# 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 3.5.0 (development version) (January 22, 2026).

## Table of Contents

## Obtaining GetDP ¶

The source code and various pre-compiled versions of GetDP (for Windows, Linux and MacOS) can be downloaded from http://getdp.info.

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://getdp.info.

## 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 (see 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 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 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.

    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 ¶

Please file issues on https://gitlab.onelab.info/getdp/getdp/issues. Provide as precise a description of the problem as you can, including sample input files that produce the bug. Don’t forget to mention both the version of GetDP and the version of your operation system (see Running GetDP to see how to get this information).

See Frequently asked questions, and the bug tracking system to see which problems we already know about.

## 2 How to read this manual ¶

After reading Overview, which depicts the general philosophy of GetDP, you might want to skip Expressions, Objects and Types for objects and directly run the demo files bundled in the distribution on your computer (see Running GetDP). You should then open these examples with a text editor and compare their structure with the examples given in Short examples and Complete examples. For each new syntax element that you fall onto, you can then go back to Expressions, Objects, and 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 (see Concept index) and for all the syntax elements (see 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 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 interface1. 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 (see 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 (see 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 (see File formats) rather than from the default problem file name (with the `.msh` extension appended).

`-msh_scaling` ¶

  value

  Multiplies the coordinates of all the nodes in the mesh by value.

`-gmshread` ¶

  filename …

  Read gmsh data files (same as GmshRead in `Resolution` operations). Allows to use such datasets outside resolutions (e.g. in pre-processing).

`-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`.

`-solver` ¶

  filename

  Specifies a solver option file (whose format varies depending on the linear algebra toolkit used).

`-slepc` ¶

  Uses SLEPc instead of Arpack as eigensolver.

`-adapt` ¶

  file

  Reads adaptation constraints from file.

`-order` ¶

  real

  Specifies the maximum interpolation order.

`-cache` ¶

  Caches network computations to disk.

`-bin` ¶

  Selects binary format for output files.

`-v2` ¶

  Creates mesh-based Gmsh output files when possible.

`-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.

`-cpu` ¶

  Reports CPU times for all operations.

`-p` ¶
`-progress` ¶

  integer

  Sets the progress update rate. This controls the refreshment rate of the counter indicating the progress of the current computation (in %).

`-onelab` ¶

  name <address>

  Communicates with OneLab (file or server address)

`-setnumber` ¶

  name value

  Sets constant number name to value

`-setstring` ¶

  name value

  Sets constant string name to value

`-info` ¶

  Displays the version information.

`-version` ¶

  Displays the version number.

`-help` ¶

  Displays a message listing basic usage and available options.

## 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 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 Short examples and 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 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 Syntactic rules used in this document):

    expression:
      ( expression ) |
      integer |
      real |
      constant-id |
      quantity |
      argument |
      current-value |
      variable-set |
      variable-get |
      register-set |
      register-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-set, register-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 > ] |
      #constant-id() |
      constant-id(expression-cst) |
      StrFind[ expression-char, expression-char ] |
      StrCmp[ expression-char, expression-char ] |
      StrLen[ expression-char ] |
      StringToName[ expression-char ] | S2N[ expression-char ] |
      Exists[ string ] | FileExists[ string ] | GroupExists[ string ] |
      GetForced[ string ] | NbrRegions [ string ] |
      GetNumber[ expression-char <, expression-cst> ]

`StrFind` searches the first expression-char for any occurrence of the second expression-char. `StrCmp` compares the two strings (returns an integer greater than, equal to, or less than 0, according as the first string is greater than, equal to, or less than the second string). `StrLen` returns the length of the string. `StringToName` creates a name from the provided string. `Exists` checks for the existence of a constant or a function. `FileExists` checks for the existence of a file. `GroupExists` checks for the existence of a group. `GetForced` gets the value of a constant (zero if does not exist). `NbrRegions` counts the numbers of elementary regions in a group. `GetNumber` allows to get the value of a ONELAB number variable (the optional second argument specifies the default value returned if the variable does not exist).

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 ] |
      List[ expression-cst-list-item ] |
      List[ { expression-cst-list } ] |
      ListAlt[ constant-id, constant-id ] |
      ListAlt[ expression-cst-list-item, expression-cst-list-item ] |
      LinSpace[ expression-cst, expression-cst, expression-cst ] |
      LogSpace[ expression-cst, expression-cst, expression-cst ] |
      - expression-cst-list-item |
      expression-cst * expression-cst-list-item |
      expression-cst-list-item * expression-cst |
      expression-cst / expression-cst-list-item |
      expression-cst-list-item / expression-cst |
      expression-cst-list-item ^ expression-cst |
      expression-cst-list-item + expression-cst-list-item |
      expression-cst-list-item - expression-cst-list-item |
      expression-cst-list-item * expression-cst-list-item |
      expression-cst-list-item / expression-cst-list-item |
      ListFromFile [ expression-char ] |
      ListFromServer [ expression-char ]

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 (see Functions). The next two cases permit to create linear and logarithmic lists of numbers, respectively. The remaining cases permit to apply arithmetic operators item-wise in lists. `ListFromFile` reads a list of numbers from a file. `ListFromServer` attemps to get a list of numbers from the ONELAB variable expression-char.

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 > <,...> ]; |
      DefineConstant [ constant-id = { expression-cst , onelab-options } <,...> ]; |
      DefineConstant [ string-id < = string-def >  <,...> ]; |
      DefineConstant [ string-id = { string-def , onelab-options } <,...> ]; |
      constant-id <()> = constant-def; |
      constant-id = DefineNumber[ constant-def, onelab-options ];
      string-id <()> = string-def; |
      string-id = DefineString[ string-def, onelab-options ]; |
      Printf [ "string" ] < > | >> string-def >; |
      Printf [ "string", expression-cst-list ] < > | >> string-def >; |
      Read [ constant-id ] ; |
      Read [ constant-id , expression-cst ]; |
      UndefineConstant | Delete [ constant-id ] ;
      UndefineFunction [ constant-id ] ;
      SetNumber[ string , expression-cst ];
      SetString[ string , string-def ];

with

    constant-id:
      string |
      string ( expression-cst-list ) |
      string ~ { expression-cst } <,...>

    constant-def:
      expression-cst-list-item |
      { expression-cst-list }

    string-id:
      string |
      string ~ { expression-cst } <,...>

    string-def:
      "string" |
      StrCat[ expression-char <,...> ] |
      Str[ 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 (see Macros, 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).

   When onelab-options are provided, the parameter is exchanged with the ONELAB server. See https://gitlab.onelab.info/doc/tutorials/wikis/ONELAB-syntax-for-Gmsh-and-GetDP for more information.

4. `DefineNumber` and `DefineString` allow to define a ONELAB parameter. In this case the affectation always takes place. `SetNumber` and `SetString` allow the direct setting of ONELAB parameters without defining local variables.

See Constant expression examples, as well as `Function` examples, for some examples.

Character expressions are defined as follows:

    expression-char:
      "string" |
      string-id |
      StrCat[ expression-char <,...> ] |
      Str[ expression-char <,...> ]
      StrChoice[ expression, expression-char, expression-char ] |
      StrSub[ expression-char, expression, expression ] |
      StrSub[ expression-char, expression ] |
      UpperCase [ expression-char ] |
      Sprintf [ expression-char ] |
      Sprintf[ expression-char, expression-cst-list ] |
      NameToString ( string ) | N2S ( string ) |
      GetString[ expression-char <, expression-char,> ] |
      Date | CurrentDirectory | CurrentDir |
      AbsolutePath [ expression-char ] |
      DirName [ expression-char ] |
      OnelabAction

`StrCat` and `Str` permit to concatenate character expressions (`Str` adds a newline character after each string except the last) when creating a string. `Str` is also used to create string lists (when string-id is followed by `()`). `StrChoice` returns the first or second expression-char depending on the value of expression. `StrSub` returns the portion of the string that starts at the character position given by the first expression and spans the number of characters given by the second expression or until the end of the string (whichever comes first; or always if the second expression is not provided). `UpperCase` converts the expression-char to upper case. `Sprintf` is equivalent to the `sprintf` C function (where expression-char is a format string that can contain floating point formatting characters: `%e`, `%g`, etc.). `NameToString` converts a variable name into a string. `GetString` allows to get the value of a ONELAB string variable (the optional second argument specifies the default value returned if the variable does not exist.) `Date` permits to access the current date. `CurrentDirectory` and `CurrentDir` return the directory of the `.pro` file. `AbsolutePath` returns the absolute path of a file. `DirName` returns the directory of a file. `OnelabAction` returns the current ONELAB action (e.g. `check` or `compute`).

List of character expressions are defined as:

    expression-char-list:
      expression-char <,...>

### 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’.

`>>`

  Bitwise right-shift operator. Shifts the bits of the first argument to the right by the number of bits specified by the second argument.

`<<`

  Bitwise left-shift operator. Shifts the bits of the first argument to the left by the number of bits specified by the second argument.

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 `Function`: defining global and piecewise expressions) 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 (see 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 Types for `Function`.
2. Classical mathematical functions (see Math functions) are the only functions allowed in a constant definition (see the definition of expression-cst in Constants).

### 4.7 Current values ¶

Current values return the current 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.

`$Breakpoint`

  In case of a breakpoint hit in TimeLoopAdaptive it is the number of the current breakpoint. In the other case when $Time corresponds not to a breakpoint the value is -1.

`$Iteration, $NLIteration`

  Current iteration in a nonlinear loop.

`$Residual, $NLResidual`

  Current residual 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` (see Types for `PostOperation`).

`$QuadraturePointIndex`

  Index of the current quadrature point.

`$KSPIteration`

  Current iteration in a Krylov subspace solver.

`$KSPResidual`

  Current residual in a Krylov subspace solver.

`$KSPIterations`

  Total number of iterations of Krylov subspace solver.

`$KSPSystemSize`

  System size of Krylov subspace solver.

Note:

1. The current X, Y and Z coordinates refer to the ‘physical world’ coordinates, i.e., coordinates in which the mesh is expressed.

Current values are “read-only”. User-defined run-time variables, which share the same syntax but whose value can be changed in an expression, are defined in Run-time variables and registers.

### 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 `Function`: defining global and piecewise expressions, and `Function` examples, for more details.

### 4.9 Run-time variables and registers ¶

Constant expressions (expression-csts) are evaluated only once during the analysis of the problem definition structure, cf. Constants. While this is perfectly fine in most situations, sometimes it is necessary to store and modify variables at run-time. For example, an iteration in a `Resolution` could depend on values computed at run-time. Also, to speed-up the evaluation of expressions (which are evaluated at runtime through GetDP’s internal stack mechanism), it can be useful to save some results in a temporary variable, at run-time, in order to reuse them later on.

Two mechanisms exit to handle such cases: run-time variables (which follow the same syntax as Current values), and registers.

Run-time variables have the following syntax:

    variable-set:
      $variable-id = expression

    variable-get:
      $variable-id

    variable-id:
      string |
      string ~ { expression-cst } <,...>

Thus, run-time variables can simply be defined anywhere in an expression and be reused later on. Current values can be seen as special cases of run-time variables, which are read-only.

Registers have the following syntax:

    register-set:
      expression#expression-cst

    register-get:
      #expression-cst

Thus, to store any 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 `Function` examples, for an example.

### 4.10 Fields ¶

A discretized quantity (defined in a function space, cf. `FunctionSpace`: building function spaces) is represented between braces `{}`, and can only appear in well-defined expressions in `Formulation` (see `Formulation`: building equations) and `PostProcessing` (see `PostProcessing`: exploiting computational results) 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

      Integral { [ f[] * Dof{T}^4 , {T} ]; ... }

  must first be linearized; and while

      Integral { [ f[] * Dof{T} , {T} ]; ... }
      Integral { [ -f[] * 12 , {T} ]; ... }

  is valid, the following, which is affine but not linear, is not:

      Integral { [ 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.

`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.

`D1` ¶

  Applies the operator specified in the first argument of `dFunction { basis-function-type, basis-function-type }` (see `FunctionSpace`: building function spaces). This is currently only used for nodal-interpolated vector fields (interpolated with `BF_Node_X`, `BF_Node_Y`, `BF_Node_Z`)

  When the first basis-function-type in `dFunction` is set to `BF_NodeX_D1` for component X, `BF_NodeY_D1` for component Y and `BF_NodeZ_D1` for component Z, then `D1` applied to a vector [u_x, u_y, u_z] gives: [du_x/dx, du_y/dy, du_z/dz] Note that in this case specifying explicitely `dFunction` is not necessary, as `BF_NodeX_D1`, `BF_NodeY_D1` and `BF_NodeZ_D1` are assigned by default as the “`D1` derivatives” of `BF_NodeX`, `BF_NodeY` and `BF_NodeZ`. This also holds for `BF_GroupOfNodes_X`, `BF_GroupOfNodes_Y` and `BF_GroupOfNodes_Z`.

  When the first basis-function-type in `dFunction` is set to `BF_NodeX_D12` for component X and `BF_NodeY_D12` for component Y, then `D1` applied to a vector [u_x, u_y] gives: [du_x/dx, du_y/dy, du_y/dx + du_x/dy]

`D2` ¶

  Applies the operator specified in the second argument of `dFunction { basis-function-type, basis-function-type }` (see `FunctionSpace`: building function spaces). This is currently only used for nodal-interpolated vector fields (interpolated with `BF_Node_X`, `BF_Node_Y`, `BF_Node_Z`)

  More specifically, when the second basis-function-type is to `BF_NodeX_D2` for component X, `BF_NodeY_D2` for component Y and `BF_NodeZ_D2` for component Z, then `D2` applied to a vector [u_x, u_y, u_z] gives: [du_y/dx + du_x/dy, du_z/dy + du_y/dz, du_x/dz + du_z/dx] Note that in this case specifying explicitely `dFunction` is not necessary, as `BF_NodeX_D2`, `BF_NodeY_D2` and `BF_NodeZ_D2` are assigned by default as the “`D2` derivatives” of `BF_NodeX`, `BF_NodeY` and `BF_NodeZ`. This also holds for `BF_GroupOfNodes_X`, `BF_GroupOfNodes_Y` and `BF_GroupOfNodes_Z`.

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 Macros, loops and conditionals ¶

Macros are defined as follows:

`Macro string | expression-char` ¶

  Begins the declaration of a user-defined file macro named string. The body of the macro starts on the line after ‘`Macro string`’, and can contain any GetDP command.

`Return` ¶

  Ends the body of the current user-defined file macro. Macro declarations cannot be imbricated, and must be made outside any GetDP object.

`Macro ( expression-char , expression-char ) ;` ¶

  Begins the declaration of a user-defined string macro. The body of the macro is given explicitly as the second argument.

Macros, 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).

loop:

`Call string | expression-char;` ¶

  Executes the body of a (previously defined) macro named string.

`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 `ElseIf`, `Else` or `EndIf`, is evaluated if expression-cst is non-zero.

`ElseIf ( expression-cst )` ¶

  The body enclosed between ‘`ElseIf ( expression-cst )`’ and the next matching `ElseIf`, `Else` or `EndIf`, is evaluated if expression-cst is non-zero and none of the expression-cst of the previous matching codes `If` and `ElseIf` were non-zero.

`Else` ¶

  The body enclosed between `Else` and the matching `EndIf` is evaluated if none of the expression-cst of the previous matching codes `If` and `ElseIf` were non-zero.

`EndIf` ¶

  Ends a matching `If` command.

`LevelTest` ¶

  Variable equal to the level of imbrication of a body in an `If`-`EndIf` test.

`Parse [ expression-char ];` ¶

  Parse the given string.

## 5 Objects ¶

This chapter presents the formal definition of the ten GetDP objects mentioned in Overview. To be concise, all the possible parameters for these objects are not given here (cf. the etc syntactic rule defined in Syntactic rules used in this document). Please refer to 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 (see 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 Constants.

The syntax for the definition of groups is:

    Group {
      < DefineGroup [ group-id <{integer}> <,...> ]; > ...
      < group-id = group-def; > ...
      < 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 <{<integer>}> |
      #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 <{<integer>}>

    group-sub-type:
      Not | StartingOn | OnPositiveSideOf | 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 (see Input file format). Ranges of integers can be specified in the same way as ranges of constant expressions in an expression-cst-list-item (see 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 Types for `Group`, for the complete list of options and `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 <, group-def > > ] = expression; > ...
      < affectation > ...
      < loop > ...
    }

with

    function-id:
      string

Note:

1. The first optional group-def in brackets must be of `Region` type, and indicates on which region the (piecewise) function is defined. The second optional group-def in brackets, also of `Region` type, defines an association with a second region for mutual contributions. A default piecewise function can be defined with `All` for group-def, for all the other non-defined regions. 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 Types for `Function`, for the complete list of built-in functions and `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 {
      { < Append < expression-cst >; >
        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 optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `Constraint` of the same `Name` with additional `Case`s.
2. 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`.
3. 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.
4. 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 Types for `Constraint`, for the complete list of options and `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 (see `FunctionSpace` examples).

The syntax for the definition of function spaces is:

    FunctionSpace {
      { < Append < expression-cst >; >
        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 { expression-cst };
                 Group group-def;
                 Resolution resolution-id { expression-cst } } >;
           < dFunction { basis-function-type, basis-function-type } ; >
           Support group-def; Entity group-def; } ...
        }
      < SubSpace {
         { < Append < expression-cst >; >
           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 Auto | 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. The optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive; its omission fixes it to a top value) permits to append an existing `FunctionSpace` of the same `Name` with additional `BasisFunction`s, `SubSpace`s, `GlobalQuantity`’s and `Constraint`s, or an existing `SubSpace` of the same `Name` with additional `NameOfBasisFunction`’s. If the `Append` `FunctionSpace` level is 2, the `Append` `SubSpace` level is automatically 1 if omitted.
2. 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).
3. The same `Name` for several `BasisFunction` fields permits to define piecewise basis functions; separate `NameOfCoef`s must be defined for those fields.
4. A constraint is associated with geometrical entities defined by an automatically created `Group` of type group-type (`Auto` automatically fixes it as the `Entity` group-def type of the related `BasisFunction`), 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.
5. 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.
6. Explicit derivatives of the basis functions can be specified using `dFunction { basis-function-type , basis-function-type }`. These derivates can be accessed using the special `D1` and `D2` operators (see Fields).

See Types for `FunctionSpace`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        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 optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `Jacobian` of the same `Name` with additional `Case`s.
2. The default case of a `Jacobian` object is defined by `Region All` and must follow all the other cases.

See Types for `Jacobian`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        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

Note:

1. The optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `Integration` of the same `Name` with additional `Case`s.

See Types for `Integration`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        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; < SubType equation-term-sub-type; > } > ...
         < 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:
      Integral | deRham

    equation-term-sub-type:
      Self (default) | Mutual | SelfAndMutual

    quantity-type:
      Local | Global | Integral

    term-op-type:
      DtDof | DtDtDof | Eig | JacNL | etc

Note:

1. The optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `Formulation` of the same `Name` with additional `Quantity`’s and `Equation`s.
2. `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 (see `Resolution`: solving systems of equations).
3. 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. The optional associated `SubType` defines either self (default) or mutual contributions, or both. Mutual contributions need piecewise functions defined on pairs or regions.
4. A `GlobalEquation` defines a global equation to be assembled in the matrix of the system.

See Types for `Formulation`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        Name resolution-id; < Hidden expression-cst; >
        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 optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `Resolution` of the same `Name` with additional `System`s and `Operation`s.
2. 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`.
3. `NameOfMesh` permits to explicitely specify the mesh to be used for the construction of the system of equations.
4. `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).
5. `DestinationSystem` permits to specify the destination system of a `TransferSolution` operation (see Types for `Resolution`).
6. `OriginSystem` permits to specify the systems from which ambiguous quantity definitions can be solved (see `Formulation`: building equations).

See Types for `Resolution`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        Name post-processing-id;
        NameOfFormulation formulation-id <{}>; < NameOfSystem system-id; >
        Quantity {
          { < Append < expression-cst >; >
            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 optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive; its omission fixes it to a top value) permits to append an existing `PostProcessing` of the same `Name` with additional `Value`s, or an existing `Quantity` of the same `Name` with additional `Quantity`’s. If the `Append` `PostProcessing` level is 2, the `Append` `Quantity` level is automatically 1 if omitted. Fixing the `Append` `Quantity` level to -n suppresses the n lastly defined `Quantity`’s before appending.
2. 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 (see `PostOperation`: exporting results).
3. 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 Types for `PostProcessing`, for the complete list of options and `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 {
      { < Append < expression-cst >; >
        Name post-operation-id; NameOfPostProcessing post-processing-id;
        < Hidden expression-cst; >
        < Format post-operation-fmt; >
        < TimeValue expression-cst-list; > < TimeImagValue expression-cst-list; >
        < LastTimeStepOnly < expression-cst >; >
        < OverrideTimeStepValue expression-cst; >
        < NoMesh expression-cst; > < AppendToExistingFile expression-cst; >
        < ResampleTime[expression-cst, expression-cst, expression-cst]; >
        < AppendTimeStepToFileName < expression-cst >; >
        Operation {
          < post-operation-op; > ...
        }
      } ...
      < affectation > ...
      < loop > ...
    } |
    PostOperation < (Append < expression-cst >) > post-operation-id UsingPost post-processing-id {
      < post-operation-op; > ...
    } ...

with

    post-operation-id:
      string |
      string ~ { expression-cst }

    post-operation-op:
      Print[ post-quantity-id <[group-def]>, print-support <,print-option> ... ] |
      Print[ expression-list, Format "string" <,print-option> ... ] |
      PrintGroup[ group-id, print-support <,print-option> ... ] |
      Echo[ "string" <,print-option> ... ] |
      CreateDir [ "string" ] |
      DeleteFile [ "string" ] |
      SendMergeFileRequest[ expression-char ] |
      < loop > ...
      etc

    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. The optional `Append < expression-cst >` (when the optional level expression-cst is strictly positive) permits to append an existing `PostOperation` of the same `Name` with additional `Operation`s.
2. Both `PostOperation` syntaxes are equivalent. The first one conforms to the overall interface, but the second one is more concise.
3. 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 (see Types for `PostOperation`). The default format is `Gmsh`.
4. The `ResampleTime` option allows equidistant resampling of the time steps by a spline interpolation. The parameters are: start time, stop time, time step.
5. The optional argument `[group-def]` of the post-quantity-id can only be used when this quantity has been defined as an integral-value (see `PostProcessing`: exploiting computational results). In this case, the sum of all elementary integrals is performed over the region group-def.

See Types for `PostOperation`, for the complete list of options and `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-sub-type-2 R3 > > ]

`group-type < group-sub-type < group-sub-type-2 > >`:

`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 (non-automatic, i.e., both sides if both in R1) > | < `OnPositiveSideOf`: only elements on positive (normal) side of R2 < , `Not`: but not those touching only its skin R3 (mandatory for free skins for correct separation of side layers) > >.

`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) >.

`GroupOfRegionsOf` ¶

  Single group of elements of regions in R1 (with basis function BF_Region just one DOF is created for all elements of R1).

`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 or C++ math library. Unless indicated otherwise, arguments to these functions can be real or complex valued when used in expressions. When used in constant expressions (expression-cst, see Constants), only real-valued arguments are accepted.

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] (real valued only).

`Cos` ¶

  `[expression]`

  Cosine of expression.

`Acos` ¶

  `[expression]`

  Arc cosine (inverse cosine) of expression in [0,Pi], expression in [-1,1] (real valued only).

`Tan` ¶

  `[expression]`

  Tangent of expression.

`Atan` ¶

  `[expression]`

  Arc tangent (inverse tangent) of expression in [-Pi/2,Pi/2] (real valued only).

`Atan2` ¶

  `[expression,expression]`

  Arc tangent (inverse tangent) of the first expression divided by the second, in [-Pi,Pi] (real valued only).

`Sinh` ¶

  `[expression]`

  Hyperbolic sine of expression.

`Cosh` ¶

  `[expression]`

  Hyperbolic cosine of expression.

`Tanh` ¶

  `[expression]`

  Hyperbolic tangent of the real valued expression.

`TanhC2` ¶

  `[expression]`

  Hyperbolic tangent of a complex valued expression.

`Fabs` ¶

  `[expression]`

  Absolute value of expression (real valued only).

`Abs` ¶

  `[expression]`

  Absolute value of expression.

`Floor` ¶

  `[expression]`

  Rounds downwards to the nearest integer that is not greater than expression (real valued only).

`Ceil` ¶

  `[expression]`

  Rounds upwards to the nearest integer that is not less than expression (real valued only).

`Fmod` ¶

  `[expression,expression]`

  Remainder of the division of the first expression by the second, with the sign of the first (real valued only).

`Min` ¶

  `[expression,expression]`

  Minimum of the two (scalar) expressions (real valued only).

`Max` ¶

  `[expression,expression]`

  Maximum of the two (scalar) expressions (real valued only).

`Sign` ¶

  `[expression]`

  -1 for expression less than zero and 1 otherwise (real valued only).

`Jn` ¶

  `[expression]`

  Returns the Bessel function of the first kind of order given by the first expression for the value of the second expression (real valued only).

`dJn` ¶

  `[expression]`

  Returns the derivative of the Bessel function of the first kind of order given by the first expression for the value of the second expression (real valued only).

`Yn` ¶

  `[expression]`

  Returns the Bessel function of the second kind of order given by the first expression for the value of the second expression (real valued only).

`dYn` ¶

  `[expression]`

  Returns the derivative of the Bessel function of the second kind of order given by the first expression for the value of the second expression (real valued only).

#### 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.

`Inv` ¶

  `[expression]`

  Inverse of the tensor expression.

`Det` ¶

  `[expression]`

  Determinant of the tensor expression.

`Rotate` ¶

  `[expression,expression,expression,expression]`

  Rotation of a vector or tensor given by the first expression by the angles in radians given by the last three expression values around the x-, y- and z-axis.

`TTrace` ¶

  `[expression]`

  Trace; expression must be a tensor.

`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[p],Sin[p]]`.

`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[p],-Cos[p]]`.

`Period` ¶

  `[expression]{expression-cst}`

  `Fmod[expression,expression-cst]` `+` `(expression<0 ? expression-cst : 0)`; the result is always in [0,expression-cst[.

`Interval` ¶

  `[expression,expression,expression]{expression-cst, expression-cst,expression-cst}`

  Not documented yet.

#### 6.2.3 Green functions ¶

The Green functions are only used in integral quantities (see `Formulation`: building equations). 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.

`Complex_MH` ¶

  `[expression-list]{expression-cst-list}`

  Not documented yet.

`Re` ¶

  `[expression]`

  Takes the real part of a complex-valued expression.

`Im` ¶

  `[expression]`

  Takes the imaginary part of a complex-valued expression.

`Conj` ¶

  `[expression]`

  Computes the conjugate of a complex-valued expression.

`Cart2Pol` ¶

  `[expression]`

  Converts the cartesian form (reale, imaginary) of a complex-valued expression into polar form (amplitude, phase [radians]).

`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, by row:

          [ scalar0 scalar1 scalar2 ]   [ CompXX CompXY CompXZ ]
      T = [ scalar3 scalar4 scalar5 ] = [ CompYX CompYY CompYZ ]
          [ scalar6 scalar7 scalar8 ]   [ CompZX CompZY CompZZ ]

`TensorV` ¶

  `[expression,expression,expression]`

  Creates a second-rank tensor of order 3 from 3 row vectors:

          [ vector0 ]
      T = [ vector1 ]
          [ vector2 ]

`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.

`SquDyadicProduct` ¶

  `[expression]`

  Dyadic product of the vector given by expression with itself.

`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.

`Cart2Sph` ¶

  `[expression]`

  Gets the tensor for transformation of vector from cartesian to spherical coordinates.

`Cart2Cyl` ¶

  `[expression]`

  Gets the tensor for transformation of vector from cartesian to cylindric coordinates. E.g. to convert a vector with (x,y,z)-components to one with (radial, tangential, axial)-components: Cart2Cyl[XYZ[]] * vector

`UnitVectorX` ¶

  `[]`

  Creates a unit vector in x-direction.

`UnitVectorY` ¶

  `[]`

  Creates a unit vector in y-direction.

`UnitVectorZ` ¶

  `[]`

  Creates a unit vector in z-direction.

#### 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. (`MPI_Printf` can be use instead, to print the message for all MPI ranks.)

`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).

`Tangent` ¶

  `[]`

  Computes the tangent to the element (only valid for line elements).

`TangentSource` ¶

  `[]`

  Computes the tangent to the source element (only valid in a quantity of Integral type and only for line elements).

`ElementVol` ¶

  `[]`

  Computes the element’s volume.

`SurfaceArea` ¶

  `[]{expression-cst-list}`

  Computes the area of the physical surfaces in expression-cst-list or of the actual surface if expression-cst-list is empty.

`GetVolume` ¶

  `[]`

  Computes the volume of the actual physical group.

`CompElementNum` ¶

  `[]`

  Returns 0 if the current element and the current source element are identical.

`GetNumElements` ¶

  `[]{expression-cst-list}`

  Counts the elements of physical numbers in expression-cst-list or of the actual region if expression-cst-list is empty.

`ElementNum` ¶

  `[]`

  Returns the tag (number) of the current element.

`QuadraturePointIndex` ¶

  `[]`

  Returns the index of the current quadrature point.

`AtIndex` ¶

  `[expression]{expression-cst-list}`

  Returns the i-th entry of expression-cst-list. This can be used to get an element in a list, using an index that is computed at runtime.

`InterpolationLinear` ¶

  `[expression]{expression-cst-list}`

  Linear interpolation of points. The number of constant expressions in expression-cst-list must be even.

`dInterpolationLinear` ¶

  `[expression]{expression-cst-list}`

  Derivative of linear interpolation of points. The number of constant expressions in expression-cst-list must be even.

`InterpolationBilinear` ¶

  `[expression,expression]{expression-cst-list}`

  Bilinear interpolation of a table based on two variables.

`dInterpolationBilinear` ¶

  `[expression,expression]{expression-cst-list}`

  Derivative of bilinear interpolation of a table based on two variables. The result is a vector.

`InterpolationAkima` ¶

  `[expression]{expression-cst-list}`

  Akima interpolation of points. The number of constant expressions in expression-cst-list must be even.

`dInterpolationAkima` ¶

  `[expression]{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.

`Field` ¶

  `[expression]`

  Evaluate the last one of the fields (“views”) loaded with `GmshRead` (see Types for `Resolution`), at the point expression. Common usage is thus `Field[XYZ[]]`.

`Field` ¶

  `[expression]{expression-cst-list}`

  Idem, but evaluate all the fields corresponding to the tags in the list, and sum all the values. A field having no value at the given position does not produce an error: its contribution to the sum is simply zero.

`ScalarField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only real-valued scalar fields. A second optional argument is the value of the time step. A third optional argument is a boolean flag to indicate that the interpolation should be performed (if possible) in the same element as the current element.

`VectorField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only real-valued vector fields. Optional arguments are treated in the same way as for `ScalarField`.

`TensorField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only real-valued tensor fields. Optional arguments are treated in the same way as for `ScalarField`.

`ComplexScalarField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only complex-valued scalar fields. Optional arguments are treated in the same way as for `ScalarField`.

`ComplexVectorField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only complex-valued vector fields. Optional arguments are treated in the same way as for `ScalarField`.

`ComplexTensorField` ¶

  `[expression]{expression-cst-list}`

  Idem, but consider only complex-valued tensor fields. Optional arguments are treated in the same way as for `ScalarField`.

`Distance` ¶

  `[expression]{expression-cst}`

  Evaluate the distance between the point (whose x, y, z coordinates are given as a vector argument) and a vector valued view, interpreted as a displacement field (i.e. compute the minimal distance between the point and the deformed mesh in the view).

`GetCpuTime` ¶

  `[]`

  Returns current CPU time, in seconds (total amount of time spent executing in user mode since GetDP was started).

`GetWallClockTime` ¶

  `[]`

  Returns the current wall clock time, in seconds (total wall clock time since GetDP was started).

`GetMemory` ¶

  `[]`

  Returns the current memory usage, in megabytes (maximum resident set size).

`SetNumberRunTime` ¶

  `[expression]{char-expression}`

  Sets the char-expression ONELAB variable at run-time to expression.

`SetNumberRunTimeWithChoices` ¶

  `[expression]{char-expression}`

  Same as `SetNumberRunTime`, but adds the value to the choices of the ONELAB variable (i.e. in the same way as `SendToServer` in `PostOperation`, which are used for plotting the history of the variable).

`GetNumberRunTime` ¶

  `[ <expression> ]{char-expression}`

  Gets the value of the char-expression ONELAB variable at run-time. If the optional expression is provided, it is used as a default value if ONELAB is not available.

`SetVariable` ¶

  `[ expression <,…> ]{ $variable-id }`

  Sets the value of the runtime variable `$variable-id` to the value of the first expression, and returns this value. If optional expressions are provided, they are appended to the variable name, separated by `_`.

`GetVariable` ¶

  `[ <expression> <,…> ]{ $variable-id }`

  Gets the value of the runtime variable `$variable-id`. If the optional expressions are provided, they are appended to the variable name, separated by `_`.

`ValueFromIndex` ¶

  `[ ]{ expression-cst-list }`

  Treats expression-cst-list as a map of (entity, value) pairs. Useful to specify nodal or element-wise constraints, where entity is the node (mesh vertex) or element number (tag).

`VectorFromIndex` ¶

  `[ ]{ expression-cst-list }`

  Same `ValueFromIndex`, but with 3 scalar values per entity.

`ValueFromTable` ¶

  `[ expression ]{ char-expression }`

  Accesses the map char-expression created by a `NodeTable` or `ElementTable` `PostOperation`. If the map is not available (e.g. in pre-processing), or if the entity is not found in the map, use expression as default value. Useful to specify nodal or element-wise constraints, where entity is the node (mesh vertex) or element number (tag).

### 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 domain analysis). If two values are provided (with `Value [ expression, expression ]`), the first value can be used using the `InitSolution1` operation. This is mainly useful for the Newmark time-stepping scheme.

`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 Pi units 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[]] ;
            < FunctionRef XYZ[]; >
          }
        }
      }

  In this example, `Function` defines the mapping that translates the geometrical elements in the region `Left` by Pi units along the X-axis, so that they correspond with the elements in the reference region `Right`. For this mapping to work, the meshes of `Left` and `Right` must be identical. (The optional `FunctionRef` function allows to transform the reference region, useful e.g. to avoid generating overlapping meshes for rotational links.)

`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` or `GroupOfRegionsOf`, 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 >.

`VolCylShell` ¶

  Volume Jacobian with cylindrical shell transformation, for n-D regions in n-D geometries, n = 2 or 3. For n=2, `VolCylShell` is identical to `VolSphShell`. For n=3, the axis of the cylinder is supposed to be along the z axis.

  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 (see Input file format).

### 6.7 Types for `Formulation` ¶

formulation-type:

`FemEquation` ¶

  Finite element method formulation (all methods of moments, integral methods).

local-term-type:

`Integral` ¶

  Integral of Galerkin or Petrov-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. (Not implemented yet.)

`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. (Not implemented yet.)

`DtDtDof` ¶

  Time derivative of 2nd order applied only to the `Dof{}` term of the equation.

`Eig` ¶

  The term is multiplied by (a certain function of) the eigenvalue. This is to be used with the `GenerateSeparate` and `EigenSolve` `Resolution` operations. An optional `Order expression;` or `Rational expression;` statement can be added in the term to specify the eigenvalue function. Full documentation of this feature is not available yet.

`JacNL` ¶

  Nonlinear part of the Jacobian matrix (tangent stiffness matrix) to be assembled for nonlinear analysis.

`DtDofJacNL` ¶

  Nonlinear part of the Jacobian matrix for the first order time derivative (tangent mass matrix) to be assembled 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.

`SolveAgain` ¶

  `[system-id]`

  Save as `Solve`, but reuses the preconditionner when called multiple times.

`SetGlobalSolverOptions` ¶

  `[char-expression]`

  Set global PETSc solver options (with the same syntax as PETSc options specified on the command line, e.g. `"-ksp_type gmres -pc_type ilu"`).

`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 matrices separately for `DtDtDof`, `DtDof` and `NoDt` terms in system-id. The separate matrices can be used with the `Update` operation (for efficient time domain analysis of linear PDEs with constant coefficients), or with the `EigenSolve` operation (for solving generalized eigenvalue problems).

`GenerateOnly` ¶

  `[system-id, expression-cst-list]`

  Not documented yet.

`GenerateOnlyJac` ¶

  `[system-id, expression-cst-list]`

  Not documented yet.

`GenerateGroup` ¶

  Not documented yet.

`GenerateRightHandSideGroup` ¶

  Not documented yet.

`Update` ¶

  `[system-id]`

  Update the system of equations system-id (built from sub-matrices generated separately with `GenerateSeparate`) with the `TimeFunction`(s) provided in `Assign` constraints. This assumes that the problem is linear, that the matrix coefficients are independent of time, and that all sources are imposed using `Assign` constraints.

`Update` ¶

  `[system-id, expression]`

  Update the system of equations system-id (built from sub-matrices generated separately with `GenerateSeparate`) with expression. This assumes that the problem is linear, that the matrix coefficients are independent of time, and that the right-hand-side of the linear system can simply be multiplied by expression at each step.

`UpdateConstraint` ¶

  `[system-id, group-id, constraint-type]`

  Recompute the constraint of type constraint-type acting on group-id during processing.

`GetResidual` ¶

  `[system-id, $variable-id]`

  Compute the residual `r = b - A x` and store its L2 norm in the run-time variable $variable-id.

`GetNormSolution | GetNormRightHandSide | GetNormResidual | GetNormIncrement` ¶

  `[system-id, $variable-id]`

  Compute the norm of the solution (resp. right-hand-side, residual or increment) and store its L2 norm in the run-time variable $variable-id.

`SwapSolutionAndResidual` ¶

  `[system-id]`

  Swap the solution `x` and residual `r` vectors.

`SwapSolutionAndRightHandSide` ¶

  `[system-id]`

  Swap the solution `x` and right-hand-side `b` vectors.

`InitSolution` ¶

  `[system-id]`

  Creates a new solution vector, adds it to the solution vector list for system-id, and initializes the solution. The values in the vector are initialized to the values given in a `Constraint` of `Init` type (if two values are given in `Init`, the second value is used). If no constraint is provided, the values are initialized to zero if the solution vector is the first in the solution list; otherwise the values are initialized using the previous solution in the list.

`InitSolution1` ¶

  `[system-id]`

  Same as `InitSolution`, but uses the first value given in the `Init` constraints.

`CreateSolution` ¶

  `[system-id]`

  Creates a new solution vector, adds it to the solution vector list for system-id, and initializes the solution to zero.

`CreateSolution` ¶

  `[system-id, expression-cst]`

  Same as `CreateSolution`, but initialize the solution by copying the expression-cstth solution in the solution list.

`Apply` ¶

  `[system-id]`

  `x <- Ax`

`SetSolutionAsRightHandSide` ¶

  `[system-id]`

  `b <- x`

`SetRightHandSideAsSolution` ¶

  `[system-id]`

  `x <- b`

`Residual` ¶

  `[system-id]`

  `res <- b - Ax`

`CopySolution` ¶

  `[system-id, char-expression | constant-id() ]`

  Copy the current solution `x` into a vector named char-expression or into a list named constant-id.

`CopySolution` ¶

  `[char-expression | constant-id(), system-id]`

  Copy the vector named char-expression or the list named constant-id into the current solution `x`.

`CopyRightHandSide` ¶

  `[system-id, char-expression | constant-id() ]`

  Copy the current right-hand side `b` into a vector named char-expression or into a list named constant-id.

`CopyRightHandSide` ¶

  `[char-expression | constant-id(), system-id]`

  Copy the vector named char-expression or the list named constant-id into the current right-hand-side `b`.

`CopyResidual` ¶

  `[system-id, char-expression | constant-id() ]`

  Copy the current residual into a vector named char-expression or into a list named constant-id.

`CopyResidual` ¶

  `[char-expression | constant-id(), system-id]`

  Copy the vector named char-expression or the list named constant-id into the current residual.

`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`.

`RemoveLastSolution` ¶

  `[system-id]`

  Removes the last solution (i.e. associated with the last time step) associated with system system-id.

`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 (see Types for `Constraint`).

`Evaluate` ¶

  `[expression <, expression>]`

  Evaluate the expression(s).

`SetTime` ¶

  `[expression]`

  Change the current time.

`SetTimeStep` ¶

  `[expression]`

  Change the current time step number (1, 2, 3, ...)

`SetDTime` ¶

  `[expression]`

  Change the current time step value (dt).

`SetFrequency` ¶

  `[system-id, expression]`

  Change the frequency of system system-id.

`SystemCommand` ¶

  `[expression-char]`

  Execute the system command given by expression-char.

`Error` ¶

  `[expression-char]`

  Output error message expression-char.

`Test` ¶

  `[expression] { resolution-op }`

  If expression is true (nonzero), perform the operations in resolution-op.

`Test` ¶

  `[expression] { resolution-op } { resolution-op }`

  If expression is true (nonzero), perform the operations in the first resolution-op, else perform the operations in the second resolution-op.

`While` ¶

  `[expression] { resolution-op }`

  While expression is true (nonzero), perform the operations in resolution-op.

`Break` ¶

  `[]`

  Aborts an iterative loop, a time loop or a While loop.

`Exit` ¶

  `[]`

  Exit, brutally.

`Sleep` ¶

  `[expression]`

  Sleeps for expression seconds;

`SetExtrapolationOrder` ¶

  `[expression-cst]`

  Chooses the extrapolation order to compute the initialization of the solution vector in time loops. Default is 0.

`Print` ¶

  `[ { expression-list } <, File expression-char > <, Format expression-char > ]`

  Print the expressions listed in expression-list. If `Format` is given, use it to format the (scalar) expressions like `Printf`.

`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 < , expression > ]`

  Eigenvalue/eigenvector computation using Arpack or SLEPc. 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). The last optional argument allows to filter which eigenvalue/eigenvector pairs will be saved. For example, `($EigenvalueReal > 0)` would only keep pairs corresponding to eigenvalues with a striclty positive real part.

`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 (incremental) computation of a 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`) and the list of frequencies to consider. The computation is an approximation that assumes that the time step is constant; it is not an actual Discrete Fourier Transform (the number of samples is unknown a priori).

`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).

  Warning: GetDP automatically handles time-dependent constraints when they are provided using the `TimeFunction` mechanism in an `Assign`-type `Constraint` (see `Constraint`: specifying constraints on function spaces and formulations). However, GetDP cannot automatically transform general time-dependent source terms in weak formulations (time-dependent functions written in a `Integral` term). Such source terms will be correctly treated only for implicit Euler, as the expression in the `Integral` term is evaluated at the current time step. For other schemes, the source term should be written explicitly, by splitting it in two (`theta f_n+1 + (1-theta) f_n`), making use of the `AtAnteriorTimeStep[]` for the second part, and specifying `NeverDt` in the `Integral` term.

`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.

  Warning: same restrictions apply for time-dependent functions in the weak formulations as for `TimeLoopTheta`.

`TimeLoopAdaptive` ¶

  `[expression-cst,expression-cst,expression-cst,expression-cst, expression-cst,integration-method,<expression-cst-list>,
  System { {system-id,expression-cst,expression-cst,norm-type} ... } |
  PostOperation { {post-operation-id,expression-cst,expression-cst,norm-type} ... } ]`
  `{ resolution-op }`
  `{ resolution-op }`

  Time loop with variable time steps. The step size is adjusted according the local truncation error (LTE) of the specified Systems/PostOperations via a predictor-corrector method.
  The parameters are: start time, end time, initial time step, min. time step, max. time step, integration method, list of breakpoints (time points to be hit). The LTE calculation can be based on all DOFs of a system and/or on a PostOperation result. The parameters here are: System/PostOperation for LTE assessment, relative LTE tolerance, absolute LTE tolerance, norm-type for LTE calculation.
  Possible choices for integration-method are: `Euler, Trapezoidal, Gear_2, Gear_3, Gear_4, Gear_5, Gear_6`. The Gear methods correspond to backward differentiation formulas of order 2..6.
  Possible choices for norm-type: `L1Norm, MeanL1Norm, L2Norm, MeanL2Norm, LinfNorm`.
  `MeanL1Norm` and `MeanL2Norm` correspond to `L1Norm` and `L2Norm` divided by the number of degrees of freedom, respectively.
  The first resolution-op is executed every time step. The second one is only executed if the actual time step is accepted (LTE is in the specified range). E.g. `SaveSolution[]` is usually placed in the 2nd resolution-op.

`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 relative error to achieve and the relaxation factor (multiplies the iterative correction dx). The optional parameter is a flag for testing purposes.

`IterativeLoopN` ¶

  `[expression-cst,expression,
  System { {system-id,expression-cst,expression-cst, assessed-object norm-type} ... } |
  PostOperation { {post-operation-id,expression-cst,expression-cst, norm-type} ... } ]`
  { resolution-op }

  Similar to `IterativeLoop[]` but allows to specify in detail the tolerances and the type of norm to be calculated for convergence assessment.
  The parameters are: the maximum number of iterations (if no convergence), the relaxation factor (multiplies the iterative correction dx). The convergence assessment can be based on all DOFs of a system and/or on a PostOperation result. The parameters here are: System/PostOperation for convergence assessment, relative tolerance, absolute tolerance, assessed object (only applicable for a specified system), norm-type for error calculation.
  Possible choices for assessed-object: `Solution, Residual, RecalcResidual`. `Residual` assesses the residual from the last iteration whereas `RecalcResidual` calculates the residual once again after each iteration. This means that with `Residual` usually one extra iteration is performed, but `RecalcResidual` causes higher computational effort per iteration. Assessing the residual can only be used for Newton’s method.
  Possible choices for norm-type: `L1Norm, MeanL1Norm, L2Norm, MeanL2Norm, LinfNorm`.
  `MeanL1Norm` and `MeanL2Norm` correspond to `L1Norm` and `L2Norm` divided by the number of degrees of freedom, respectively.

`IterativeLinearSolver` ¶

  Generic iterative linear solver. To be documented.

`PostOperation` ¶

  `[post-operation-id]`

  Perform the specified `PostOperation`.

`GmshRead` ¶

  `[expression-char]`

  When GetDP is linked with the Gmsh library, read a file using Gmsh. This file can be in any format recognized by Gmsh. If the file contains one or multiple post-processing fields, these fields will be evaluated using the built-in `Field[]`, `ScalarField[]`, `VectorField[]`, etc., functions (see Miscellaneous functions).

  (Note that `GmshOpen` and `GmshMerge` can be used instead of `GmshRead` to force Gmsh to do classical “open” and “merge” operations, instead of trying to “be intelligent” when reading post-processing datasets, i.e., creating new models on the fly if necessary.)

`GmshRead` ¶

  `[expression-char, expression-cst]`

  Same thing as the `GmshRead` command above, except that the field is forced to be stored with the given tag. The tag can be used to retrieve the given field with the built-in `Field[]`, `ScalarField[]`, `VectorField[]`, etc., functions (see Miscellaneous functions).

`GmshRead` ¶

  `[expression-char, `$`string]`

  Same as the `GmshRead`, but evaluates expression-char by replacing a double precision format specifier with the value of the runtime variable `$`string.

`GmshWrite` ¶

  `[expression-char, expression-cst]`

  Writes the a Gmsh field to disk. (The format is guessed from the file extension.)

`GmshClearAll` ¶

  `[]`

  Clears all Gmsh data (loaded with `GmshRead` and friends).

`DeleteFile` ¶

  `[expression-char]`

  Delete a file.

`RenameFile` ¶

  `[expression-char, expression-char]`

  Rename a file.

`CreateDir | CreateDirectory` ¶

  `[expression-char]`

  Create a directory.

`MPI_SetCommSelf` ¶

  `[]`

  Changes MPI communicator to self.

`MPI_SetCommWorld` ¶

  `[]`

  Changes MPI communicator to world.

`MPI_Barrier` ¶

  `[]`

  MPI barrier (blocks until all processes have reached this call).

`MPI_BroadcastFields` ¶

  `[ < expression-list > ]`

  Broadcast all fields over MPI (except those listed in the list).

`MPI_BroadcastVariables` ¶

  `[]`

  Broadcast all runtime variables over MPI.

### 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.

`AppendToExistingFile` ¶

  expression-cst

  Appends the result to the file specified with `File`. (Same behavior as `>` if expression-cst = 1; same behavior as `>>` if expression-cst = 2.)

`Name | Label` ¶

  `expression-char`

  For formats that support it, sets the label of the output field to expression-char (also used with with `SendToServer` to force the label).

`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.

`AtGaussPoints` ¶

  expression-cst

  Print result at the specified number of Gauss points.

`Skin` ¶

  Computes the result on the boundary of the region.

`Smoothing` ¶

  < expression-cst >

  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.

`TimeValue` ¶

  `expression-cst-list-item | { expression-cst-list }`

  Outputs results for the specified time value(s) only.

`TimeImagValue` ¶

  `expression-cst-list-item | { expression-cst-list }`

  Outputs results for the specified imaginary time value(s) only.

`LastTimeStepOnly` ¶

  Outputs results for the last time step only (useful when calling a `PostOperation` directly in a `Resolution`, for example).

`AppendExpressionToFileName` ¶

  expression

  Evaluate the given expression at run-time and append it to the filename.

`AppendExpressionFormat` ¶

  expression-char

  C-style format string for printing the expression provided in `AppendExpressionToFileName`. Default is `"%.16g"`.

`AppendTimeStepToFileName` ¶

  < expression-cst >

  Appends the time step to the output file; only makes sense with `LastTimeStepOnly`.

`AppendStringToFileName` ¶

  expression-char

  Append the given expression-char to the filename.

`OverrideTimeStepValue` ¶

  expression-cst

  Overrides the value of the current time step with the given value.

`NoMesh` ¶

  < expression-cst >

  Prevents the mesh from being written in the output file (useful with new mesh-based solution formats).

`SendToServer` ¶

  expression-char

  Send the value to the Onelab server, using expression-char as the parameter name.

`SendToServer` ¶

  expression-char { expression-cst-list }

  Send the requested harmonics of the value to the Onelab server, using expression-char as the parameter name.

`Color` ¶

  expression-char

  Used with `SendToServer`, sets the color of the parameter in the Onelab server.

`Hidden` ¶

  < expression-cst >

  Used with `SendToServer`, selects the visibility of the exchanged value.

`Closed` ¶

  expression-char

  Used with `SendToServer`, closes (or opens) the subtree containing the parameter.

`Units` ¶

  expression-char

  Used with `SendToServer`, sets the units of the parameter in the Onelab server.

`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).

`StoreInVariable` ¶

  `$expression-char`

  Stores the result of a point-wise evaluation or an `OnRegion` post-processing operation in the run-time variable $code[$]expression-char.

`StoreInRegister` ¶

  `expression-cst`

  Stores the result of point-wise evaluation or an `OnRegion` post-processing operation in the register expression-cst.

`StoreMinInRegister` ¶
`StoreMaxInRegister` ¶

  `expression-cst`

  Stores the minimum or maximum value of an `OnElementsOf` post-processing operation in the register expression-cst.

`StoreMinXinRegister` ¶
`StoreMinYinRegister` ¶
`StoreMinZinRegister` ¶
`StoreMaxXinRegister` ¶
`StoreMaxYinRegister` ¶
`StoreMaxZinRegister` ¶

  `expression-cst`

  Stores the X, Y or Z coordinate of the location, where the minimum or maximum of an `OnElementsOf` post-processing operation occurs, in the register expression-cst.

`StoreInField` ¶

  `expression-cst`

  Stores the result of a post-processing operation in the field (Gmsh list-based post-processing view) with tag expression-cst.

`StoreInMeshBasedField` ¶

  `expression-cst`

  Stores the result of a post-processing operation in the mesh-based field (Gmsh mesh-based post-processing view) with tag 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 Input file format and the documentation of Gmsh (http://gmsh.info) 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 <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 <x-coord y-coord z-coord> … value.

`NodeTable` ¶

  Table of nodal values, in the form node-number node-value(s). When exported to a file, the total number of nodal values is printed first. The data is automatically exported as a run-time accessible list as well as a ONELAB variable, with the name of the `PostOperation` quantity. The values are also directly usable by the `ValueFromTable` function, which allows to use them as values in a nodal `Constraint`.

`ElementTable` ¶

  Table of element values, in the form element-number element-node-value(s). When exported to a file, the total number of element values is printed first. The data is automatically exported as a run-time accessible list as well as a ONELAB variable, with the name of the `PostOperation` quantity. The values are also directly usable by the `ValueFromTable` function, which allows to use them as values in an element-wise `Constraint`.

`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 (see `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` (see Constants, as well as 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 (see 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 (see Run-time variables and 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.

The same results can be obtained using a run-time variable `$v`:

    g[] = ($v = f[$1]) * Sin[$v^2];

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[] = 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. `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` (see `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 {
          Integral { [ epsr[] * Dof{Grad v} , {Grad v} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
        }
      }
    }

The density of the `Integral` 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 {
          Integral { [ 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 {
          Integral { [ nu[] * Dof{Curl a} , {Curl a} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
          Integral { [ - 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 {
          Integral { Dt [ mu[] * Dof{h} , {h} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
          Integral { [ 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 (see `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 (see `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 (see `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 domain 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 (see `Formulation` examples; see `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 `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 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 (see Running GetDP).

Many other ready-to-use examples are available on the website of the ONELAB project: http://onelab.info.

### 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` (see Gmsh examples). The geometry is two-dimensional and by symmetry only one half of the structure is modeled.

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 (see 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 (see 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` (see Gmsh examples). The geometry is two-dimensional and, by symmetry, one fourth of the structure is modeled.

The jacobian and integration methods used are the same as for the electrostatic problem presented in 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 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 Objects, and Types for objects. The format of the post-processing files is explained in Types for `PostOperation`.

### A.1 Input file format ¶

The native mesh format read by GetDP is the mesh file format produced by Gmsh (http://gmsh.info). 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 versions of the ‘msh’ format (2.0 and above), as well as binary meshes. See the Gmsh documentation for more information about these formats.

### 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://gmsh.info.

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 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 Compiling the source code ¶

Stable releases and source snapshots are available from http://getdp.info/src/. You can also access the Git repository directly:

1. The first time you want to download the latest full source, type:

       git clone http://gitlab.onelab.info/getdp/getdp.git

2. To update your local version to the latest and greatest, go in the getdp directory and type:

       git pull

Once you have the source code, you need to run CMake to configure your build (see the README.txt file in the top-level source directory for detailed information on how to run CMake).

Each build can be configured using a series of options, to selectively enable optional modules or features. Here is the list of CMake options:

`ENABLE_ARPACK` ¶

  Enable Arpack eigensolver (requires Fortran) (default: ON)

`ENABLE_BLAS_LAPACK` ¶

  Enable BLAS/Lapack for linear algebra (e.g. for Arpack) (default: ON)

`ENABLE_BUILD_LIB` ¶

  Enable ’lib’ target for building static GetDP library (default: OFF)

`ENABLE_BUILD_SHARED` ¶

  Enable ’shared’ target for building shared GetDP library (default: OFF)

`ENABLE_BUILD_DYNAMIC` ¶

  Enable dynamic GetDP executable (linked with shared lib) (default: OFF)

`ENABLE_BUILD_ANDROID` ¶

  Enable Android NDK library target (experimental) (default: OFF)

`ENABLE_BUILD_IOS` ¶

  Enable iOS (ARM) library target (experimental) (default: OFF)

`ENABLE_FORTRAN` ¶

  Enable Fortran (needed for Arpack/Sparskit/Zitsol & Bessel) (default: ON)

`ENABLE_GMSH` ¶

  Enable Gmsh functions (for field interpolation) (default: ON)

`ENABLE_GSL` ¶

  Enable GSL functions (for some built-in functions) (default: ON)

`ENABLE_KERNEL` ¶

  Enable kernel (required for actual computations) (default: ON)

`ENABLE_MMA` ¶

  Enable MMA optimizer (default: ON)

`ENABLE_MPI` ¶

  Enable MPI parallelization (with PETSc/SLEPc) (default: OFF)

`ENABLE_MULTIHARMONIC` ¶

  Enable multi-harmonic support (default: OFF)

`ENABLE_NR` ¶

  Enable NR functions (if GSL is unavailable) (default: ON)

`ENABLE_NX` ¶

  Enable proprietary NX extension (default: OFF)

`ENABLE_OCTAVE` ¶

  Enable Octave functions (default: OFF)

`ENABLE_OPENMP` ¶

  Enable OpenMP parallelization of some functions (experimental) (default: OFF)

`ENABLE_PETSC` ¶

  Enable PETSc linear solver (default: ON)

`ENABLE_PRIVATE_API` ¶

  Enable private API (default: OFF)

`ENABLE_PYTHON` ¶

  Enable Python functions (default: ON)

`ENABLE_SLEPC` ¶

  Enable SLEPc eigensolver (default: ON)

`ENABLE_SMALLFEM` ¶

  Enable experimental SmallFem assembler (default: OFF)

`ENABLE_SPARSKIT` ¶

  Enable Sparskit solver instead of PETSc (requires Fortran) (default: ON)

`ENABLE_SYSTEM_CONTRIB` ¶

  Use system versions of contrib libraries, when possible (default: OFF)

`ENABLE_PEWE` ¶

  Enable PeWe exact solutions (requires Fortran) (default: ON)

`ENABLE_WRAP_PYTHON` ¶

  Build Python wrappers (default: OFF)

`ENABLE_ZITSOL` ¶

  Enable Zitsol solvers (requires PETSc and Fortran) (default: OFF)

## Appendix D Frequently asked questions ¶

### D.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 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://getdp.info is the primary location to obtain information about GetDP. There you will for example find the complete reference manual and the bug tracking database.

### D.2 Installation ¶

1. Which OSes does GetDP run on?

   Gmsh runs on Windows, MacOS X, Linux and most Unix variants.

2. What do I need to compile GetDP from the sources?

   You need a C++ and a Fortran 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?

   You need cmake (http://www.cmake.org) and a C++ compiler (and a Fortran compiler depending on the modules/solvers you want to compile). See Compiling the source code and the README.txt file in the top-level source directory for more information.

4. GetDP [from a binary distribution] complains about missing libraries.

   Try `ldd getdp` (or `otool -L getdp` on MacOS 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.

### D.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://gmsh.info. 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 (PETSc or Sparskit).

   With PETSc-based linear solvers you can either specify options on the command line (e.g. with `-ksp_type gmres -pc_type ilu`), through a specific option file (with `-solver file`), through the `.petscrc` file located in your home directly, or directly in the `Resolution` field using the `SetGlobalSolverOptions[]` command.

   With Sparskit-based linear solvers can either specify options directly on command line (e.g. with `-Nb_Fill 200`), specify an option file explicitly (with `-solver file`), or edit the `solver.par` file in the current working directory. If no `solver.par` file exists in the current directory, GetDP will give create it the next time you perform a linear system solution.

## Appendix E 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 (see 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 F Version history ¶

    3.5.0 (May 13, 2022): generalized Trace operator on non-conforming meshes; new
    Errf function; source code reorganization.

    3.4.0 (September 23, 2021): new Min and Max functions on constants (at parse
    time); fixed regression in trees of edges in 2D; added support for non-ASCII
    paths on command line on Windows; GetDP now requires C++11 and CMake 3.3; small
    bug fixes.

    3.3.0 (December 21, 2019): improved support for curved elements; added support
    for auto-similar trees of edges (e.g. for sliding surfaces in 3D); update for
    latest Gmsh version.

    3.2.0 (July 1, 2019): improved node and edge link constraints search using
    rtree; added support for BF_Edge basis functions on curved elements; small
    fixes.

    3.1.0 (April 19, 2019): added support for high-order (curved) Lagrange elements
    (P2, P3 and P4); added support for latest Gmsh version; code refactoring.

    3.0.4 (December 9, 2018): allow general groups in Jacobian definitions; fixed
    string parser regression.

    3.0.3 (October 18, 2018): new AtGaussPoint PostOperation option; bug fixes.

    3.0.2 (September 10, 2018): small compilation fixes.

    3.0.1 (September 7, 2018): small bug fixes.

    3.0.0 (August 22, 2018): new extrapolation (see SetExtrapolationOrder) in
    time-domain resolutions; new string macros; added support for Gmsh MSH4 file
    format; new file handling operations and ElementTable format in PostOperation;
    added support for curved (2nd order) simplices; enhanced communication of
    post-processing data with ONELAB; many new functions (Atanh, JnSph, YnSph,
    ValueFromTable, ListFromServer, GroupExists, ...); various small bug fixes.

    2.11.3 (November 5, 2017): new 'Eig' operator for general eigenvalue problems
    (polynomial, rational); small improvements and bug fixes.

    2.11.2 (June 23, 2017): minor build system changes.

    2.11.1 (May 13, 2017): small bug fixes and improvements.

    2.11.0 (January 3, 2017): small improvements (complex math functions, mutual
    terms, one side of, get/save runtime variables) and bug fixes.

    2.10.0 (October 9, 2016): ONELAB 1.3 with usability and performance
    improvements.

    2.9.2 (August 21, 2016): small bug fixes.

    2.9.1 (August 18, 2016): small improvements (CopySolution[], -cpu) and bug
    fixes.

    2.9.0 (July 11, 2016): new ONELAB 1.2 protocol with native support for lists;
    simple C++ and Python API for exchanging (lists of) numbers and strings;
    extended .pro language for the construction of extensible problem definitions
    ("Append"); new VolCylShell transformation; new functions (Min, Max, SetDTime,
    ...); small fixes.

    2.8.0 (March 5, 2016): new Parse[], {Set,Get}{Number,String}[] and
    OnPositiveSideOf commands; added support for lists of strings; various
    improvements and bug fixes for better interactive use with ONELAB.

    2.7.0 (November 7, 2015): new Else/ElseIf commands; new timing and memory
    reporting functions.

    2.6.1 (July 30, 2015): enhanced Print[] command; minor fixes.

    2.6.0 (July 21, 2015): new ability to define and use Macros in .pro files; new
    run-time variables (act as registers, but with user-defined names starting with
    '$') and run-time ONELAB Get/Set functions; new Append*ToFileName PostOperation
    options; new GetResdidual and associated operations; fixes and extended format
    support in MSH file reader; fixed UpdateConstraint for complex-simulated-real
    and multi-harmonic calculations.

    2.5.1 (April 18, 2015): enhanced Python[] and DefineFunction[].

    2.5.0 (March 12, 2015): added option to embed Octave and Python interpreters;
    extended "Field" functions with gradient; extended string and list handling
    functions; new resolution and postprocessing functions (RenameFile, While, ...);
    extended EigenSolve with eigenvalue filter and high order polynomial EV
    problems; small bug fixes.

    2.4.4 (July 9, 2014): better stability, updated onelab API version and inline
    parameter definitions, fixed UpdateConstraint in harmonic case, improved
    performance of multi-harmonic assembly, fixed memory leak in parallel MPI
    version, improved EigenSolve (quadratic EVP with SLEPC, EVP on real matrices),
    new CosineTransform, MPI_Printf, SendMergeFileRequest parser commands, small
    improvements and bug fixes.

    2.4.3 (February 7, 2014): new mandatory 'Name' attribute to define onelab
    variables in DefineConstant[] & co; minor bug fixes.

    2.4.2 (Septembre 27, 2013): fixed function arguments in nested expressions;
    minor improvements.

    2.4.1 (July 16, 2013): minor improvements and bug fixes.

    2.4.0 (July 9, 2013): new two-step Init constraints; faster network computation
    (with new -cache); improved Update operation; better cpu/memory reporting; new
    -setnumber, -setstring and -gmshread command line options; accept unicode file
    paths on Windows; small bug fixes.

    2.3.1 (May 11, 2013): updated onelab; small bug fixes.

    2.3.0 (March 9, 2013): moved build system from autoconf to cmake; new family of
    Field functions to use data imported from Gmsh; improved list handling; general
    code cleanup.

    2.2.1 (July 15, 2012): cleaned up nonlinear convergence tests and integrated
    experimental adaptive time loop code; small bug fixes.

    2.2.0 (June 19, 2012): new solver interface based on ONELAB; parallel SLEPC
    eigensolvers; cleaned up syntax for groups, moving band and global basis
    functions; new Field[] functions to interpolate post-processing datasets from
    Gmsh; fixed bug in Sur/Lin transformation of 2 forms; fixed bug for periodic
    constraints on high-order edge elements.

    2.1.1 (April 12, 2011): default direct solver using MUMPS.

    2.1.0 (October 24, 2010): parallel resolution using PETSc solvers; new Gmsh2
    output format; new experimental SLEPc-based eigensolvers; various bug and
    performance fixes (missing face basis functions, slow PETSc assembly with global
    quantities, ...)

    2.0.0 (March 16, 2010): general code cleanup (separated interface from kernel
    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).

    1.2.1 (March 18, 2006): Small fixes.

    1.2.0 (March 10, 2006): Windows versions do not depend on Cygwin anymore; major
    parser cleanup (loops & co).

    1.1.2 (September 3, 2005): Small fixes.

    1.1.0 (August 21, 2005): 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.

    1.0.1 (February 6, 2005): Small fixes.

    1.0.0 (April 24, 2004): New license (GNU GPL); added support for latest Gmsh
    mesh file format; more code cleanups.

    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.

    0.89 (March 26, 2003): Code cleanup.

    0.88: Integrated FMM code.

    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.

    0.86 (January 25, 2003): Updated Gmsh output format; many small bug fixes.

    0.85 (January 21, 2002): Upgraded communication interface with Gmsh; new
    ChangeOfValues option in PostOperation; many internal changes.

    0.84 (September 6, 2001): 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.

    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.

    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[].

    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.

    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.

    0.79: Various bug fixes (mainly for the post-processing of intergal quantities);
    automatic treatment of degenerated cases in axisymmetrical problems.

    0.78: Various bug fixes.

    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.

    0.76: Reorganized high order shape functions; optimization of the
    post-processing (faster and less bloated); lots of internal cleanups.

    0.74: High order shape functions; lots of small bug fixes.

    0.73: Eigen value problems (Lanczos); minor corrections.

    0.7: constraint syntax; fourier transform; unary minus correction; complex
    integral quantity correction; separate iteration matrix generation.

    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.

    0.3: First distributed version.

## Appendix G Copyright and credits ¶

                     GetDP is copyright (C) 1997-2022

                              Patrick Dular

                                   and

                           Christophe Geuzaine
                         <cgeuzaine at uliege.be>

                           University of Liege

    Major code contributions to GetDP have been provided by Johan Gyselinck, Ruth
    Sabariego, Michael Asam, Bertrand Thierry, Francois Henrotte, Guillaume
    Dem\'esy. Other code contributors include: David Colignon, Tuan Ledinh, Patrick
    Lefevre, Andre Nicolet, Jean-Francois Remacle, Timo Tarhasaari, Christophe
    Trophime, Marc Ume and Peter Binde. See the source code for more details.

    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, 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.

    The AVL tree code (src/common/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 KissFFT code (src/numeric/kissfft.hh) is copyright (c) 2003-2010 Mark
    Borgerding. Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions are met: *
    Redistributions of source code must retain the above copyright notice, this list
    of conditions and the following disclaimer. * Redistributions in binary form
    must reproduce the above copyright notice, this list of conditions and the
    following disclaimer in the documentation and/or other materials provided with
    the distribution. * Neither the author nor the names of any contributors may be
    used to endorse or promote products derived from this software without specific
    prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
    CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
    PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
    CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY,
    OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
    SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
    INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
    CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING
    IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY
    OF SUCH DAMAGE.

    The picojson code (src/common/picojson.h) is Copyright 2009-2010 Cybozu Labs,
    Inc., Copyright 2011-2014 Kazuho Oku, All rights reserved. Redistribution and
    use in source and binary forms, with or without modification, are permitted
    provided that the following conditions are met: 1. Redistributions of source
    code must retain the above copyright notice, this list of conditions and the
    following disclaimer. 2. Redistributions in binary form must reproduce the above
    copyright notice, this list of conditions and the following disclaimer in the
    documentation and/or other materials provided with the distribution. THIS
    SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
    EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
    WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
    DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
    ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
    (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
    LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
    ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
    (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

    This version of GetDP may contain code (in the contrib/Arpack subdirectory)
    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 contrib/Sparskit subdirectory)
    copyright (C) 1990 Yousef Saad: check the configuration options.

## Appendix H License ¶

    GetDP is provided under the terms of the GNU General Public License
    (GPL), Version 2 or later.

    		    GNU GENERAL PUBLIC LICENSE
    		       Version 2, June 1991

     Copyright (C) 1989, 1991 Free Software Foundation, Inc.
     51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

    			    Preamble

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## Concept index ¶

Jump to: | . A B C D E F G H I J K L M N O P Q R S T U V W

| Index Entry | Section

.
| `.msh` file | Input file format
| `.pre` file | File pre
| `.res` file | File res

A
| Acknowledgments | Copyright and credits
| Analytical integration | Integration
| Approximation spaces | FunctionSpace
| Arguments | Functions
| Arguments, definition | Arguments
| Axisymmetric, transformation | Jacobian

B
| Basis Functions | FunctionSpace
| Binary operators | Operator types
| Boundary conditions | Constraint
| Boundary Element Method | Which problems can GetDP actually solve?
| Bugs, reporting | Bug reports
| Built-in functions | Functions

C
| Change of coordinates | Jacobian
| Changelog | Version history
| Circuit equations | Constraint
| Command line options | Running GetDP
| Comments | Comments
| Complete examples | Complete examples
| Complex-valued, system | Resolution
| Concepts, index | Concept index
| Conditionals | Macros loops and conditionals
| Constant, definition | Constants
| Constant, evaluation | Constants
| Constraint, definition | Constraint
| Constraint, examples | Constraint examples
| Constraint, types | Types for Constraint
| Contributors, list | Copyright and credits
| Coordinate change | Jacobian
| Copyright | Copying conditions
| Copyright | Copyright and credits
| Credits | Copyright and credits
| Curl | Fields
| Current values | Current values

D
| Dependences, objects | Numerical tools as objects
| Derivative, exterior | Fields
| Derivative, time | Formulation
| Developments, future | Which problems can GetDP actually solve?
| Differential operators | Fields
| Discrete function spaces | FunctionSpace
| Discrete quantities | Fields
| Discretized Geometry | Group
| Divergence | Fields
| Document syntax | Syntactic rules
| Download | Obtaining GetDP
| Download | Copying conditions

E
| Edge element space, example | Magnetostatics
| Efficiency, tips | Tips and tricks
| Electromagnetism | Which problems can GetDP actually solve?
| Electrostatic formulation | Electrostatics
| Elementary matrices | Formulation
| Entities, topological | Group
| Equations | Formulation
| Evaluation mechanism | Constants
| Evaluation, order | Evaluation order
| Examples, complete | Complete examples
| Examples, short | Short examples
| Exporting results | PostOperation
| Expression, definition | Expression definition
| Exterior derivative | Fields

F
| FAQ | Frequently asked questions
| Fields | Fields
| File, `.msh` | Input file format
| File, `.pre` | File pre
| File, `.res` | File res
| File, comment | Comments
| File, include | Includes
| File, mesh | Input file format
| File, pre-processing | File pre
| File, result | File res
| Finite Difference Method | Which problems can GetDP actually solve?
| Finite Element Method | Which problems can GetDP actually solve?
| Finite Volume Method | Which problems can GetDP actually solve?
| Floating point numbers | Constants
| Floating potential, example | Electrostatics 2
| Format, output | PostOperation
| Formulation, definition | Formulation
| Formulation, electrostatics | Electrostatics
| Formulation, examples | Formulation examples
| Formulation, types | Types for Formulation
| Frequency | Resolution
| Frequently asked questions | Frequently asked questions
| Function groups | Group
| Function space, definition | FunctionSpace
| Function space, examples | FunctionSpace examples
| Function space, types | Types for FunctionSpace
| Function, definition | Functions
| Function, definition | Function
| Function, examples | Function examples
| Future developments | Which problems can GetDP actually solve?

G
| Gauss, integration | Integration
| Geometric transformations | Jacobian
| Global quantity | Formulation
| Global quantity, example | Electrostatics 2
| Gmsh, examples | Gmsh examples
| Gmsh, file format | Input file format
| Gradient | Fields
| Grid | Group
| Group, definition | Group
| Group, examples | Group examples
| Group, types | Types for Group

H
| Hierarchical basis functions | FunctionSpace
| History, versions | Version history

I
| Includes | Includes
| Index, concepts | Concept index
| Index, metasyntactic variables | Metasyntactic variable index
| Index, syntax | Syntax index
| Input file format | Input file format
| Integer numbers | Constants
| Integral Equation Method | Which problems can GetDP actually solve?
| Integral quantity | Formulation
| Integration, definition | Integration
| Integration, examples | Integration examples
| Integration, types | Types for Integration
| Internet address | Obtaining GetDP
| Internet address | Copying conditions
| Interpolation | Fields
| Interpolation | FunctionSpace
| Introduction | Overview
| Iterative loop | Resolution

J
| Jacobian, definition | Jacobian
| Jacobian, examples | Jacobian examples
| Jacobian, types | Types for Jacobian

K
| Keywords, index | Syntax index

L
| License | Copying conditions
| License | License
| Linear system solving | Resolution
| Linking, objects | Numerical tools as objects
| Local quantity | Formulation
| Loops | Macros loops and conditionals

M
| Macros | Macros loops and conditionals
| Maps | PostOperation
| Matrices, elementary | Formulation
| Mechanics | Which problems can GetDP actually solve?
| Mesh | Group
| Mesh, examples | Gmsh examples
| Mesh, file format | Input file format
| Metasyntactic variables, index | Metasyntactic variable index
| Method of Moments | Which problems can GetDP actually solve?

N
| Networks | Constraint
| Newmark, time scheme | Resolution
| Newton, nonlinear scheme | Resolution
| Nodal function space, example | Electrostatics
| Nonlinear system solving | Resolution
| Numbers, integer | Constants
| Numbers, real | Constants
| Numerical integration | Integration

O
| Objects, definition | Objects
| Objects, dependences | Numerical tools as objects
| Objects, types | Types for objects
| Operating system | Running GetDP
| Operation, priorities | Evaluation order
| Operators, definition | Operator types
| Operators, differential | Fields
| Options, command line | Running GetDP
| Order of evaluation | Evaluation order
| Output file format | Output file format
| Overview | Overview

P
| Parameters | Functions
| Parse | Macros loops and conditionals
| Philosophy, general | Numerical tools as objects
| Physical problems | Which problems can GetDP actually solve?
| Picard, nonlinear scheme | Resolution
| Piecewise functions | Functions
| Piecewise functions | Function
| Platforms | Running GetDP
| Post-operation, definition | PostOperation
| Post-operation, examples | PostOperation examples
| Post-operation, types | Types for PostOperation
| Post-processing, definition | PostProcessing
| Post-processing, examples | PostProcessing examples
| Post-processing, types | Types for PostProcessing
| Priorities, operations | Evaluation order
| Processing cycle | Numerical tools as objects

Q
| Quantities, discrete | Fields
| Quantity, global | Formulation
| Quantity, integral | Formulation
| Quantity, local | Formulation
| Quantity, post-processing | PostProcessing
| Questions, frequently asked | Frequently asked questions

R
| Reading, guidelines | How to Read this Manual
| Real numbers | Constants
| Region groups | Group
| Registers, definition | Run-time variables and registers
| Relaxation factor | Resolution
| Reporting bugs | Bug reports
| Resolution, definition | Resolution
| Resolution, examples | Resolution examples
| Resolution, types | Types for Resolution
| Results, exploitation | PostProcessing
| Results, export | PostOperation
| Rules, syntactic | Syntactic rules
| Run-time variables, definition | Run-time variables and registers
| Running GetDP | Running GetDP

S
| Scope of GetDP | Which problems can GetDP actually solve?
| Sections | PostOperation
| Short examples | Short examples
| Solving, system | Resolution
| Spaces, discrete | FunctionSpace
| String | Constants
| Symmetry, integral kernel | Formulation
| Syntax, index | Syntax index
| Syntax, rules | Syntactic rules
| System, definition | Resolution

T
| Ternary operators | Operator types
| Thermics | Which problems can GetDP actually solve?
| Theta, time scheme | Resolution
| Time derivative | Formulation
| Time stepping | Resolution
| Time, discretization | Resolution
| Tips | Tips and tricks
| Tools, order of definition | Numerical tools as objects
| Topology | Group
| Transformations, geometric | Jacobian
| Tree | Group
| Tricks | Tips and tricks
| Types, definition | Types for objects

U
| Unary operators | Operator types
| User-defined functions | Function

V
| Values, current | Current values
| Variables, index | Metasyntactic variable index
| Versions | Version history

W
| Web site | Obtaining GetDP
| Web site | Copying conditions
| Wiki | Complete examples

## Metasyntactic variable index ¶

Jump to: | : … < | A B C E F G I J L M O P Q R S T V

| Index Entry | Section

:
| `:` | Syntactic rules

…
| `…` | Syntactic rules

<
| `<, >` | Syntactic rules

| `|` | Syntactic rules

A
| `affectation` | Constants
| `argument` | Arguments

B
| `basis-function-id` | FunctionSpace
| `basis-function-list` | FunctionSpace
| `basis-function-type` | FunctionSpace
| `basis-function-type` | Types for FunctionSpace
| `built-in-function-id` | Functions

C
| `coef-id` | FunctionSpace
| `constant-def` | Constants
| `constant-id` | Constants
| `constraint-case-id` | Constraint
| `constraint-case-val` | Constraint
| `constraint-id` | Constraint
| `constraint-type` | Constraint
| `constraint-type` | Types for Constraint
| `constraint-val` | Constraint
| `coord-function-id` | Coordinate functions

E
| `element-type` | Integration
| `element-type` | Types for Integration
| `etc` | Syntactic rules
| `expression` | Expression definition
| `expression-char` | Constants
| `expression-cst` | Constants
| `expression-cst-list` | Constants
| `expression-cst-list-item` | Constants
| `expression-list` | Expression definition
| `extended-math-function-id` | Extended math functions

F
| `formulation-id` | Formulation
| `formulation-list` | Resolution
| `formulation-type` | Formulation
| `formulation-type` | Types for Formulation
| `function-id` | Function
| `function-space-id` | FunctionSpace
| `function-space-type` | FunctionSpace
| `function-space-type` | Types for FunctionSpace

G
| `global-quantity-id` | FunctionSpace
| `global-quantity-type` | FunctionSpace
| `global-quantity-type` | Types for FunctionSpace
| `green-function-id` | Green functions
| `group-def` | Group
| `group-id` | Group
| `group-list` | Group
| `group-list-item` | Group
| `group-sub-type` | Group
| `group-type` | Group
| `group-type` | Types for Group

I
| `integer` | Constants
| `integral-value` | PostProcessing
| `integration-id` | Integration
| `integration-type` | Integration
| `integration-type` | Types for Integration

J
| `jacobian-id` | Jacobian
| `jacobian-type` | Jacobian
| `jacobian-type` | Types for Jacobian

L
| `local-term-type` | Formulation
| `local-term-type` | Types for Formulation
| `local-value` | PostProcessing
| `loop` | Macros loops and conditionals

M
| `math-function-id` | Math functions
| `misc-function-id` | Miscellaneous functions

O
| `operator-binary` | Operator types
| `operator-ternary-left` | Operator types
| `operator-ternary-right` | Operator types
| `operator-unary` | Operator types

P
| `post-operation-fmt` | PostOperation
| `post-operation-fmt` | Types for PostOperation
| `post-operation-id` | PostOperation
| `post-operation-op` | PostOperation
| `post-processing-id` | PostProcessing
| `post-quantity-id` | PostProcessing
| `post-quantity-type` | PostProcessing
| `post-value` | PostProcessing
| `post-value` | Types for PostProcessing
| `print-option` | PostOperation
| `print-option` | Types for PostOperation
| `print-support` | PostOperation
| `print-support` | Types for PostOperation

Q
| `quantity` | Fields
| `quantity-dof` | Fields
| `quantity-id` | Fields
| `quantity-operator` | Fields
| `quantity-type` | Formulation
| `quantity-type` | Types for Formulation

R
| `real` | Constants
| `register-get` | Run-time variables and registers
| `register-set` | Run-time variables and registers
| `resolution-id` | Resolution
| `resolution-op` | Resolution
| `resolution-op` | Types for Resolution

S
| `string` | Constants
| `string-id` | Constants
| `sub-space-id` | FunctionSpace
| `system-id` | Resolution
| `system-type` | Resolution

T
| `term-op-type` | Formulation
| `term-op-type` | Types for Formulation
| `type-function-id` | Type manipulation functions

V
| `variable-get` | Run-time variables and registers
| `variable-set` | Run-time variables and registers

## Syntax index ¶

Jump to: | - ! ? ( * / & # % ^ + < = > | ~ $ 0 1 2 3 A B C D E F G H I J L M N O P Q R S T U V W X Y Z

| Index Entry | Section

-
| `-` | Operator types
| `-` | Operator types
| `-adapt` | Running GetDP
| `-bin` | Running GetDP
| `-cache` | Running GetDP
| `-cal` | Running GetDP
| `-check` | Running GetDP
| `-cpu` | Running GetDP
| `-gmshread` | Running GetDP
| `-help` | Running GetDP
| `-info` | Running GetDP
| `-msh` | Running GetDP
| `-msh_scaling` | Running GetDP
| `-name` | Running GetDP
| `-onelab` | Running GetDP
| `-order` | Running GetDP
| `-p` | Running GetDP
| `-pos` | Running GetDP
| `-pre` | Running GetDP
| `-progress` | Running GetDP
| `-res` | Running GetDP
| `-restart` | Running GetDP
| `-setnumber` | Running GetDP
| `-setstring` | Running GetDP
| `-slepc` | Running GetDP
| `-solve` | Running GetDP
| `-solver` | Running GetDP
| `-split` | Running GetDP
| `-v` | Running GetDP
| `-v2` | Running GetDP
| `-verbose` | Running GetDP
| `-version` | Running GetDP

!
| `!` | Operator types
| `!=` | Operator types

?
| `?:` | Operator types

(
| `()` | Evaluation order

*
| `*` | Operator types

/
| `/` | Operator types
| `/*, */` | Comments
| `//` | Comments
| `/\` | Operator types

&
| `&` | Operator types
| `&&` | Operator types

#
| `#expression-cst` | Run-time variables and registers
| `#include` | Includes

%
| `%` | Operator types

^
| `^` | Operator types

+
| `+` | Operator types

<
| `<` | Operator types
| `<<` | Operator types
| `<=` | Operator types

=
| `=` | Constants
| `=` | Group
| `=` | Function
| `==` | Operator types

>
| `>` | Operator types
| `>=` | Operator types
| `>>` | Operator types

| `|` | Operator types
| `||` | Operator types

~
| `~` | Constants

$
| `$A` | Current values
| `$B` | Current values
| `$Breakpoint` | Current values
| `$C` | Current values
| `$DTime` | Current values
| `$EigenvalueImag` | Current values
| `$EigenvalueReal` | Current values
| `$integer` | Arguments
| `$Iteration` | Current values
| `$Theta` | Current values
| `$Time` | Current values
| `$TimeStep` | Current values
| `$X` | Current values
| `$XS` | Current values
| `$Y` | Current values
| `$YS` | Current values
| `$Z` | Current values
| `$ZS` | Current values

0
| `0D` | Constants

1
| `1D` | Constants

2
| `2D` | Constants

3
| `3D` | Constants

A
| `Abs` | Math functions
| `Acos` | Math functions
| `Adapt` | Types for PostOperation
| `Adaptation` | Types for PostOperation
| `AliasOf` | Types for FunctionSpace
| `All` | Group
| `All` | Jacobian
| `Analytic` | Integration
| `Append` | Constraint
| `Append` | FunctionSpace
| `Append` | Jacobian
| `Append` | Integration
| `Append` | Formulation
| `Append` | Resolution
| `Append` | PostProcessing
| `Append` | PostOperation
| `AppendExpressionFormat` | Types for PostOperation
| `AppendExpressionToFileName` | Types for PostOperation
| `AppendStringToFileName` | Types for PostOperation
| `AppendTimeStepToFileName` | Types for PostOperation
| `AppendToExistingFile` | Types for PostOperation
| `Apply` | Types for Resolution
| `Asin` | Math functions
| `Assign` | Types for Constraint
| `AssignFromResolution` | Types for Constraint
| `AssociatedWith` | Types for FunctionSpace
| `Atan` | Math functions
| `Atan2` | Math functions
| `AtGaussPoints` | Types for PostOperation
| `AtIndex` | Miscellaneous functions

B
| `BasisFunction` | FunctionSpace
| `BF` | Fields
| `BF_CurlEdge` | Types for FunctionSpace
| `BF_CurlGroupOfEdges` | Types for FunctionSpace
| `BF_CurlGroupOfPerpendicularEdge` | Types for FunctionSpace
| `BF_CurlPerpendicularEdge` | Types for FunctionSpace
| `BF_dGlobal` | Types for FunctionSpace
| `BF_DivFacet` | Types for FunctionSpace
| `BF_DivPerpendicularFacet` | Types for FunctionSpace
| `BF_Edge` | Types for FunctionSpace
| `BF_Facet` | Types for FunctionSpace
| `BF_Global` | Types for FunctionSpace
| `BF_GradGroupOfNodes` | Types for FunctionSpace
| `BF_GradNode` | Types for FunctionSpace
| `BF_GroupOfEdges` | Types for FunctionSpace
| `BF_GroupOfNodes` | Types for FunctionSpace
| `BF_GroupOfPerpendicularEdge` | Types for FunctionSpace
| `BF_Node` | Types for FunctionSpace
| `BF_NodeX` | Types for FunctionSpace
| `BF_NodeY` | Types for FunctionSpace
| `BF_NodeZ` | Types for FunctionSpace
| `BF_One` | Types for FunctionSpace
| `BF_PerpendicularEdge` | Types for FunctionSpace
| `BF_PerpendicularFacet` | Types for FunctionSpace
| `BF_Region` | Types for FunctionSpace
| `BF_RegionX` | Types for FunctionSpace
| `BF_RegionY` | Types for FunctionSpace
| `BF_RegionZ` | Types for FunctionSpace
| `BF_Volume` | Types for FunctionSpace
| `BF_Zero` | Types for FunctionSpace
| `Break` | Types for Resolution

C
| `Call string | expression-char;` | Macros loops and conditionals
| `Cart2Cyl` | Type manipulation functions
| `Cart2Pol` | Type manipulation functions
| `Cart2Sph` | Type manipulation functions
| `Case` | Constraint
| `Case` | Jacobian
| `Case` | Integration
| `Ceil` | Math functions
| `ChangeOfCoordinates` | Types for PostOperation
| `ChangeOfValues` | Types for PostOperation
| `Closed` | Types for PostOperation
| `Color` | Types for PostOperation
| `CompElementNum` | Miscellaneous functions
| `Complex` | Type manipulation functions
| `Complex_MH` | Type manipulation functions
| `ComplexScalarField` | Miscellaneous functions
| `ComplexTensorField` | Miscellaneous functions
| `ComplexVectorField` | Miscellaneous functions
| `CompX` | Type manipulation functions
| `CompXX` | Type manipulation functions
| `CompXY` | Type manipulation functions
| `CompXZ` | Type manipulation functions
| `CompY` | Type manipulation functions
| `CompYX` | Type manipulation functions
| `CompYY` | Type manipulation functions
| `CompYZ` | Type manipulation functions
| `CompZ` | Type manipulation functions
| `CompZX` | Type manipulation functions
| `CompZY` | Type manipulation functions
| `CompZZ` | Type manipulation functions
| `Conj` | Type manipulation functions
| `Constraint` | Constraint
| `Constraint` | FunctionSpace
| `CopyResidual` | Types for Resolution
| `CopyResidual` | Types for Resolution
| `CopyRightHandSide` | Types for Resolution
| `CopyRightHandSide` | Types for Resolution
| `CopySolution` | Types for Resolution
| `CopySolution` | Types for Resolution
| `Cos` | Math functions
| `Cos_wt_p` | Extended math functions
| `Cosh` | Math functions
| `CreateDir | CreateDirectory` | Types for Resolution
| `CreateSolution` | Types for Resolution
| `CreateSolution` | Types for Resolution
| `Criterion` | Integration
| `Cross` | Extended math functions
| `Curl` | Fields

D
| `d` | Fields
| `D1` | Fields
| `D2` | Fields
| `DecomposeInSimplex` | Types for PostOperation
| `DefineConstant` | Constants
| `DefineFunction` | Function
| `DefineGroup` | Group
| `DeleteFile` | Types for Resolution
| `Depth` | Types for PostOperation
| `deRham` | Types for Formulation
| `DestinationSystem` | Resolution
| `Det` | Extended math functions
| `dFunction` | FunctionSpace
| `Dimension` | Types for PostOperation
| `dInterpolationAkima` | Miscellaneous functions
| `dInterpolationBilinear` | Miscellaneous functions
| `dInterpolationLinear` | Miscellaneous functions
| `Distance` | Miscellaneous functions
| `Div` | Fields
| `dJn` | Math functions
| `Dof` | Fields
| `Dt` | Types for Formulation
| `DtDof` | Types for Formulation
| `DtDofJacNL` | Types for Formulation
| `DtDt` | Types for Formulation
| `DtDtDof` | Types for Formulation
| `DualEdgesOf` | Types for Group
| `DualFacetsOf` | Types for Group
| `DualNodesOf` | Types for Group
| `DualVolumesOf` | Types for Group
| `dYn` | Math functions

E
| `EdgesOf` | Types for Group
| `EdgesOfTreeIn` | Types for Group
| `Eig` | Types for Formulation
| `EigenSolve` | Types for Resolution
| `EigenvalueLegend` | Types for PostOperation
| `ElementNum` | Miscellaneous functions
| `ElementsOf` | Types for Group
| `ElementTable` | Types for PostOperation
| `ElementVol` | Miscellaneous functions
| `Else` | Macros loops and conditionals
| `ElseIf ( expression-cst )` | Macros loops and conditionals
| `ENABLE_ARPACK` | Compiling the source code
| `ENABLE_BLAS_LAPACK` | Compiling the source code
| `ENABLE_BUILD_ANDROID` | Compiling the source code
| `ENABLE_BUILD_DYNAMIC` | Compiling the source code
| `ENABLE_BUILD_IOS` | Compiling the source code
| `ENABLE_BUILD_LIB` | Compiling the source code
| `ENABLE_BUILD_SHARED` | Compiling the source code
| `ENABLE_FORTRAN` | Compiling the source code
| `ENABLE_GMSH` | Compiling the source code
| `ENABLE_GSL` | Compiling the source code
| `ENABLE_KERNEL` | Compiling the source code
| `ENABLE_MMA` | Compiling the source code
| `ENABLE_MPI` | Compiling the source code
| `ENABLE_MULTIHARMONIC` | Compiling the source code
| `ENABLE_NR` | Compiling the source code
| `ENABLE_NX` | Compiling the source code
| `ENABLE_OCTAVE` | Compiling the source code
| `ENABLE_OPENMP` | Compiling the source code
| `ENABLE_PETSC` | Compiling the source code
| `ENABLE_PEWE` | Compiling the source code
| `ENABLE_PRIVATE_API` | Compiling the source code
| `ENABLE_PYTHON` | Compiling the source code
| `ENABLE_SLEPC` | Compiling the source code
| `ENABLE_SMALLFEM` | Compiling the source code
| `ENABLE_SPARSKIT` | Compiling the source code
| `ENABLE_SYSTEM_CONTRIB` | Compiling the source code
| `ENABLE_WRAP_PYTHON` | Compiling the source code
| `ENABLE_ZITSOL` | Compiling the source code
| `EndFor` | Macros loops and conditionals
| `EndIf` | Macros loops and conditionals
| `Entity` | FunctionSpace
| `EntitySubType` | FunctionSpace
| `EntityType` | FunctionSpace
| `Equation` | Formulation
| `Error` | Types for Resolution
| `Evaluate` | Types for Resolution
| `Exit` | Types for Resolution
| `Exp` | Math functions

F
| `Fabs` | Math functions
| `FacetsOf` | Types for Group
| `FacetsOfTreeIn` | Types for Group
| `FemEquation` | Types for Formulation
| `Field` | Miscellaneous functions
| `Field` | Miscellaneous functions
| `File` | Types for PostOperation
| `File` | Types for PostOperation
| `File` | Types for PostOperation
| `Floor` | Math functions
| `Fmod` | Math functions
| `For ( expression-cst : expression-cst : expression-cst )` | Macros loops and conditionals
| `For ( expression-cst : expression-cst )` | Macros loops and conditionals
| `For string In { expression-cst : expression-cst : expression-cst }` | Macros loops and conditionals
| `For string In { expression-cst : expression-cst }` | Macros loops and conditionals
| `Form0` | Types for FunctionSpace
| `Form1` | Types for FunctionSpace
| `Form1P` | Types for FunctionSpace
| `Form2` | Types for FunctionSpace
| `Form2P` | Types for FunctionSpace
| `Form3` | Types for FunctionSpace
| `Format` | PostOperation
| `Format` | Types for PostOperation
| `Formulation` | FunctionSpace
| `Formulation` | Formulation
| `FourierTransform` | Types for Resolution
| `Frequency` | Resolution
| `Frequency` | Types for PostOperation
| `FrequencyLegend` | Types for PostOperation
| `Function` | Function
| `Function` | FunctionSpace
| `FunctionSpace` | FunctionSpace

G
| `Gauss` | Types for Integration
| `GaussLegendre` | Types for Integration
| `Generate` | Types for Resolution
| `GenerateGroup` | Types for Resolution
| `GenerateJac` | Types for Resolution
| `GenerateOnly` | Types for Resolution
| `GenerateOnlyJac` | Types for Resolution
| `GenerateRightHandSideGroup` | Types for Resolution
| `GenerateSeparate` | Types for Resolution
| `GeoElement` | Integration
| `GetCpuTime` | Miscellaneous functions
| `GetMemory` | Miscellaneous functions
| `GetNormSolution | GetNormRightHandSide | GetNormResidual | GetNormIncrement` | Types for Resolution
| `GetNumberRunTime` | Miscellaneous functions
| `GetNumElements` | Miscellaneous functions
| `GetResidual` | Types for Resolution
| `GetVariable` | Miscellaneous functions
| `GetVolume` | Miscellaneous functions
| `GetWallClockTime` | Miscellaneous functions
| `Global` | Types for Group
| `Global` | Types for Formulation
| `GlobalEquation` | Formulation
| `GlobalQuantity` | FunctionSpace
| `GlobalTerm` | Formulation
| `Gmsh` | Types for PostOperation
| `GmshClearAll` | Types for Resolution
| `GmshParsed` | Types for PostOperation
| `GmshRead` | Types for Resolution
| `GmshRead` | Types for Resolution
| `GmshRead` | Types for Resolution
| `GmshWrite` | Types for Resolution
| `Gnuplot` | Types for PostOperation
| `Grad` | Fields
| `GradHelmholtz` | Green functions
| `GradLaplace` | Green functions
| `Group` | Group
| `Group` | FunctionSpace
| `GroupOfRegionsOf` | Types for Group
| `GroupsOfEdgesOf` | Types for Group
| `GroupsOfEdgesOnNodesOf` | Types for Group
| `GroupsOfNodesOf` | Types for Group

H
| `HarmonicToTime` | Types for PostOperation
| `Helmholtz` | Green functions
| `Hexahedron` | Types for Integration
| `Hidden` | Types for PostOperation
| `Hypot` | Extended math functions

I
| `If ( expression-cst )` | Macros loops and conditionals
| `Im` | Type manipulation functions
| `In` | Formulation
| `In` | PostProcessing
| `Include` | Includes
| `IndexOfSystem` | Formulation
| `Init` | Types for Constraint
| `InitFromResolution` | Types for Constraint
| `InitSolution` | Types for Resolution
| `InitSolution1` | Types for Resolution
| `Integral` | PostProcessing
| `Integral` | Types for Formulation
| `Integral` | Types for Formulation
| `Integral` | Types for PostProcessing
| `Integration` | Integration
| `Integration` | Formulation
| `Integration` | PostProcessing
| `InterpolationAkima` | Miscellaneous functions
| `InterpolationBilinear` | Miscellaneous functions
| `InterpolationLinear` | Miscellaneous functions
| `Interval` | Extended math functions
| `Inv` | Extended math functions
| `Iso` | Types for PostOperation
| `Iso` | Types for PostOperation
| `IterativeLinearSolver` | Types for Resolution
| `IterativeLoop` | Types for Resolution
| `IterativeLoopN` | Types for Resolution

J
| `JacNL` | Types for Formulation
| `Jacobian` | Jacobian
| `Jacobian` | Formulation
| `Jacobian` | PostProcessing
| `Jn` | Math functions

L
| `Lanczos` | Types for Resolution
| `Laplace` | Green functions
| `LastTimeStepOnly` | Types for PostOperation
| `LevelTest` | Macros loops and conditionals
| `Lin` | Types for Jacobian
| `Line` | Types for Integration
| `Link` | Types for Constraint
| `LinkCplx` | Types for Constraint
| `List` | Constants
| `ListAlt` | Constants
| `Local` | PostProcessing
| `Local` | Types for Formulation
| `Local` | Types for PostProcessing
| `Log` | Math functions
| `Log10` | Math functions
| `Loop` | Formulation

M
| `Macro ( expression-char , expression-char ) ;` | Macros loops and conditionals
| `Macro string | expression-char` | Macros loops and conditionals
| `Max` | Math functions
| `Min` | Math functions
| `MPI_Barrier` | Types for Resolution
| `MPI_BroadcastFields` | Types for Resolution
| `MPI_BroadcastVariables` | Types for Resolution
| `MPI_SetCommSelf` | Types for Resolution
| `MPI_SetCommWorld` | Types for Resolution

N
| `Name` | Constraint
| `Name` | FunctionSpace
| `Name` | Jacobian
| `Name` | Integration
| `Name` | Formulation
| `Name` | Resolution
| `Name` | PostProcessing
| `Name` | PostOperation
| `Name | Label` | Types for PostOperation
| `NameOfBasisFunction` | FunctionSpace
| `NameOfCoef` | FunctionSpace
| `NameOfConstraint` | FunctionSpace
| `NameOfConstraint` | Formulation
| `NameOfFormulation` | Resolution
| `NameOfFormulation` | PostProcessing
| `NameOfMesh` | Resolution
| `NameOfPostProcessing` | PostOperation
| `NameOfSpace` | Formulation
| `NameOfSystem` | PostProcessing
| `Network` | Formulation
| `Network` | Types for Constraint
| `NeverDt` | Types for Formulation
| `Node` | Formulation
| `NodesOf` | Types for Group
| `NodeTable` | Types for PostOperation
| `NoMesh` | Types for PostOperation
| `NoNewLine` | Types for PostOperation
| `Norm` | Extended math functions
| `Normal` | Miscellaneous functions
| `NormalSource` | Miscellaneous functions
| `NumberOfPoints` | Integration

O
| `OnBox` | Types for PostOperation
| `OnElementsOf` | Types for PostOperation
| `OnGlobal` | Types for PostOperation
| `OnGrid` | Types for PostOperation
| `OnGrid` | Types for PostOperation
| `OnLine` | Types for PostOperation
| `OnPlane` | Types for PostOperation
| `OnPoint` | Types for PostOperation
| `OnRegion` | Types for PostOperation
| `OnSection` | Types for PostOperation
| `Operation` | Resolution
| `Operation` | PostOperation
| `Order` | Miscellaneous functions
| `OriginSystem` | Resolution
| `OverrideTimeStepValue` | Types for PostOperation

P
| `Parse [ expression-char ];` | Macros loops and conditionals
| `Period` | Extended math functions
| `Pi` | Constants
| `Point` | Types for Integration
| `PostOperation` | PostOperation
| `PostOperation` | Types for Resolution
| `PostProcessing` | PostProcessing
| `Print` | PostOperation
| `Print` | Types for Resolution
| `Print` | Types for Resolution
| `Printf` | Miscellaneous functions
| `Prism` | Types for Integration
| `Pyramid` | Types for Integration

Q
| `Quadrangle` | Types for Integration
| `QuadraturePointIndex` | Miscellaneous functions
| `Quantity` | FunctionSpace
| `Quantity` | Formulation
| `Quantity` | PostProcessing

R
| `Rand` | Miscellaneous functions
| `Re` | Type manipulation functions
| `Region` | Types for Group
| `Region` | Constraint
| `Region` | Jacobian
| `RemoveLastSolution` | Types for Resolution
| `RenameFile` | Types for Resolution
| `Residual` | Types for Resolution
| `Resolution` | FunctionSpace
| `Resolution` | Resolution
| `Return` | Macros loops and conditionals
| `Rot` | Fields
| `Rotate` | Extended math functions

S
| `SaveSolution` | Types for Resolution
| `SaveSolutions` | Types for Resolution
| `Scalar` | Types for FunctionSpace
| `ScalarField` | Miscellaneous functions
| `SendToServer` | Types for PostOperation
| `SendToServer` | Types for PostOperation
| `SetDTime` | Types for Resolution
| `SetExtrapolationOrder` | Types for Resolution
| `SetFrequency` | Types for Resolution
| `SetGlobalSolverOptions` | Types for Resolution
| `SetNumberRunTime` | Miscellaneous functions
| `SetNumberRunTimeWithChoices` | Miscellaneous functions
| `SetRightHandSideAsSolution` | Types for Resolution
| `SetSolutionAsRightHandSide` | Types for Resolution
| `SetTime` | Types for Resolution
| `SetTimeStep` | Types for Resolution
| `SetVariable` | Miscellaneous functions
| `Sign` | Math functions
| `SimpleTable` | Types for PostOperation
| `Sin` | Math functions
| `Sin_wt_p` | Extended math functions
| `Sinh` | Math functions
| `Skin` | Types for PostOperation
| `Sleep` | Types for Resolution
| `Smoothing` | Types for PostOperation
| `Solve` | Types for Resolution
| `SolveAgain` | Types for Resolution
| `SolveJac` | Types for Resolution
| `Solver` | Resolution
| `Sort` | Types for PostOperation
| `Sqrt` | Math functions
| `SquDyadicProduct` | Type manipulation functions
| `SquNorm` | Extended math functions
| `StoreInField` | Types for PostOperation
| `StoreInMeshBasedField` | Types for PostOperation
| `StoreInRegister` | Types for PostOperation
| `StoreInVariable` | Types for PostOperation
| `StoreMaxInRegister` | Types for PostOperation
| `StoreMaxXinRegister` | Types for PostOperation
| `StoreMaxYinRegister` | Types for PostOperation
| `StoreMaxZinRegister` | Types for PostOperation
| `StoreMinInRegister` | Types for PostOperation
| `StoreMinXinRegister` | Types for PostOperation
| `StoreMinYinRegister` | Types for PostOperation
| `StoreMinZinRegister` | Types for PostOperation
| `SubRegion` | Constraint
| `SubSpace` | FunctionSpace
| `Support` | FunctionSpace
| `Sur` | Types for Jacobian
| `SurAxi` | Types for Jacobian
| `SurfaceArea` | Miscellaneous functions
| `SwapSolutionAndResidual` | Types for Resolution
| `SwapSolutionAndRightHandSide` | Types for Resolution
| `Symmetry` | Formulation
| `System` | Resolution
| `SystemCommand` | Types for Resolution

T
| `Table` | Types for PostOperation
| `Tan` | Math functions
| `Tangent` | Miscellaneous functions
| `TangentSource` | Miscellaneous functions
| `Tanh` | Math functions
| `TanhC2` | Math functions
| `Target` | Types for PostOperation
| `Tensor` | Type manipulation functions
| `TensorDiag` | Type manipulation functions
| `TensorField` | Miscellaneous functions
| `TensorSym` | Type manipulation functions
| `TensorV` | Type manipulation functions
| `Test` | Types for Resolution
| `Test` | Types for Resolution
| `Tetrahedron` | Types for Integration
| `TimeFunction` | Constraint
| `TimeImagValue` | Types for PostOperation
| `TimeLegend` | Types for PostOperation
| `TimeLoopAdaptive` | Types for Resolution
| `TimeLoopNewmark` | Types for Resolution
| `TimeLoopTheta` | Types for Resolution
| `TimeStep` | Types for PostOperation
| `TimeTable` | Types for PostOperation
| `TimeValue` | Types for PostOperation
| `TransferSolution` | Types for Resolution
| `Transpose` | Extended math functions
| `Triangle` | Types for Integration
| `TTrace` | Extended math functions
| `Type` | Constraint
| `Type` | FunctionSpace
| `Type` | Integration
| `Type` | Formulation
| `Type` | Resolution

U
| `Unit` | Extended math functions
| `Units` | Types for PostOperation
| `UnitVectorX` | Type manipulation functions
| `UnitVectorY` | Type manipulation functions
| `UnitVectorZ` | Type manipulation functions
| `Update` | Types for Resolution
| `Update` | Types for Resolution
| `UpdateConstraint` | Types for Resolution
| `UsingPost` | PostOperation

V
| `Value` | PostProcessing
| `Value` | Types for PostOperation
| `ValueFromIndex` | Miscellaneous functions
| `ValueFromTable` | Miscellaneous functions
| `Vector` | Type manipulation functions
| `Vector` | Types for FunctionSpace
| `VectorField` | Miscellaneous functions
| `VectorFromIndex` | Miscellaneous functions
| `Vol` | Types for Jacobian
| `VolAxi` | Types for Jacobian
| `VolAxiRectShell` | Types for Jacobian
| `VolAxiSphShell` | Types for Jacobian
| `VolAxiSqu` | Types for Jacobian
| `VolAxiSquRectShell` | Types for Jacobian
| `VolAxiSquSphShell` | Types for Jacobian
| `VolCylShell` | Types for Jacobian
| `VolRectShell` | Types for Jacobian
| `VolSphShell` | Types for Jacobian
| `VolumesOf` | Types for Group

W
| `While` | Types for Resolution

X
| `X` | Coordinate functions
| `XYZ` | Coordinate functions

Y
| `Y` | Coordinate functions
| `Yn` | Math functions

Z
| `Z` | Coordinate functions

#### Footnotes

##### (1)

If you are looking for a graphical front-end to GetDP, you may consider using Gmsh (available at http://gmsh.info). 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 Input file format, and Types for `PostOperation`).