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# GetDP 1.0

Patrick Dular and Christophe Geuzaine

GetDP is a scientific computation software for the numerical solution of integro-differential equations, using finite element and integral type methods. This is edition 4.1 (April, 7 2005) of the GetDP Reference Manual, for GetDP 1.0.

> Copying conditions |  | Terms and conditions of use.
> Introduction |  | What is GetDP?
> 1. Overview |  | Quick overview of the general philosophy of GetDP.
> 2. Expressions |  | Definition of basic expressions in GetDP.
> 3. Objects |  | Definition of the 10 GetDP objects.
> 4. Types for objects |  | Definition of all available types for the 10 objects.
> 5. Short examples |  | Simple object examples.
> 6. Complete examples |  | Some simple complete examples.
> 7. Running GetDP |  | How to run GetDP on your operating system.
> 8. File formats |  | Input and output file formats.
> 9. Bugs, versions and credits |  | Contact information and ChangeLog
> A. Tips and tricks |  | Some tips to make your life easier with GetDP.
> B. Frequently asked questions |  | The GetDP FAQ
> C. Gmsh examples |  | Sample Gmsh input files.
> D. License |  | Complete copy of the license.
> Concept index |  | Index of concepts.
> Metasyntactic variable index |  | Index of metasyntactic variables used in this manual.
> Syntax index |  | Index of reserved keywords in the GetDP language.

> -- The Detailed Node Listing ---
> Introduction

> Research and collaboration activities |  |
> Education and training activities |  |
> Industrial studies |  |
> How to read this manual |  |

> Overview

> 1.1 Numerical tools as objects |  |
> 1.2 Syntactic rules used in this document |  |
> 1.3 Comments |  |
> 1.4 Includes |  |
> 1.5 Which problems can GetDP actually solve? |  |

> Expressions

> 2.1 Definition |  |
> 2.2 Constants |  |
> 2.3 Operators |  |
> 2.4 Functions |  |
> 2.5 Current values |  |
> 2.6 Arguments |  |
> 2.7 Registers |  |
> 2.8 Fields |  |

> Definition

> 2.3 Operators |  |
> 2.2 Constants |  |
> 2.4 Functions |  |
> 2.5 Current values |  |
> 2.8 Fields |  |

> Operators

> 2.3.1 Operator types |  |
> 2.3.2 Evaluation order |  |

> Objects

> 3.1 `Group`: defining topological entities |  |
> 3.2 `Function`: defining global and piecewise expressions |  |
> 3.3 `Constraint`: specifying constraints on function spaces and formulations |  |
> 3.4 `FunctionSpace`: building function spaces |  |
> 3.5 `Jacobian`: defining jacobian methods |  |
> 3.6 `Integration`: defining integration methods |  |
> 3.7 `Formulation`: building equations |  |
> 3.8 `Resolution`: solving systems of equations |  |
> 3.9 `PostProcessing`: exploiting computational results |  |
> 3.10 `PostOperation`: exporting results |  |

> Types for objects

> 4.1 Types for `Group` |  |
> 4.2 Types for `Function` |  |
> 4.3 Types for `Constraint` |  |
> 4.4 Types for `FunctionSpace` |  |
> 4.5 Types for `Jacobian` |  |
> 4.6 Types for `Integration` |  |
> 4.7 Types for `Formulation` |  |
> 4.8 Types for `Resolution` |  |
> 4.9 Types for `PostProcessing` |  |
> 4.10 Types for `PostOperation` |  |

> Types for `Function`

> 4.2.1 Math functions |  |
> 4.2.2 Extended math functions |  |
> 4.2.3 Green functions |  |
> 4.2.4 Type manipulation functions |  |
> 4.2.5 Coordinate functions |  |
> 4.2.6 Miscellaneous functions |  |

> Short examples

> 5.1 Constant expression examples |  |
> 5.2 `Group` examples |  |
> 5.3 `Function` examples |  |
> 5.4 `Constraint` examples |  |
> 5.5 `FunctionSpace` examples |  |
> 5.6 `Jacobian` examples |  |
> 5.7 `Integration` examples |  |
> 5.8 `Formulation` examples |  |
> 5.9 `Resolution` examples |  |
> 5.10 `PostProcessing` examples |  |
> 5.11 `PostOperation` examples |  |

> `FunctionSpace` examples

> 5.5.1 Nodal finite element spaces |  |
> 5.5.2 High order nodal finite element space |  |
> 5.5.3 Nodal finite element space with floating potentials |  |
> 5.5.4 Edge finite element space |  |
> 5.5.5 Edge finite element space with gauge condition |  |
> 5.5.6 Coupled edge and nodal finite element spaces |  |
> 5.5.7 Coupled edge and nodal finite element spaces for multiply connected domains |  |

> `Formulation` examples

> 5.8.1 Electrostatic scalar potential formulation |  |
> 5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges |  |
> 5.8.3 Magnetostatic 3D vector potential formulation |  |
> 5.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation |  |
> 5.8.5 Nonlinearities, Mixed formulations, ... |  |

> `Resolution` examples

> 5.9.1 Static resolution (electrostatic problem) |  |
> 5.9.2 Frequency domain resolution (magnetodynamic problem) |  |
> 5.9.3 Time domain resolution (magnetodynamic problem) |  |
> 5.9.4 Nonlinear time domain resolution (magnetodynamic problem) |  |
> 5.9.5 Coupled formulations |  |

> Complete examples

> 6.1 Electrostatic problem |  |
> 6.2 Magnetostatic problem |  |
> 6.3 Magnetodynamic problem |  |

> File formats

> 8.1 Input file format |  |
> 8.2 Output file format |  |

> Output file format

> 8.2.1 File ``.pre'` |  |
> 8.2.2 File ``.res'` |  |

> Bugs, versions and credits

> 9.1 Bugs |  |
> 9.2 Versions |  |
> 9.3 Credits |  |

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# 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 section D. 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 9.3 Credits.

The source code and various pre-compiled versions of GetDP (for Unix, Windows and Mac OS) can be downloaded from the web site http://www.geuz.org/getdp/.

If you use GetDP, we would appreciate that you mention it in your work. References and the latest news about GetDP are always available on http://www.geuz.org/getdp/. Please send all GetDP-related questions to the public GetDP mailing list at getdp@geuz.org.

If you want to integrate GetDP into a closed-source software, or want to sell a modified closed-source version of GetDP, please contact one of the authors. You can purchase a version of GetDP under a different license, with "no strings attached" (for example allowing you to take parts of GetDP and integrate them into your own proprietary code).

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# Introduction

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.

> Research and collaboration activities |  |
> Education and training activities |  |
> Industrial studies |  |
> How to read this manual |  |

## Research and collaboration activities

The internal structure of the software is very close to the structure used to define discrete problems in the input data files. As a result, a unicity and conciseness of both development and application levels is obtained without any interface between them, which facilitates work of any kind--particularly validations. GetDP permits to rapidly develop tools for the comparison of methods and the exchange of solutions between research teams, which is one of the main aims of collaboration. Moreover, after a short training, GetDP could be used by teams as a basis for their own developments, while allowing a large freedom in the latter.

## Education and training activities

The software environment consists of modeling tools applicable to various physical problems. Education and training can therefore be offered in various domains and at different levels. The closeness between the definition of discrete problems and their symbolic mathematical expressions entails that the theory and mathematical bases of numerical methods, which are essential to anyone who wants to tackle the solving of discrete problems, can be directly followed by their practical applications. These applications can be evolutionary, in the sense that the offered tools are of various levels of complexity. Everyone can tackle, step by step, according to his apprenticeship level, the tools adapted to the solving of more and more complex problems, as well as various methods for the solving of the same problem.

## Industrial studies

The treatment of industrial problems can also be facilitated because GetDP is adapted to the study of a wide range of physical problems using various numerical methods. In particular, adapted made to measure and ready-to-use software could be rapidly developed for given problems.

## How to read this manual

After reading 1. Overview, and 2. Expressions, which depict the general concepts and the way to construct simple expressions in GetDP, you can safely (during a first reading) skip 3. Objects and 4. Types for objects and directly go to 5. Short examples. This chapter gives simple examples of the problem definition syntax used to define the discrete problems. For each example, you should then go back to 3. Objects, and 4. Types for objects, and have a detailed view of the syntax of the objects appearing in it. Note that indexes for many concepts and for all the syntax elements are available at the end of this manual.

Once the examples presented in 5. Short examples, are understood, you may start to use GetDP on your computer (see section 7. Running GetDP), for example by solving the complete examples presented in 6. Complete examples.

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# 1. Overview

> 1.1 Numerical tools as objects |  |
> 1.2 Syntactic rules used in this document |  |
> 1.3 Comments |  |
> 1.4 Includes |  |
> 1.5 Which problems can GetDP actually solve? |  |

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## 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 treatments of defined problems.

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## 1.2 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.

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## 1.3 Comments

Both C and C++ style comments are supported and can be used in the input data file 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.

These commands won't have the described effects inside double quotes or inside GetDP keywords.

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## 1.4 Includes

Files can be included by placing one of the following commands (expression-char represents a file name) on a separate line, and outside any GetDP object. Any text placed after an include command on the same line is ignored.

    Include expression-char
    #include expression-char

See 2.2 Constants, for the definition of the character expression expression-char.

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## 1.5 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
  eigen value problems (experimental)

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.

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# 2. Expressions

> 2.1 Definition |  |
> 2.2 Constants |  |
> 2.3 Operators |  |
> 2.4 Functions |  |
> 2.5 Current values |  |
> 2.6 Arguments |  |
> 2.7 Registers |  |
> 2.8 Fields |  |

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## 2.1 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 1.2 Syntactic rules used in this document):

    expression:
      integer |
      real |
      constant-id |
      quantity |
      argument |
      current-value |
      register-value-set |
      register-value-get |
      operator-unary expression |
      expression operator-binary expression |
      expression operator-ternary-left expression operator-ternary-right expression |
      built-in-function-id [ < expression-list > ] < { expression-cst-list } > |
      function-id [ < expression-list > ]

The following sections introduce the quantities that can appear in expressions, i.e. constant terminals (integer, real) and constant expression identifiers (constant-id, expression-cst-list), discretized fields (quantity), arguments (argument), current values (current-value), register values (register-value-set, register-value-get), operators (operator-unary, operator-binary, operator-ternary-left, operator-ternary-right) and built-in or user-defined functions (built-in-function-id, function-id). List of expressions are defined as:

    expression-list:
      expression <,...>

> 2.3 Operators |  |
> 2.2 Constants |  |
> 2.4 Functions |  |
> 2.5 Current values |  |
> 2.8 Fields |  |

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## 2.2 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:
      integer |
      real |
      constant-id |
      operator-unary expression-cst |
      expression-cst operator-binary expression-cst |
      expression-cst operator-ternary-left expression-cst operator-ternary-right
          expression-cst |
      math-function-id [ < expression-cst-list > ]

List of constant expressions are defined as:

    expression-cst-list:
      expression-cst-list-item <,...>

with

    expression-cst-list-item:
      expression-cst |
      expression-cst : expression-cst |
      expression-cst : expression-cst : expression-cst |
      constant-id {} |
      constant-id { expression-cst-list } |
      List[ constant-id ] |
      ListAlt[ constant-id, constant-id ]

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 last 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 section 2.4 Functions).

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 > <,...> ]; |
      constant-id = constant-def;

with

    constant-id:
      string

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

Notes:

1. Five constants are predefined in GetDP: `Pi` (3.1415926535897932), `0D` (0), `1D` (1), `2D` (2) and `3D` (3).
2. The assignment in `DefineConstant` (zero if no expression-cst is given) is performed only if constant-id has not yet been defined. This kind of explicit default definition mechanism is most useful in general problem definition structures making use of a large number of generic constants, functions or groups. When exploiting only a part of a complex problem definition structure, the default definition mechanism allows to define the quantities of interest only, the others being assigned a default value (that will not be used during the processing but that avoids the error messages produced when references to undefined quantities are made).

See 5.1 Constant expression examples, as well as 5.3 `Function` examples, for some examples.

Constant character strings can also be defined. The syntax for their definition is the following:

    string-id = "string";

with

    string-id:
      string

Character expressions can be defined, using these constant strings:

    expression-char:
      "string" |
      string-id |
      StrCat[ expression-char , expression-char ]

The third case in this definition permits to concatenate two character expressions.

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## 2.3 Operators

> 2.3.1 Operator types |  |
> 2.3.2 Evaluation order |  |

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### 2.3.1 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.

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.

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### 2.3.2 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), !`
`/\`
`*, /, %`
`+, -`
`<, >, <=, >=`
`!=, ==`
`&&, ||`
`?:`

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## 2.4 Functions

Two types of functions coexist in GetDP: user-defined functions (function-id, see 3.2 `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 section 2.6 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 4.2 Types for `Function`.
2. Classical mathematical functions (see section 4.2.1 Math functions) are the only functions allowed in a constant definition (see the definition of expression-cst in 2.2 Constants).

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## 2.5 Current values

Current values are a special kind of arguments (see section 2.6 Arguments) which return the current integer or floating point value of an internal GetDP variable:

`$Time`
  Value of the current time. This value is set to zero for non time dependent analyses.
`$DTime`
  Value of the current time increment used in a time stepping algorithm.
`$Theta`
  Current theta value in a theta time stepping algorithm.
`$TimeStep`
  Number of the current time step in a time stepping algorithm.
`$Iteration`
  Number of the current iteration in a nonlinear loop.
`$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 section 4.10 Types for `PostOperation`).

Note:

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

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## 2.6 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 3.2 `Function`: defining global and piecewise expressions, and 5.3 `Function` examples, for more details.

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## 2.7 Registers

In many situations, identical parts of expressions are used more than once. If this is not a problem with constant expressions (since expression-csts are evaluated only once during the analysis of the problem definition structure, cf. 2.2 Constants), it may introduce some important overhead while evaluating complex expressions (which are evaluated at runtime, thanks to an internal stack mechanism). In order to circumvent this problem, the evaluation result of any part of an expression can be saved in a register: a memory location where this partial result will be accessible without any costly reevaluation of the partial expression.

Registers have the following syntax:

    register-value-set:
      expression#integer

    register-value-get:
      #integer

Thus, to store any part of an expression in the register 5, one should add `#5` directly after the expression. To reuse the value stored in this register, one simply uses `#5` instead of the expression it should replace.

See 5.3 `Function` examples, for an example.

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## 2.8 Fields

A discretized quantity (defined in a function space, cf. 3.4 `FunctionSpace`: building function spaces) is represented between braces `{}`, and can only appear in well-defined expressions in `Formulation` (see section 3.7 `Formulation`: building equations) and `PostProcessing` (see section 3.9 `PostProcessing`: exploiting computational results) objects:

    quantity:
      < quantity-dof > { < quantity-operator > quantity-id }

with

    quantity-id:
      string

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.

`BF`
  Indicates that only a basis function will be used (only valid with basis functions associated with regions).

quantity-operator:

`d`
  Exterior derivative (d): applied to a p-form, gives a (p+1)-form.

`Grad`
  Gradient: applied to a scalar field, gives a vector.

`Curl`
`Rot`
  Curl: applied to a vector field, gives a vector.

`Div`
  Divergence (div): applied to a vector field, gives a scalar.

`dInv`
  d^(-1): applied to a p-form, gives a (p-1)-form.

`GradInv`
  Inverse grad: applied to a gradient field, gives a scalar.

`CurlInv`
`RotInv`
  Inverse curl: applied to a curl field, gives a vector.

`DivInv`
  Inverse div: applied to a divergence field.

Note:

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.

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# 3. Objects

This chapter presents the formal definition of the ten GetDP objects mentioned in 1. Overview. To be concise, all the possible parameters for these objects are not given here (cf. the etc syntactic rule defined in 1.2 Syntactic rules used in this document). Please refer to 4. Types for objects, for this.

> 3.1 `Group`: defining topological entities |  |
> 3.2 `Function`: defining global and piecewise expressions |  |
> 3.3 `Constraint`: specifying constraints on function spaces and formulations |  |
> 3.4 `FunctionSpace`: building function spaces |  |
> 3.5 `Jacobian`: defining jacobian methods |  |
> 3.6 `Integration`: defining integration methods |  |
> 3.7 `Formulation`: building equations |  |
> 3.8 `Resolution`: solving systems of equations |  |
> 3.9 `PostProcessing`: exploiting computational results |  |
> 3.10 `PostOperation`: exporting results |  |

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## 3.1 `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 section 8.1 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 2.2 Constants.

The syntax for the definition of groups is:

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

with

    group-id:
      string

    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 | OnOneSideOf | etc

Notes:

1. integer as a group-list-item is the only interface with the mesh; with each element is associated a region number, being this integer, and a geometrical type (see section 8.1 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 section 2.2 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. Multidefinition: `group-id{#n} = group-def` defines the groups `group-id{i}`, i=1, ..., n, by replacing for each i any `group-id{}` in group-def with `group-id{i}`.
4. `#group-list` is an abbreviation of `Region[group-list]`.

See 4.1 Types for `Group`, for the complete list of options and 5.2 `Group` examples, for some examples.

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## 3.2 `Function`: defining global and piecewise expressions

A user-defined function can be global in space or piecewise defined in region groups. A physical characteristic is an example of a piecewise defined function (e.g. magnetic permeability, electric conductivity, etc.) and can be simply a constant, for linear materials, or a function of one or several arguments for nonlinear materials. Such functions can of course depend on space coordinates or time, which can be needed to express complex constraints.

A definition of initially empty functions can be made thanks to the `DefineFunction` command so that their identifiers exist and can be referred to (but cannot be used) in other objects. The syntax for the definition of functions is:

    Function {
      < DefineFunction [ function-id <,...> ]; > ...
      < function-id [ < group-def > ] = expression; > ...
    }

with

    function-id:
      string

Note:

1. The optional group-def in brackets must be of `Region` type, and indicates on which region the (piecewise) function is defined. Warning: it is incorrect to write `f[reg1]=1; g[reg2]=f[]+1;` since the domains of definition of `f[]` and `g[]` don't match.

See 4.2 Types for `Function`, for the complete list of built-in functions and 5.3 `Function` examples, for some examples.

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## 3.3 `Constraint`: specifying constraints on function spaces and formulations

Constraints can be referred to in `FunctionSpace` objects to be used for boundary conditions, to impose global quantities or to initialize quantities. These constraints can be expressed with functions or be imposed by the pre-resolution of another discrete problem. Other constraints can also be defined, e.g. constraints of network type for the definition of circuit connections, to be used in `Formulation` objects.

The syntax for the definition of constraints is:

    Constraint {
      { Name constraint-id <{#integer}>; Type constraint-type;
        Case {
          { Region group-def; < Type constraint-type; >
            < SubRegion group-def; > < TimeFunction expression; >
            constraint-val; } ...
        }
      | Case constraint-case-id {
          { Region group-def; < Type constraint-type; >
            constraint-case-val; } ...
        } ...
      } ...
    }

with

    constraint-id:
    constraint-case-id:
      string

    constraint-type:
      Assign | Init | Network | etc

    constraint-val:
      Value expression | NameOfResolution resolution-id | etc

    constraint-case-val:
      Branch { integer, integer } | etc

Notes:

1. Multidefinition: `Name constraint-id{#n}` defines the constraints `constraint-id{i}`, i=1, ..., n, replacing for each i any `group-id{}` in group-defs of `Case` fields with group-id{i}.
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 4.3 Types for `Constraint`, for the complete list of options and 5.4 `Constraint` examples, for some examples.

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## 3.4 `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 section 5.5 `FunctionSpace` examples).

The syntax for the definition of function spaces is:

    FunctionSpace {
      { Name function-space-id <{#integer}>;
        Type function-space-type;
        BasisFunction {
         { Name basis-function-id; NameOfCoef coef-id;
           Function basis-function-type
             < { Quantity quantity-id;
                 Formulation formulation-id {#integer};
                 Group group-def; Resolution resolution-id {} } >;
           Support group-def; Entity group-def; } ...
        }
      < SubSpace {
         { Name sub-space-id;
           NameOfBasisFunction basis-function-list; } ...
        } >
      < GlobalQuantity {
         { Name global-quantity-id; Type global-quantity-type;
           NameOfCoef coef-id; } ...
        } >
      < Constraint {
         { NameOfCoef coef-id;
           EntityType group-type; < EntitySubType group-sub-type; >
           NameOfConstraint constraint-id <{}>; } ...
        } >
      } ...
    }

with

    function-space-id:
    basis-function-id:
    coef-id:
    sub-space-id:
    global-quantity-id:
    formulation-id:
    resolution-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. Multidefinition: `Name function-space-id{#n}` defines the function spaces `function-space-id{i}`, i=1, ..., n, replacing for each i any `{}` in group-defs of `BasisFunction` field and in constraint-id`{}` of `Constraint` field with {i}.
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. Piecewise defined basis functions: the same `Name` for several `BasisFunction` fields permits to define piecewise basis functions; separate `NameOfCoef`s must be defined for those fields.
4. Constraint: a constraint is associated with geometrical entities defined by an automatically created `Group` of type group-type, using the `Region` defined in a `Constraint` object as its main argument, and the optional `SubRegion` in the same object as a group-sub-type argument.
5. Function: a global basis function (`BF_Global` or `BF_dGlobal`) needs parameters, i.e. it is given by the quantity (quantity-id) pre-computed from multiresolutions performed on multiformulations.

See 4.4 Types for `FunctionSpace`, for the complete list of options and 5.5 `FunctionSpace` examples, for some examples.

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## 3.5 `Jacobian`: defining jacobian methods

Jacobian methods can be referred to in `Formulation` and `PostProcessing` objects to be used in the computation of integral terms and for changes of coordinates. They are based on `Group` objects and define the geometrical transformations applied to the reference elements (i.e. lines, triangles, quadrangles, tetrahedra, prisms, hexahedra, etc.). Besides the classical lineic, surfacic and volume Jacobians, the `Jacobian` object allows the construction of various transformation methods (e.g. infinite transformations for unbounded domains) thanks to dedicated jacobian methods.

The syntax for the definition of Jacobian methods is:

    Jacobian {
      { Name jacobian-id;
        Case {
          { Region group-def | All;
            Jacobian jacobian-type < { expression-cst-list } >; } ...
        }
      } ...
    }

with

    jacobian-id:
      string

    jacobian-type:
      Vol | Sur | VolAxi | etc

Note:

1. The default case of a `Jacobian` object is defined by `Region All` and must follow all the other cases.

See 4.5 Types for `Jacobian`, for the complete list of options and 5.6 `Jacobian` examples, for some examples.

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## 3.6 `Integration`: defining integration methods

Various numerical or analytical integration methods can be referred to in `Formulation` and `PostProcessing` objects to be used in the computation of integral terms, each with a set of particular options (number of integration points for quadrature methods--which can be linked to an error criterion for adaptative methods, definition of transformations for singular integrations, etc.). Moreover, a choice can be made between several integration methods according to a criterion (e.g. on the proximity between the source and computation points in integral formulations).

The syntax for the definition of integration methods is:

    Integration {
      { Name integration-id; < Criterion expression; >
        Case {
        < { Type integration-type;
            Case {
              { GeoElement element-type; NumberOfPoints expression-cst } ...
            }
          } ... >
        < { Type Analytic; } ... >
        }
      } ...
    }

with

    integration-id:
      string

    integration-type:
      Gauss | etc

    element-type:
      Line | Triangle | Tetrahedron etc

See 4.6 Types for `Integration`, for the complete list of options and 5.7 `Integration` examples, for some examples.

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## 3.7 `Formulation`: building equations

The `Formulation` tool permits to deal with volume, surface and line integrals with many kinds of densities to integrate, written in a form that is similar to their symbolic expressions (it uses the same expression syntax as elsewhere in GetDP), which therefore permits to directly take into account various kinds of elementary matrices (e.g. with scalar or cross products, anisotropies, nonlinearities, time derivatives, various test functions, etc.). In case nonlinear physical characteristics are considered, arguments are used for associated functions. In that way, many formulations can be directly written in the data file, as they are written symbolically. Fields involved in each formulation are declared as belonging to beforehand defined function spaces. The uncoupling between formulations and function spaces allows to maintain a generality in both their definitions.

A `Formulation` is characterized by its type, the involved quantities (of local, global or integral type) and a list of equation terms. Global equations can also be considered, e.g. for the coupling with network relations.

The syntax for the definition of formulations is:

    Formulation {
      { Name formulation-id <{#integer}>; Type formulation-type;
        Quantity {
          { Name quantity-id; Type quantity-type;
            NameOfSpace function-space-id <{}>
                      < [ sub-space-id | global-quantity-id ] >;
            < Symmetry expression-cst; >
            < [ expression ]; In group-def;
              Jacobian jacobian-id; Integration integration-id; >
            < IndexOfSystem integer; >  } ...
        }
        Equation {
         < local-term-type
             { < term-op-type > [ expression, expression ];
               In group-def; Jacobian jacobian-id;
               Integration integration-id; } > ...
         < GlobalTerm
             { < term-op-type > [ expression, expression ];
               In group-def; } > ...
         < GlobalEquation
             { Type Network; NameOfConstraint constraint-id;
               { Node expression; Loop expression; Equation expression;
                 In group-def; } ...
             } > ...
        }
      } ...
    }

with

    formulation-id:
      string

    formulation-type:
      FemEquation | etc

    local-term-type:
      Galerkin | deRham

    quantity-type:
      Local | Global | Integral

    term-op-type:
      Dt | DtDt | JacNL | etc

Note:

1. Multidefinition: `Name formulation-id{#n}` defines the formulations `formulation-id{i}`, i=1, ..., n, replacing for each i any `{}` in function-space-id`{}` of `Quantity` field and in group-defs of `Equation` field with {i}.
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 section 3.8 `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.
4. A `GlobalEquation` defines a global equation to be assembled in the matrix of the system.

See 4.7 Types for `Formulation`, for the complete list of options and 5.8 `Formulation` examples, for some examples.

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## 3.8 `Resolution`: solving systems of equations

The operations available in a `Resolution` include: the generation of a linear system, its solving with various kinds of linear solvers, the saving of the solution or its transfer to another system, the definition of various time stepping methods, the construction of iterative loops for nonlinear problems (Newton-Raphson and fixed point methods), etc. Multi-harmonic resolutions, coupled problems (e.g. magneto-thermal) or linked problems (e.g. pre-computations of source fields) are thus easily defined in GetDP.

The `Resolution` object is characterized by a list of systems to build and their associated formulations, using time or frequency domain, and a list of elementary operations:

    Resolution {
      { Name resolution-id <{#integer}>;
        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 > } ...
        }
        Operation { < resolution-op; > ... }
      } ...
    }

with

    resolution-id:
    system-id:
      string

    formulation-list:
      formulation-id <{}> | { formulation-id <{}> <,...> }

    system-type:
      Real | Complex

    resolution-op:
      Generate[system-id] | Solve[system-id] | etc

Notes:

1. Multidefinition: `Name resolution-id{#n}` defines the resolutions `resolution-id{i}`, i=1, ..., n, replacing for each i any `{}` in formulation-id`{}` of `System` field with {i}.
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 section 4.8 Types for `Resolution`).
6. `OriginSystem` permits to specify the systems from which ambiguous quantity definitions can be solved (see section 3.7 `Formulation`: building equations).

See 4.8 Types for `Resolution`, for the complete list of options and 5.9 `Resolution` examples, for some examples.

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## 3.9 `PostProcessing`: exploiting computational results

The `PostProcessing` object is based on the quantities defined in a `Formulation` and permits the construction (thanks to the expression syntax) of any useful piecewise defined quantity of interest:

    PostProcessing {
      { Name post-processing-id <{#integer}>;
        NameOfFormulation formulation-id <{}>; < NameOfSystem system-id; >
        Quantity {
          { Name post-quantity-id; Value { post-value ... } } ...
        }
      } ...
    }

with

    post-processing-id:
    post-quantity-id:
      string

    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. Multidefinition: `Name post-processing-id{#n}` defines the post-processings `post-processing-id{i}`, i=1, ..., n, replacing for each i any `{}` in formulation-id`{}` with {i}.
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 section 3.10 `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 4.9 Types for `PostProcessing`, for the complete list of options and 5.10 `PostProcessing` examples, for some examples.

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## 3.10 `PostOperation`: exporting results

The `PostOperation` is the bridge between results obtained with GetDP and the external world. It defines several elementary operations on `PostProcessing` quantities (e.g. plot on a region, section on a user-defined plane, etc.), and outputs the results in several file formats.

    PostOperation {
      { Name post-operation-id; NameOfPostProcessing post-processing-id;
        < Format post-operation-fmt; >
        Operation { < post-operation-op; > ... }
      } ...
    } |
    PostOperation post-operation-id UsingPost post-processing-id {
      < post-operation-op; > ...
    } ...

with

    post-operation-id:
      string

    post-operation-op:
      Print[ post-quantity-term, print-support <,print-option> ... ] |
      etc

    post-quantity-term:
      post-quantity-id <[group-def]> |
      post-quantity-id post-quantity-op post-quantity-id[group-def] |
      post-quantity-id[group-def] post-quantity-op post-quantity-id

    post-quantity-op:
      + | - | * | /

    print-support:
      OnElementsOf group-def | OnRegion group-def | OnGlobal | etc

    print-option:
      File expression-char | Format post-operation-fmt | etc

    post-operation-fmt:
      Table | TimeTable | etc

Notes:

1. Both `PostOperation` syntaxes are equivalent. The first one conforms to the overall interface, but the second one is more concise.
2. The format post-operation-fmt defined outside the `Operation` field is applied to all the post-processing operations, unless other formats are explicitly given in these operations with the `Format` option (see section 4.10 Types for `PostOperation`). The default format is `Gmsh`.
3. The optional argument `[group-def]` of the post-quantity-id can only be used when this quantity has been defined as an integral-value (see section 3.9 `PostProcessing`: exploiting computational results). In this case, the sum of all elementary integrals is performed over the region group-def.
4. The post-quantity-op allows the simple combination of space-dependent quantities (post-quantity-id) with global integral quantities (`post-quantity-id[group-def]`).

See 4.10 Types for `PostOperation`, for the complete list of options and 5.11 `PostOperation` examples, for some examples.

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# 4. Types for objects

This chapter presents the complete list of choices associated with metasyntactic variables introduced for the ten GetDP objects.

> 4.1 Types for `Group` |  |
> 4.2 Types for `Function` |  |
> 4.3 Types for `Constraint` |  |
> 4.4 Types for `FunctionSpace` |  |
> 4.5 Types for `Jacobian` |  |
> 4.6 Types for `Integration` |  |
> 4.7 Types for `Formulation` |  |
> 4.8 Types for `Resolution` |  |
> 4.9 Types for `PostProcessing` |  |
> 4.10 Types for `PostOperation` |  |

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## 4.1 Types for `Group`

Types in

    group-type [ R1 <, group-sub-type R2 > ]

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

`Region`
  Regions in R1.

`Global`
  Regions in R1 (variant of `Region` used with global `BasisFunction`s `BF_Global` and `BF_dGlobal`).

`NodesOf`
  Nodes of elements of R1

  < `Not`: but not those of R2 >.

`EdgesOf`
  Edges of elements of R1

  < `Not`: but not those of R2 >.

`FacetsOf`
  Facets of elements of R1

  < `Not`: but not those of R2 >.

`VolumesOf`
  Volumes of elements of R1

  < `Not`: but not those of R2 >.

`ElementsOf`
  Elements of regions in R1

  < `OnOneSideOf`: only elements on one side of R2) >.

`GroupsOfNodesOf`
  Groups of nodes of elements of R1 (a group is associated with each region).

`GroupsOfEdgesOf`
  Groups of edges of elements of R1 (a group is associated with each region).

  < `InSupport`: in a support R2 being a group of type `ElementOf`, i.e. containing elements >.

`GroupsOfEdgesOnNodesOf`
  Groups of edges incident to nodes of elements of R1 (a group is associated with each node).

  < `Not`: but not those of R2) >.

`EdgesOfTreeIn`
  Edges of a tree of edges of R1

  < `StartingOn`: a complete tree is first built on R2 >.

`FacetsOfTreeIn`
  Facets of a tree of facets of R1

  < `StartingOn`: a complete tree is first built on R2 >.

`DualNodesOf`
  Dual nodes of elements of R1.

`DualEdgesOf`
  Dual edges of elements of R1.

`DualFacetsOf`
  Dual facets of elements of R1.

`DualVolumesOf`
  Dual volumes of elements of R1.

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## 4.2 Types for `Function`

> 4.2.1 Math functions |  |
> 4.2.2 Extended math functions |  |
> 4.2.3 Green functions |  |
> 4.2.4 Type manipulation functions |  |
> 4.2.5 Coordinate functions |  |
> 4.2.6 Miscellaneous functions |  |

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### 4.2.1 Math functions

The following functions are the equivalent of the functions of the C math library, and always return real-valued expressions. These are the only functions allowed in constant expressions (expression-cst, see 2.2 Constants).

math-function-id:

`Exp`
  `[expression]`

  Exponential function: e^expression.

`Log`
  `[expression]`

  Natural logarithm: ln(expression), expression>0.

`Log10`
  `[expression]`

  Base 10 logarithm: log10(expression), expression>0.

`Sqrt`
  `[expression]`

  Square root, expression>=0.

`Sin`
  `[expression]`

  Sine of expression.

`Asin`
  `[expression]`

  Arc sine (inverse sine) of expression in [-Pi/2,Pi/2], expression in [-1,1].

`Cos`
  `[expression]`

  Cosine of expression.

`Acos`
  `[expression]`

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

`Tan`
  `[expression]`

  Tangent of expression.

`Atan`
  `[expression]`

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

`Atan2`
  `[expression,expression]`

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

`Sinh`
  `[expression]`

  Hyperbolic sine of expression.

`Cosh`
  `[expression]`

  Hyperbolic cosine of expression.

`Tanh`
  `[expression]`

  Hyperbolic tangent of expression.

`Fabs`
  `[expression]`

  Absolute value of expression.

`Fmod`
  `[expression,expression]`

  Remainder of the division of the first expression by the second, with the sign of the first.

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### 4.2.2 Extended math functions

extended-math-function-id:

`Cross`
  `[expression,expression]`

  Cross product of the two arguments; expression must be a vector.

`Hypot`
  `[expression,expression]`

  Square root of the sum of the squares of its arguments.

`Norm`
  `[expression]`

  Absolute value if expression is a scalar; euclidian norm if expression is a vector.

`SquNorm`
  `[expression]`

  Square norm: `Norm[expression]^2`.

`Unit`
  `[expression]`

  Normalization: `expression/Norm[expression]`. Returns 0 if the norm is smaller than 1.e-30.

`Transpose`
  `[expression]`

  Transposition; expression must be a tensor.

`TTrace`
  `[expression]`

  Trace; expression must be a tensor.

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

  The first parameter represents the angular frequency and the second represents the phase. If the type of the current system is `Real`, `F_Cos_wt_p[]{w,p}` is identical to `Cos[w*$Time+p]`. If the type of the current system is `Complex`, it is identical to `Complex[Cos[w],Sin[w]]`.

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

  The first parameter represents the angular frequency and the second represents the phase. If the type of the current system is `Real`, `F_Sin_wt_p[]{w,p}` is identical to `Sin[w*$Time+p]`. If the type of the current system is `Complex`, it is identical to `Complex[Sin[w],-Cos[w]]`.

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

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

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### 4.2.3 Green functions

The Green functions are only used in integral quantities (see section 3.7 `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`).

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### 4.2.4 Type manipulation functions

type-function-id:

`Complex`
  `[expression-list]`

  Creates a (multi-harmonic) complex expression from an number of real-valued expressions. The number of expressions in expression-list must be even.

`Re`
  `[expression]`

  Takes the real part of a complex-valued expression.

`Im`
  `[expression]`

  Takes the imaginary part of a complex-valued expression.

`Vector`
  `[expression,expression,expression]`

  Creates a vector from 3 scalars.

`Tensor`
  `[expression,expression,expression,expression,expression,expression,`

  `expression,expression,expression]`

  Creates a second-rank tensor of order 3 from 9 scalars.

`TensorV`
  `[expression,expression,expression]`

  Creates a second-rank tensor of order 3 from 3 vectors.

`TensorSym`
  `[expression,expression,expression,expression,expression,expression]`

  Creates a symmetrical second-rank tensor of order 3 from 6 scalars.

`TensorDiag`
  `[expression,expression,expression]`

  Creates a diagonal second-rank tensor of order 3 from 3 scalars.

`CompX`
  `[expression]`

  Gets the X component of a vector.

`CompY`
  `[expression]`

  Gets the Y component of a vector.

`CompZ`
  `[expression]`

  Gets the Z component of a vector.

`CompXX`
  `[expression]`

  Gets the XX component of a tensor.

`CompXY`
  `[expression]`

  Gets the XY component of a tensor.

`CompXZ`
  `[expression]`

  Gets the XZ component of a tensor.

`CompYX`
  `[expression]`

  Gets the YX component of a tensor.

`CompYY`
  `[expression]`

  Gets the YY component of a tensor.

`CompYZ`
  `[expression]`

  Gets the YZ component of a tensor.

`CompZX`
  `[expression]`

  Gets the ZX component of a tensor.

`CompZY`
  `[expression]`

  Gets the ZY component of a tensor.

`CompZZ`
  `[expression]`

  Gets the ZZ component of a tensor.

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### 4.2.5 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.

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### 4.2.6 Miscellaneous functions

misc-function-id:

`Printf`
  `[expression]`

  Prints the value of expression when evaluated.

`Normal`
  `[]`

  Computes the normal to the element.

`NormalSource`
  `[]`

  Computes the normal to the source element (only valid in a quantity of Integral type).

`F_CompElementNum`
  `[]`

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

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

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

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

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

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

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

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

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

`Order`
  `[quantity]`

  Returns the interpolation order of the quantity.

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## 4.3 Types for `Constraint`

constraint-type:

`Assign`
  To assign a value (e.g. for boundary condition).

`Init`
  To give an initial value (e.g. initial value in a time analysis).

`AssignFromResolution`
  To assign a value to be computed by a pre-resolution.

`InitFromResolution`
  To give an initial value to be computed by a pre-resolution.

`Network`
  To describe the node connections of branches in a network.

`Link`
  To define links between degrees of freedom. This is not documented yet.

`LinkCplx`
  To define complex-valued links between degrees of freedom. This is not documented yet either.

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## 4.4 Types for `FunctionSpace`

function-space-type:

`Form0`
  0-form, i.e. scalar field of potential type.

`Form1`
  1-form, i.e. curl-conform field (associated with a curl).

`Form2`
  2-form, i.e. div-conform field (associated with a divergence).

`Form3`
  3-form, i.e. scalar field of density type.

`Form1P`
  1-form perpendicular to the z=0 plane, i.e. perpendicular curl-conform field (associated with a curl).

`Form2P`
  2-form in the z=0 plane, i.e. parallel div-conform field (associated with a divergence).

`Scalar`
  Scalar field.

`Vector`
  Vector field.

basis-function-type:

`BF_Node`
  Nodal function (on `NodesOf`, value `Form0`).

`BF_Edge`
  Edge function (on `EdgesOf`, value `Form1`).

`BF_Facet`
  Facet function (on `FacetsOf`, value `Form2`).

`BF_Volume`
  Volume function (on `VolumesOf`, value `Form3`).

`BF_GradNode`
  Gradient of nodal function (on `NodesOf`, value `Form1`).

`BF_CurlEdge`
  Curl of edge function (on `EdgesOf`, value `Form2`).

`BF_DivFacet`
  Divergence of facet function (on `FacetsOf`, value `Form3`).

`BF_GroupOfNodes`
  Sum of nodal functions (on `GroupsOfNodesOf`, value `Form0`).

`BF_GradGroupOfNodes`
  Gradient of sum of nodal functions (on `GroupsOfNodesOf`, value `Form1`).

`BF_GroupOfEdges`
  Sum of edge functions (on `GroupsOfEdgesOf`, value `Form1`).

`BF_CurlGroupOfEdges`
  Curl of sum of edge functions (on `GroupsOfEdgesOf`, value `Form2`).

`BF_PerpendicularEdge`
  1-form (0, 0, `BF_Node`) (on `NodesOf`, value `Form1P`).

`BF_CurlPerpendicularEdge`
  Curl of 1-form (0, 0, `BF_Node`) (on `NodesOf`, value `Form2P`).

`BF_GroupOfPerpendicularEdge`
  Sum of 1-forms (0, 0, `BF_Node`) (on `NodesOf`, value `Form1P`).

`BF_CurlGroupOfPerpendicularEdge`
  Curl of sum of 1-forms (0, 0, `BF_Node`) (on `NodesOf`, value `Form2P`).

`BF_PerpendicularFacet`
  2-form (90 degree rotation of `BF_Edge`) (on `EdgesOf`, value `Form2P`).

`BF_DivPerpendicularFacet`
  Div of 2-form (90 degree rotation of `BF_Edge`) (on `EdgesOf`, value `Form3`).

`BF_Region`
  Unit value 1 (on `Region`, value `Scalar`).

`BF_RegionX`
  Unit vector (1, 0, 0) (on `Region`, value `Vector`).

`BF_RegionY`
  Unit vector (0, 1, 0) (on `Region`, value `Vector`).

`BF_RegionZ`
  Unit vector (0, 0, 1) (on `Region`, value `Vector`).

`BF_Global`
  Global pre-computed quantity (on `Global`, value depends on parameters).

`BF_dGlobal`
  Exterior derivative of global pre-computed quantity (on `Global`, value depends on parameters).

`BF_NodeX`
  Vector (`BF_Node`, 0, 0) (on `NodesOf`, value `Vector`).

`BF_NodeY`
  Vector (0, `BF_Node`, 0) (on `NodesOf`, value `Vector`).

`BF_NodeZ`
  Vector (0, 0, `BF_Node`) (on `NodesOf`, value `Vector`).

`BF_Zero`
  Zero value 0 (on all regions, value `Scalar`).

`BF_One`
  Unit value 1 (on all regions, value `Scalar`).

global-quantity-type:

`AliasOf`
  Another name for a name of coefficient of basis function.

`AssociatedWith`
  A global quantity associated with a name of coefficient of basis function, and therefore with this basis function.

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## 4.5 Types for `Jacobian`

jacobian-type:

`Vol`
  Volume Jacobian, for n-D regions in n-D geometries, n = 1, 2 or 3.

`Sur`
  Surface Jacobian, for (n-1)-D regions in n-D geometries, n = 1, 2 or 3.

`Lin`
  Line Jacobian, for (n-2)-D regions in n-D geometries, n = 2 or 3.

`VolAxi`
  Axisymmetrical volume Jacobian (1st type: r), for 2-D regions in axisymmetrical geometries.

`SurAxi`
  Axisymmetrical surface Jacobian (1st type: r), for 1-D regions in axisymmetrical geometries.

`VolAxiSqu`
  Axisymmetrical volume Jacobian (2nd type: r^2), for 2-D regions in axisymmetrical geometries.

`VolSphShell`
  Volume Jacobian with spherical shell transformation, for n-D regions in n-D geometries, n = 2 or 3.

  Parameters: radius-internal, radius-external <, center-X, center-Y, center-Z, power, 1/infinity >.

`VolAxiSphShell`
  Same as `VolAxi`, but with spherical shell transformation.

  Parameters: radius-internal, radius-external <, center-X, center-Y, center-Z, power, 1/infinity >.

`VolAxiSquSphShell`
  Same as `VolAxiSqu`, but with spherical shell transformation.

  Parameters: radius-internal, radius-external <, center-X, center-Y, center-Z, power, 1/infinity >.

`VolRectShell`
  Volume Jacobian with rectangular shell transformation, for n-D regions in n-D geometries, n = 2 or 3.

  Parameters: radius-internal, radius-external <, direction, center-X, center-Y, center-Z, power, 1/infinity >.

`VolAxiRectShell`
  Same as `VolAxi`, but with rectangular shell transformation.

  Parameters: radius-internal, radius-external <, direction, center-X, center-Y, center-Z, power, 1/infinity >.

`VolAxiSquRectShell`
  Same as `VolAxiSqu`, but with rectangular shell transformation.

  Parameters: radius-internal, radius-external <, direction, center-X, center-Y, center-Z, power, 1/infinity >.

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## 4.6 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 section 8.1 Input file format).

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## 4.7 Types for `Formulation`

formulation-type:

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

local-term-type:

`Galerkin`
  Integral of Galerkin type.

`deRham`
  deRham projection (collocation).

quantity-type:

`Local`
  Local quantity defining a field in a function space. In case a subspace is considered, its identifier has to be given between the brackets following the `NameOfSpace function-space-id`.

`Global`
  Global quantity defining a global quantity from a function space. The identifier of this quantity has to be given between the brackets following the `NameOfSpace function-space-id`.

`Integral`
  Integral quantity obtained by the integration of a `LocalQuantity` before its use in an `Equation` term.

term-op-type:

`Dt`
  Time derivative applied to the whole term of the equation.

`DtDof`
  Time derivative applied only to the `Dof{}` term of the equation.

`DtDt`
  Time derivative of 2nd order applied to the whole term of the equation.

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

`JacNL`
  Jacobian term to be assembled in the Jacobian matrix for nonlinear analysis.

`NeverDt`
  No time scheme applied to the term (e.g. Theta is always 1 even if a theta scheme is applied).

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## 4.8 Types for `Resolution`

resolution-op:

`Generate`
  `[system-id]`

  Generate the system of equations system-id.

`Solve`
  `[system-id]`

  Solve the system of equations system-id.

`GenerateJac`
  `[system-id]`

  Generate the system of equations system-id using a jacobian matrix (of which the unknowns are corrections dx of the current solution x).

`SolveJac`
  `[system-id]`

  Solve the system of equations system-id using a jacobian matrix (of which the unknowns are corrections dx of the current solution x). Then, Increment the solution (x=x+dx) and compute the relative error dx/x.

`GenerateSeparate`
  `[system-id]`

  Generate iteration matrices separately for system system-id. It is destined to be used with `Update` in order to create more efficiently the actual system to solve (this is only useful in linear transient problems with one single excitation) or with `Lanczos` in order to generate the matrices of a (generalized) eigen value problem.

`Update`
  `[system-id, expression]`

  Update the system of equations system-id (built from iteration matrices generated separately with `GenerateSeparate`) with expression

`InitSolution`
  `[system-id]`

  Initialize the solution of system-id to zero (default) or to the values given in a `Constraint` of `Init` type.

`SaveSolution`
  `[system-id]`

  Save the solution of the system of equations system-id.

`SaveSolutions`
  `[system-id]`

  Save all the solutions available for the system of equations system-id. This should be used with algorithms that generate more than one solution at once, e.g. `Lanczos` or `FourierTransform`.

`TransferSolution`
  `[system-id]`

  Transfer the solution of system system-id, as an `Assign` constraint, to the system of equations defined with a `DestinationSystem` command. This is used with the `AssignFromResolution` constraint type (see section 4.3 Types for `Constraint`).

`TransferInitSolution`
  `[system-id]`

  Transfer the solution of system system-id, as an `Init` constraint, to the system of equations defined with a `DestinationSystem` command. This is used with the `InitFromResolution` constraint type (see section 4.3 Types for `Constraint`).

`SetTime`
  `[expression]`

  Change the current time.

`SetFrequency`
  `[system-id, expression]`

  Change the frequency of system system-id.

`SystemCommand`
  `[expression-char]`

  Execute the system command given by expression-char.

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

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

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

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

`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.

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

  Print the current values of the expressions listed in expression-list.

`Lanczos`
  `[system-id, expression-cst, { expression-cst-list } , expression-cst]`

  Eigen value computation by 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 eigen values/vectors to store, the spectral shift.

`FourierTransform`
  `[system-id, system-id, { expression-cst-list }]`

  On-the-fly computation of a discrete Fourier transform. The parameters are: the (time domain) system, the destination system in which the result of the Fourier tranform is to be saved (it should be declared with `Type Complex`), the list of frequencies to consider in the discrete Fourier transform.

`TimeLoopTheta`
  `[expression-cst,expression-cst,expression,expression-cst]` `{ resolution-op }`

  Time loop of a theta scheme. The parameters are: the initial time, the end time, the time step and the theta parameter (e.g. 1 for implicit Euler, 0.5 for Crank-Nicholson).

`TimeLoopNewmark`
  `[expression-cst,expression-cst,expression,expression-cst,expression-cst]`
  { resolution-op }

  Time loop of a Newmark scheme. The parameters are: the initial time, the end time, the time step, the beta and the gamma parameter.

`IterativeLoop`
  `[expression-cst,expression,expression-cst<,expression-cst>]` { resolution-op }

  Iterative loop for nonlinear analysis. The parameters are: the maximum number of iterations (if no convergence), the relaxation factor (multiplies the iterative correction dx) and the relative error to achieve. The optional parameter is a flag for testing purposes.

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## 4.9 Types for `PostProcessing`

post-value:

`Local`
  { local-value }

  To compute a local quantity.

`Integral`
  { integral-value }

  To integrate the expression over each element.

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## 4.10 Types for `PostOperation`

print-support:

`OnElementsOf`
  group-def

  To compute a quantity on the elements belonging to the region group-def, where the solution was computed during the processing stage.

`OnRegion`
  group-def

  To compute a global quantity associated with the region group-def.

`OnGlobal`
  To compute a global integral quantity, with no associated region.

`OnSection`
  { { expression-cst-list } { expression-cst-list } { expression-cst-list } }

  To compute a quantity on a section of the mesh defined by three points (i.e. on the intersection of the mesh with a cutting a plane, specified by three points). Each expression-cst-list must contain exactly three elements (the coordinates of the points).

`OnGrid`
  group-def

  To compute a quantity in elements of a mesh which differs from the real support of the solution. `OnGrid group-def` differs from `OnElementsOf group-def` by the reinterpolation that must be performed.

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

  To compute a quantity on a parametric grid. The three expressions represent the three cartesian coordinates x, y and z, and can be functions of the current values `$A`, `$B` and `$C`. The values for `$A`, `$B` and `$C` are specified by each expression-cst-list-item or expression-cst-list. For example, `OnGrid {Cos[$A], Sin[$A], 0} { 0:2*Pi:Pi/180, 0, 0 }` will compute the quantity on 360 points equally distributed on a circle in the z=0 plane, and centered on the origin.

`OnPoint`
  { expression-cst-list }

  To compute a quantity at a point. The expression-cst-list must contain exactly three elements (the coordinates of the point).

`OnLine`
  { { expression-cst-list } { expression-cst-list } } { expression-cst }

  To compute a quantity along a line (given by its two end points), with an associated number of divisions equal to expression-cst. The interpolation points on the line are equidistant. Each expression-cst-list must contain exactly three elements (the coordinates of the points).

`OnPlane`
  { { expression-cst-list } { expression-cst-list } { expression-cst-list } }
  `{ expression-cst, expression-cst }`

  To compute a quantity on a plane (specified by three points), with an associated number of divisions equal to each expression-cst along both generating directions. Each expression-cst-list must contain exactly three elements (the coordinates of the points).

`OnBox`
  { { expression-cst-list } { expression-cst-list } { expression-cst-list }
  `{ expression-cst-list } } { expression-cst, expression-cst, expression-cst }`

  To compute a quantity in a box (specified by four points), with an associated number of divisions equal to each expression-cst along the three generating directions. Each expression-cst-list must contain exactly three elements (the coordinates of the points).

print-option:

`File`
  `expression-char`

  Outputs the result in a file named expression-char.

`File`
  `> expression-char`

  Same as `File expression-char`, except that, if several `File > expression-char` options appear in the same `PostOperation`, the results are concatenated in the file expression-char.

`File`
  `>> expression-char`

  Appends the result to a file named expression-char.

`Depth`
  expression-cst

  Recursive division of the elements if expression-cst is greater than zero, derefinement if expression-cst is smaller than zero. If expression-cst is equal to zero, evaluation at the barycenter of the elements.

`Skin`
  Computes the result on the boundary of the region.

`Smoothing`
  Smoothes the solution at the nodes.

`HarmonicToTime`
  expression-cst

  Converts a harmonic solution into a time-dependent one (with expression-cst steps).

`Dimension`
  expression-cst

  Forces the dimension of the elements to consider in an element search. Specifies the problem dimension during an adaptation (h- or p-refinement).

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

  Outputs results for the specified time steps only.

`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.).

post-operation-fmt:

`Gmsh`
`GmshParsed`
  Gmsh output. See the documentation of Gmsh (http://www.geuz.org/gmsh/) for a description of the file formats.

`Table`
  Space oriented column output, e.g. suitable for Gnuplot, Excel, Caleida Graph, etc. The columns are: element-type element-index x-coord y-coord z-coord <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.

`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.

`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.

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# 5. Short examples

> 5.1 Constant expression examples |  |
> 5.2 `Group` examples |  |
> 5.3 `Function` examples |  |
> 5.4 `Constraint` examples |  |
> 5.5 `FunctionSpace` examples |  |
> 5.6 `Jacobian` examples |  |
> 5.7 `Integration` examples |  |
> 5.8 `Formulation` examples |  |
> 5.9 `Resolution` examples |  |
> 5.10 `PostProcessing` examples |  |
> 5.11 `PostOperation` examples |  |

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## 5.1 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;

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## 5.2 `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`.

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## 5.3 `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 section 5.8 `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 section 2.2 Constants, as well as 2.4 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 section 2.2 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 section 2.7 Registers). For example, the evaluation of `g[] = f[$1]*Sin[f[$1]^2]` would require two (costly) linear interpolations. But the result of the evaluation of `f[]` may be stored in a register (for example the register 0) with

    g[] = f[$1]#0 * Sin[#0^2];

thus reducing the number of evaluations of `f[]` (and of the argument `$1`) to one.

A function can also be time dependent, e.g.

    Function {
      Freq = 50.; Phase = 30./180.*Pi; // Constants
      TimeFct_Sin[] = Sin [ 2.*Pi*Freq * $Time + Phase ];
      TimeFct_Exp[] = Exp [ - $Time / 0.0119 ];
      TimeFct_ExtSin[] = F_Sin_wt_p [] {2.*Pi*Freq, Phase};
    }

Note that `TimeFct_ExtSin[]` is identical to `TimeFct_Sin[]` in a time domain analysis, but also permits to define phasors implicitely in the case of harmonic analyses.

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## 5.4 `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. 5.3 `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`.

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## 5.5 `FunctionSpace` examples

Various discrete function spaces can be defined in the frame of the finite element method.

> 5.5.1 Nodal finite element spaces |  |
> 5.5.2 High order nodal finite element space |  |
> 5.5.3 Nodal finite element space with floating potentials |  |
> 5.5.4 Edge finite element space |  |
> 5.5.5 Edge finite element space with gauge condition |  |
> 5.5.6 Coupled edge and nodal finite element spaces |  |
> 5.5.7 Coupled edge and nodal finite element spaces for multiply connected domains |  |

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### 5.5.1 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 section 5.4 `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.

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### 5.5.2 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]; }
        }
        ...

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### 5.5.3 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`.

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### 5.5.4 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 { ... }
      }
    }

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### 5.5.5 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; }
          ...
        }
      }
    }

The same gauge could also be defined in the BasisFunction field itself, by limiting the set of entities supporting the unknowns: `Entity EdgesOf[All]` would then be replaced by `Entity EdgesOfTreeIn[Domain, StartingOn Surf]`.

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### 5.5.6 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.

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### 5.5.7 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.

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## 5.6 `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; }
        }
      }
    }

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## 5.7 `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.

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## 5.8 `Formulation` examples

> 5.8.1 Electrostatic scalar potential formulation |  |
> 5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges |  |
> 5.8.3 Magnetostatic 3D vector potential formulation |  |
> 5.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation |  |
> 5.8.5 Nonlinearities, Mixed formulations, ... |  |

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### 5.8.1 Electrostatic scalar potential formulation

An electrostatic formulation using an electric scalar potential v, i.e.

( epsr grad v, grad vp ) W = 0, for all vp in S0(W),

is expressed by

    Formulation {
      { Name Electrostatics_v; Type FemEquation;
        Quantity {
          { Name v; Type Local; NameOfSpace Hgrad_v; }
        }
        Equation {
          Galerkin { [ epsr[] * Dof{Grad v} , {Grad v} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
        }
      }
    }

The density of the `Galerkin` term is a copy of the symbolic form of the formulation, i.e. the product of a relative permittivity function `epsr[]` by a vector of degrees of freedom (`Dof{.}`); the scalar product of this with the gradient of test function `v` results in a symmetrical matrix.

Note that another `Quantity` could be defined for test functions, e.g. `vp` defined by `{ Name vp; Type Local; NameOfSpace Hgrad_v; }`. However, its use would result in the computation of a full matrix and consequently in a loss of efficiency.

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### 5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges

An extension of the formulation above can be made to take floating potentials and electrical charges into account (the latter being defined in `FunctionSpace Hgrad_v_floating`), i.e.

    Formulation {
      { Name Electrostatics_v_floating; Type FemEquation;
        Quantity {
          { Name v; Type Local; NameOfSpace Hgrad_v_floating; }
          { Name V; Type Global;
            NameOfSpace Hgrad_v_floating [GlobalElectricPotential]; }
          { Name Q; Type Global;
            NameOfSpace Hgrad_v_floating [GlobalElectricCharge]; }
        }
        Equation {
          Galerkin { [ epsr[] * Dof{Grad v} , {Grad v} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
          GlobalTerm { [ - Dof{Q}/eps0 , {V} ]; In SkinDomainC; }
        }
      }
    }

with the predefinition `Function { eps0 = 8.854187818e-12; }`.

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### 5.8.3 Magnetostatic 3D vector potential formulation

A magnetostatic 3D vector potential formulation

( nu curl a , curl ap ) W - ( js , ap ) Ws = 0, for all ap in S1(W) with gauge condition,

with a source current density js in inductors Ws, is expressed by

    Formulation {
      { Name Magnetostatics_a_3D; Type FemEquation;
        Quantity {
          { Name a; Type Local; NameOfSpace Hcurl_a_Gauge; }
        }
        Equation {
          Galerkin { [ nu[] * Dof{Curl a} , {Curl a} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
          Galerkin { [ - SourceCurrentDensity[] , {a} ];
                     In DomainWithSourceCurrentDensity;
                     Jacobian Vol; Integration Int_1; }
        }
      }
    }

Note that js is here given by a function `SourceCurrentDensity[]`, but could also be given by data computed from another problem, e.g. from an electrokinetic problem (coupling of formulations) or from a fully fixed function space (constraints fixing the density, which is usually more efficient in time domain analyses).

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### 5.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation

A magnetodynamic 3D or 2D h-phi formulation, i.e. coupling the magnetic field h with a magnetic scalar potential phi,

Dt ( mu h , hp ) W + ( ro curl h , curl hp ) Wc = 0, for all hp in S1(W),

can be expressed by

    Formulation {
      { Name Magnetodynamics_hphi; Type FemEquation;
        Quantity {
          { Name h; Type Local; NameOfSpace Hcurl_hphi; }
        }
        Equation {
          Galerkin { Dt [ mu[] * Dof{h} , {h} ];
                     In Domain; Jacobian Vol; Integration Int_1; }
          Galerkin { [ rho[] * Dof{Curl h} , {Curl h} ];
                     In DomainC; Jacobian Vol; Integration Int_1; }
        }
      }
    }

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### 5.8.5 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.

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## 5.9 `Resolution` examples

> 5.9.1 Static resolution (electrostatic problem) |  |
> 5.9.2 Frequency domain resolution (magnetodynamic problem) |  |
> 5.9.3 Time domain resolution (magnetodynamic problem) |  |
> 5.9.4 Nonlinear time domain resolution (magnetodynamic problem) |  |
> 5.9.5 Coupled formulations |  |

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### 5.9.1 Static resolution (electrostatic problem)

A static resolution, e.g. for the electrostatic formulation (see section 5.8 `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`).

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### 5.9.2 Frequency domain resolution (magnetodynamic problem)

A frequency domain resolution, e.g. for the magnetodynamic h-phi formulation (see section 5.8 `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.;
    }

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### 5.9.3 Time domain resolution (magnetodynamic problem)

A time domain resolution, e.g. for the same magnetodynamic h-phi formulation (see section 5.8 `Formulation` examples), is given by

    Resolution {
      { Name Magnetodynamics_hphi_Time;
        System {
          { Name Sys_Mag; NameOfFormulation Magnetodynamics_hphi; }
        }
        Operation {
          InitSolution[Sys_Mag]; SaveSolution[Sys_Mag];
          TimeLoopTheta[Mag_Time0, Mag_TimeMax, Mag_DTime[], Mag_Theta[]] {
            Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag];
          }
        }
      }
    }

If, e.g. the `Resolution` above is preceded by the constant and function definitions below

    Function {
      Tc = 10.e-3;
      Mag_Time0 = 0.; Mag_TimeMax = 2.*Tc; Mag_DTime[] = Tc/20.;
      Mag_Theta[] = 1./2.;
    }

the performed time analysis will be a Crank-Nicolson scheme (theta-scheme with `Theta = 0.5`) with initial time 0 ms, end time 20 ms and time step 1 ms.

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### 5.9.4 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;
    }

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### 5.9.5 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.

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## 5.10 `PostProcessing` examples

The quantities to be post-computed based on a solution of a resolution are defined e.g. for the electrostatic problem (see section 5.8 `Formulation` examples; see section 5.9 `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.

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## 5.11 `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 5.10 `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 4.10 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 ];
         }
      }
    }

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# 6. Complete examples

> 6.1 Electrostatic problem |  |
> 6.2 Magnetostatic problem |  |
> 6.3 Magnetodynamic problem |  |

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## 6.1 Electrostatic problem

An elementary electrostatic problem is first considered. 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'`), of which the geometry is defined in the file ``mStrip.geo'`: see C. Gmsh examples. The geometry is two-dimensional; one half of the structure is considered by symmetry.

The structure of the following files points out the separation of the data describing the particular problem and the method used to solve it, and therefore how it is possible to build black boxes adapted to well defined categories of problems. The files are commented (see section 1.3 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 ; } }
               }
             }
      }
    }

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## 6.2 Magnetostatic problem

A magnetostatic problem is considered. 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'`), of which the geometry is defined in file ``Core.geo'` (see section C. Gmsh examples). The geometry is two-dimensional; one fourth of the structure is considered by symmetry.

The jacobian and integration methods used are the same as for the electrostatic problem presented in 6.1 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; }
            }
          }
        }
      }
    }

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## 6.3 Magnetodynamic problem

A magnetodynamic problem is considered. 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 the previous section). It is applied to a core-inductor system (defined in file ``CoreMassive.pro'`), of which the geometry has already been defined in file ``Core.geo'` (giving file ``Core.msh'` with Gmsh; see section 8.1 Input file format).

The jacobian and integration methods used are defined in the same file as in the electrostatic problem in 6.1 Electrostatic problem.

    /* -------------------------------------------------------------------
       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; } } }
        }
      }
    }

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# 7. Running GetDP

GetDP has no graphical interface(1). It is a command-line driven program that reads a problem definition file once at the beginning of the processing. This problem definition file is a regular ASCII text file (see section 1.1 Numerical tools as objects), hence created with whatever 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 an 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 section 1.4 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.

`-ipos`
  post-processing-id ...

  Enters an interactive post-processing mode, permetting to manually type `PostOperation`-like commands. These commands are based on the PostProcessing object(s) selected by the post-processing-id(s). If post-processing-id is omitted, the list of available choices is displayed.

`-msh`
  filename

  Reads the mesh database (in `.msh` format) from filename (see section 8. File formats) rather than reading it from the default problem file name (with the ``.msh'` extension appended).

`-split`

  Saves processing results in separate files (one for each timestep).

`-res`
  filename ...

  Loads processing results from file(s).

`-name`
  string

  Uses string as the default generic file name for input or output of mesh, pre-processing and processing files.

`-restart`

  Restarts processing of a transient problem interrupted before being complete.

`-solve`
  resolution-id

  Same as `-pre resolution-id -cal`.

`-adapt`
  file

  Reads adaptation constraints from file.

`-order`
  real

  Specifies the maximum interpolation order.

`-bin`

  Selects binary format for output files.

`-log`

  Saves all processing history in a log file (the input file name with an appended `.log` extension).

`-socket`
  string

  Communicates through socket string.

`-check`

  Lets you check the problem structure interactively.

`-v`
`-verbose`
  integer

  Sets the verbosity level. A value of 0 means that no information will be displayed during the processing.

`-p`
`-progress`
  integer

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

`-info`

  Displays the version information.

`-version`

  Displays the version number.

`-help`

  Displays a message listing basic usage and available options.

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# 8. 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 3. Objects, and 4. Types for objects. The format of the post-processing files is explained in 4.10 Types for `PostOperation`.

> 8.1 Input file format |  |
> 8.2 Output file format |  |

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## 8.1 Input file format

The native mesh format read by GetDP is the mesh file format produced by Gmsh (http://www.geuz.org/gmsh/). The 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).

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## 8.2 Output file format

> 8.2.1 File ``.pre'` |  |
> 8.2.2 File ``.res'` |  |

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### 8.2.1 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.

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### 8.2.2 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 */
    getdp-version-number file-res-format
    $EndResFormat
    $Solution  /* DofData #dofdata-number */
    dofdata-number time-value time-step-number
    solution-value
    ...
    $EndSolution
    ...

Notes:

1. A `$Solution` field contains the solution associated with a `$DofData` field.
2. There is one `$Solution` field for each time step, of which the time is time-value (0 for non time dependent analyses).
3. The order of the solution-values in a `$Solution` field follows the numbering of the equations given in the ``.pre'` file (one floating point value for each degree of freedom).

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# 9. Bugs, versions and credits

> 9.1 Bugs |  |
> 9.2 Versions |  |
> 9.3 Credits |  |

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## 9.1 Bugs

If you think you have found a bug in GetDP, you can report it by electronic mail to the GetDP mailing list at getdp@geuz.org. Please send as precise a description of the problem as you can, including sample input files that produce the bug (problem definition and mesh files). Don't forget to mention both the version of GetDP and the version of your operation system (see section 7. Running GetDP to see how to get this information).

See the ``TODO'` file in the distribution to check the problems we already know about.

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## 9.2 Versions

    $Id: VERSIONS,v 1.51 2005/02/05 20:38:19 geuzaine Exp $

    New since 1.0: Lanczos with GSL+lapack; various small improvements and
    bug fixes;

    1.0: 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: 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: Updated Gmsh output format; many small bug fixes;

    0.85: Upgraded communication interface with Gmsh; New ChangeOfValues
    option in PostOperation; Many internal changes;

    0.84: New ChangeOfCoordinate option in PostOperation; Fixed crash in
    InterpolationAkima; Improved interactive postprocessing (-ipos);
    Changed syntax of parametric OnGrid ($S, $T -> $A, $B, $C); Corrected
    Skin for non simplicial meshes; Fixed floating point exception in
    diagonal matrix scaling; Many other small fixes and cleanups;

    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|K;

    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;

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## 9.3 Credits

    $Id: CREDITS,v 1.12 2004/09/30 04:58:18 geuzaine Exp $

                 GetDP is copyright (C) 1997-2004

                          Patrick Dular
                  <patrick.dular at ulg.ac.be>

                               and

                       Christophe Geuzaine
                  <geuzaine at acm.caltech.edu>

    Major code contributions to GetDP have been provided by Johan
    Gyselinck <johan.gyselinck at ulg.ac.be> and Ruth Sabariego
    <ruth.sabariego at student.ulg.ac.be>.

    Other code contributors include: David Colignon <david.colignon at
    ulg.ac.be>, Tuan Ledinh <tuan.ledinh at student.ulg.ac.be>, Patrick
    Lefevre <P.Lefevre at ulg.ac.be>, Andre Nicolet <andre.nicolet at
    fresnel.fr>, <ohyeahq at yahoo.co.jp>, Jean-Francois Remacle <remacle
    at scorec.rpi.edu>, Timo Tarhasaari <timo.tarhasaari at tut.fi>,
    Christophe Trophime <trophime at labs.polycnrs-gre.fr> and Marc Ume
    <Marc.Ume at digitalgraphics.be>. See the source code for more
    details.

    The AVL tree code (DataStr/avl.*) is copyright (C) 1988-1993, 1995 The
    Regents of the University of California. Permission to use, copy,
    modify, and distribute this software and its documentation for any
    purpose and without fee is hereby granted, provided that the above
    copyright notice appear in all copies and that both that copyright
    notice and this permission notice appear in supporting documentation,
    and that the name of the University of California not be used in
    advertising or publicity pertaining to distribution of the software
    without specific, written prior permission.  The University of
    California makes no representations about the suitability of this
    software for any purpose.  It is provided "as is" without express or
    implied warranty.

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

    Thanks to the following folks who have contributed by providing fresh
    ideas on theoretical or programming topics, who have sent patches,
    requests for changes or improvements, or who gave us access to exotic
    machines for testing GetDP: Olivier Adam <o.adam at ulg.ac.be>,
    Alejandro Angulo <aacusm at yahoo.com>, Geoffrey Deliege <deliege at
    mailserv.esat.kuleuven.ac.be>, Mark Evans <evans at gte.net>, Philippe
    Geuzaine <geuzaine at gnat.colorado.edu>, Eric Godard <godard at
    montefiore.ulg.ac.be>, Sebastien Guenneau <guenneau at
    liverpool.ac.uk>, Francois Henrotte <fhenrott at esat.kuleuven.ac.be>,
    Daniel Kedzierski <kedzierski at uol.com.br>, Samuel Kvasnica
    <kvasnica at iaee.tuwien.ac.at>, Benoit Meys <bmeys at
    techspace-aero.be>, Uwe Pahner <uwe.pahner at esat.kuleuven.ac.be>,
    Georgia Psoni and Robert Struijs <gpsoni at free.fr>, Ahmed Rassili
    <a.rassili at ulg.ac.be>, Thierry Scordilis <t.scordilis at atral.fr>
    and Herve Tortel <tortel at loe.u-3mrs.fr>, Jose Geraldo A. Brito Neto
    <jgabrito at terra.com.br>, Matthias Fenner <m.fenner at gmx.net>.

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# A. 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 section 1.4 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.

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# B. Frequently asked questions

    $Id: FAQ,v 1.9 2004/03/08 05:38:37 geuzaine Exp $

    This is the GetDP FAQ

    ********************************************************************

    Section 1: The basics

    * 1.1 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.

    * 1.2 What are the terms and conditions of use?

    GetDP is distributed under the terms of the GNU General Public
    License. See the file doc/LICENSE for more information, or go to the
    GNU foundation's web site at http://www.gnu.org.

    * 1.3 What does 'GetDP' mean?

    General environment for the treatment of Discrete Problems

    * 1.4 Where can I find more information?

    <http://www.geuz.org/getdp/> is the primary site to obtain information
    about GetDP. You will find a short presentation, a complete reference
    guide as well as a searchable archive of the GetDP mailing list
    (<getdp@geuz.org>) on this site.

    ********************************************************************

    Section 2: Installation

    * 2.1 Which OSes does GetDP run on?

    GetDP is known to run on Windows 95/98/NT/2000/XP, Linux, Mac OS X,
    Compaq Tru64 Unix (aka OSF1, aka Digital Unix), Sun OS, IBM AIX, SGI
    IRIX, FreeBSD and HP-UX. It should compile on any Unix-like operating
    system, provided that you have access to a recent C and Fortran 77
    compiler.

    * 2.2 What do I need to compile GetDP from the sources?

    You need a C and a Fortran 77 compiler (e.g. the GNU compilers gcc and
    g77) as well as the GSL (version 1.2 or higher; freely available from
    http://sources.redhat.com/gsl/).

    Under Windows, you will need the Cygwin tools and compilers (freely
    available from http://www.cygwin.com).

    * 2.3 How to I compile GetDP?

    Just type

    ./configure; make; make install

    If you change some configuration options (type ./configure --help to
    get the list of all available choices), don't forget to do 'make
    clean' before rebuilding GetDP.

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

    Try 'ldd getdp' (or 'otool -L getdp' on Mac OS X) to check if all the
    required shared libraries are installed on your system. If not,
    install them. If it still doesn't work, recompile GetDP from the
    sources.

    ********************************************************************

    Section 3: Usage

    * 3.1 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://www.geuz.org/gmsh>). This format being
    very simple (see the reference GetDP manual for more details), it
    should be straightforward to write a converter from your mesh format
    to the 'msh' format.

    * 3.2 How can I visualize the results produced by GetDP?

    You can specify a format in all post-processing operations. Available
    formats include 'Table', 'TimeTable' and 'Gmsh'. 'Table' and
    'TimeTable' output lists of numbers easily readable by
    Excel/gnuplot/Caleida Graph/etc. 'Gmsh' outputs post-processing views
    directly loadable by Gmsh.

    * 3.3 How do I change the linear solver used by GetDP?

    Edit the 'solver.par' file in the current working directory. You can
    also remove the file: GetDP will give you the opportunity to create it
    dynamically next time you perform a linear system solving.

    If you don't like the default linear solvers (based on Yousef Saad's
    Sparskit 2.0), you can configure and compile GetDP to use PETSc
    (http://www.mcs.anl.gov/petsc) instead: run ./configure --help for
    more info.

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# C. 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://www.geuz.org/gmsh/.

This appendix reproduces verbatim the input files needed by Gmsh to produce the mesh files ``mStrip.msh'` and ``Core.msh'` used in the examples of 6. 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 */

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# D. License

GNU General Public License

Version 2, June 1991

    Copyright © 1989, 1991 Free Software Foundation, Inc.
    59 Temple Place - Suite 330, Boston, MA  02111-1307, USA

    Everyone is permitted to copy and distribute verbatim copies
    of this license document, but changing it is not allowed.

## Preamble

The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.

Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.

The precise terms and conditions for copying, distribution and modification follow.

## TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

1. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program", below, refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification".) Each licensee is addressed as "you".

   Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.

2. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program.

   You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.

3. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:

   1. You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.

   2. You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.

   3. If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)

   These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.

   Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.

   In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.

4. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:

   1. Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

   2. Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

   3. Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)

   The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.

   If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.

5. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

6. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it.

7. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License.

8. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program.

   If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.

   It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.

   This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.

9. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.

10. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

    Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.

11. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.

    ## NO WARRANTY

12. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

13. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

## END OF TERMS AND CONDITIONS

## How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found.

    one line to give the program's name and a brief idea of what it does.
    Copyright (C) yyyy  name of author

    This program is free software; you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation; either version 2 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place - Suite 330, Boston,
    MA  02111-1307, USA.

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

    Gnomovision version 69, Copyright (C) 19yy name of author
    Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type
    `show w'.
    This is free software, and you are welcome to redistribute it
    under certain conditions; type `show c' for details.

The hypothetical commands ``show w'` and ``show c'` should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than ``show w'` and ``show c'`; they could even be mouse-clicks or menu items--whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names:

    Yoyodyne, Inc., hereby disclaims all copyright interest in the program
    `Gnomovision' (which makes passes at compilers) written by James Hacker.

    signature of Ty Coon, 1 April 1989
    Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.

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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 | 8.1 Input file format
| ``.pre'` file | 8.2.1 File ``.pre'`
| ``.res'` file | 8.2.2 File ``.res'`
A |  |
| Acknowledgments | 9.3 Credits
| Analytical integration | 3.6 `Integration`: defining integration methods
| Approximation spaces | 3.4 `FunctionSpace`: building function spaces
| Arguments | 2.4 Functions
| Arguments, definition | 2.6 Arguments
| Authors, e-mail | 9.1 Bugs
| Axisymmetric, transformation | 3.5 `Jacobian`: defining jacobian methods
B |  |
| Basis Functions | 3.4 `FunctionSpace`: building function spaces
| Binary operators | 2.3.1 Operator types
| Boundary conditions | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| Boundary Element Method | 1.5 Which problems can GetDP actually solve?
| Bugs, known | 9.1 Bugs
| Bugs, reporting | 9.1 Bugs
| Built-in functions | 2.4 Functions
C |  |
| Change of coordinates | 3.5 `Jacobian`: defining jacobian methods
| Changelog | 9.2 Versions
| Circuit equations | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| Command line options | 7. Running GetDP
| Comments | 1.3 Comments
| Complete examples | 6. Complete examples
| Complex-valued, system | 3.8 `Resolution`: solving systems of equations
| Concepts, index | Concept index
| Constant, definition | 2.2 Constants
| Constant, evaluation | 2.2 Constants
| Constraint, definition | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| Constraint, examples | 5.4 `Constraint` examples
| Constraint, types | 4.3 Types for `Constraint`
| Contributors, list | 9.3 Credits
| Coordinate change | 3.5 `Jacobian`: defining jacobian methods
| Copyright | Copying conditions
| Credits | 9.3 Credits
| Curl | 2.8 Fields
| Current values | 2.5 Current values
D |  |
| Dependences, objects | 1.1 Numerical tools as objects
| Derivative, exterior | 2.8 Fields
| Derivative, time | 3.7 `Formulation`: building equations
| Developments, future | 1.5 Which problems can GetDP actually solve?
| Differential operators | 2.8 Fields
| Discrete function spaces | 3.4 `FunctionSpace`: building function spaces
| Discrete quantities | 2.8 Fields
| Discretized Geometry | 3.1 `Group`: defining topological entities
| Divergence | 2.8 Fields
| Document syntax | 1.2 Syntactic rules used in this document
| Download | Copying conditions
E |  |
| E-mail, authors | 9.1 Bugs
| Edge element space, example | 5.8.3 Magnetostatic 3D vector potential formulation
| Efficiency, tips | A. Tips and tricks
| Electromagnetism | 1.5 Which problems can GetDP actually solve?
| Electrostatic formulation | 5.8.1 Electrostatic scalar potential formulation
| Elementary matrices | 3.7 `Formulation`: building equations
| Entities, topological | 3.1 `Group`: defining topological entities
| Equations | 3.7 `Formulation`: building equations
| Evaluation mechanism | 2.2 Constants
| Evaluation, order | 2.3.2 Evaluation order
| Examples, complete | 6. Complete examples
| Examples, short | 5. Short examples
| Exporting results | 3.10 `PostOperation`: exporting results
| Expression, definition | 2.1 Definition
| Exterior derivative | 2.8 Fields
F |  |
| FAQ | B. Frequently asked questions
| Fields | 2.8 Fields
| File, ``.msh'` | 8.1 Input file format
| File, ``.pre'` | 8.2.1 File ``.pre'`
| File, ``.res'` | 8.2.2 File ``.res'`
| File, comment | 1.3 Comments
| File, include | 1.4 Includes
| File, mesh | 8.1 Input file format
| File, pre-processing | 8.2.1 File ``.pre'`
| File, result | 8.2.2 File ``.res'`
| Finite Difference Method | 1.5 Which problems can GetDP actually solve?
| Finite Element Method | 1.5 Which problems can GetDP actually solve?
| Finite Volume Method | 1.5 Which problems can GetDP actually solve?
| Floating point numbers | 2.2 Constants
| Floating potential, example | 5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges
| Format, output | 3.10 `PostOperation`: exporting results
| Formulation, definition | 3.7 `Formulation`: building equations
| Formulation, electrostatics | 5.8.1 Electrostatic scalar potential formulation
| Formulation, examples | 5.8 `Formulation` examples
| Formulation, types | 4.7 Types for `Formulation`
| Frequency | 3.8 `Resolution`: solving systems of equations
| Frequently asked questions | B. Frequently asked questions
| Function groups | 3.1 `Group`: defining topological entities
| Function space, definition | 3.4 `FunctionSpace`: building function spaces
| Function space, examples | 5.5 `FunctionSpace` examples
| Function space, types | 4.4 Types for `FunctionSpace`
| Function, definition | 2.4 Functions
| Function, definition | 3.2 `Function`: defining global and piecewise expressions
| Function, examples | 5.3 `Function` examples
| Future developments | 1.5 Which problems can GetDP actually solve?
G |  |
| Gauss, integration | 3.6 `Integration`: defining integration methods
| Geometric transformations | 3.5 `Jacobian`: defining jacobian methods
| Global quantity | 3.7 `Formulation`: building equations
| Global quantity, example | 5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges
| Gmsh, examples | C. Gmsh examples
| Gmsh, file format | 8.1 Input file format
| GNU General Public License | D. License
| Gradient | 2.8 Fields
| Grid | 3.1 `Group`: defining topological entities
| Group, definition | 3.1 `Group`: defining topological entities
| Group, examples | 5.2 `Group` examples
| Group, types | 4.1 Types for `Group`
H |  |
| Hierarchical basis functions | 3.4 `FunctionSpace`: building function spaces
| History, versions | 9.2 Versions
I |  |
| Includes | 1.4 Includes
| Index, concepts | Concept index
| Index, metasyntactic variables | Metasyntactic variable index
| Index, syntax | Syntax index
| Input file format | 8.1 Input file format
| Integer numbers | 2.2 Constants
| Integral Equation Method | 1.5 Which problems can GetDP actually solve?
| Integral quantity | 3.7 `Formulation`: building equations
| Integration, definition | 3.6 `Integration`: defining integration methods
| Integration, examples | 5.7 `Integration` examples
| Integration, types | 4.6 Types for `Integration`
| Internet address | Copying conditions
| Interpolation | 2.8 Fields
| Interpolation | 3.4 `FunctionSpace`: building function spaces
| Introduction | Introduction
| Iterative loop | 3.8 `Resolution`: solving systems of equations
J |  |
| Jacobian, definition | 3.5 `Jacobian`: defining jacobian methods
| Jacobian, examples | 5.6 `Jacobian` examples
| Jacobian, types | 4.5 Types for `Jacobian`
K |  |
| Keywords, index | Syntax index
| Known bugs | 9.1 Bugs
L |  |
| License | Copying conditions
| License | D. License
| Linear system solving | 3.8 `Resolution`: solving systems of equations
| Linking, objects | 1.1 Numerical tools as objects
| Local quantity | 3.7 `Formulation`: building equations
M |  |
| Mailing list | Copying conditions
| Maps | 3.10 `PostOperation`: exporting results
| Matrices, elementary | 3.7 `Formulation`: building equations
| Mechanics | 1.5 Which problems can GetDP actually solve?
| Mesh | 3.1 `Group`: defining topological entities
| Mesh, examples | C. Gmsh examples
| Mesh, file format | 8.1 Input file format
| Metasyntactic variables, index | Metasyntactic variable index
| Method of Moments | 1.5 Which problems can GetDP actually solve?
N |  |
| Networks | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| Newmark, time scheme | 3.8 `Resolution`: solving systems of equations
| Newton, nonlinear scheme | 3.8 `Resolution`: solving systems of equations
| Nodal function space, example | 5.8.1 Electrostatic scalar potential formulation
| Nonlinear system solving | 3.8 `Resolution`: solving systems of equations
| Numbers, integer | 2.2 Constants
| Numbers, real | 2.2 Constants
| Numerical integration | 3.6 `Integration`: defining integration methods
| Numerical tools, overview | 1. Overview
O |  |
| Objects, definition | 3. Objects
| Objects, dependences | 1.1 Numerical tools as objects
| Objects, overview | 1. Overview
| Objects, types | 4. Types for objects
| Operating system | 7. Running GetDP
| Operation, priorities | 2.3.2 Evaluation order
| Operators, definition | 2.3.1 Operator types
| Operators, differential | 2.8 Fields
| Options, command line | 7. Running GetDP
| Order of evaluation | 2.3.2 Evaluation order
| Output file format | 8.2 Output file format
| Overview | 1. Overview
P |  |
| Parameters | 2.4 Functions
| Philosophy, general | 1.1 Numerical tools as objects
| Physical problems | 1.5 Which problems can GetDP actually solve?
| Picard, nonlinear scheme | 3.8 `Resolution`: solving systems of equations
| Piecewise functions | 2.4 Functions
| Piecewise functions | 3.2 `Function`: defining global and piecewise expressions
| Platforms | 7. Running GetDP
| Post-operation, definition | 3.10 `PostOperation`: exporting results
| Post-operation, examples | 5.11 `PostOperation` examples
| Post-operation, types | 4.10 Types for `PostOperation`
| Post-processing, definition | 3.9 `PostProcessing`: exploiting computational results
| Post-processing, examples | 5.10 `PostProcessing` examples
| Post-processing, types | 4.9 Types for `PostProcessing`
| Priorities, operations | 2.3.2 Evaluation order
| Processing cycle | 1.1 Numerical tools as objects
Q |  |
| Quantities, discrete | 2.8 Fields
| Quantity, global | 3.7 `Formulation`: building equations
| Quantity, integral | 3.7 `Formulation`: building equations
| Quantity, local | 3.7 `Formulation`: building equations
| Quantity, post-processing | 3.9 `PostProcessing`: exploiting computational results
| Questions, frequently asked | B. Frequently asked questions
R |  |
| Reading, guidelines | How to read this manual
| Real numbers | 2.2 Constants
| Region groups | 3.1 `Group`: defining topological entities
| Registers, definition | 2.7 Registers
| Relaxation factor | 3.8 `Resolution`: solving systems of equations
| Reporting bugs | 9.1 Bugs
| Resolution, definition | 3.8 `Resolution`: solving systems of equations
| Resolution, examples | 5.9 `Resolution` examples
| Resolution, types | 4.8 Types for `Resolution`
| Results, exploitation | 3.9 `PostProcessing`: exploiting computational results
| Results, export | 3.10 `PostOperation`: exporting results
| Rules, syntactic | 1.2 Syntactic rules used in this document
| Running GetDP | 7. Running GetDP
S |  |
| Scope of GetDP | 1.5 Which problems can GetDP actually solve?
| Sections | 3.10 `PostOperation`: exporting results
| Short examples | 5. Short examples
| Solving, system | 3.8 `Resolution`: solving systems of equations
| Spaces, discrete | 3.4 `FunctionSpace`: building function spaces
| String | 2.2 Constants
| Symmetry, integral kernel | 3.7 `Formulation`: building equations
| Syntax, index | Syntax index
| Syntax, rules | 1.2 Syntactic rules used in this document
| System, definition | 3.8 `Resolution`: solving systems of equations
T |  |
| Ternary operators | 2.3.1 Operator types
| Thermics | 1.5 Which problems can GetDP actually solve?
| Theta, time scheme | 3.8 `Resolution`: solving systems of equations
| Time derivative | 3.7 `Formulation`: building equations
| Time stepping | 3.8 `Resolution`: solving systems of equations
| Time, discretization | 3.8 `Resolution`: solving systems of equations
| Tips | A. Tips and tricks
| Tools, order of definition | 1.1 Numerical tools as objects
| Tools, overview | 1. Overview
| Topology | 3.1 `Group`: defining topological entities
| Transformations, geometric | 3.5 `Jacobian`: defining jacobian methods
| Tree | 3.1 `Group`: defining topological entities
| Tricks | A. Tips and tricks
| Types, definition | 4. Types for objects
U |  |
| Unary operators | 2.3.1 Operator types
| User-defined functions | 3.2 `Function`: defining global and piecewise expressions
V |  |
| Values, current | 2.5 Current values
| Variables, index | Metasyntactic variable index
| Versions | 9.2 Versions
W |  |
| Web site | Copying conditions

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# Metasyntactic variable index

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

| Index Entry | Section
: |  |
| `:` | 1.2 Syntactic rules used in this document
< |  |
| `<, >` | 1.2 Syntactic rules used in this document
| `|` | 1.2 Syntactic rules used in this document
A |  |
| `argument` | 2.6 Arguments
B |  |
| `basis-function-id` | 3.4 `FunctionSpace`: building function spaces
| `basis-function-list` | 3.4 `FunctionSpace`: building function spaces
| `basis-function-type` | 3.4 `FunctionSpace`: building function spaces
| `basis-function-type` | 4.4 Types for `FunctionSpace`
| `built-in-function-id` | 2.4 Functions
C |  |
| `coef-id` | 3.4 `FunctionSpace`: building function spaces
| `constant-def` | 2.2 Constants
| `constant-id` | 2.2 Constants
| `constraint-case-id` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `constraint-case-val` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `constraint-id` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `constraint-type` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `constraint-type` | 4.3 Types for `Constraint`
| `constraint-val` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `coord-function-id` | 4.2.5 Coordinate functions
D |  |
| `...` | 1.2 Syntactic rules used in this document
E |  |
| `element-type` | 3.6 `Integration`: defining integration methods
| `element-type` | 4.6 Types for `Integration`
| `etc` | 1.2 Syntactic rules used in this document
| `expression` | 2.1 Definition
| `expression-char` | 2.2 Constants
| `expression-cst` | 2.2 Constants
| `expression-cst-list` | 2.2 Constants
| `expression-cst-list-item` | 2.2 Constants
| `expression-list` | 2.1 Definition
| `extended-math-function-id` | 4.2.2 Extended math functions
F |  |
| `formulation-id` | 3.7 `Formulation`: building equations
| `formulation-list` | 3.8 `Resolution`: solving systems of equations
| `formulation-type` | 3.7 `Formulation`: building equations
| `formulation-type` | 4.7 Types for `Formulation`
| `function-id` | 3.2 `Function`: defining global and piecewise expressions
| `function-space-id` | 3.4 `FunctionSpace`: building function spaces
| `function-space-type` | 3.4 `FunctionSpace`: building function spaces
| `function-space-type` | 4.4 Types for `FunctionSpace`
G |  |
| `global-quantity-id` | 3.4 `FunctionSpace`: building function spaces
| `global-quantity-type` | 3.4 `FunctionSpace`: building function spaces
| `global-quantity-type` | 4.4 Types for `FunctionSpace`
| `green-function-id` | 4.2.3 Green functions
| `group-def` | 3.1 `Group`: defining topological entities
| `group-id` | 3.1 `Group`: defining topological entities
| `group-list` | 3.1 `Group`: defining topological entities
| `group-list-item` | 3.1 `Group`: defining topological entities
| `group-sub-type` | 3.1 `Group`: defining topological entities
| `group-type` | 3.1 `Group`: defining topological entities
| `group-type` | 4.1 Types for `Group`
I |  |
| `integer` | 2.2 Constants
| `integral-value` | 3.9 `PostProcessing`: exploiting computational results
| `integration-id` | 3.6 `Integration`: defining integration methods
| `integration-type` | 3.6 `Integration`: defining integration methods
| `integration-type` | 4.6 Types for `Integration`
J |  |
| `jacobian-id` | 3.5 `Jacobian`: defining jacobian methods
| `jacobian-type` | 3.5 `Jacobian`: defining jacobian methods
| `jacobian-type` | 4.5 Types for `Jacobian`
L |  |
| `local-term-type` | 3.7 `Formulation`: building equations
| `local-term-type` | 4.7 Types for `Formulation`
| `local-value` | 3.9 `PostProcessing`: exploiting computational results
M |  |
| `math-function-id` | 4.2.1 Math functions
| `misc-function-id` | 4.2.6 Miscellaneous functions
O |  |
| `operator-binary` | 2.3.1 Operator types
| `operator-ternary-left` | 2.3.1 Operator types
| `operator-ternary-right` | 2.3.1 Operator types
| `operator-unary` | 2.3.1 Operator types
P |  |
| `post-operation-fmt` | 3.10 `PostOperation`: exporting results
| `post-operation-fmt` | 4.10 Types for `PostOperation`
| `post-operation-id` | 3.10 `PostOperation`: exporting results
| `post-operation-op` | 3.10 `PostOperation`: exporting results
| `post-processing-id` | 3.9 `PostProcessing`: exploiting computational results
| `post-quantity-id` | 3.9 `PostProcessing`: exploiting computational results
| `post-quantity-type` | 3.9 `PostProcessing`: exploiting computational results
| `post-value` | 3.9 `PostProcessing`: exploiting computational results
| `post-value` | 4.9 Types for `PostProcessing`
| `print-option` | 3.10 `PostOperation`: exporting results
| `print-option` | 4.10 Types for `PostOperation`
| `print-support` | 3.10 `PostOperation`: exporting results
| `print-support` | 4.10 Types for `PostOperation`
Q |  |
| `quantity` | 2.8 Fields
| `quantity-dof` | 2.8 Fields
| `quantity-id` | 2.8 Fields
| `quantity-operator` | 2.8 Fields
| `quantity-type` | 3.7 `Formulation`: building equations
| `quantity-type` | 4.7 Types for `Formulation`
R |  |
| `real` | 2.2 Constants
| `register-value-get` | 2.7 Registers
| `register-value-set` | 2.7 Registers
| `resolution-id` | 3.8 `Resolution`: solving systems of equations
| `resolution-op` | 3.8 `Resolution`: solving systems of equations
| `resolution-op` | 4.8 Types for `Resolution`
S |  |
| `string` | 2.2 Constants
| `string-id` | 2.2 Constants
| `sub-space-id` | 3.4 `FunctionSpace`: building function spaces
| `system-id` | 3.8 `Resolution`: solving systems of equations
| `system-type` | 3.8 `Resolution`: solving systems of equations
T |  |
| `term-op-type` | 3.7 `Formulation`: building equations
| `term-op-type` | 4.7 Types for `Formulation`
| `type-function-id` | 4.2.4 Type manipulation functions

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# Syntax index

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

| Index Entry | Section
! |  |
| `!` | 2.3.1 Operator types
| `!=` | 2.3.1 Operator types
# |  |
| `#include` | 1.4 Includes
| `#integer` | 2.7 Registers
$ |  |
| `$A` | 2.5 Current values
| `$B` | 2.5 Current values
| `$C` | 2.5 Current values
| `$DTime` | 2.5 Current values
| `$integer` | 2.6 Arguments
| `$Iteration` | 2.5 Current values
| `$Theta` | 2.5 Current values
| `$Time` | 2.5 Current values
| `$TimeStep` | 2.5 Current values
| `$X` | 2.5 Current values
| `$XS` | 2.5 Current values
| `$Y` | 2.5 Current values
| `$YS` | 2.5 Current values
| `$Z` | 2.5 Current values
| `$ZS` | 2.5 Current values
% |  |
| `%` | 2.3.1 Operator types
& |  |
| `&&` | 2.3.1 Operator types
( |  |
| `()` | 2.3.2 Evaluation order
* |  |
| `*` | 2.3.1 Operator types
+ |  |
| `+` | 2.3.1 Operator types
- |  |
| `-` | 2.3.1 Operator types
| `-` | 2.3.1 Operator types
| `-adapt` | 7. Running GetDP
| `-adapt` | 7. Running GetDP
| `-bin` | 7. Running GetDP
| `-bin` | 7. Running GetDP
| `-cal` | 7. Running GetDP
| `-cal` | 7. Running GetDP
| `-check` | 7. Running GetDP
| `-check` | 7. Running GetDP
| `-help` | 7. Running GetDP
| `-help` | 7. Running GetDP
| `-info` | 7. Running GetDP
| `-info` | 7. Running GetDP
| `-ipos` | 7. Running GetDP
| `-ipos` | 7. Running GetDP
| `-log` | 7. Running GetDP
| `-log` | 7. Running GetDP
| `-msh` | 7. Running GetDP
| `-msh` | 7. Running GetDP
| `-name` | 7. Running GetDP
| `-name` | 7. Running GetDP
| `-order` | 7. Running GetDP
| `-order` | 7. Running GetDP
| `-p` | 7. Running GetDP
| `-p` | 7. Running GetDP
| `-pos` | 7. Running GetDP
| `-pos` | 7. Running GetDP
| `-pre` | 7. Running GetDP
| `-pre` | 7. Running GetDP
| `-progress` | 7. Running GetDP
| `-progress` | 7. Running GetDP
| `-res` | 7. Running GetDP
| `-res` | 7. Running GetDP
| `-restart` | 7. Running GetDP
| `-restart` | 7. Running GetDP
| `-socket` | 7. Running GetDP
| `-socket` | 7. Running GetDP
| `-solve` | 7. Running GetDP
| `-solve` | 7. Running GetDP
| `-split` | 7. Running GetDP
| `-split` | 7. Running GetDP
| `-v` | 7. Running GetDP
| `-v` | 7. Running GetDP
| `-verbose` | 7. Running GetDP
| `-verbose` | 7. Running GetDP
| `-version` | 7. Running GetDP
| `-version` | 7. Running GetDP
/ |  |
| `/` | 2.3.1 Operator types
| `/*, */` | 1.3 Comments
| `//` | 1.3 Comments
| `/\` | 2.3.1 Operator types
0 |  |
| `0D` | 2.2 Constants
1 |  |
| `1D` | 2.2 Constants
2 |  |
| `2D` | 2.2 Constants
3 |  |
| `3D` | 2.2 Constants
< |  |
| `<` | 2.3.1 Operator types
| `<=` | 2.3.1 Operator types
= |  |
| `=` | 2.2 Constants
| `=` | 3.1 `Group`: defining topological entities
| `=` | 3.2 `Function`: defining global and piecewise expressions
| `==` | 2.3.1 Operator types
> |  |
| `>` | 2.3.1 Operator types
| `>=` | 2.3.1 Operator types
? |  |
| `?:` | 2.3.1 Operator types
^ |  |
| `^` | 2.3.1 Operator types
| `||` | 2.3.1 Operator types
A |  |
| `Acos` | 4.2.1 Math functions
| `Acos` | 4.2.1 Math functions
| `Adapt` | 4.10 Types for `PostOperation`
| `Adapt` | 4.10 Types for `PostOperation`
| `Adaptation` | 4.10 Types for `PostOperation`
| `Adaptation` | 4.10 Types for `PostOperation`
| `AliasOf` | 4.4 Types for `FunctionSpace`
| `AliasOf` | 4.4 Types for `FunctionSpace`
| `All` | 3.5 `Jacobian`: defining jacobian methods
| `Analytic` | 3.6 `Integration`: defining integration methods
| `Asin` | 4.2.1 Math functions
| `Asin` | 4.2.1 Math functions
| `Assign` | 4.3 Types for `Constraint`
| `Assign` | 4.3 Types for `Constraint`
| `AssignFromResolution` | 4.3 Types for `Constraint`
| `AssignFromResolution` | 4.3 Types for `Constraint`
| `AssociatedWith` | 4.4 Types for `FunctionSpace`
| `AssociatedWith` | 4.4 Types for `FunctionSpace`
| `Atan` | 4.2.1 Math functions
| `Atan` | 4.2.1 Math functions
| `Atan2` | 4.2.1 Math functions
| `Atan2` | 4.2.1 Math functions
B |  |
| `BasisFunction` | 3.4 `FunctionSpace`: building function spaces
| `BF` | 2.8 Fields
| `BF` | 2.8 Fields
| `BF_CurlEdge` | 4.4 Types for `FunctionSpace`
| `BF_CurlEdge` | 4.4 Types for `FunctionSpace`
| `BF_CurlGroupOfEdges` | 4.4 Types for `FunctionSpace`
| `BF_CurlGroupOfEdges` | 4.4 Types for `FunctionSpace`
| `BF_CurlGroupOfPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_CurlGroupOfPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_CurlPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_CurlPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_dGlobal` | 4.4 Types for `FunctionSpace`
| `BF_dGlobal` | 4.4 Types for `FunctionSpace`
| `BF_DivFacet` | 4.4 Types for `FunctionSpace`
| `BF_DivFacet` | 4.4 Types for `FunctionSpace`
| `BF_DivPerpendicularFacet` | 4.4 Types for `FunctionSpace`
| `BF_DivPerpendicularFacet` | 4.4 Types for `FunctionSpace`
| `BF_Edge` | 4.4 Types for `FunctionSpace`
| `BF_Edge` | 4.4 Types for `FunctionSpace`
| `BF_Facet` | 4.4 Types for `FunctionSpace`
| `BF_Facet` | 4.4 Types for `FunctionSpace`
| `BF_Global` | 4.4 Types for `FunctionSpace`
| `BF_Global` | 4.4 Types for `FunctionSpace`
| `BF_GradGroupOfNodes` | 4.4 Types for `FunctionSpace`
| `BF_GradGroupOfNodes` | 4.4 Types for `FunctionSpace`
| `BF_GradNode` | 4.4 Types for `FunctionSpace`
| `BF_GradNode` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfEdges` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfEdges` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfNodes` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfNodes` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_GroupOfPerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_Node` | 4.4 Types for `FunctionSpace`
| `BF_Node` | 4.4 Types for `FunctionSpace`
| `BF_NodeX` | 4.4 Types for `FunctionSpace`
| `BF_NodeX` | 4.4 Types for `FunctionSpace`
| `BF_NodeY` | 4.4 Types for `FunctionSpace`
| `BF_NodeY` | 4.4 Types for `FunctionSpace`
| `BF_NodeZ` | 4.4 Types for `FunctionSpace`
| `BF_NodeZ` | 4.4 Types for `FunctionSpace`
| `BF_One` | 4.4 Types for `FunctionSpace`
| `BF_One` | 4.4 Types for `FunctionSpace`
| `BF_PerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_PerpendicularEdge` | 4.4 Types for `FunctionSpace`
| `BF_PerpendicularFacet` | 4.4 Types for `FunctionSpace`
| `BF_PerpendicularFacet` | 4.4 Types for `FunctionSpace`
| `BF_Region` | 4.4 Types for `FunctionSpace`
| `BF_Region` | 4.4 Types for `FunctionSpace`
| `BF_RegionX` | 4.4 Types for `FunctionSpace`
| `BF_RegionX` | 4.4 Types for `FunctionSpace`
| `BF_RegionY` | 4.4 Types for `FunctionSpace`
| `BF_RegionY` | 4.4 Types for `FunctionSpace`
| `BF_RegionZ` | 4.4 Types for `FunctionSpace`
| `BF_RegionZ` | 4.4 Types for `FunctionSpace`
| `BF_Volume` | 4.4 Types for `FunctionSpace`
| `BF_Volume` | 4.4 Types for `FunctionSpace`
| `BF_Zero` | 4.4 Types for `FunctionSpace`
| `BF_Zero` | 4.4 Types for `FunctionSpace`
C |  |
| `Case` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `Case` | 3.5 `Jacobian`: defining jacobian methods
| `Case` | 3.6 `Integration`: defining integration methods
| `ChangeOfCoordinates` | 4.10 Types for `PostOperation`
| `ChangeOfCoordinates` | 4.10 Types for `PostOperation`
| `ChangeOfValues` | 4.10 Types for `PostOperation`
| `ChangeOfValues` | 4.10 Types for `PostOperation`
| `Complex` | 4.2.4 Type manipulation functions
| `Complex` | 4.2.4 Type manipulation functions
| `CompX` | 4.2.4 Type manipulation functions
| `CompX` | 4.2.4 Type manipulation functions
| `CompXX` | 4.2.4 Type manipulation functions
| `CompXX` | 4.2.4 Type manipulation functions
| `CompXY` | 4.2.4 Type manipulation functions
| `CompXY` | 4.2.4 Type manipulation functions
| `CompXZ` | 4.2.4 Type manipulation functions
| `CompXZ` | 4.2.4 Type manipulation functions
| `CompY` | 4.2.4 Type manipulation functions
| `CompY` | 4.2.4 Type manipulation functions
| `CompYX` | 4.2.4 Type manipulation functions
| `CompYX` | 4.2.4 Type manipulation functions
| `CompYY` | 4.2.4 Type manipulation functions
| `CompYY` | 4.2.4 Type manipulation functions
| `CompYZ` | 4.2.4 Type manipulation functions
| `CompYZ` | 4.2.4 Type manipulation functions
| `CompZ` | 4.2.4 Type manipulation functions
| `CompZ` | 4.2.4 Type manipulation functions
| `CompZX` | 4.2.4 Type manipulation functions
| `CompZX` | 4.2.4 Type manipulation functions
| `CompZY` | 4.2.4 Type manipulation functions
| `CompZY` | 4.2.4 Type manipulation functions
| `CompZZ` | 4.2.4 Type manipulation functions
| `CompZZ` | 4.2.4 Type manipulation functions
| `Constraint` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `Constraint` | 3.4 `FunctionSpace`: building function spaces
| `Cos` | 4.2.1 Math functions
| `Cos` | 4.2.1 Math functions
| `Cosh` | 4.2.1 Math functions
| `Cosh` | 4.2.1 Math functions
| `Criterion` | 3.6 `Integration`: defining integration methods
| `Cross` | 4.2.2 Extended math functions
| `Cross` | 4.2.2 Extended math functions
| `Curl` | 2.8 Fields
| `Curl` | 2.8 Fields
| `CurlInv` | 2.8 Fields
| `CurlInv` | 2.8 Fields
D |  |
| `d` | 2.8 Fields
| `d` | 2.8 Fields
| `DefineConstant` | 2.2 Constants
| `DefineFunction` | 3.2 `Function`: defining global and piecewise expressions
| `DefineGroup` | 3.1 `Group`: defining topological entities
| `Depth` | 4.10 Types for `PostOperation`
| `Depth` | 4.10 Types for `PostOperation`
| `deRham` | 4.7 Types for `Formulation`
| `deRham` | 4.7 Types for `Formulation`
| `DestinationSystem` | 3.8 `Resolution`: solving systems of equations
| `Dimension` | 4.10 Types for `PostOperation`
| `Dimension` | 4.10 Types for `PostOperation`
| `dInterpolationAkima` | 4.2.6 Miscellaneous functions
| `dInterpolationAkima` | 4.2.6 Miscellaneous functions
| `dInterpolationLinear` | 4.2.6 Miscellaneous functions
| `dInterpolationLinear` | 4.2.6 Miscellaneous functions
| `dInv` | 2.8 Fields
| `dInv` | 2.8 Fields
| `Div` | 2.8 Fields
| `Div` | 2.8 Fields
| `DivInv` | 2.8 Fields
| `DivInv` | 2.8 Fields
| `Dof` | 2.8 Fields
| `Dof` | 2.8 Fields
| `Dt` | 4.7 Types for `Formulation`
| `Dt` | 4.7 Types for `Formulation`
| `DtDof` | 4.7 Types for `Formulation`
| `DtDof` | 4.7 Types for `Formulation`
| `DtDt` | 4.7 Types for `Formulation`
| `DtDt` | 4.7 Types for `Formulation`
| `DtDtDof` | 4.7 Types for `Formulation`
| `DtDtDof` | 4.7 Types for `Formulation`
| `DualEdgesOf` | 4.1 Types for `Group`
| `DualEdgesOf` | 4.1 Types for `Group`
| `DualFacetsOf` | 4.1 Types for `Group`
| `DualFacetsOf` | 4.1 Types for `Group`
| `DualNodesOf` | 4.1 Types for `Group`
| `DualNodesOf` | 4.1 Types for `Group`
| `DualVolumesOf` | 4.1 Types for `Group`
| `DualVolumesOf` | 4.1 Types for `Group`
E |  |
| `EdgesOf` | 4.1 Types for `Group`
| `EdgesOf` | 4.1 Types for `Group`
| `EdgesOfTreeIn` | 4.1 Types for `Group`
| `EdgesOfTreeIn` | 4.1 Types for `Group`
| `ElementsOf` | 4.1 Types for `Group`
| `ElementsOf` | 4.1 Types for `Group`
| `Else` | 4.8 Types for `Resolution`
| `Else` | 4.8 Types for `Resolution`
| `Entity` | 3.4 `FunctionSpace`: building function spaces
| `EntitySubType` | 3.4 `FunctionSpace`: building function spaces
| `EntityType` | 3.4 `FunctionSpace`: building function spaces
| `Equation` | 3.7 `Formulation`: building equations
| `Exp` | 4.2.1 Math functions
| `Exp` | 4.2.1 Math functions
F |  |
| `F_CompElementNum` | 4.2.6 Miscellaneous functions
| `F_CompElementNum` | 4.2.6 Miscellaneous functions
| `F_Cos_wt_p` | 4.2.2 Extended math functions
| `F_Cos_wt_p` | 4.2.2 Extended math functions
| `F_Period` | 4.2.2 Extended math functions
| `F_Period` | 4.2.2 Extended math functions
| `F_Sin_wt_p` | 4.2.2 Extended math functions
| `F_Sin_wt_p` | 4.2.2 Extended math functions
| `Fabs` | 4.2.1 Math functions
| `Fabs` | 4.2.1 Math functions
| `FacetsOf` | 4.1 Types for `Group`
| `FacetsOf` | 4.1 Types for `Group`
| `FacetsOfTreeIn` | 4.1 Types for `Group`
| `FacetsOfTreeIn` | 4.1 Types for `Group`
| `FemEquation` | 4.7 Types for `Formulation`
| `FemEquation` | 4.7 Types for `Formulation`
| `File` | 4.10 Types for `PostOperation`
| `File` | 4.10 Types for `PostOperation`
| `File` | 4.10 Types for `PostOperation`
| `File` | 4.10 Types for `PostOperation`
| `File` | 4.10 Types for `PostOperation`
| `File` | 4.10 Types for `PostOperation`
| `Fmod` | 4.2.1 Math functions
| `Fmod` | 4.2.1 Math functions
| `Form0` | 4.4 Types for `FunctionSpace`
| `Form0` | 4.4 Types for `FunctionSpace`
| `Form1` | 4.4 Types for `FunctionSpace`
| `Form1` | 4.4 Types for `FunctionSpace`
| `Form1P` | 4.4 Types for `FunctionSpace`
| `Form1P` | 4.4 Types for `FunctionSpace`
| `Form2` | 4.4 Types for `FunctionSpace`
| `Form2` | 4.4 Types for `FunctionSpace`
| `Form2P` | 4.4 Types for `FunctionSpace`
| `Form2P` | 4.4 Types for `FunctionSpace`
| `Form3` | 4.4 Types for `FunctionSpace`
| `Form3` | 4.4 Types for `FunctionSpace`
| `Format` | 3.10 `PostOperation`: exporting results
| `Format` | 4.10 Types for `PostOperation`
| `Formulation` | 3.4 `FunctionSpace`: building function spaces
| `Formulation` | 3.7 `Formulation`: building equations
| `FourierTransform` | 4.8 Types for `Resolution`
| `FourierTransform` | 4.8 Types for `Resolution`
| `Frequency` | 3.8 `Resolution`: solving systems of equations
| `Frequency` | 4.10 Types for `PostOperation`
| `Function` | 3.2 `Function`: defining global and piecewise expressions
| `Function` | 3.4 `FunctionSpace`: building function spaces
| `FunctionSpace` | 3.4 `FunctionSpace`: building function spaces
G |  |
| `Galerkin` | 4.7 Types for `Formulation`
| `Galerkin` | 4.7 Types for `Formulation`
| `Gauss` | 4.6 Types for `Integration`
| `Gauss` | 4.6 Types for `Integration`
| `GaussLegendre` | 4.6 Types for `Integration`
| `GaussLegendre` | 4.6 Types for `Integration`
| `Generate` | 4.8 Types for `Resolution`
| `Generate` | 4.8 Types for `Resolution`
| `GenerateJac` | 4.8 Types for `Resolution`
| `GenerateJac` | 4.8 Types for `Resolution`
| `GenerateSeparate` | 4.8 Types for `Resolution`
| `GenerateSeparate` | 4.8 Types for `Resolution`
| `GeoElement` | 3.6 `Integration`: defining integration methods
| `Global` | 4.1 Types for `Group`
| `Global` | 4.1 Types for `Group`
| `Global` | 4.7 Types for `Formulation`
| `Global` | 4.7 Types for `Formulation`
| `GlobalEquation` | 3.7 `Formulation`: building equations
| `GlobalQuantity` | 3.4 `FunctionSpace`: building function spaces
| `GlobalTerm` | 3.7 `Formulation`: building equations
| `Gmsh` | 4.10 Types for `PostOperation`
| `Gmsh` | 4.10 Types for `PostOperation`
| `GmshParsed` | 4.10 Types for `PostOperation`
| `GmshParsed` | 4.10 Types for `PostOperation`
| `Gnuplot` | 4.10 Types for `PostOperation`
| `Gnuplot` | 4.10 Types for `PostOperation`
| `Grad` | 2.8 Fields
| `Grad` | 2.8 Fields
| `GradHelmholtz` | 4.2.3 Green functions
| `GradHelmholtz` | 4.2.3 Green functions
| `GradInv` | 2.8 Fields
| `GradInv` | 2.8 Fields
| `GradLaplace` | 4.2.3 Green functions
| `GradLaplace` | 4.2.3 Green functions
| `Group` | 3.1 `Group`: defining topological entities
| `Group` | 3.4 `FunctionSpace`: building function spaces
| `GroupsOfEdgesOf` | 4.1 Types for `Group`
| `GroupsOfEdgesOf` | 4.1 Types for `Group`
| `GroupsOfEdgesOnNodesOf` | 4.1 Types for `Group`
| `GroupsOfEdgesOnNodesOf` | 4.1 Types for `Group`
| `GroupsOfNodesOf` | 4.1 Types for `Group`
| `GroupsOfNodesOf` | 4.1 Types for `Group`
H |  |
| `HarmonicToTime` | 4.10 Types for `PostOperation`
| `HarmonicToTime` | 4.10 Types for `PostOperation`
| `Helmholtz` | 4.2.3 Green functions
| `Helmholtz` | 4.2.3 Green functions
| `Hexahedron` | 4.6 Types for `Integration`
| `Hexahedron` | 4.6 Types for `Integration`
| `Hypot` | 4.2.2 Extended math functions
| `Hypot` | 4.2.2 Extended math functions
I |  |
| `If` | 4.8 Types for `Resolution`
| `If` | 4.8 Types for `Resolution`
| `If` | 4.8 Types for `Resolution`
| `If` | 4.8 Types for `Resolution`
| `Im` | 4.2.4 Type manipulation functions
| `Im` | 4.2.4 Type manipulation functions
| `In` | 3.7 `Formulation`: building equations
| `In` | 3.9 `PostProcessing`: exploiting computational results
| `Include` | 1.4 Includes
| `IndexOfSystem` | 3.7 `Formulation`: building equations
| `Init` | 4.3 Types for `Constraint`
| `Init` | 4.3 Types for `Constraint`
| `InitFromResolution` | 4.3 Types for `Constraint`
| `InitFromResolution` | 4.3 Types for `Constraint`
| `InitSolution` | 4.8 Types for `Resolution`
| `InitSolution` | 4.8 Types for `Resolution`
| `Integral` | 3.9 `PostProcessing`: exploiting computational results
| `Integral` | 4.7 Types for `Formulation`
| `Integral` | 4.9 Types for `PostProcessing`
| `Integral` | 4.9 Types for `PostProcessing`
| `Integration` | 3.6 `Integration`: defining integration methods
| `Integration` | 3.7 `Formulation`: building equations
| `Integration` | 3.9 `PostProcessing`: exploiting computational results
| `InterpolationAkima` | 4.2.6 Miscellaneous functions
| `InterpolationAkima` | 4.2.6 Miscellaneous functions
| `InterpolationLinear` | 4.2.6 Miscellaneous functions
| `InterpolationLinear` | 4.2.6 Miscellaneous functions
| `Iso` | 4.10 Types for `PostOperation`
| `Iso` | 4.10 Types for `PostOperation`
| `Iso` | 4.10 Types for `PostOperation`
| `Iso` | 4.10 Types for `PostOperation`
| `IterativeLoop` | 4.8 Types for `Resolution`
| `IterativeLoop` | 4.8 Types for `Resolution`
J |  |
| `JacNL` | 4.7 Types for `Formulation`
| `JacNL` | 4.7 Types for `Formulation`
| `Jacobian` | 3.5 `Jacobian`: defining jacobian methods
| `Jacobian` | 3.7 `Formulation`: building equations
| `Jacobian` | 3.9 `PostProcessing`: exploiting computational results
L |  |
| `Lanczos` | 4.8 Types for `Resolution`
| `Lanczos` | 4.8 Types for `Resolution`
| `Laplace` | 4.2.3 Green functions
| `Laplace` | 4.2.3 Green functions
| `Lin` | 4.5 Types for `Jacobian`
| `Lin` | 4.5 Types for `Jacobian`
| `Line` | 4.6 Types for `Integration`
| `Line` | 4.6 Types for `Integration`
| `Link` | 4.3 Types for `Constraint`
| `Link` | 4.3 Types for `Constraint`
| `LinkCplx` | 4.3 Types for `Constraint`
| `LinkCplx` | 4.3 Types for `Constraint`
| `List` | 2.2 Constants
| `ListAlt` | 2.2 Constants
| `Local` | 3.9 `PostProcessing`: exploiting computational results
| `Local` | 4.7 Types for `Formulation`
| `Local` | 4.9 Types for `PostProcessing`
| `Local` | 4.9 Types for `PostProcessing`
| `Log` | 4.2.1 Math functions
| `Log` | 4.2.1 Math functions
| `Log10` | 4.2.1 Math functions
| `Log10` | 4.2.1 Math functions
| `Loop` | 3.7 `Formulation`: building equations
N |  |
| `Name` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `Name` | 3.4 `FunctionSpace`: building function spaces
| `Name` | 3.5 `Jacobian`: defining jacobian methods
| `Name` | 3.6 `Integration`: defining integration methods
| `Name` | 3.7 `Formulation`: building equations
| `Name` | 3.8 `Resolution`: solving systems of equations
| `Name` | 3.9 `PostProcessing`: exploiting computational results
| `Name` | 3.10 `PostOperation`: exporting results
| `NameOfBasisFunction` | 3.4 `FunctionSpace`: building function spaces
| `NameOfCoef` | 3.4 `FunctionSpace`: building function spaces
| `NameOfConstraint` | 3.4 `FunctionSpace`: building function spaces
| `NameOfConstraint` | 3.7 `Formulation`: building equations
| `NameOfFormulation` | 3.8 `Resolution`: solving systems of equations
| `NameOfFormulation` | 3.9 `PostProcessing`: exploiting computational results
| `NameOfMesh` | 3.8 `Resolution`: solving systems of equations
| `NameOfPostProcessing` | 3.10 `PostOperation`: exporting results
| `NameOfSpace` | 3.7 `Formulation`: building equations
| `NameOfSystem` | 3.9 `PostProcessing`: exploiting computational results
| `Network` | 3.7 `Formulation`: building equations
| `Network` | 4.3 Types for `Constraint`
| `NeverDt` | 4.7 Types for `Formulation`
| `NeverDt` | 4.7 Types for `Formulation`
| `Node` | 3.7 `Formulation`: building equations
| `NodesOf` | 4.1 Types for `Group`
| `NodesOf` | 4.1 Types for `Group`
| `NoNewLine` | 4.10 Types for `PostOperation`
| `NoNewLine` | 4.10 Types for `PostOperation`
| `Norm` | 4.2.2 Extended math functions
| `Norm` | 4.2.2 Extended math functions
| `Normal` | 4.2.6 Miscellaneous functions
| `Normal` | 4.2.6 Miscellaneous functions
| `NormalSource` | 4.2.6 Miscellaneous functions
| `NormalSource` | 4.2.6 Miscellaneous functions
| `NumberOfPoints` | 3.6 `Integration`: defining integration methods
O |  |
| `OnBox` | 4.10 Types for `PostOperation`
| `OnBox` | 4.10 Types for `PostOperation`
| `OnElementsOf` | 4.10 Types for `PostOperation`
| `OnElementsOf` | 4.10 Types for `PostOperation`
| `OnGlobal` | 4.10 Types for `PostOperation`
| `OnGlobal` | 4.10 Types for `PostOperation`
| `OnGrid` | 4.10 Types for `PostOperation`
| `OnGrid` | 4.10 Types for `PostOperation`
| `OnGrid` | 4.10 Types for `PostOperation`
| `OnGrid` | 4.10 Types for `PostOperation`
| `OnLine` | 4.10 Types for `PostOperation`
| `OnLine` | 4.10 Types for `PostOperation`
| `OnPlane` | 4.10 Types for `PostOperation`
| `OnPlane` | 4.10 Types for `PostOperation`
| `OnPoint` | 4.10 Types for `PostOperation`
| `OnPoint` | 4.10 Types for `PostOperation`
| `OnRegion` | 4.10 Types for `PostOperation`
| `OnRegion` | 4.10 Types for `PostOperation`
| `OnSection` | 4.10 Types for `PostOperation`
| `OnSection` | 4.10 Types for `PostOperation`
| `Operation` | 3.8 `Resolution`: solving systems of equations
| `Operation` | 3.10 `PostOperation`: exporting results
| `Order` | 4.2.6 Miscellaneous functions
| `Order` | 4.2.6 Miscellaneous functions
| `OriginSystem` | 3.8 `Resolution`: solving systems of equations
P |  |
| `Pi` | 2.2 Constants
| `Point` | 4.6 Types for `Integration`
| `Point` | 4.6 Types for `Integration`
| `PostOperation` | 3.10 `PostOperation`: exporting results
| `PostProcessing` | 3.9 `PostProcessing`: exploiting computational results
| `Print` | 3.10 `PostOperation`: exporting results
| `Print` | 4.8 Types for `Resolution`
| `Print` | 4.8 Types for `Resolution`
| `Print` | 4.8 Types for `Resolution`
| `Printf` | 4.2.6 Miscellaneous functions
| `Printf` | 4.2.6 Miscellaneous functions
| `Prism` | 4.6 Types for `Integration`
| `Prism` | 4.6 Types for `Integration`
| `Pyramid` | 4.6 Types for `Integration`
| `Pyramid` | 4.6 Types for `Integration`
Q |  |
| `Quadrangle` | 4.6 Types for `Integration`
| `Quadrangle` | 4.6 Types for `Integration`
| `Quantity` | 3.4 `FunctionSpace`: building function spaces
| `Quantity` | 3.7 `Formulation`: building equations
| `Quantity` | 3.9 `PostProcessing`: exploiting computational results
R |  |
| `Re` | 4.2.4 Type manipulation functions
| `Re` | 4.2.4 Type manipulation functions
| `Region` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `Region` | 3.5 `Jacobian`: defining jacobian methods
| `Resolution` | 3.4 `FunctionSpace`: building function spaces
| `Resolution` | 3.8 `Resolution`: solving systems of equations
| `Rot` | 2.8 Fields
| `Rot` | 2.8 Fields
| `RotInv` | 2.8 Fields
| `RotInv` | 2.8 Fields
S |  |
| `SaveSolution` | 4.8 Types for `Resolution`
| `SaveSolution` | 4.8 Types for `Resolution`
| `SaveSolutions` | 4.8 Types for `Resolution`
| `SaveSolutions` | 4.8 Types for `Resolution`
| `Scalar` | 4.4 Types for `FunctionSpace`
| `Scalar` | 4.4 Types for `FunctionSpace`
| `SetFrequency` | 4.8 Types for `Resolution`
| `SetFrequency` | 4.8 Types for `Resolution`
| `SetTime` | 4.8 Types for `Resolution`
| `SetTime` | 4.8 Types for `Resolution`
| `Sin` | 4.2.1 Math functions
| `Sin` | 4.2.1 Math functions
| `Sinh` | 4.2.1 Math functions
| `Sinh` | 4.2.1 Math functions
| `Skin` | 4.10 Types for `PostOperation`
| `Skin` | 4.10 Types for `PostOperation`
| `Smoothing` | 4.10 Types for `PostOperation`
| `Smoothing` | 4.10 Types for `PostOperation`
| `Solve` | 4.8 Types for `Resolution`
| `Solve` | 4.8 Types for `Resolution`
| `SolveJac` | 4.8 Types for `Resolution`
| `SolveJac` | 4.8 Types for `Resolution`
| `Solver` | 3.8 `Resolution`: solving systems of equations
| `Sort` | 4.10 Types for `PostOperation`
| `Sort` | 4.10 Types for `PostOperation`
| `Sqrt` | 4.2.1 Math functions
| `Sqrt` | 4.2.1 Math functions
| `SquNorm` | 4.2.2 Extended math functions
| `SquNorm` | 4.2.2 Extended math functions
| `SubRegion` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `SubSpace` | 3.4 `FunctionSpace`: building function spaces
| `Support` | 3.4 `FunctionSpace`: building function spaces
| `Sur` | 4.5 Types for `Jacobian`
| `Sur` | 4.5 Types for `Jacobian`
| `SurAxi` | 4.5 Types for `Jacobian`
| `SurAxi` | 4.5 Types for `Jacobian`
| `Symmetry` | 3.7 `Formulation`: building equations
| `System` | 3.8 `Resolution`: solving systems of equations
| `SystemCommand` | 4.8 Types for `Resolution`
| `SystemCommand` | 4.8 Types for `Resolution`
T |  |
| `Table` | 4.10 Types for `PostOperation`
| `Table` | 4.10 Types for `PostOperation`
| `Tan` | 4.2.1 Math functions
| `Tan` | 4.2.1 Math functions
| `Tanh` | 4.2.1 Math functions
| `Tanh` | 4.2.1 Math functions
| `Target` | 4.10 Types for `PostOperation`
| `Target` | 4.10 Types for `PostOperation`
| `Tensor` | 4.2.4 Type manipulation functions
| `Tensor` | 4.2.4 Type manipulation functions
| `TensorDiag` | 4.2.4 Type manipulation functions
| `TensorDiag` | 4.2.4 Type manipulation functions
| `TensorSym` | 4.2.4 Type manipulation functions
| `TensorSym` | 4.2.4 Type manipulation functions
| `TensorV` | 4.2.4 Type manipulation functions
| `TensorV` | 4.2.4 Type manipulation functions
| `Tetrahedron` | 4.6 Types for `Integration`
| `Tetrahedron` | 4.6 Types for `Integration`
| `TimeFunction` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `TimeLoopNewmark` | 4.8 Types for `Resolution`
| `TimeLoopNewmark` | 4.8 Types for `Resolution`
| `TimeLoopTheta` | 4.8 Types for `Resolution`
| `TimeLoopTheta` | 4.8 Types for `Resolution`
| `TimeStep` | 4.10 Types for `PostOperation`
| `TimeStep` | 4.10 Types for `PostOperation`
| `TimeTable` | 4.10 Types for `PostOperation`
| `TimeTable` | 4.10 Types for `PostOperation`
| `TransferInitSolution` | 4.8 Types for `Resolution`
| `TransferInitSolution` | 4.8 Types for `Resolution`
| `TransferSolution` | 4.8 Types for `Resolution`
| `TransferSolution` | 4.8 Types for `Resolution`
| `Transpose` | 4.2.2 Extended math functions
| `Transpose` | 4.2.2 Extended math functions
| `Triangle` | 4.6 Types for `Integration`
| `Triangle` | 4.6 Types for `Integration`
| `TTrace` | 4.2.2 Extended math functions
| `TTrace` | 4.2.2 Extended math functions
| `Type` | 3.3 `Constraint`: specifying constraints on function spaces and formulations
| `Type` | 3.4 `FunctionSpace`: building function spaces
| `Type` | 3.6 `Integration`: defining integration methods
| `Type` | 3.7 `Formulation`: building equations
| `Type` | 3.8 `Resolution`: solving systems of equations
U |  |
| `Unit` | 4.2.2 Extended math functions
| `Unit` | 4.2.2 Extended math functions
| `Update` | 4.8 Types for `Resolution`
| `Update` | 4.8 Types for `Resolution`
| `UsingPost` | 3.10 `PostOperation`: exporting results
V |  |
| `Value` | 3.9 `PostProcessing`: exploiting computational results
| `Value` | 4.10 Types for `PostOperation`
| `Vector` | 4.2.4 Type manipulation functions
| `Vector` | 4.2.4 Type manipulation functions
| `Vector` | 4.4 Types for `FunctionSpace`
| `Vector` | 4.4 Types for `FunctionSpace`
| `Vol` | 4.5 Types for `Jacobian`
| `Vol` | 4.5 Types for `Jacobian`
| `VolAxi` | 4.5 Types for `Jacobian`
| `VolAxi` | 4.5 Types for `Jacobian`
| `VolAxiRectShell` | 4.5 Types for `Jacobian`
| `VolAxiRectShell` | 4.5 Types for `Jacobian`
| `VolAxiSphShell` | 4.5 Types for `Jacobian`
| `VolAxiSphShell` | 4.5 Types for `Jacobian`
| `VolAxiSqu` | 4.5 Types for `Jacobian`
| `VolAxiSqu` | 4.5 Types for `Jacobian`
| `VolAxiSquRectShell` | 4.5 Types for `Jacobian`
| `VolAxiSquRectShell` | 4.5 Types for `Jacobian`
| `VolAxiSquSphShell` | 4.5 Types for `Jacobian`
| `VolAxiSquSphShell` | 4.5 Types for `Jacobian`
| `VolRectShell` | 4.5 Types for `Jacobian`
| `VolRectShell` | 4.5 Types for `Jacobian`
| `VolSphShell` | 4.5 Types for `Jacobian`
| `VolSphShell` | 4.5 Types for `Jacobian`
| `VolumesOf` | 4.1 Types for `Group`
| `VolumesOf` | 4.1 Types for `Group`
X |  |
| `X` | 4.2.5 Coordinate functions
| `X` | 4.2.5 Coordinate functions
| `XYZ` | 4.2.5 Coordinate functions
| `XYZ` | 4.2.5 Coordinate functions
Y |  |
| `Y` | 4.2.5 Coordinate functions
| `Y` | 4.2.5 Coordinate functions
Z |  |
| `Z` | 4.2.5 Coordinate functions
| `Z` | 4.2.5 Coordinate functions

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

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# Footnotes

### (1)

If you are looking for a graphical front-end to GetDP, you may consider using Gmsh (available at http://www.geuz.org/gmsh/). Gmsh permits to construct geometries, generate meshes, launch computations and visualize results directly from within a user-friendly graphical interface. The file formats used by Gmsh for mesh generation and post-processing are the default file formats accepted by GetDP (see 8.1 Input file format, and 4.10 Types for `PostOperation`).

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# Table of Contents

Copying conditions
Introduction
1. Overview

1.1 Numerical tools as objects
1.2 Syntactic rules used in this document
1.3 Comments
1.4 Includes
1.5 Which problems can GetDP actually solve?

2. Expressions

2.1 Definition
2.2 Constants
2.3 Operators

2.3.1 Operator types
2.3.2 Evaluation order

2.4 Functions
2.5 Current values
2.6 Arguments
2.7 Registers
2.8 Fields

3. Objects

3.1 `Group`: defining topological entities
3.2 `Function`: defining global and piecewise expressions
3.3 `Constraint`: specifying constraints on function spaces and formulations
3.4 `FunctionSpace`: building function spaces
3.5 `Jacobian`: defining jacobian methods
3.6 `Integration`: defining integration methods
3.7 `Formulation`: building equations
3.8 `Resolution`: solving systems of equations
3.9 `PostProcessing`: exploiting computational results
3.10 `PostOperation`: exporting results

4. Types for objects

4.1 Types for `Group`
4.2 Types for `Function`

4.2.1 Math functions
4.2.2 Extended math functions
4.2.3 Green functions
4.2.4 Type manipulation functions
4.2.5 Coordinate functions
4.2.6 Miscellaneous functions

4.3 Types for `Constraint`
4.4 Types for `FunctionSpace`
4.5 Types for `Jacobian`
4.6 Types for `Integration`
4.7 Types for `Formulation`
4.8 Types for `Resolution`
4.9 Types for `PostProcessing`
4.10 Types for `PostOperation`

5. Short examples

5.1 Constant expression examples
5.2 `Group` examples
5.3 `Function` examples
5.4 `Constraint` examples
5.5 `FunctionSpace` examples

5.5.1 Nodal finite element spaces
5.5.2 High order nodal finite element space
5.5.3 Nodal finite element space with floating potentials
5.5.4 Edge finite element space
5.5.5 Edge finite element space with gauge condition
5.5.6 Coupled edge and nodal finite element spaces
5.5.7 Coupled edge and nodal finite element spaces for multiply connected domains

5.6 `Jacobian` examples
5.7 `Integration` examples
5.8 `Formulation` examples

5.8.1 Electrostatic scalar potential formulation
5.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges
5.8.3 Magnetostatic 3D vector potential formulation
5.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation
5.8.5 Nonlinearities, Mixed formulations, ...

5.9 `Resolution` examples

5.9.1 Static resolution (electrostatic problem)
5.9.2 Frequency domain resolution (magnetodynamic problem)
5.9.3 Time domain resolution (magnetodynamic problem)
5.9.4 Nonlinear time domain resolution (magnetodynamic problem)
5.9.5 Coupled formulations

5.10 `PostProcessing` examples
5.11 `PostOperation` examples

6. Complete examples

6.1 Electrostatic problem
6.2 Magnetostatic problem
6.3 Magnetodynamic problem

7. Running GetDP
8. File formats

8.1 Input file format
8.2 Output file format

8.2.1 File ``.pre'`
8.2.2 File ``.res'`

9. Bugs, versions and credits

9.1 Bugs
9.2 Versions
9.3 Credits

A. Tips and tricks
B. Frequently asked questions
C. Gmsh examples
D. License
Concept index
Metasyntactic variable index
Syntax index

[Top] | [Contents] | [Index] | [ ? ]

# Short Table of Contents

> Copying conditions
> Introduction
> 1. Overview
> 2. Expressions
> 3. Objects
> 4. Types for objects
> 5. Short examples
> 6. Complete examples
> 7. Running GetDP
> 8. File formats
> 9. Bugs, versions and credits
> A. Tips and tricks
> B. Frequently asked questions
> C. Gmsh examples
> D. License
> Concept index
> Metasyntactic variable index
> Syntax index

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# About this document

This document was generated by Charlie & on April, 7 2005 using texi2html

The buttons in the navigation panels have the following meaning:

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