GetDP 3.5.0

Table of Contents

GetDP

Patrick Dular and Christophe Geuzaine

GetDP is a general finite element solver that uses mixed finite elements to discretize de Rham-type complexes in one, two and three dimensions. This is the GetDP Reference Manual for GetDP 3.5.0 (May 13, 2022).

Obtaining GetDP

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

If you use GetDP, we would appreciate that you mention it in your work. References and the latest news about GetDP are always available on http://getdp.info.

Copying conditions

GetDP is “free software”; this means that everyone is free to use it and to redistribute it on a free basis. GetDP is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of GetDP that they might get from you.

Specifically, we want to make sure that you have the right to give away copies of GetDP, that you receive source code or else can get it if you want it, that you can change GetDP or use pieces of GetDP in new free programs, and that you know you can do these things.

To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of GetDP, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.

Also, for our own protection, we must make certain that everyone finds out that there is no warranty for GetDP. If GetDP is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.

The precise conditions of the license for GetDP are found in the General Public License that accompanies the source code (see License). Further information about this license is available from the GNU Project webpage http://www.gnu.org/copyleft/gpl-faq.html. Detailed copyright information can be found in Copyright and credits.

If you want to integrate parts of GetDP into a closed-source software, or want to sell a modified closed-source version of GetDP, you will need to obtain a different license. Please contact us directly for more information.

1 Overview

GetDP (a “General environment for the treatment of Discrete Problems”) is a scientific software environment for the numerical solution of integro-differential equations, open to the coupling of physical problems (electromagnetic, thermal, etc.) as well as of numerical methods (finite element method, integral methods, etc.). It can deal with such problems of various dimensions (1D, 2D or 3D) and time states (static, transient or harmonic).

The main feature of GetDP is the closeness between its internal structure (written in C), the organization of data defining discrete problems (written by the user in ASCII data files) and the symbolic mathematical expressions of these problems. Its aim is to be welcoming and of easy use for both development and application levels: it consists of a working environment in which the definition of any problem makes use of a limited number of objects, which makes the environment structured and concise. It therefore gives researchers advanced developing tools and a large freedom in adding new functionalities.

The modeling tools provided by GetDP can be tackled at various levels of complexity: this opens the software to a wide range of activities, such as research, collaboration, education, training and industrial studies.

1.1 Numerical tools as objects

An assembly of computational tools (or objects) in GetDP leads to a problem definition structure, which is a transcription of the mathematical expression of the problem, and forms a text data file: the equations describing a phenomenon, written in a mathematical form adapted to a chosen numerical method, directly constitute data for GetDP.

The resolution of a discrete problem with GetDP requires the definition, in a text data file, of the GetDP objects listed (together with their dependencies) in the following figure and table.

Group           ---
Function        Group
Constraint      Group, Function, (Resolution)
FunctionSpace   Group, Constraint, (Formulation), (Resolution)
Jacobian        Group
Integration     ---
Formulation     Group, Function, (Constraint), FunctionSpace,
                Jacobian, Integration
Resolution      Function, Formulation
PostProcessing  Group, Function, Jacobian, Integration,
                Formulation, Resolution
PostOperation   Group, PostProcessing

The gathering of all these objects constitutes the problem definition structure, which is a copy of the formal mathematical formulation of the problem. Reading the first column of the table from top to bottom pictures the working philosophy and the linking of operations peculiar to GetDP, from group definition to results visualization. The decomposition highlighted in the figure points out the separation between the objects defining the method of resolution, which may be isolated in a “black box” (bottom) and those defining the data peculiar to a given problem (top).

The computational tools which are in the center of a problem definition structure are formulations (Formulation) and function spaces (FunctionSpace). Formulations define systems of equations that have to be built and solved, while function spaces contain all the quantities, i.e., functions, fields of vectors or covectors, known or not, involved in formulations.

Each object of a problem definition structure must be defined before being referred to by others. A linking which always respects this property is the following: it first contains the objects defining particular data of a problem, such as geometry, physical characteristics and boundary conditions (i.e., Group, Function and Constraint) followed by those defining a resolution method, such as unknowns, equations and related objects (i.e., Jacobian, Integration, FunctionSpace, Formulation, Resolution and PostProcessing). The processing cycle ends with the presentation of the results (i.e., lists of numbers in various formats), defined in PostOperation fields. This decomposition points out the possibility of building black boxes, containing objects of the second group, adapted to treatment of general classes of problems that share the same resolution methods.

1.2 Which problems can GetDP actually solve?

The preceding explanations may seem very (too) general. Which are the problems that GetDP can actually solve? To answer this question, here is a list of methods that we have considered and coupled until now:

Numerical methods

finite element method
boundary element method (experimental, undocumented)
volume integral methods (experimental, undocumented)

Geometrical models

one-dimensional models (1D)
two-dimensional models (2D), plane and axisymmetric
three-dimensional models (3D)

Time states

static states
sinusoidal and harmonic states
transient states
eigenvalue problems

These methods have been successfully applied to build coupled physical models involving electromagnetic phenomena (magnetostatics, magnetodynamics, electrostatics, electrokinetics, electrodynamics, wave propagation, lumped electric circuits), acoustic phenomena, thermal phenomena and mechanical phenomena (elasticity, rigid body movement).

As can be guessed from the preceding list, GetDP has been initially developed in the field of computational electromagnetics, which fully uses all the offered coupling features. We believe that this does not interfere with the expected generality of the software because a particular modeling forms a problem definition structure which is totally external to the software: GetDP offers computational tools; the user freely applies them to define and solve his problem.

Nevertheless, specific numerical tools will always need to be implemented to solve specific problems in areas other than those mentionned above. If you think the general phisosophy of GetDP is right for you and your problem, but you discover that GetDP lacks the tools necessary to handle it, let us know: we would love to discuss it with you. For example, at the time of this writing, many areas of GetDP would need to be improved to make GetDP as useful for computational mechanics or computational fluid dynamics as it is for computational electromagnetics... So if you have the skills and some free time, feel free to join the project: we gladly accept all code contributions!

1.3 Bug reports

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

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

2 How to read this manual

After reading Overview, which depicts the general philosophy of GetDP, you might want to skip Expressions, Objects and Types for objects and directly run the demo files bundled in the distribution on your computer (see Running GetDP). You should then open these examples with a text editor and compare their structure with the examples given in Short examples and Complete examples. For each new syntax element that you fall onto, you can then go back to Expressions, Objects, and Types for objects, and find in these chapters the detailed description of the syntactic rules as well as all the available options.

Indexes for many concepts (see Concept index) and for all the syntax elements (see Syntax index) are available at the end of this manual.

2.1 Syntactic rules used in this document

Here are the rules we tried to follow when writing this user’s guide. Note that metasyntactic variable definitions stay valid throughout all the manual (and not only in the sections where the definitions appear). See Metasyntactic variable index, for an index of all metasyntactic variables.

1. Keywords and literal symbols are printed like this.
2. Metasyntactic variables (i.e., text bits that are not part of the syntax, but stand for other text bits) are printed like this.
3. A colon (:) after a metasyntactic variable separates the variable from its definition.
4. Optional rules are enclosed in < > pairs.
5. Multiple choices are separated by |.
6. Three dots (…) indicate a possible repetition of the preceding rule.
7. For conciseness, the notation rule <, rule > … is replaced by rule <,…>.
8. The etc symbol replaces nonlisted rules.

3 Running GetDP

GetDP has no graphical interface¹. It is a command-line driven program that reads a problem definition file once at the beginning of the processing. This problem definition file is a regular ASCII text file (see Numerical tools as objects), hence created with whatever text editor you like.

If you just type the program name at your shell prompt (without any argument), you will get a short help on how to run GetDP. All GetDP calls look like

getdp filename options

where filename is the ASCII file containing the problem definition, i.e., the structures this user’s guide has taught you to create. This file can include other files (see Includes), so that only one problem definition file should always be given on the command line. The input files containing the problem definition structure are usually given the .pro extension (if so, there is no need to specify the extension on the command line). The name of this file (without the extension) is used as a basis for the creation of intermediate files during the pre-processing and the processing stages.

The options are a combination of the following commands (in any order):

-pre

resolution-id

Performs the pre-processing associated with the resolution resolution-id. In the pre-processing stage, GetDP creates the geometric database (from the mesh file), identifies the degrees of freedom (the unknowns) of the problem and sets up the constraints on these degrees of freedom. The pre-processing creates a file with a .pre extension. If resolution-id is omitted, the list of available choices is displayed.

-cal

Performs the processing. This requires that a pre-processing has been performed previously, or that a -pre option is given on the same command line. The performed resolution is the one given as an argument to the -pre option. In the processing stage, GetDP executes all the commands given in the Operation field of the selected Resolution object (such as matrix assemblies, system resolutions, …).

-pos

post-operation-id …

Performs the operations in the PostOperation(s) selected by the post-operation-id(s). This requires that a processing has been performed previously, or that a -cal option is given on the same command line. If post-operation-id is omitted, the list of available choices is displayed.

-msh

filename

Reads the mesh (in .msh format) from filename (see File formats) rather than from the default problem file name (with the .msh extension appended).

-msh_scaling

value

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

-gmshread

filename …

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

-split

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

-res

filename …

Loads processing results from file(s).

-name

string

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

-restart

Restarts processing of a time stepping resolution interrupted before being complete.

-solve

resolution-id

Same as -pre resolution-id -cal.

-solver

filename

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

-slepc

Uses SLEPc instead of Arpack as eigensolver.

-adapt

file

Reads adaptation constraints from file.

-order

real

Specifies the maximum interpolation order.

-cache

Caches network computations to disk.

-bin

Selects binary format for output files.

-v2

Creates mesh-based Gmsh output files when possible.

-check

Lets you check the problem structure interactively.

-v

-verbose

integer

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

-cpu

Reports CPU times for all operations.

-p

-progress

integer

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

-onelab

name <address>

Communicates with OneLab (file or server address)

-setnumber

name value

Sets constant number name to value

-setstring

name value

Sets constant string name to value

-info

Displays the version information.

-version

Displays the version number.

-help

Displays a message listing basic usage and available options.

4 Expressions

This chapter and the next two describe in a rather formal way all the commands that can be used in the ASCII text input files. If you are just beginning to use GetDP, or just want to see what GetDP is all about, you should skip this chapter and the next two for now, have a quick look at Running GetDP, and run the demo problems bundled in the distribution on your computer. You should then open the .pro files in a text editor and compare their structure with the examples given in Short examples and Complete examples. Once you have a general idea of how the files are organized, you might want to come back here to learn more about the specific syntax of all the objects, and all the available options.

4.1 Comments

Both C and C++ style comments are supported and can be used in the input data files to comment selected text regions:

1. the text region comprised between /* and */ pairs is ignored;
2. the rest of a line after a double slash // is ignored.

Comments cannot be used inside double quotes or inside GetDP keywords.

4.2 Includes

An input data file can be included in another input data file by placing one of the following commands (expression-char represents a file name) on a separate line, outside the GetDP objects. Any text placed after an include command on the same line is ignored.

Include expression-char
#include expression-char

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

4.3 Expressions definition

Expressions are the basic tool of GetDP. They cover a wide range of functional expressions, from constants to formal expressions containing functions (built-in or user-defined, depending on space and time, etc.), arguments, discrete quantities and their associated differential operators, etc. Note that ‘white space’ (spaces, tabs, new line characters) is ignored inside expressions (as well as inside all GetDP objects).

Expressions are denoted by the metasyntactic variable expression (remember the definition of the syntactic rules in Syntactic rules):

expression:
  ( expression ) |
  integer |
  real |
  constant-id |
  quantity |
  argument |
  current-value |
  variable-set |
  variable-get |
  register-set |
  register-get |
  operator-unary expression |
  expression operator-binary expression |
  expression operator-ternary-left expression operator-ternary-right expression |
  built-in-function-id [ < expression-list > ] < { expression-cst-list } > |
  function-id [ < expression-list > ] |
  < Real | Complex > [ expression ] |
  Dt [ expression ] |
  AtAnteriorTimeStep [ expression, integer ] |
  Order [ quantity ] |
  Trace [ expression, group-id ] |
  expression ##integer

The following sections introduce the quantities that can appear in expressions, i.e., constant terminals (integer, real) and constant expression identifiers (constant-id, expression-cst-list), discretized fields (quantity), arguments (argument), current values (current-value), register values (register-set, register-get), operators (operator-unary, operator-binary, operator-ternary-left, operator-ternary-right) and built-in or user-defined functions (built-in-function-id, function-id). The last seven cases in this definition permit to cast an expression as real or complex, get the time derivative or evaluate an expression at an anterior time step, retrieve the interpolation order of a discretized quantity, evaluate the trace of an expression, and print the value of an expression for debugging purposes.

List of expressions are defined as:

expression-list:
  expression <,…>

4.4 Constants

The three constant types used in GetDP are integer, real and string. These types have the same meaning and syntax as in the C or C++ programming languages. Besides general expressions (expression), purely constant expressions, denoted by the metasyntactic variable expression-cst, are also used:

expression-cst:
  ( expression-cst ) |
  integer |
  real |
  constant-id |
  operator-unary expression-cst |
  expression-cst operator-binary expression-cst |
  expression-cst operator-ternary-left expression-cst operator-ternary-right
      expression-cst |
  math-function-id [ < expression-cst-list > ] |
  #constant-id() |
  constant-id(expression-cst) |
  StrFind[ expression-char, expression-char ] |
  StrCmp[ expression-char, expression-char ] |
  StrLen[ expression-char ] |
  StringToName[ expression-char ] | S2N[ expression-char ] |
  Exists[ string ] | FileExists[ string ] | GroupExists[ string ] |
  GetForced[ string ] | NbrRegions [ string ] |
  GetNumber[ expression-char <, expression-cst> ]

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

List of constant expressions are defined as:

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

with

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

The second case in this last definition permits to create a list containing the range of numbers comprised between the two expression-cst, with a unit incrementation step. The third case also permits to create a list containing the range of numbers comprised between the two expression-cst, but with a positive or negative incrementation step equal to the third expression-cst. The fourth and fifth cases permit to reference constant identifiers (constant-ids) of lists of constants and constant identifiers of sublists of constants (see below for the definition of constant identifiers) . The sixth case is a synonym for the fourth. The seventh case permits to create alternate lists: the arguments of ListAlt must be constant-ids of lists of constants of the same dimension. The result is an alternate list of these constants: first constant of argument 1, first constant of argument 2, second constant of argument 1, etc. These kinds of lists of constants are for example often used for function parameters (see Functions). The next two cases permit to create linear and logarithmic lists of numbers, respectively. The remaining cases permit to apply arithmetic operators item-wise in lists. ListFromFile reads a list of numbers from a file. ListFromServer attemps to get a list of numbers from the ONELAB variable expression-char.

Contrary to a general expression which is evaluated at runtime (thanks to an internal stack mechanism), an expression-cst is completely evaluated during the syntactic analysis of the problem (when GetDP reads the .pro file). The definition of such constants or lists of constants with identifiers can be made outside or inside any GetDP object. The syntax for the definition of constants is:

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

with

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

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

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

string-def:
  "string" |
  StrCat[ expression-char <,…> ] |
  Str[ expression-char <,…> ]

Notes:

1. Five constants are predefined in GetDP: Pi (3.1415926535897932), 0D (0), 1D (1), 2D (2) and 3D (3).
2. When ~{expression-cst} is appended to a string string, the result is a new string formed by the concatenation of string, _ (an underscore) and the value of the expression-cst. This is most useful in loops (see Macros loops and conditionals), where it permits to define unique strings automatically. For example,

   For i In {1:3}
     x~{i} = i;
   EndFor
   

   is the same as

   x_1 = 1;
   x_2 = 2;
   x_3 = 3;
   

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

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

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

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

Character expressions are defined as follows:

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

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

List of character expressions are defined as:

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

4.5 Operators

4.5.1 Operator types

The operators in GetDP are similar to the corresponding operators in the C or C++ programming languages.

operator-unary:

-

Unary minus.

!

Logical not.

operator-binary:

^

Exponentiation. The evaluation of the both arguments must result in a scalar value.

*

Multiplication or scalar product, depending on the type of the arguments.

/\

Cross product. The evaluation of both arguments must result in vectors.

/

Division.

%

Modulo. The evaluation of the second argument must result in a scalar value.

+

Addition.

-

Subtraction.

==

Equality.

!=

Inequality.

>

Greater. The evaluation of both arguments must result in scalar values.

>=

Greater or equality. The evaluation of both arguments must result in scalar values.

<

Less. The evaluation of both arguments must result in scalar values.

<=

Less or equality. The evaluation of both arguments must result in scalar values.

&&

Logical ‘and’. The evaluation of both arguments must result in scalar values.

||

Logical ‘or’. The evaluation of both arguments must result in floating point values. Warning: the logical ‘or’ always (unlike in C or C++) implies the evaluation of both arguments. That is, the second operand of || is evaluated even if the first one is true.

&

Binary ‘and’.

|

Binary ‘or’.

>>

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

<<

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

operator-ternary-left:

?

operator-ternary-right:

:

The only ternary operator, formed by operator-ternary-left and operator-ternary-right is defined as in the C or C++ programming languages. The ternary operator first evaluates its first argument (the expression-cst located before the ?), which must result in a scalar value. If it is true (non-zero) the second argument (located between ? and :) is evaluated and returned; otherwise the third argument (located after :) is evaluated and returned.

4.5.2 Evaluation order

The evaluation priorities are summarized below (from stronger to weaker, i.e., ^ has the highest evaluation priority). Parentheses () may be used anywhere to change the order of evaluation.

^

- (unary), !

| &

/\

*, /, %

+, -

<, >, <=, >=, <<, >>

!=, ==

&&, ||

?:

4.6 Functions

Two types of functions coexist in GetDP: user-defined functions (function-id, see Function) and built-in functions (built-in-function-id, defined in this section).

Both types of functions are always followed by a pair of brackets [] that can possibly contain arguments (see Arguments). This makes it simple to distinguish a function-id or a built-in-function-id from a constant-id. As shown below, built-in functions might also have parameters, given between braces {}, and which are completely evaluated during the analysis of the syntax (since they are of expression-cst-list type):

built-in-function-id [ < expression-list > ] < { expression-cst-list } >

with

built-in-function-id:
  math-function-id |
  extended-math-function-id |
  green-function-id |
  type-function-id |
  coord-function-id |
  misc-function-id

Notes:

1. All possible values for built-in-function-id are listed in Types for Function.
2. Classical mathematical functions (see Math functions) are the only functions allowed in a constant definition (see the definition of expression-cst in Constants).

4.7 Current values

Current values return the current floating point value of an internal GetDP variable:

$Time

Value of the current time. This value is set to zero for non time dependent analyses.

$DTime

Value of the current time increment used in a time stepping algorithm.

$Theta

Current theta value in a theta time stepping algorithm.

$TimeStep

Number of the current time step in a time stepping algorithm.

$Breakpoint

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

$Iteration, $NLIteration

Current iteration in a nonlinear loop.

$Residual, $NLResidual

Current residual in a nonlinear loop.

$EigenvalueReal

Real part of the current eigenvalue.

$EigenvalueImag

Imaginary part of the current eigenvalue.

$X, $XS

Value of the current (destination or source) X-coordinate.

$Y, $YS

Value of the current (destination or source) Y-coordinate.

$Z, $ZS

Value of the current (destination or source) Z-coordinate.

$A, $B, $C

Value of the current parametric coordinates used in the parametric OnGrid PostOperation (see Types for PostOperation).

$QuadraturePointIndex

Index of the current quadrature point.

$KSPIteration

Current iteration in a Krylov subspace solver.

$KSPResidual

Current residual in a Krylov subspace solver.

$KSPIterations

Total number of iterations of Krylov subspace solver.

$KSPSystemSize

System size of Krylov subspace solver.

Note:

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

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

4.8 Arguments

Function arguments can be used in expressions and have the following syntax (integer indicates the position of the argument in the expression-list of the function, starting from 1):

argument:
  $integer

See Function, and Function examples, for more details.

4.9 Run-time variables and registers

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

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

Run-time variables have the following syntax:

variable-set:
  $variable-id = expression

variable-get:
  $variable-id

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

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

Registers have the following syntax:

register-set:
  expression#expression-cst

register-get:
  #expression-cst

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

See Function examples, for an example.

4.10 Fields

A discretized quantity (defined in a function space, cf. FunctionSpace) is represented between braces {}, and can only appear in well-defined expressions in Formulation (see Formulation) and PostProcessing (see PostProcessing) objects:

quantity:
  < quantity-dof > { < quantity-operator > quantity-id } |
  { < quantity-operator > quantity-id } [ expression-cst-list ]

with

quantity-id:
  string |
  string ~ { expression-cst }

and

quantity-dof:

Dof

Defines a vector of discrete quantities (vector of Degrees of freedom), to be used only in Equation terms of formulations to define (elementary) matrices. Roughly said, the Dof symbol in front of a discrete quantity indicates that this quantity is an unknown quantity, and should therefore not be considered as already computed.

An Equation term must be linear with respect to the Dof. Thus, for example, a nonlinear term like

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

must first be linearized; and while

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

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

Integral { [ f[] * (Dof{T} - 12) , {T} ]; … }

GetDP supports two linearization techniques. The first is functional iteration (or Picard method), where one simply plugs the value obtained at the previous iteration into the nonlinear equation (the previous value is known, and is accessed e.g. with {T} instead Dof{T}). The second is the Newton-Raphson iteration, where the Jacobian is specified with a JacNL equation term.

BF

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

quantity-operator:

d

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

Grad

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

Curl

Rot

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

Div

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

D1

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

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

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

D2

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

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

Notes:

1. While the operators Grad, Curl and Div can be applied to 0, 1 and 2-forms respectively, the exterior derivative operator d is usually preferred with such fields.
2. The second case permits to evaluate a discretized quantity at a certain position X, Y, Z (when expression-cst-list contains three items) or at a specific time, N time steps ago (when expression-cst-list contains a single item).

4.11 Macros, loops and conditionals

Macros are defined as follows:

Macro string | expression-char

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

Return

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

Macro ( expression-char , expression-char ) ;

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

Macros, loops and conditionals can be used in any of the following objects: Group, Function, Constraint (as well as in a contraint-case), FunctionSpace, Formulation (as well as in the quantity and equation defintions), Resolution (as well as resolution-term, system defintion and operations), PostProcessing (in the definition of the PostQuantities) and PostOperation (as well as in the operation list).

loop:

Call string | expression-char;

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

For ( expression-cst : expression-cst )

Iterates from the value of the first expression-cst to the value of the second expression-cst, with a unit incrementation step. At each iteration, the commands comprised between ‘For ( expression-cst : expression-cst )’ and the matching EndFor are executed.

For ( expression-cst : expression-cst : expression-cst )

Iterates from the value of the first expression-cst to the value of the second expression-cst, with a positive or negative incrementation step equal to the third expression-cst. At each iteration, the commands comprised between ‘For ( expression-cst : expression-cst : expression-cst )’ and the matching EndFor are executed.

For string In { expression-cst : expression-cst }

Iterates from the value of the first expression-cst to the value of the second expression-cst, with a unit incrementation step. At each iteration, the value of the iterate is affected to an expression named string, and the commands comprised between ‘For string In { expression-cst : expression-cst }’ and the matching EndFor are executed.

For string In { expression-cst : expression-cst : expression-cst }

Iterates from the value of the first expression-cst to the value of the second expression-cst, with a positive or negative incrementation step equal to the third expression-cst. At each iteration, the value of the iterate is affected to an expression named string, and the commands comprised between ‘For string In { expression-cst : expression-cst : expression-cst }’ and the matching EndFor are executed.

EndFor

Ends a matching For command.

If ( expression-cst )

The body enclosed between ‘If ( expression-cst )’ and the matching ElseIf, Else or EndIf, is evaluated if expression-cst is non-zero.

ElseIf ( expression-cst )

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

Else

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

EndIf

Ends a matching If command.

LevelTest

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

Parse [ expression-char ];

Parse the given string.

5 Objects

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

5.1 Group: defining topological entities

Meshes (grids) constitute the input data of GetDP. All that is needed by GetDP as a mesh is a file containing a list of nodes (with their coordinates) and a list of geometrical elements with, for each one, a number characterizing its geometrical type (i.e., line, triangle, quadrangle, tetrahedron, hexahedron, prism, etc.), a number characterizing the physical region to which it belongs and the list of its nodes. This minimal input set should be easy to extract from most of the classical mesh file formats (see Input file format, for a complete description of the mesh file format read by GetDP).

Groups of geometrical entities of various types can be considered and are used in many objects. There are region groups, of which the entities are regions, and function groups, with nodes, edges, facets, volumes, groups of nodes, edges of tree, facets of tree, … of regions.

Amongst region groups, elementary and global groups can be distinguished: elementary groups are relative to single regions (e.g., physical regions in which piecewise defined functions or constraints can be defined) while global groups are relative to sets of regions for which given treatments have to be performed (e.g., domain of integration, support of a function space, etc.).

Groups of function type contain lists of entities built on some region groups (e.g., nodes for nodal elements, edges for edge elements, edges of tree for gauge conditions, groups of nodes for floating potentials, elements on one side of a surface for cuts, etc.).

A definition of initially empty groups can be obtained thanks to a DefineGroup command, so that their identifiers exist and can be referred to in other objects, even if these groups are not explicitly defined. This procedure is similar to the DefineConstant procedure introduced for constants in Constants.

The syntax for the definition of groups is:

Group {
  < DefineGroup [ group-id <{integer}> <,…> ]; > …
  < group-id = group-def; > …
  < group-id += group-def; > …
  < group-id -= group-def; > …
  < affectation > …
  < loop > …
}

with

group-id:
  string |
  string ~ { expression-cst }

group-def:
  group-type [ group-list <, group-sub-type group-list > ] |
  group-id <{<integer>}> |
  #group-list

group-type:
  Region | Global | NodesOf | EdgesOf | etc

group-list:
  All | group-list-item | { group-list-item <,…> }

group-list-item:
  integer |
  integer : integer |
  integer : integer : integer |
  group-id <{<integer>}>

group-sub-type:
  Not | StartingOn | OnPositiveSideOf | etc

Notes:

1. integer as a group-list-item is the only interface with the mesh; with each element is associated a region number, being this integer, and a geometrical type (see Input file format). Ranges of integers can be specified in the same way as ranges of constant expressions in an expression-cst-list-item (see Constants). For example, i:j replaces the list of consecutive integers i, i+1, …, j-1, j.
2. Array of groups: DefineGroup[group-id{n}] defines the empty groups group-id{i}, i=1, …, n. Such a definition is optional, i.e., each group-id{i} can be separately defined, in any order.
3. #group-list is an abbreviation of Region[group-list].

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

5.2 Function: defining global and piecewise expressions

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

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

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

with

function-id:
  string

Note:

1. The first optional group-def in brackets must be of Region type, and indicates on which region the (piecewise) function is defined. The second optional group-def in brackets, also of Region type, defines an association with a second region for mutual contributions. A default piecewise function can be defined with All for group-def, for all the other non-defined regions. Warning: it is incorrect to write f[reg1]=1; g[reg2]=f[]+1; since the domains of definition of f[] and g[] don’t match.
2. One can also define initially empty functions inline by replacing the expression with ***.

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

5.3 Constraint: specifying constraints on function spaces and formulations

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

The syntax for the definition of constraints is:

Constraint {
  { < Append < expression-cst >; >
    Name constraint-id; Type constraint-type;
    Case {
      { Region group-def; < Type constraint-type; >
        < SubRegion group-def; > < TimeFunction expression; >
        < RegionRef group-def; > < SubRegionRef group-def; >
        < Coefficient expression; > < Function expression; >
        < Filter expression; >
        constraint-val; } …
      < loop > …
    }
  | Case constraint-case-id {
      { Region group-def; < Type constraint-type; >
        constraint-case-val; } …
      < loop > …
    } …
  } …
  < affectation > …
  < loop > …
}

with

constraint-id:
constraint-case-id:
  string |
  string ~ { expression-cst }

constraint-type:
  Assign | Init | Network | Link | etc

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

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

Notes:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive) permits to append an existing Constraint of the same Name with additional Cases.
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 FunctionSpaces. The optional region type SubRegion group-def will be the argument of the associated group-sub-type.
4. expression in Value of constraint-val cannot be time dependent ($Time) because it is evaluated only once during the pre-processing (for efficiency reasons). Time dependences must be defined in TimeFunction expression.

See Types for Constraint, for the complete list of options and Constraint examples, for some examples.

5.4 FunctionSpace: building function spaces

A FunctionSpace is characterized by the type of its interpolated fields, one or several basis functions and optional constraints (in space and time). Subspaces of a function space can be defined (e.g., for the use with hierarchical elements), as well as direct associations of global quantities (e.g., floating potential, electric charge, current, voltage, magnetomotive force, etc.).

A key point is that basis functions are defined by any number of subsets of functions, being added. Each subset is characterized by associated built-in functions for evaluation, a support of definition and a set of associated supporting geometrical entities (e.g., nodes, edges, facets, volumes, groups of nodes, edges incident to a node, etc.). The freedom in defining various kinds of basis functions associated with different geometrical entities to interpolate a field permits to build made-to-measure function spaces adapted to a wide variety of field approximations (see FunctionSpace examples).

The syntax for the definition of function spaces is:

FunctionSpace {
  { < Append < expression-cst >; >
    Name function-space-id;
    Type function-space-type;
    BasisFunction {
     { Name basis-function-id; NameOfCoef coef-id;
       Function basis-function-type
         < { Quantity quantity-id;
             Formulation formulation-id { expression-cst };
             Group group-def;
             Resolution resolution-id { expression-cst } } >;
       < dFunction { basis-function-type, basis-function-type } ; >
       Support group-def; Entity group-def; } …
    }
  < SubSpace {
     { < Append < expression-cst >; >
       Name sub-space-id;
       NameOfBasisFunction basis-function-list; } …
    } >
  < GlobalQuantity {
     { Name global-quantity-id; Type global-quantity-type;
       NameOfCoef coef-id; } …
    } >
  < Constraint {
     { NameOfCoef coef-id;
       EntityType Auto | group-type; < EntitySubType group-sub-type; >
       NameOfConstraint constraint-id <{}>; } …
    } >
  } …
  < affectation > …
  < loop > …
}

with

function-space-id:
formulation-id:
resolution-id:
  string |
  string ~ { expression-cst }

basis-function-id:
coef-id:
sub-space-id:
global-quantity-id:
  string

function-space-type:
  Scalar | Vector | Form0 | Form1 | etc

basis-function-type:
  BF_Node | BF_Edge | etc

basis-function-list:
  basis-function-id | { basis-function-id <,…> }

global-quantity-type:
  AliasOf | AssociatedWith

Notes:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive; its omission fixes it to a top value) permits to append an existing FunctionSpace of the same Name with additional BasisFunctions, SubSpaces, GlobalQuantity’s and Constraints, or an existing SubSpace of the same Name with additional NameOfBasisFunction’s. If the Append FunctionSpace level is 2, the Append SubSpace level is automatically 1 if omitted.
2. When the definition region of a function type group used as an Entity of a BasisFunction is the same as that of the associated Support, it is replaced by All for more efficient treatments during the computation process (this prevents the construction and the analysis of a list of geometrical entities).
3. The same Name for several BasisFunction fields permits to define piecewise basis functions; separate NameOfCoefs must be defined for those fields.
4. A constraint is associated with geometrical entities defined by an automatically created Group of type group-type (Auto automatically fixes it as the Entity group-def type of the related BasisFunction), using the Region defined in a Constraint object as its main argument, and the optional SubRegion in the same object as a group-sub-type argument.
5. A global basis function (BF_Global or BF_dGlobal) needs parameters, i.e., it is given by the quantity (quantity-id) pre-computed from multiresolutions performed on multiformulations.
6. Explicit derivatives of the basis functions can be specified using dFunction { basis-function-type , basis-function-type }. These derivates can be accessed using the special D1 and D2 operators (see Fields).

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

5.5 Jacobian: defining jacobian methods

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

The syntax for the definition of Jacobian methods is:

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

with

jacobian-id:
  string

jacobian-type:
  Vol | Sur | VolAxi | etc

Note:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive) permits to append an existing Jacobian of the same Name with additional Cases.
2. The default case of a Jacobian object is defined by Region All and must follow all the other cases.

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

5.6 Integration: defining integration methods

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

The syntax for the definition of integration methods is:

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

with

integration-id:
  string

integration-type:
  Gauss | etc

element-type:
  Line | Triangle | Tetrahedron etc

Note:

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

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

5.7 Formulation: building equations

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

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

The syntax for the definition of formulations is:

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

with

formulation-id:
  string |
  string ~ { expression-cst }

formulation-type:
  FemEquation | etc

local-term-type:
  Integral | deRham

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

quantity-type:
  Local | Global | Integral

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

Note:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive) permits to append an existing Formulation of the same Name with additional Quantity’s and Equations.
2. IndexOfSystem permits to resolve ambiguous cases when several quantities belong to the same function space, but to different systems of equations. The integer parameter then specifies the index in the list of an OriginSystem command (see Resolution).
3. A GlobalTerm defines a term to be assembled in an equation associated with a global quantity. This equation is a finite element equation if that global quantity is linked with local quantities. The optional associated SubType defines either self (default) or mutual contributions, or both. Mutual contributions need piecewise functions defined on pairs or regions.
4. A GlobalEquation defines a global equation to be assembled in the matrix of the system.

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

5.8 Resolution: solving systems of equations

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

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

Resolution {
  { < Append < expression-cst >; >
    Name resolution-id; < Hidden expression-cst; >
    System {
      { Name system-id; NameOfFormulation formulation-list;
        < Type system-type; >
        < Frequency expression-cst-list-item |
          Frequency { expression-cst-list }; >
        < DestinationSystem system-id; >
        < OriginSystem system-id; | OriginSystem { system-id <,…> }; >
        < NameOfMesh expression-char > < Solver expression-char >
        < loop > } …
      < loop > …
    }
    Operation {
      < resolution-op; > …
      < loop > …
    }
  } …
  < affectation > …
  < loop > …
}

with

resolution-id:
system-id:
  string |
  string ~ { expression-cst }

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

system-type:
  Real | Complex

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

Notes:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive) permits to append an existing Resolution of the same Name with additional Systems and Operations.
2. The default type for a system of equations is Real. A frequency domain analysis is defined through the definition of one or several frequencies (Frequency expression-cst-list-item | Frequency { expression-cst-list }). Complex systems of equations with no predefined list of frequencies (e.g., in modal analyses) can be explicitely defined with Type Complex.
3. NameOfMesh permits to explicitely specify the mesh to be used for the construction of the system of equations.
4. Solver permits to explicitely specify the name of the solver parameter file to use for the solving of the system of equations. This is ony valid if GetDP was compiled against the default solver library (it is the case if you downloaded a pre-compiled copy of GetDP from the internet).
5. DestinationSystem permits to specify the destination system of a TransferSolution operation (see Types for Resolution).
6. OriginSystem permits to specify the systems from which ambiguous quantity definitions can be solved (see Formulation).

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

5.9 PostProcessing: exploiting computational results

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

PostProcessing {
  { < Append < expression-cst >; >
    Name post-processing-id;
    NameOfFormulation formulation-id <{}>; < NameOfSystem system-id; >
    Quantity {
      { < Append < expression-cst >; >
        Name post-quantity-id; Value { post-value … } } …
      < loop > …
    }
  } …
  < affectation > …
  < loop > …
}

with

post-processing-id:
post-quantity-id:
  string |
  string ~ { expression-cst }

post-value:
  Local { local-value } | Integral { integral-value }

local-value:
  [ expression ]; In group-def; Jacobian jacobian-id;

integral-value:
  [ expression ]; In group-def;
  Integration integration-id; Jacobian jacobian-id;

Notes:

1. The optional Append < expression-cst > (when the optional level expression-cst is strictly positive; its omission fixes it to a top value) permits to append an existing PostProcessing of the same Name with additional Values, or an existing Quantity of the same Name with additional Quantity’s. If the Append PostProcessing level is 2, the Append Quantity level is automatically 1 if omitted. Fixing the Append Quantity level to -n suppresses the n lastly defined Quantity’s before appending.
2. The quantity defined with integral-value is piecewise defined over the elements of the mesh of group-def, and takes, in each element, the value of the integration of expression over this element. The global integral of expression over a whole region (being either group-def or a subset of group-def) has to be defined in the PostOperation with the post-quantity-id[group-def] command (see PostOperation).
3. If NameOfSystem system-id is not given, the system is automatically selected as the one to which the first quantity listed in the Quantity field of formulation-id is associated.

See Types for PostProcessing, for the complete list of options and PostProcessing examples, for some examples.

5.10 PostOperation: exporting results

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

PostOperation {
  { < Append < expression-cst >; >
    Name post-operation-id; NameOfPostProcessing post-processing-id;
    < Hidden expression-cst; >
    < Format post-operation-fmt; >
    < TimeValue expression-cst-list; > < TimeImagValue expression-cst-list; >
    < LastTimeStepOnly < expression-cst >; >
    < OverrideTimeStepValue expression-cst; >
    < NoMesh expression-cst; > < AppendToExistingFile expression-cst; >
    < ResampleTime[expression-cst, expression-cst, expression-cst]; >
    < AppendTimeStepToFileName < expression-cst >; >
    Operation {
      < post-operation-op; > …
    }
  } …
  < affectation > …
  < loop > …
} |
PostOperation < (Append < expression-cst >) > post-operation-id UsingPost post-processing-id {
  < post-operation-op; > …
} …

with

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

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

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

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

post-operation-fmt:
  Table | TimeTable | etc

Notes:

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

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

6 Types for objects

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

6.1 Types for Group

Types in

group-type [ R1 <, group-sub-type R2 <, group-sub-type-2 R3 > > ]

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

Region

Regions in R1.

Global

Regions in R1 (variant of Region used with global BasisFunctions BF_Global and BF_dGlobal).

NodesOf

Nodes of elements of R1

< Not: but not those of R2 >.

EdgesOf

Edges of elements of R1

< Not: but not those of R2 >.

FacetsOf

Facets of elements of R1

< Not: but not those of R2 >.

VolumesOf

Volumes of elements of R1

< Not: but not those of R2 >.

ElementsOf

Elements of regions in R1

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

GroupsOfNodesOf

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

GroupsOfEdgesOf

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

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

GroupsOfEdgesOnNodesOf

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

< Not: but not those of R2) >.

GroupOfRegionsOf

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

EdgesOfTreeIn

Edges of a tree of edges of R1

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

FacetsOfTreeIn

Facets of a tree of facets of R1

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

DualNodesOf

Dual nodes of elements of R1.

DualEdgesOf

Dual edges of elements of R1.

DualFacetsOf

Dual facets of elements of R1.

DualVolumesOf

Dual volumes of elements of R1.

6.2 Types for Function

6.2.1 Math functions

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

math-function-id:

Exp

[expression]

Exponential function: e^expression.

Log

[expression]

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

Log10

[expression]

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

Sqrt

[expression]

Square root, expression>=0.

Sin

[expression]

Sine of expression.

Asin

[expression]

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

Cos

[expression]

Cosine of expression.

Acos

[expression]

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

Tan

[expression]

Tangent of expression.

Atan

[expression]

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

Atan2

[expression,expression]

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

Sinh

[expression]

Hyperbolic sine of expression.

Cosh

[expression]

Hyperbolic cosine of expression.

Tanh

[expression]

Hyperbolic tangent of the real valued expression.

TanhC2

[expression]

Hyperbolic tangent of a complex valued expression.

Fabs

[expression]

Absolute value of expression (real valued only).

Abs

[expression]

Absolute value of expression.

Floor

[expression]

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

Ceil

[expression]

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

Fmod

[expression,expression]

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

Min

[expression,expression]

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

Max

[expression,expression]

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

Sign

[expression]

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

Jn

[expression]

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

dJn

[expression]

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

Yn

[expression]

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

dYn

[expression]

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

6.2.2 Extended math functions

extended-math-function-id:

Cross

[expression,expression]

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

Hypot

[expression,expression]

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

Norm

[expression]

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

SquNorm

[expression]

Square norm: Norm[expression]^2.

Unit

[expression]

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

Transpose

[expression]

Transposition; expression must be a tensor.

Inv

[expression]

Inverse of the tensor expression.

Det

[expression]

Determinant of the tensor expression.

Rotate

[expression,expression,expression,expression]

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

TTrace

[expression]

Trace; expression must be a tensor.

Cos_wt_p

[]{expression-cst,expression-cst}

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

Sin_wt_p

[]{expression-cst,expression-cst}

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

Period

[expression]{expression-cst}

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

Interval

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

Not documented yet.

6.2.3 Green functions

The Green functions are only used in integral quantities (see Formulation). The first parameter represents the dimension of the problem:

• 1D: r = Fabs[$X-$XS]
• 2D: r = Sqrt[($X-$XS)^2+($Y-$YS)^2]
• 3D: r = Sqrt[($X-$XS)^2+($Y-$YS)^2+($Z-$ZS)^2]

The triplets of values given in the definitions below correspond to the 1D, 2D and 3D cases.

green-function-id:

Laplace

[]{expression-cst}

r/2, 1/(2*Pi)*ln(1/r), 1/(4*Pi*r).

GradLaplace

[]{expression-cst}

Gradient of Laplace relative to the destination point ($X, $Y, $Z).

Helmholtz

[]{expression-cst, expression-cst}

exp(j*k0*r)/(4*Pi*r), where k0 is given by the second parameter.

GradHelmholtz

[]{expression-cst, expression-cst}

Gradient of Helmholtz relative to the destination point ($X, $Y, $Z).

6.2.4 Type manipulation functions

type-function-id:

Complex

[expression-list]

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

Complex_MH

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

Not documented yet.

Re

[expression]

Takes the real part of a complex-valued expression.

Im

[expression]

Takes the imaginary part of a complex-valued expression.

Conj

[expression]

Computes the conjugate of a complex-valued expression.

Cart2Pol

[expression]

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

Vector

[expression,expression,expression]

Creates a vector from 3 scalars.

Tensor

[expression,expression,expression,expression,expression,expression,

expression,expression,expression]

Creates a second-rank tensor of order 3 from 9 scalars, by row:

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

TensorV

[expression,expression,expression]

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

    [ vector0 ]
T = [ vector1 ]
    [ vector2 ]

TensorSym

[expression,expression,expression,expression,expression,expression]

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

TensorDiag

[expression,expression,expression]

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

SquDyadicProduct

[expression]

Dyadic product of the vector given by expression with itself.

CompX

[expression]

Gets the X component of a vector.

CompY

[expression]

Gets the Y component of a vector.

CompZ

[expression]

Gets the Z component of a vector.

CompXX

[expression]

Gets the XX component of a tensor.

CompXY

[expression]

Gets the XY component of a tensor.

CompXZ

[expression]

Gets the XZ component of a tensor.

CompYX

[expression]

Gets the YX component of a tensor.

CompYY

[expression]

Gets the YY component of a tensor.

CompYZ

[expression]

Gets the YZ component of a tensor.

CompZX

[expression]

Gets the ZX component of a tensor.

CompZY

[expression]

Gets the ZY component of a tensor.

CompZZ

[expression]

Gets the ZZ component of a tensor.

Cart2Sph

[expression]

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

Cart2Cyl

[expression]

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

UnitVectorX

[]

Creates a unit vector in x-direction.

UnitVectorY

[]

Creates a unit vector in y-direction.

UnitVectorZ

[]

Creates a unit vector in z-direction.

6.2.5 Coordinate functions

coord-function-id:

X

[]

Gets the X coordinate.

Y

[]

Gets the Y coordinate.

Z

[]

Gets the Z coordinate.

XYZ

[]

Gets X, Y and Z in a vector.

6.2.6 Miscellaneous functions

misc-function-id:

Printf

[expression]

Prints the value of expression when evaluated. (MPI_Printf can be use instead, to print the message for all MPI ranks.)

Rand

[expression]

Returns a pseudo-random number in [0, expression].

Normal

[]

Computes the normal to the element.

NormalSource

[]

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

Tangent

[]

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

TangentSource

[]

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

ElementVol

[]

Computes the element’s volume.

SurfaceArea

[]{expression-cst-list}

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

GetVolume

[]

Computes the volume of the actual physical group.

CompElementNum

[]

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

GetNumElements

[]{expression-cst-list}

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

ElementNum

[]

Returns the tag (number) of the current element.

QuadraturePointIndex

[]

Returns the index of the current quadrature point.

AtIndex

[expression]{expression-cst-list}

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

InterpolationLinear

[expression]{expression-cst-list}

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

dInterpolationLinear

[expression]{expression-cst-list}

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

InterpolationBilinear

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

Bilinear interpolation of a table based on two variables.

dInterpolationBilinear

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

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

InterpolationAkima

[expression]{expression-cst-list}

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

dInterpolationAkima

[expression]{expression-cst-list}

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

Order

[quantity]

Returns the interpolation order of the quantity.

Field

[expression]

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

Field

[expression]{expression-cst-list}

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

ScalarField

[expression]{expression-cst-list}

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

VectorField

[expression]{expression-cst-list}

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

TensorField

[expression]{expression-cst-list}

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

ComplexScalarField

[expression]{expression-cst-list}

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

ComplexVectorField

[expression]{expression-cst-list}

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

ComplexTensorField

[expression]{expression-cst-list}

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

Distance

[expression]{expression-cst}

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

GetCpuTime

[]

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

GetWallClockTime

[]

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

GetMemory

[]

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

SetNumberRunTime

[expression]{char-expression}

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

SetNumberRunTimeWithChoices

[expression]{char-expression}

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

GetNumberRunTime

[ <expression> ]{char-expression}

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

SetVariable

[ expression <,…> ]{ $variable-id }

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

GetVariable

[ <expression> <,…> ]{ $variable-id }

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

ValueFromIndex

[ ]{ expression-cst-list }

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

VectorFromIndex

[ ]{ expression-cst-list }

Same ValueFromIndex, but with 3 scalar values per entity.

ValueFromTable

[ expression ]{ char-expression }

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

6.3 Types for Constraint

constraint-type:

Assign

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

Init

To give an initial value (e.g., initial value in a time domain analysis). If two values are provided (with Value [ expression, expression ]), the first value can be used using the InitSolution1 operation. This is mainly useful for the Newmark time-stepping scheme.

AssignFromResolution

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

InitFromResolution

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

Network

To describe the node connections of branches in a network.

Link

To define links between degrees of freedom in the constrained region with degrees of freedom in a “reference” region, with some coefficient. For example, to link the degrees of freedom in the contrained region Left with the degrees of freedom in the reference region Right, located Pi units to the right of the region Left along the X-axis, with the coeficient -1, one could write:

{ Name periodic;
  Case {
    { Region Left; Type Link ; RegionRef Right;
      Coefficient -1; Function Vector[X[]+Pi, Y[], Z[]] ;
      < FunctionRef XYZ[]; >
    }
  }
}

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

LinkCplx

To define complex-valued links between degrees of freedom. The syntax is the same as for constraints of type Link, but Coeficient can be complex.

6.4 Types for FunctionSpace

function-space-type:

Form0

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

Form1

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

Form2

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

Form3

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

Form1P

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

Form2P

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

Scalar

Scalar field.

Vector

Vector field.

basis-function-type:

BF_Node

Nodal function (on NodesOf, value Form0).

BF_Edge

Edge function (on EdgesOf, value Form1).

BF_Facet

Facet function (on FacetsOf, value Form2).

BF_Volume

Volume function (on VolumesOf, value Form3).

BF_GradNode

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

BF_CurlEdge

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

BF_DivFacet

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

BF_GroupOfNodes

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

BF_GradGroupOfNodes

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

BF_GroupOfEdges

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

BF_CurlGroupOfEdges

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

BF_PerpendicularEdge

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

BF_CurlPerpendicularEdge

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

BF_GroupOfPerpendicularEdge

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

BF_CurlGroupOfPerpendicularEdge

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

BF_PerpendicularFacet

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

BF_DivPerpendicularFacet

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

BF_Region

Unit value 1 (on Region or GroupOfRegionsOf, value Scalar).

BF_RegionX

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

BF_RegionY

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

BF_RegionZ

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

BF_Global

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

BF_dGlobal

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

BF_NodeX

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

BF_NodeY

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

BF_NodeZ

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

BF_Zero

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

BF_One

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

global-quantity-type:

AliasOf

Another name for a name of coefficient of basis function.

AssociatedWith

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

6.5 Types for Jacobian

jacobian-type:

Vol

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

Sur

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

Lin

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

VolAxi

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

SurAxi

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

VolAxiSqu

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

VolSphShell

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

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

VolCylShell

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

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

VolAxiSphShell

Same as VolAxi, but with spherical shell transformation.

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

VolAxiSquSphShell

Same as VolAxiSqu, but with spherical shell transformation.

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

VolRectShell

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

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

VolAxiRectShell

Same as VolAxi, but with rectangular shell transformation.

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

VolAxiSquRectShell

Same as VolAxiSqu, but with rectangular shell transformation.

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

6.6 Types for Integration

integration-type:

Gauss

Numerical Gauss integration.

GaussLegendre

Numerical Gauss integration obtained by application of a multiplicative rule on the one-dimensional Gauss integration.

element-type:

Line

Line (2 nodes, 1 edge, 1 volume) (#1).

Triangle

Triangle (3 nodes, 3 edges, 1 facet, 1 volume) (#2).

Quadrangle

Quadrangle (4 nodes, 4 edges, 1 facet, 1 volume) (#3).

Tetrahedron

Tetrahedron (4 nodes, 6 edges, 4 facets, 1 volume) (#4).

Hexahedron

Hexahedron (8 nodes, 12 edges, 6 facets, 1 volume) (#5).

Prism

Prism (6 nodes, 9 edges, 5 facets, 1 volume) (#6).

Pyramid

Pyramid (5 nodes, 8 edges, 5 facets, 1 volume) (#7).

Point

Point (1 node) (#15).

Note:

1. n in (#n) is the type number of the element (see Input file format).

6.7 Types for Formulation

formulation-type:

FemEquation

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

local-term-type:

Integral

Integral of Galerkin or Petrov-Galerkin type.

deRham

deRham projection (collocation).

quantity-type:

Local

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

Global

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

Integral

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

term-op-type:

Dt

Time derivative applied to the whole term of the equation. (Not implemented yet.)

DtDof

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

DtDt

Time derivative of 2nd order applied to the whole term of the equation. (Not implemented yet.)

DtDtDof

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

Eig

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

JacNL

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

DtDofJacNL

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

NeverDt

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

6.8 Types for Resolution

resolution-op:

Generate

[system-id]

Generate the system of equations system-id.

Solve

[system-id]

Solve the system of equations system-id.

SolveAgain

[system-id]

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

SetGlobalSolverOptions

[char-expression]

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

GenerateJac

[system-id]

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

SolveJac

[system-id]

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

GenerateSeparate

[system-id]

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

GenerateOnly

[system-id, expression-cst-list]

Not documented yet.

GenerateOnlyJac

[system-id, expression-cst-list]

Not documented yet.

GenerateGroup

Not documented yet.

GenerateRightHandSideGroup

Not documented yet.

Update

[system-id]

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

Update

[system-id, expression]

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

UpdateConstraint

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

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

GetResidual

[system-id, $variable-id]

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

GetNormSolution | GetNormRightHandSide | GetNormResidual | GetNormIncrement

[system-id, $variable-id]

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

SwapSolutionAndResidual

[system-id]

Swap the solution x and residual r vectors.

SwapSolutionAndRightHandSide

[system-id]

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

InitSolution

[system-id]

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

InitSolution1

[system-id]

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

CreateSolution

[system-id]

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

CreateSolution

[system-id, expression-cst]

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

Apply

[system-id]

x <- Ax

SetSolutionAsRightHandSide

[system-id]

b <- x

SetRightHandSideAsSolution

[system-id]

x <- b

Residual

[system-id]

res <- b - Ax

CopySolution

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

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

CopySolution

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

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

CopyRightHandSide

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

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

CopyRightHandSide

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

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

CopyResidual

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

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

CopyResidual

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

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

SaveSolution

[system-id]

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

SaveSolutions

[system-id]

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

RemoveLastSolution

[system-id]

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

TransferSolution

[system-id]

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

Evaluate

[expression <, expression>]

Evaluate the expression(s).

SetTime

[expression]

Change the current time.

SetTimeStep

[expression]

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

SetDTime

[expression]

Change the current time step value (dt).

SetFrequency

[system-id, expression]

Change the frequency of system system-id.

SystemCommand

[expression-char]

Execute the system command given by expression-char.

Error

[expression-char]

Output error message expression-char.

Test

[expression] { resolution-op }

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

Test

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

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

While

[expression] { resolution-op }

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

Break

[]

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

Exit

[]

Exit, brutally.

Sleep

[expression]

Sleeps for expression seconds;

SetExtrapolationOrder

[expression-cst]

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

Print

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

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

Print

[ system-id <, File expression-char > <, { expression-cst-list } >
<, TimeStep { expression-cst-list } >]

Print the system system-id. If the expression-cst-list is given, print only the values of the degrees of freedom given in that list. If the TimeStep option is present, limit the printing to the selected time steps.

EigenSolve

[system-id, expression-cst, expression-cst, expression-cst < , expression > ]

Eigenvalue/eigenvector computation using Arpack or SLEPc. The parameters are: the system (which has to be generated with GenerateSeparate[]), the number of eigenvalues/eigenvectors to compute and the real and imaginary spectral shift (around which to look for eigenvalues). The last optional argument allows to filter which eigenvalue/eigenvector pairs will be saved. For example, ($EigenvalueReal > 0) would only keep pairs corresponding to eigenvalues with a striclty positive real part.

Lanczos

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

Eigenvalue/eigenvector computation using the Lanczos algorithm. The parameters are: the system (which has to be generated with GenerateSeparate[]), the size of the Lanczos space, the indices of the eigenvalues/eigenvectors to store, the spectral shift. This routine is deprecated: use EigenSolve instead.

FourierTransform

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

On-the-fly (incremental) computation of a Fourier transform. The parameters are: the (time domain) system, the destination system in which the result of the Fourier tranform is to be saved (it should be declared with Type Complex) and the list of frequencies to consider. The computation is an approximation that assumes that the time step is constant; it is not an actual Discrete Fourier Transform (the number of samples is unknown a priori).

TimeLoopTheta

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

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

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

TimeLoopNewmark

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

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

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

TimeLoopAdaptive

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

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

IterativeLoop

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

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

IterativeLoopN

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

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

IterativeLinearSolver

Generic iterative linear solver. To be documented.

PostOperation

[post-operation-id]

Perform the specified PostOperation.

GmshRead

[expression-char]

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

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

GmshRead

[expression-char, expression-cst]

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

GmshRead

[expression-char, $string]

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

GmshWrite

[expression-char, expression-cst]

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

GmshClearAll

[]

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

DeleteFile

[expression-char]

Delete a file.

RenameFile

[expression-char, expression-char]

Rename a file.

CreateDir | CreateDirectory

[expression-char]

Create a directory.

MPI_SetCommSelf

[]

Changes MPI communicator to self.

MPI_SetCommWorld

[]

Changes MPI communicator to world.

MPI_Barrier

[]

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

MPI_BroadcastFields

[ < expression-list > ]

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

MPI_BroadcastVariables

[]

Broadcast all runtime variables over MPI.

6.9 Types for PostProcessing

post-value:

Local

{ local-value }

To compute a local quantity.

Integral

{ integral-value }

To integrate the expression over each element.

6.10 Types for PostOperation

print-support:

OnElementsOf

group-def

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

OnRegion

group-def

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

OnGlobal

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

OnSection

{ { expression-cst-list } { expression-cst-list } { expression-cst-list } }

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

OnGrid

group-def

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

OnGrid

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

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

OnPoint

{ expression-cst-list }

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

OnLine

{ { expression-cst-list } { expression-cst-list } } { expression-cst }

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

OnPlane

{ { expression-cst-list } { expression-cst-list } { expression-cst-list } }
{ expression-cst, expression-cst }

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

OnBox

{ { expression-cst-list } { expression-cst-list } { expression-cst-list }
{ expression-cst-list } } { expression-cst, expression-cst, expression-cst }

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

print-option:

File

expression-char

Outputs the result in a file named expression-char.

File

> expression-char

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

File

>> expression-char

Appends the result to a file named expression-char.

AppendToExistingFile

expression-cst

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

Name | Label

expression-char

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

Depth

expression-cst

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

AtGaussPoints

expression-cst

Print result at the specified number of Gauss points.

Skin

Computes the result on the boundary of the region.

Smoothing

< expression-cst >

Smoothes the solution at the nodes.

HarmonicToTime

expression-cst

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

Dimension

expression-cst

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

TimeStep

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

Outputs results for the specified time steps only.

TimeValue

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

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

TimeImagValue

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

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

LastTimeStepOnly

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

AppendExpressionToFileName

expression

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

AppendExpressionFormat

expression-char

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

AppendTimeStepToFileName

< expression-cst >

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

AppendStringToFileName

expression-char

Append the given expression-char to the filename.

OverrideTimeStepValue

expression-cst

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

NoMesh

< expression-cst >

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

SendToServer

expression-char

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

SendToServer

expression-char { expression-cst-list }

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

Color

expression-char

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

Hidden

< expression-cst >

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

Closed

expression-char

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

Units

expression-char

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

Frequency

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

Outputs results for the specified frequencies only.

Format

post-operation-fmt

Outputs results in the specified format.

Adapt

P1 | H1 | H2

Performs p- or h-refinement on the post-processing result, considered as an error map.

Target

expression-cst

Specifies the target for the optimizer during adaptation (error for P1|H1, number of elements for H2).

Value

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

Specifies acceptable output values for discrete optimization (e.g. the available interpolation orders with Adapt P1).

Sort

Position | Connection

Sorts the output by position (x, y, z) or by connection (for LINE elements only).

Iso

expression-cst

Outputs directly contour prints (with expression-cst values) instead of elementary values.

Iso

{ expression-cst-list }

Outputs directly contour prints for the values specified in the expression-cst-list instead of elementary values.

NoNewLine

Suppresses the new lines in the output when printing global quantities (i.e., with Print OnRegion or Print OnGlobal).

ChangeOfCoordinates

{ expression, expression, expression }

Changes the coordinates of the results according to the three expressions given in argument. The three expressions represent the three new cartesian coordinates x, y and z, and can be functions of the current values of the cartesian coordinates $X, $Y and $Z.

ChangeOfValues

{ expression-list }

Changes the values of the results according to the expressions given in argument. The expressions represent the new values (x-compoment, y-component, etc.), and can be functions of the current values of the solution ($Val0, $Val1, etc.).

DecomposeInSimplex

Decomposes all output elements in simplices (points, lines, triangles or tetrahedra).

StoreInVariable

$expression-char

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

StoreInRegister

expression-cst

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

StoreMinInRegister

StoreMaxInRegister

expression-cst

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

StoreMinXinRegister

StoreMinYinRegister

StoreMinZinRegister

StoreMaxXinRegister

StoreMaxYinRegister

StoreMaxZinRegister

expression-cst

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

StoreInField

expression-cst

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

StoreInMeshBasedField

expression-cst

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

TimeLegend

< { expression, expression, expression } >

Includes a time legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot.

FrequencyLegend

< { expression, expression, expression } >

Includes a frequency legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot.

EigenvalueLegend

< { expression, expression, expression } >

Includes an eigenvalue legend in Gmsh plots. If the three optional expressions giving the position of the legend are not specified, the legend is centered on top of the plot.

post-operation-fmt:

Gmsh

GmshParsed

Gmsh output. See Input file format and the documentation of Gmsh (http://gmsh.info) for a description of the file formats.

Table

Space oriented column output, e.g., suitable for Gnuplot, Excel, Caleida Graph, etc. The columns are: element-type element-index x-coord y-coord z-coord <x-coord y-coord z-coord> … real real real values. The three real numbers preceding the values contain context-dependent information, depending on the type of plot: curvilinear abscissa for OnLine plots, normal to the plane for OnPlane plots, parametric coordinates for parametric OnGrid plots, etc.

SimpleTable

Like Table, but with only the x-coord y-coord z-coord and values columns.

TimeTable

Time oriented column output, e.g., suitable for Gnuplot, Excel, Caleida Graph, etc. The columns are: time-step time x-coord y-coord z-coord <x-coord y-coord z-coord> … value.

NodeTable

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

ElementTable

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

Gnuplot

Space oriented column output similar to the Table format, except that a new line is created for each node of each element, with a repetition of the first node if the number of nodes in the element is greater than 2. This permits to draw unstructured meshes and nice three-dimensional elevation plots in Gnuplot. The columns are: element-type element-index x-coord y-coord z-coord real real real values. The three real numbers preceding the values contain context-dependent information, depending on the type of plot: curvilinear abscissa for OnLine plots, normal to the plane for OnPlane plots, parametric coordinates for parametric OnGrid plots, etc.

Adaptation

Adaptation map, suitable for the GetDP -adapt command line option.

7 Short examples

7.1 Constant expression examples

The simplest constant expression consists of an integer or a real number as in

21
-3

or

-3.1415
27e3
-290.53e-12

Using operators and the classic math functions, constant-ids can be defined:

c1 = Sin[2/3*3.1415] * 5000^2;
c2 = -1/c1;

7.2 Group examples

Let us assume that some elements in the input mesh have the region numbers 1000, 2000 and 3000. In the definitions

Group {
  Air = Region[1000]; Core = Region[2000]; Inductor = Region[3000];
  NonConductingDomain = Region[{Air, Core}];
  ConductingDomain    = Region[{Inductor}];
}

Air, Core, Inductor are identifiers of elementary region groups while NonConductingDomain and ConductingDomain are global region groups.

Groups of function type contain lists of entities built on the region groups appearing in their arguments. For example,

NodesOf[NonConductingDomain]

represents the group of nodes of geometrical elements belonging to the regions in NonConductingDomain and

EdgesOf[DomainC, Not SkinDomainC]

represents the group of edges of geometrical elements belonging to the regions in DomainC but not to those of SkinDomainC.

7.3 Function examples

A physical characteristic is a piecewise defined function. The magnetic permeability mu[] can for example be defined in the considered regions by

Function {
  mu[Air] = 4.e-7*Pi;
  mu[Core] = 1000.*4.e-7*Pi;
}

A nonlinear characteristic can be defined through an expression with arguments, e.g.,

Function {
  mu0 = 4.e-7*Pi;
  a1 = 1000.; b1 = 100.; // Constants
  mu[NonlinearCore] = mu0 + 1./(a1+b1*Norm[$1]^6);
}

where function mu[] in region NonLinearCore has one argument $1 which has to be the magnetic flux density. This function is actually called when writing the equations of a formulation, which permits to directly extend it to a nonlinear form by adding only the necessary arguments. For example, in a magnetic vector potential formulation, one may write mu[{Curl a}] instead of mu[] in Equation terms (see Formulation examples). Multiple arguments can be specified in a similar way: writing mu[{Curl a},{T}] in an Equation term will provide the function mu[] with two usable arguments, $1 (the magnetic flux density) and $2 (the temperature).

It is also possible to directly interpolate one-dimensional functions from tabulated data. In the following example, the function f(x) as well as its derivative f’(x) are interpolated from the (x,f(x)) couples (0,0.65), (1,0.72), (2,0.98) and (3,1.12):

Function {
  couples = {0, 0.65 , 1, 0.72 , 2, 0.98 , 3, 1.12};
  f[] = InterpolationLinear[$1]{List[couples]};
  dfdx[] = dInterpolationLinear[$1]{List[couples]};
}

The function f[] may then be called in an Equation term of a Formulation with one argument, x. Notice how the list of constants List[couples] is supplied as a list of parameters to the built-in function InterpolationLinear (see Constants, as well as Functions). In order to facilitate the construction of such interpolations, the couples can also be specified in two separate lists, merged with the alternate list ListAlt command (see Constants):

Function {
  data_x = {0, 1, 2, 3};
  data_f = {0.65, 0.72, 0.98, 1.12};
  f[] = InterpolationLinear[$1]{ListAlt[data_x, data_f]};
  dfdx[] = dInterpolationLinear[$1]{ListAlt[data_x, data_f]};
}

In order to optimize the evaluation time of complex expressions, registers may be used (see Run-time variables and registers). For example, the evaluation of g[] = f[$1]*Sin[f[$1]^2] would require two (costly) linear interpolations. But the result of the evaluation of f[] may be stored in a register (for example the register 0) with

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

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

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

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

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

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

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

7.4 Constraint examples

Constraints are referred to in FunctionSpaces 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 Values are associated with Regions (with their nodes, their edges, their global regions, …) is defined in the FunctionSpaces which use the Constraint. In other words, a Constraint defines data but does not define the method to process them. A time dependent essential boundary condition on Surf1 would be introduced as (cf. Function examples for the definition of TimeFct_Exp[]):

      { Region Surf1; Value 1.; TimeFunction 3*TimeFct_Exp[] }

It is important to notice that the time dependence cannot be introduced in the Value field, since the Value is only evaluated once during the pre-processing.

Other constraints can be referred to in Formulations. It is the case of those defining electrical circuit connections (Network type), e.g.,

Constraint {
  { Name ElectricalCircuit; Type Network;
    Case Circuit1 {
      { Region VoltageSource; Branch {1,2}; }
      { Region PrimaryCoil; Branch {1,2}; }
    }
    Case Circuit2 {
      { Region SecondaryCoil; Branch {1,2}; }
      { Region Charge; Branch {1,2}; }
    }
  }
}

which defines two non-connected circuits (Circuit1 and Circuit2), with an independent numbering of nodes: region VoltageSource is connected in parallel with region PrimaryCoil, and region SecondaryCoil is connected in parallel with region Charge.

7.5 FunctionSpace examples

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

7.5.1 Nodal finite element spaces

The most elementary function space is the nodal finite element space, defined on a mesh of a domain W and denoted S0(W) (associated finite elements can be of various geometries), and associated with essential boundary conditions (Dirichlet conditions). It contains 0-forms, i.e., scalar fields of potential type:

v = Sum [ vn * sn, for all n in N ], v in S0(W)

where N is the set of nodes of W, sn is the nodal basis function associated with node n and vn is the value of v at node n. It is defined by

FunctionSpace {
  { Name Hgrad_v; Type Form0;
    BasisFunction {
      { Name sn; NameOfCoef vn; Function BF_Node;
        Support Domain; Entity NodesOf[All]; }
    }
    Constraint {
      { NameOfCoef vn; EntityType NodesOf;
        NameOfConstraint DirichletBoundaryCondition1; }
    }
  }
}

Function sn is the built-in basis function BF_Node associated with all nodes (NodesOf) in the mesh of W (Domain). Previously defined Constraint DirichletBoundaryCondition1 (see Constraint examples) is used as boundary condition.

In the example above, Entity NodesOf[All] is preferred to Entity NodesOf[Domain]. In this way, the list of all the nodes of Domain will not have to be generated. All the nodes of each geometrical element in Support Domain will be directly taken into account.

7.5.2 High order nodal finite element space

Higher order finite elements can be directly taken into account by BF_Node. Hierarchical finite elements for 0-forms can be used by simply adding other basis functions (associated with other geometrical entities, e.g., edges and facets) to BasisFunction, e.g.,

    …
    BasisFunction {
      { Name sn; NameOfCoef vn; Function BF_Node;
        Support Domain; Entity NodesOf[All]; }
      { Name s2; NameOfCoef v2; Function BF_Node_2E;
        Support Domain; Entity EdgesOf[All]; }
    }
    …

7.5.3 Nodal finite element space with floating potentials

A scalar potential with floating values vf on certain boundaries Gf, f in Cf, e.g., for electrostatic problems, can be expressed as

v = Sum [ vn * sn, for all n in Nv ] + Sum [ vf * sf, for all f in Cf ], v in S0(W)

where Nv is the set of inner nodes of W and each function sf is associated with the group of nodes of boundary Gf, f in Cf (SkinDomainC); sf is the sum of the nodal basis functions of all the nodes of Cf. Its function space is defined by

FunctionSpace {
  { Name Hgrad_v_floating; Type Form0;
    BasisFunction {
      { Name sn; NameOfCoef vn; Function BF_Node;
        Support Domain; Entity NodesOf[All, Not SkinDomainC]; }
      { Name sf; NameOfCoef vf; Function BF_GroupOfNodes;
        Support Domain; Entity GroupsOfNodesOf[SkinDomainC]; }
    }
    GlobalQuantity {
      { Name GlobalElectricPotential; Type AliasOf; NameOfCoef vf; }
      { Name GlobalElectricCharge; Type AssociatedWith;
        NameOfCoef vf; }
    }
    Constraint { … }
  }
}

Two global quantities have been associated with this space: the electric potential GlobalElectricPotential, being an alias of coefficient vf, and the electric charge GlobalElectricCharge, being associated with coefficient vf.

7.5.4 Edge finite element space

Another space is the edge finite element space, denoted S1(W), containing 1-forms, i.e., curl-conform fields:

h = Sum [ he * se, for all e in E ], h in S1(W)

where E is the set of edges of W, se is the edge basis function for edge e and he is the circulation of h along edge e. It is defined by

FunctionSpace {
  { Name Hcurl_h; Type Form1;
    BasisFunction {
      { Name se; NameOfCoef he; Function BF_Edge;
        Support Domain; Entity EdgesOf[All]; }
    }
    Constraint { … }
  }
}

7.5.5 Edge finite element space with gauge condition

A 1-form function space containing vector potentials can be associated with a gauge condition, which can be defined as a constraint, e.g., a zero value is fixed for all circulations along edges of a tree (EdgesOfTreeIn) built in the mesh (Domain), having to be complete on certain boundaries (StartingOn Surf):

Constraint {
  { Name GaugeCondition_a_Mag_3D; Type Assign;
    Case {
      { Region Domain; SubRegion Surf; Value 0.; }
    }
  }
}

FunctionSpace {
  { Name Hcurl_a_Gauge; Type Form1;
    BasisFunction {
      { Name se; NameOfCoef ae; Function BF_Edge;
        Support Domain; Entity EdgesOf[All]; }
    }
    Constraint {
      { NameOfCoef ae;
        EntityType EdgesOfTreeIn; EntitySubType StartingOn;
        NameOfConstraint GaugeCondition_a_Mag_3D; }
      …
    }
  }
}

7.5.6 Coupled edge and nodal finite element spaces

A 1-form function space, containing curl free fields in certain regions WcC (DomainCC) of W, which are the complementary part of Wc (DomainC) in W, can be explicitly characterized by

h = Sum [ hk * sk, for all e in Ec ] + Sum [ phin * vn, for all n in NcC ], h in S1(W)

where Ec is the set of inner edges of W, NcC is the set of nodes inside WcC and on its boundary dWcC, sk is an edge basis function and vn is a vector nodal function. Such a space, coupling a vector field with a scalar potential, can be defined by

FunctionSpace {
  { Name Hcurl_hphi; Type Form1;
    BasisFunction {
      { Name sk; NameOfCoef hk; Function BF_Edge;
        Support DomainC; Entity EdgesOf[All, Not SkinDomainC]; }
      { Name vn; NameOfCoef phin; Function BF_GradNode;
        Support DomainCC; Entity NodesOf[All]; }
      { Name vn; NameOfCoef phic; Function BF_GroupOfEdges;
        Support DomainC; Entity GroupsOfEdgesOnNodesOf[SkinDomainC];}
    }
    Constraint {
      { NameOfCoef hk;
        EntityType EdgesOf; NameOfConstraint MagneticField; }
      { NameOfCoef phin;
        EntityType NodesOf; NameOfConstraint MagneticScalarPotential; }
      { NameOfCoef phic;
        EntityType NodesOf; NameOfConstraint MagneticScalarPotential; }
    }
  }
}

This example points out the definition of a piecewise defined basis function, e.g., function vn being defined with BF_GradNode in DomainCC and BF_GroupOfEdges in DomainC. This leads to an easy coupling between these regions.

7.5.7 Coupled edge and nodal finite element spaces for multiply connected domains

In case a multiply connected domain WcC is considered, basis functions associated with cuts (SurfaceCut) have to be added to the previous basis functions, which gives the function space below:

Group {
  _TransitionLayer_SkinDomainC_ =
    ElementsOf[SkinDomainC, OnOneSideOf SurfaceCut];
}

FunctionSpace {
  { Name Hcurl_hphi; Type Form1;
    BasisFunction {

      … same as above …

      { Name sc; NameOfCoef Ic; Function BF_GradGroupOfNodes;
        Support ElementsOf[DomainCC, OnOneSideOf SurfaceCut];
        Entity GroupsOfNodesOf[SurfaceCut]; }
      { Name sc; NameOfCoef Icc; Function BF_GroupOfEdges;
        Support DomainC;
        Entity GroupsOfEdgesOf
                 [SurfaceCut,
                  InSupport _TransitionLayer_SkinDomainC_]; }
    }
    GlobalQuantity {
      { Name I; Type AliasOf       ; NameOfCoef Ic; }
      { Name U; Type AssociatedWith; NameOfCoef Ic; }
    }
    Constraint {

      … same as above …

      { NameOfCoef Ic;
        EntityType GroupsOfNodesOf; NameOfConstraint Current; }
      { NameOfCoef Icc;
        EntityType GroupsOfNodesOf; NameOfConstraint Current; }
      { NameOfCoef U;
        EntityType GroupsOfNodesOf; NameOfConstraint Voltage; }
    }
  }
}

Global quantities associated with the cuts, i.e., currents and voltages if h is the magnetic field, have also been defined.

7.6 Jacobian examples

A simple Jacobian method is for volume transformations (of n-D regions in n-D geometries; n = 1, 2 or 3), e.g., in region Domain,

Jacobian {
  { Name Vol;
    Case {
      { Region Domain; Jacobian Vol; }
    }
  }
}

Jacobian VolAxi would define a volume Jacobian for axisymmetrical problems.

A Jacobian method can also be piecewise defined, in DomainInf, where an infinite geometrical transformation has to be made using two constant parameters (inner and outer radius of a spherical shell), and in all the other regions (All, being the default); in each case, a volume Jacobian is used. This method is defined by:

Jacobian {
  { Name Vol;
    Case {
      { Region DomainInf; Jacobian VolSphShell {Val_Rint, Val_Rext}; }
      { Region All; Jacobian Vol; }
    }
  }
}

7.7 Integration examples

A commonly used numerical integration method is the Gauss integration, with a number of integration points (NumberOfPoints) depending on geometrical element types (GeoElement), i.e.

Integration {
  { Name Int_1;
    Case { {Type Gauss;
            Case { { GeoElement Triangle   ; NumberOfPoints 4; }
                   { GeoElement Quadrangle ; NumberOfPoints 4; }
                   { GeoElement Tetrahedron; NumberOfPoints 4; }
                   { GeoElement Hexahedron ; NumberOfPoints 6; }
                   { GeoElement Prism      ; NumberOfPoints 9; } }
           }
         }
  }
}

The method above is valid for both 2D and 3D problems, for different kinds of elements.

7.8 Formulation examples

7.8.1 Electrostatic scalar potential formulation

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

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

is expressed by

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

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

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

7.8.2 Electrostatic scalar potential formulation with floating potentials and electric charges

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

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

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

7.8.3 Magnetostatic 3D vector potential formulation

A magnetostatic 3D vector potential formulation

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

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

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

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

7.8.4 Magnetodynamic 3D or 2D magnetic field and magnetic scalar potential formulation

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

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

can be expressed by

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

7.8.5 Nonlinearities, Mixed formulations, …

In case nonlinear physical characteristics are considered, arguments are used for associated functions, e.g., mu[{h}]. Several test functions can be considered in an Equation field. Consequently, mixed formulations can be defined.

7.9 Resolution examples

7.9.1 Static resolution (electrostatic problem)

A static resolution, e.g., for the electrostatic formulation (see Formulation examples), can be defined by

Resolution {
  { Name Electrostatics_v;
    System {
      { Name Sys_Ele; NameOfFormulation Electrostatics_v; }
    }
    Operation {
      Generate[Sys_Ele]; Solve[Sys_Ele]; SaveSolution[Sys_Ele];
    }
  }
}

The generation (Generate) of the matrix of the system Sys_Ele will be made with the formulation Electrostatics_v, followed by its solving (Solve) and the saving of the solution (SaveSolution).

7.9.2 Frequency domain resolution (magnetodynamic problem)

A frequency domain resolution, e.g., for the magnetodynamic h-phi formulation (see Formulation examples), is given by

Resolution {
  { Name Magnetodynamics_hphi;
    System {
      { Name Sys_Mag; NameOfFormulation Magnetodynamics_hphi;
        Frequency Freq; }
    }
    Operation {
      Generate[Sys_Mag]; Solve[Sys_Mag]; SaveSolution[Sys_Mag];
    }
  }
}

preceded by the definition of constant Freq, e.g.,

Function {
  Freq = 50.;
}

7.9.3 Time domain resolution (magnetodynamic problem)

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