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# 4. Types for objects

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

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

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## 4.1 Types for `Group`

Types in

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

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

`Region`
  Regions in R1.

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

`NodesOf`
  Nodes of elements of R1

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

`EdgesOf`
  Edges of elements of R1

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

`FacetsOf`
  Facets of elements of R1

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

`VolumesOf`
  Volumes of elements of R1

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

`ElementsOf`
  Elements of regions in R1

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

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

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

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

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

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

`EdgesOfTreeIn`
  Edges of a tree of edges of R1

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

`FacetsOfTreeIn`
  Facets of a tree of facets of R1

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

`DualNodesOf`
  Dual nodes of elements of R1.

`DualEdgesOf`
  Dual edges of elements of R1.

`DualFacetsOf`
  Dual facets of elements of R1.

`DualVolumesOf`
  Dual volumes of elements of R1.

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## 4.2 Types for `Function`

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

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### 4.2.1 Math functions

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

math-function-id:

`Exp`
  `[expression]`

  Exponential function: e^expression.

`Log`
  `[expression]`

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

`Log10`
  `[expression]`

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

`Sqrt`
  `[expression]`

  Square root, expression>=0.

`Sin`
  `[expression]`

  Sine of expression.

`Asin`
  `[expression]`

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

`Cos`
  `[expression]`

  Cosine of expression.

`Acos`
  `[expression]`

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

`Tan`
  `[expression]`

  Tangent of expression.

`Atan`
  `[expression]`

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

`Atan2`
  `[expression,expression]`

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

`Sinh`
  `[expression]`

  Hyperbolic sine of expression.

`Cosh`
  `[expression]`

  Hyperbolic cosine of expression.

`Tanh`
  `[expression]`

  Hyperbolic tangent of expression.

`Fabs`
  `[expression]`

  Absolute value of expression.

`Fmod`
  `[expression,expression]`

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

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### 4.2.2 Extended math functions

extended-math-function-id:

`Cross`
  `[expression,expression]`

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

`Hypot`
  `[expression,expression]`

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

`Norm`
  `[expression]`

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

`SquNorm`
  `[expression]`

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

`Unit`
  `[expression]`

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

`Transpose`
  `[expression]`

  Transposition; expression must be a tensor.

`TTrace`
  `[expression]`

  Trace; expression must be a tensor.

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

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

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

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

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

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

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### 4.2.3 Green functions

The Green functions are only used in integral quantities (see section 3.7 `Formulation`: building equations). The first parameter represents the dimension of the problem:

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

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

green-function-id:

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

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

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

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

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

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

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

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

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### 4.2.4 Type manipulation functions

type-function-id:

`Complex`
  `[expression-list]`

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

`Re`
  `[expression]`

  Takes the real part of a complex-valued expression.

`Im`
  `[expression]`

  Takes the imaginary part of a complex-valued expression.

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

  Creates a vector from 3 scalars.

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

  `expression,expression,expression]`

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

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

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

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

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

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

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

`CompX`
  `[expression]`

  Gets the X component of a vector.

`CompY`
  `[expression]`

  Gets the Y component of a vector.

`CompZ`
  `[expression]`

  Gets the Z component of a vector.

`CompXX`
  `[expression]`

  Gets the XX component of a tensor.

`CompXY`
  `[expression]`

  Gets the XY component of a tensor.

`CompXZ`
  `[expression]`

  Gets the XZ component of a tensor.

`CompYX`
  `[expression]`

  Gets the YX component of a tensor.

`CompYY`
  `[expression]`

  Gets the YY component of a tensor.

`CompYZ`
  `[expression]`

  Gets the YZ component of a tensor.

`CompZX`
  `[expression]`

  Gets the ZX component of a tensor.

`CompZY`
  `[expression]`

  Gets the ZY component of a tensor.

`CompZZ`
  `[expression]`

  Gets the ZZ component of a tensor.

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### 4.2.5 Coordinate functions

coord-function-id:

`X`
  `[]`

  Gets the X coordinate.

`Y`
  `[]`

  Gets the Y coordinate.

`Z`
  `[]`

  Gets the Z coordinate.

`XYZ`
  `[]`

  Gets X, Y and Z in a vector.

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### 4.2.6 Miscellaneous functions

misc-function-id:

`Printf`
  `[expression]`

  Prints the value of expression when evaluated.

`Normal`
  `[]`

  Computes the normal to the element.

`NormalSource`
  `[]`

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

`F_CompElementNum`
  `[]`

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

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

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

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

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

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

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

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

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

`Order`
  `[quantity]`

  Returns the interpolation order of the quantity.

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## 4.3 Types for `Constraint`

constraint-type:

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

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

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

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

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

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

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

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## 4.4 Types for `FunctionSpace`

function-space-type:

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

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

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

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

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

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

`Scalar`
  Scalar field.

`Vector`
  Vector field.

basis-function-type:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

global-quantity-type:

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

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

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## 4.5 Types for `Jacobian`

jacobian-type:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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## 4.6 Types for `Integration`

integration-type:

`Gauss`
  Numerical Gauss integration.

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

element-type:

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

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

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

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

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

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

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

`Point`
  Point (1 node) (#15).

Note:

1. n in (#n) is the type number of the element (see section 8.1 Input file format).

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## 4.7 Types for `Formulation`

formulation-type:

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

local-term-type:

`Galerkin`
  Integral of Galerkin type.

`deRham`
  deRham projection (collocation).

quantity-type:

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

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

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

term-op-type:

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

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

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

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

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

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

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## 4.8 Types for `Resolution`

resolution-op:

`Generate`
  `[system-id]`

  Generate the system of equations system-id.

`Solve`
  `[system-id]`

  Solve the system of equations system-id.

`GenerateJac`
  `[system-id]`

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

`SolveJac`
  `[system-id]`

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

`GenerateSeparate`
  `[system-id]`

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

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

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

`InitSolution`
  `[system-id]`

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

`SaveSolution`
  `[system-id]`

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

`SaveSolutions`
  `[system-id]`

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

`TransferSolution`
  `[system-id]`

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

`TransferInitSolution`
  `[system-id]`

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

`SetTime`
  `[expression]`

  Change the current time.

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

  Change the frequency of system system-id.

`SystemCommand`
  `[expression-char]`

  Execute the system command given by expression-char.

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

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

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

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

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

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

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

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

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

  Eigen value computation by the Lanczos algorithm. The parameters are: the system (which has to be generated with `GenerateSeparate[]`), the size of the Lanczos space, the indices of the eigen values/vectors to store, the spectral shift.

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

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

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

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

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

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

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

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

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## 4.9 Types for `PostProcessing`

post-value:

`Local`
  { local-value }

  To compute a local quantity.

`Integral`
  { integral-value }

  To integrate the expression over each element.

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## 4.10 Types for `PostOperation`

print-support:

`OnElementsOf`
  group-def

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

`OnRegion`
  group-def

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

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

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

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

`OnGrid`
  group-def

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

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

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

`OnPoint`
  { expression-cst-list }

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

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

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

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

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

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

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

print-option:

`File`
  `expression-char`

  Outputs the result in a file named expression-char.

`File`
  `> expression-char`

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

`File`
  `>> expression-char`

  Appends the result to a file named expression-char.

`Depth`
  expression-cst

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

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

`Smoothing`
  Smoothes the solution at the nodes.

`HarmonicToTime`
  expression-cst

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

`Dimension`
  expression-cst

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

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

  Outputs results for the specified time steps only.

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

  Outputs results for the specified frequencies only.

`Format`
  post-operation-fmt

  Outputs results in the specified format.

`Adapt`
  `P1 | H1 | H2`

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

`Target`
  expression-cst

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

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

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

`Sort`
  `Position | Connection`

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

`Iso`
  expression-cst

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

`Iso`
  `{ expression-cst-list }`

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

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

`ChangeOfCoordinates`
  `{ expression, expression, expression }`

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

`ChangeOfValues`
  `{ expression-list }`

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

post-operation-fmt:

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

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

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

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

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

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