANTITY-ID: See 5.9. (line 1750) POST-QUANTITY-TYPE: See 5.9. (line 1750) POST-VALUE <1>: See 6.9. (line 2926) POST-VALUE: See 5.9. (line 1750) PRINT-OPTION <1>: See 6.10. (line 3027) PRINT-OPTION: See 5.10. (line 1802) PRINT-SUPPORT <1>: See 6.10. (line 2941) PRINT-SUPPORT: See 5.10. (line 1802) QUANTITY: See 4.10. (line 1044) QUANTITY-DOF: See 4.10. (line 1044) QUANTITY-ID: See 4.10. (line 1044) QUANTITY-OPERATOR: See 4.10. (line 1044) QUANTITY-TYPE <1>: See 6.7. (line 2657) QUANTITY-TYPE: See 5.7. (line 1578) REAL: See 4.4. (line 664) REGISTER-VALUE-GET: See 4.9. (line 1016) REGISTER-VALUE-SET: See 4.9. (line 1016) RESOLUTION-ID: See 5.8. (line 1670) RESOLUTION-OP <1>: See 6.8. (line 2714) RESOLUTION-OP: See 5.8. (line 1670) STRING: See 4.4. (line 664) STRING-ID: See 4.4. (line 664) SUB-SPACE-ID: See 5.4. (line 1396) SYSTEM-ID: See 5.8. (line 1670) SYSTEM-TYPE: See 5.8. (line 1670) TERM-OP-TYPE <1>: See 6.7. (line 2657) TERM-OP-TYPE: See 5.7. (line 1578) TYPE-FUNCTION-ID: See 6.2.4. (line 2168) |: See 2.1. (line 398) Syntax index ************ !: See 4.5.1. (line 809) !=: See 4.5.1. (line 809) #include: See 4.2. (line 594) #INTEGER: See 4.9. (line 1016) $A: See 4.7. (line 957) $B: See 4.7. (line 957) $C: See 4.7. (line 957) $DTime: See 4.7. (line 957) $EigenvalueImag: See 4.7. (line 957) $EigenvalueReal: See 4.7. (line 957) $INTEGER: See 4.8. (line 1004) $Iteration: See 4.7. (line 957) $Theta: See 4.7. (line 957) $Time: See 4.7. (line 957) $TimeStep: See 4.7. (line 957) $X: See 4.7. (line 957) $XS: See 4.7. (line 957) $Y: See 4.7. (line 957) $YS: See 4.7. (line 957) $Z: See 4.7. (line 957) $ZS: See 4.7. (line 957) %: See 4.5.1. (line 809) &: See 4.5.1. (line 809) &&: See 4.5.1. (line 809) (): See 4.5.2. (line 895) *: See 4.5.1. (line 809) +: See 4.5.1. (line 809) -: See 4.5.1. (line 809) -adapt: See 3. (line 510) -bin: See 3. (line 520) -cal: See 3. (line 462) -check: See 3. (line 528) -help: See 3. (line 552) -info: See 3. (line 546) -msh: See 3. (line 480) -name: See 3. (line 495) -order: See 3. (line 515) -p: See 3. (line 538) -pos: See 3. (line 471) -pre: See 3. (line 451) -progress: See 3. (line 539) -res: See 3. (line 490) -restart: See 3. (line 501) -socket: See 3. (line 523) -solve: See 3. (line 505) -split: See 3. (line 487) -v: See 3. (line 531) -verbose: See 3. (line 532) -version: See 3. (line 549) /: See 4.5.1. (line 809) /*, */: See 4.1. (line 582) //: See 4.1. (line 582) /\: See 4.5.1. (line 809) 0D: See 4.4. (line 664) 1D: See 4.4. (line 664) 2D: See 4.4. (line 664) 3D: See 4.4. (line 664) <: See 4.5.1. (line 809) <=: See 4.5.1. (line 809) = <1>: See 5.2. (line 1287) = <2>: See 5.1. (line 1197) =: See 4.4. (line 664) ==: See 4.5.1. (line 809) >: See 4.5.1. (line 809) >=: See 4.5.1. (line 809) ?:: See 4.5.1. (line 809) ^: See 4.5.1. (line 809) Acos: See 6.2.1. (line 2011) Adapt: See 6.10. (line 3091) Adaptation: See 6.10. (line 3219) AliasOf: See 6.4. (line 2533) All: See 5.5. (line 1497) Analytic: See 5.6. (line 1536) Asin: See 6.2.1. (line 2000) Assign: See 6.3. (line 2364) AssignFromResolution: See 6.3. (line 2370) AssociatedWith: See 6.4. (line 2536) Atan: See 6.2.1. (line 2022) Atan2: See 6.2.1. (line 2027) BasisFunction: See 5.4. (line 1396) BF: See 4.10. (line 1087) BF_CurlEdge: See 6.4. (line 2455) BF_CurlGroupOfEdges: See 6.4. (line 2471) BF_CurlGroupOfPerpendicularEdge: See 6.4. (line 2484) BF_CurlPerpendicularEdge: See 6.4. (line 2478) BF_dGlobal: See 6.4. (line 2512) BF_DivFacet: See 6.4. (line 2458) BF_DivPerpendicularFacet: See 6.4. (line 2492) BF_Edge: See 6.4. (line 2443) BF_Facet: See 6.4. (line 2446) BF_Global: See 6.4. (line 2508) BF_GradGroupOfNodes: See 6.4. (line 2464) BF_GradNode: See 6.4. (line 2452) BF_GroupOfEdges: See 6.4. (line 2468) BF_GroupOfNodes: See 6.4. (line 2461) BF_GroupOfPerpendicularEdge: See 6.4. (line 2481) BF_Node: See 6.4. (line 2440) BF_NodeX: See 6.4. (line 2516) BF_NodeY: See 6.4. (line 2519) BF_NodeZ: See 6.4. (line 2522) BF_One: See 6.4. (line 2528) BF_PerpendicularEdge: See 6.4. (line 2475) BF_PerpendicularFacet: See 6.4. (line 2488) BF_Region: See 6.4. (line 2496) BF_RegionX: See 6.4. (line 2499) BF_RegionY: See 6.4. (line 2502) BF_RegionZ: See 6.4. (line 2505) BF_Volume: See 6.4. (line 2449) BF_Zero: See 6.4. (line 2525) Break: See 6.8. (line 2845) Case <1>: See 5.6. (line 1536) Case <2>: See 5.5. (line 1497) Case: See 5.3. (line 1327) ChangeOfCoordinates: See 6.10. (line 3131) ChangeOfValues: See 6.10. (line 3140) Complex: See 6.2.4. (line 2169) CompX: See 6.2.4. (line 2213) CompXX: See 6.2.4. (line 2228) CompXY: See 6.2.4. (line 2233) CompXZ: See 6.2.4. (line 2238) CompY: See 6.2.4. (line 2218) CompYX: See 6.2.4. (line 2243) CompYY: See 6.2.4. (line 2248) CompYZ: See 6.2.4. (line 2253) CompZ: See 6.2.4. (line 2223) CompZX: See 6.2.4. (line 2258) CompZY: See 6.2.4. (line 2263) CompZZ: See 6.2.4. (line 2268) Constraint <1>: See 5.4. (line 1396) Constraint: See 5.3. (line 1327) Cos: See 6.2.1. (line 2006) Cosh: See 6.2.1. (line 2038) Criterion: See 5.6. (line 1536) Cross: See 6.2.2. (line 2064) Curl: See 4.10. (line 1099) CurlInv: See 4.10. (line 1112) d: See 4.10. (line 1093) DecomposeInSimplex: See 6.10. (line 3148) DefineConstant: See 4.4. (line 664) DefineFunction: See 5.2. (line 1287) DefineGroup: See 5.1. (line 1197) Depth: See 6.10. (line 3045) deRham: See 6.7. (line 2667) DestinationSystem: See 5.8. (line 1670) Dimension: See 6.10. (line 3065) dInterpolationAkima: See 6.2.6. (line 2348) dInterpolationLinear: See 6.2.6. (line 2336) dInv: See 4.10. (line 1106) Div: See 4.10. (line 1103) DivInv: See 4.10. (line 1116) Dof: See 4.10. (line 1062) Dt: See 6.7. (line 2688) DtDof: See 6.7. (line 2691) DtDt: See 6.7. (line 2694) DtDtDof: See 6.7. (line 2698) DualEdgesOf: See 6.1. (line 1953) DualFacetsOf: See 6.1. (line 1956) DualNodesOf: See 6.1. (line 1950) DualVolumesOf: See 6.1. (line 1959) EdgesOf: See 6.1. (line 1903) EdgesOfTreeIn: See 6.1. (line 1940) EigenSolve: See 6.8. (line 2863) EigenvalueLegend: See 6.10. (line 3172) ElementsOf: See 6.1. (line 1918) Else: See 6.8. (line 2838) EndFor: See 4.11. (line 1168) EndIf: See 4.11. (line 1175) Entity: See 5.4. (line 1396) EntitySubType: See 5.4. (line 1396) EntityType: See 5.4. (line 1396) Equation: See 5.7. (line 1578) Evaluate: See 6.8. (line 2809) Exp: See 6.2.1. (line 1975) F_CompElementNum: See 6.2.6. (line 2324) F_Cos_wt_p: See 6.2.2. (line 2101) F_Period: See 6.2.2. (line 2119) F_Sin_wt_p: See 6.2.2. (line 2110) Fabs: See 6.2.1. (line 2048) FacetsOf: See 6.1. (line 1908) FacetsOfTreeIn: See 6.1. (line 1945) FemEquation: See 6.7. (line 2658) File: See 6.10. (line 3028) Fmod: See 6.2.1. (line 2053) For ( EXPRESSION-CST : EXPRESSION-CST ): See 4.11. (line 1137) For ( EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST ):See 4.11. (line 1143) For STRING In { EXPRESSION-CST : EXPRESSION-CST : EXPRESSION-CST }:See 4.11. (line 1159) For STRING In { EXPRESSION-CST : EXPRESSION-CST }:See 4.11. (line 1151) Form0: See 6.4. (line 2412) Form1: See 6.4. (line 2415) Form1P: See 6.4. (line 2424) Form2: See 6.4. (line 2418) Form2P: See 6.4. (line 2428) Form3: See 6.4. (line 2421) Format <1>: See 6.10. (line 3086) Format: See 5.10. (line 1802) Formulation <1>: See 5.7. (line 1578) Formulation: See 5.4. (line 1396) FourierTransform: See 6.8. (line 2882) Frequency <1>: See 6.10. (line 3081) Frequency: See 5.8. (line 1670) FrequencyLegend: See 6.10. (line 3165) Function <1>: See 5.4. (line 1396) Function: See 5.2. (line 1287) FunctionSpace: See 5.4. (line 1396) Galerkin: See 6.7. (line 2664) Gauss: See 6.6. (line 2616) GaussLegendre: See 6.6. (line 2619) Generate: See 6.8. (line 2715) GenerateJac: See 6.8. (line 2725) GenerateOnly: See 6.8. (line 2750) GenerateOnlyJac: See 6.8. (line 2755) GenerateSeparate: See 6.8. (line 2740) GeoElement: See 5.6. (line 1536) Global <1>: See 6.7. (line 2677) Global: See 6.1. (line 1894) GlobalEquation: See 5.7. (line 1578) GlobalQuantity: See 5.4. (line 1396) GlobalTerm: See 5.7. (line 1578) Gmsh: See 6.10. (line 3181) GmshParsed: See 6.10. (line 3182) Gnuplot: See 6.10. (line 3206) Grad: See 4.10. (line 1096) GradHelmholtz: See 6.2.3. (line 2158) GradInv: See 4.10. (line 1109) GradLaplace: See 6.2.3. (line 2146) Group <1>: See 5.4. (line 1396) Group: See 5.1. (line 1197) GroupsOfEdgesOf: See 6.1. (line 1927) GroupsOfEdgesOnNodesOf: See 6.1. (line 1934) GroupsOfNodesOf: See 6.1. (line 1923) HarmonicToTime: See 6.10. (line 3059) Helmholtz: See 6.2.3. (line 2152) Hexahedron: See 6.6. (line 2637) Hypot: See 6.2.2. (line 2069) If: See 6.8. (line 2829) If ( EXPRESSION-CST ): See 4.11. (line 1171) Im: See 6.2.4. (line 2181) In <1>: See 5.9. (line 1750) In: See 5.7. (line 1578) Include: See 4.2. (line 594) IndexOfSystem: See 5.7. (line 1578) Init: See 6.3. (line 2367) InitFromResolution: See 6.3. (line 2373) InitSolution: See 6.8. (line 2772) Integral <1>: See 6.9. (line 2932) Integral <2>: See 6.7. (line 2682) Integral: See 5.9. (line 1750) Integration <1>: See 5.9. (line 1750) Integration <2>: See 5.7. (line 1578) Integration: See 5.6. (line 1536) InterpolationAkima: See 6.2.6. (line 2342) InterpolationLinear: See 6.2.6. (line 2330) Iso: See 6.10. (line 3115) IterativeLoop: See 6.8. (line 2907) JacNL: See 6.7. (line 2702) Jacobian <1>: See 5.9. (line 1750) Jacobian <2>: See 5.7. (line 1578) Jacobian: See 5.5. (line 1497) Lanczos: See 6.8. (line 2872) Laplace: See 6.2.3. (line 2141) LastTimeStepOnly: See 6.10. (line 3077) Lin: See 6.5. (line 2552) Line: See 6.6. (line 2625) Link: See 6.3. (line 2379) LinkCplx: See 6.3. (line 2402) List: See 4.4. (line 664) ListAlt: See 4.4. (line 664) Local <1>: See 6.9. (line 2927) Local <2>: See 6.7. (line 2672) Local: See 5.9. (line 1750) Log: See 6.2.1. (line 1980) Log10: See 6.2.1. (line 1985) Loop: See 5.7. (line 1578) Name <1>: See 5.10. (line 1802) Name <2>: See 5.9. (line 1750) Name <3>: See 5.8. (line 1670) Name <4>: See 5.7. (line 1578) Name <5>: See 5.6. (line 1536) Name <6>: See 5.5. (line 1497) Name <7>: See 5.4. (line 1396) Name: See 5.3. (line 1327) NameOfBasisFunction: See 5.4. (line 1396) NameOfCoef: See 5.4. (line 1396) NameOfConstraint <1>: See 5.7. (line 1578) NameOfConstraint: See 5.4. (line 1396) NameOfFormulation <1>: See 5.9. (line 1750) NameOfFormulation: See 5.8. (line 1670) NameOfMesh: See 5.8. (line 1670) NameOfPostProcessing: See 5.10. (line 1802) NameOfSpace: See 5.7. (line 1578) NameOfSystem: See 5.9. (line 1750) Network <1>: See 6.3. (line 2376) Network: See 5.7. (line 1578) NeverDt: See 6.7. (line 2706) Node: See 5.7. (line 1578) NodesOf: See 6.1. (line 1898) NoNewLine: See 6.10. (line 3127) Norm: See 6.2.2. (line 2074) Normal: See 6.2.6. (line 2313) NormalSource: See 6.2.6. (line 2318) NumberOfPoints: See 5.6. (line 1536) OnBox: See 6.10. (line 3015) OnElementsOf: See 6.10. (line 2942) OnGlobal: See 6.10. (line 2954) OnGrid: See 6.10. (line 2966) OnLine: See 6.10. (line 2995) OnPlane: See 6.10. (line 3005) OnPoint: See 6.10. (line 2989) OnRegion: See 6.10. (line 2949) OnSection: See 6.10. (line 2957) Operation <1>: See 5.10. (line 1802) Operation: See 5.8. (line 1670) Order: See 6.2.6. (line 2354) OriginSystem: See 5.8. (line 1670) Pi: See 4.4. (line 664) Point: See 6.6. (line 2646) PostOperation <1>: See 6.8. (line 2917) PostOperation: See 5.10. (line 1802) PostProcessing: See 5.9. (line 1750) Print <1>: See 6.8. (line 2848) Print: See 5.10. (line 1802) Printf: See 6.2.6. (line 2303) Prism: See 6.6. (line 2640) Pyramid: See 6.6. (line 2643) Quadrangle: See 6.6. (line 2631) Quantity <1>: See 5.9. (line 1750) Quantity <2>: See 5.7. (line 1578) Quantity: See 5.4. (line 1396) Rand: See 6.2.6. (line 2308) Re: See 6.2.4. (line 2176) Region <1>: See 6.1. (line 1891) Region <2>: See 5.5. (line 1497) Region: See 5.3. (line 1327) Resolution <1>: See 5.8. (line 1670) Resolution: See 5.4. (line 1396) Rot: See 4.10. (line 1100) RotInv: See 4.10. (line 1113) SaveSolution: See 6.8. (line 2778) SaveSolutions: See 6.8. (line 2783) Scalar: See 6.4. (line 2432) SetFrequency: See 6.8. (line 2819) SetTime: See 6.8. (line 2814) SimpleTable: See 6.10. (line 3197) Sin: See 6.2.1. (line 1995) Sinh: See 6.2.1. (line 2033) Skin: See 6.10. (line 3053) Smoothing: See 6.10. (line 3056) Solve: See 6.8. (line 2720) SolveJac: See 6.8. (line 2732) Solver: See 5.8. (line 1670) Sort: See 6.10. (line 3109) Sqrt: See 6.2.1. (line 1990) SquNorm: See 6.2.2. (line 2080) Store: See 6.10. (line 3152) SubRegion: See 5.3. (line 1327) SubSpace: See 5.4. (line 1396) Support: See 5.4. (line 1396) Sur: See 6.5. (line 2548) SurAxi: See 6.5. (line 2559) Symmetry: See 5.7. (line 1578) System: See 5.8. (line 1670) SystemCommand: See 6.8. (line 2824) Table: See 6.10. (line 3187) Tan: See 6.2.1. (line 2017) Tanh: See 6.2.1. (line 2043) Target: See 6.10. (line 3097) Tensor: See 6.2.4. (line 2191) TensorDiag: See 6.2.4. (line 2208) TensorSym: See 6.2.4. (line 2203) TensorV: See 6.2.4. (line 2198) Tetrahedron: See 6.6. (line 2634) TimeFunction: See 5.3. (line 1327) TimeLegend: See 6.10. (line 3158) TimeLoopNewmark: See 6.8. (line 2899) TimeLoopTheta: See 6.8. (line 2891) TimeStep: See 6.10. (line 3072) TimeTable: See 6.10. (line 3201) TransferInitSolution: See 6.8. (line 2800) TransferSolution: See 6.8. (line 2791) Transpose: See 6.2.2. (line 2091) Triangle: See 6.6. (line 2628) TTrace: See 6.2.2. (line 2096) Type <1>: See 5.8. (line 1670) Type <2>: See 5.7. (line 1578) Type <3>: See 5.6. (line 1536) Type <4>: See 5.4. (line 1396) Type: See 5.3. (line 1327) Unit: See 6.2.2. (line 2085) Update: See 6.8. (line 2760) UpdateConstraint: See 6.8. (line 2767) UsingPost: See 5.10. (line 1802) Value <1>: See 6.10. (line 3103) Value: See 5.9. (line 1750) Vector <1>: See 6.4. (line 2435) Vector: See 6.2.4. (line 2186) Vol: See 6.5. (line 2545) VolAxi: See 6.5. (line 2555) VolAxiRectShell: See 6.5. (line 2597) VolAxiSphShell: See 6.5. (line 2575) VolAxiSqu: See 6.5. (line 2563) VolAxiSquRectShell: See 6.5. (line 2604) VolAxiSquSphShell: See 6.5. (line 2582) VolRectShell: See 6.5. (line 2589) VolSphShell: See 6.5. (line 2567) VolumesOf: See 6.1. (line 1913) X: See 6.2.5. (line 2278) XYZ: See 6.2.5. (line 2293) Y: See 6.2.5. (line 2283) Z: See 6.2.5. (line 2288) |: See 4.5.1. (line 809) ||: See 4.5.1. (line 809) ~: See 4.4. (line 664)