Function Operators
StandardFunctionOperators
StandardFunctionOperators are abstract types that encode primitive (linear) operators (like Identity, Gradient etc.) used to dispatch different evaluations of finite element basis functions.
List of primitive operators
| StandardFunctionOperator | Description | Mathematically |
|---|---|---|
| Identity | identity | $v \rightarrow v$ |
| IdentityComponent{c} | identity of c-th component | $v \rightarrow v_c$ |
| NormalFlux | normal flux (function times normal) | $v \rightarrow v \cdot \vec{n}$ (only ON_FACES) |
| TangentFlux | tangent flux (function times tangent) | $v \rightarrow v \cdot \vec{t}$ (only ON_EDGES) |
| Gradient | gradient/Jacobian (as a vector) | $v \rightarrow \nabla v$ |
| SymmetricGradient | symmetric part of the gradient | $v \rightarrow Voigt(\mathrm{sym}(\nabla v))$ |
| Divergence | divergence | $v \rightarrow \mathrm{div}(v) = \nabla \cdot v$ |
| CurlScalar | curl operator 1D to 2D (rotated gradient) | $v \rightarrow [-dv/dx_2,dv/dx_1]$ |
| Curl2D | curl operator 2D to 1D | $v \rightarrow dv_1/dx_2 - dv_2/dx_1$ |
| Curl3D | curl operator 3D to 3D | $v \rightarrow \nabla \times v$ |
| Hessian | Hesse matrix = all 2nd order derivatives (as a vector) | $v \rightarrow D^2 v$ (e.g. in 2D: xx,xy,yx,yy for each component) |
| SymmetricHessian{a} | symmetric part of Hesse matrix, offdiagonals scaled by a | $v \rightarrow sym(D^2 v)$ (e.g. in 2D: xx,yy,a*xy for each component) |
| Laplacian | Laplace Operator (diagonal of Hessian) | $v \rightarrow \Delta v$ (e.g. in 2D: xx,yy for each component) |
As each finite element type is transformed differently from the reference domain to the general domain, the evaluation of each function operator has to be implemented for each finite element class. Currently, not every function operator works in any dimension and for any finite element. More evaluations are added as soon as they are needed (and possibly upon request). Also, the function operators can be combined with user-defined actions to evaluate other operators that can be build from the ones available (e.g. the deviator).
ExtendableFEMBase.AbstractFunctionOperator — Typeabstract type AbstractFunctionOperatorroot type for FunctionOperators.
ExtendableFEMBase.Curl2D — Typeabstract type Curl2D <: StandardFunctionOperatorevaluates the curl of some two-dimensional vector field, i.e. Curl2D((u1,u2)) = du2/dx1 - du1/dx2
ExtendableFEMBase.Curl3D — Typeabstract type Curl3D <: StandardFunctionOperatorevaluates the curl of some three-dimensional vector field, i.e. Curl3D(u) = abla imes u
ExtendableFEMBase.CurlScalar — Typeabstract type CurlScalar <: StandardFunctionOperatorevaluates the curl of some scalar function in 2D, i.e. the rotated gradient.
ExtendableFEMBase.Divergence — Typeabstract type Divergence <: StandardFunctionOperatorevaluates the divergence of the finite element function.
ExtendableFEMBase.Gradient — Typeabstract type Gradient <: StandardFunctionOperatorevaluates the gradient of the finite element function.
ExtendableFEMBase.Hessian — Typeabstract type Hessian <: StandardFunctionOperatorevaluates the full Hessian of some (possibly vector-valued) finite element function.
ExtendableFEMBase.Identity — Typeabstract type Identity <: StandardFunctionOperatoridentity operator: evaluates finite element function.
ExtendableFEMBase.IdentityComponent — Typeabstract type IdentityComponent{c} <: StandardFunctionOperatoridentity operator: evaluates only the c-th component of the finite element function.
ExtendableFEMBase.Laplacian — Typeabstract type Laplacian <: StandardFunctionOperatorevaluates the Laplacian of some (possibly vector-valued) finite element function.
ExtendableFEMBase.NormalFlux — Typeabstract type NormalFlux <: StandardFunctionOperatorevaluates the normal-flux of the finite element function.
only available on FACES/BFACES and currently only for H1 and Hdiv elements
ExtendableFEMBase.OperatorPair — Typeabstract type OperatorPair{<:StandardFunctionOperator,<:StandardFunctionOperator} <: StandardFunctionOperatorallows to evaluate two operators in place of one, e.g. OperatorPair{Identity,Gradient}.
ExtendableFEMBase.OperatorTriple — Typeabstract type OperatorTriple{<:StandardFunctionOperator,<:StandardFunctionOperator} <: StandardFunctionOperatorallows to evaluate three operators in place of one, e.g. OperatorTriple{Identity,Gradient,Hessian}.
ExtendableFEMBase.SymmetricGradient — Typeabstract type SymmetricGradient{offdiagval} <: StandardFunctionOperatorevaluates the symmetric part of the gradient of the finite element function and returns its Voigt compression (off-diagonals on position [j,k] and [k,j] are added together and weighted with offdiagval).
ExtendableFEMBase.SymmetricHessian — Typeabstract type SymmetricHessian{offdiagval} <: StandardFunctionOperatorevaluates only the symmetric part of the Hessian of some (possibly vector-valued) finite element function. A concatenation of Voigt compressed Hessians for each component of the finite element functions is returned. The weighting of the mixed derivatives can be steered with offdiagval (e.g. √2 or 1 depending on the use case).
ExtendableFEMBase.TangentFlux — Typeabstract type TangentFlux <: StandardFunctionOperatorevaluates the tangent-flux of the finite element function.
ExtendableFEMBase.TangentialGradient — Typeabstract type TangentialGradient <: StandardFunctionOperatorevaluates the gradient of the tangential part of some vector-valued finite element function.
ReconstructionOperators
There are special operators that allow to evaluate a primitive operator of some discrete reconstructed version of a testfunction.
ExtendableFEMBase.Reconstruct — Typeabstract type Reconstruct{FETypeR, O} <: ExtendableFEMBase.ReconstructionOperatorreconstruction operator: evaluates a reconstructed version of the finite element function.
FETypeR specifies the reconstruction space (needs to be defined for the finite element that it is applied to). O specifies the StandardFunctionOperator that shall be evaluated.
Divergence-free reconstruction operators
For gradient-robust discretisations of certain classical non divergence-conforming ansatz spaces, reconstruction operators are available that map a discretely divergence-free H1 function to a pointwise divergence-free Hdiv function. So far such operators are available for the vector-valued Crouzeix-Raviart (H1CR) and Bernardi–Raugel (H1BR) finite element types, as well as for the P2-bubble (H1P2B) finite element type in two dimensions.
Example: Reconst{HDIVRT0{d}, Identity} gives the reconstruction of the Identity operator into HDIVRT0 (and is available for H1BR{d} and H1CR{d} for d = 1,2)
Operator Pairs (experimental)
Two function operators can be put into an OperatorPair so that one can provide effectively two operators in each argument of an assembly pattern. However, the user should make sure that both operators can be evaluated together reasonably (meaning both should be well-defined on the element geometries and the finite element space where the argument will be evaluated, and the action of the operator has to operate with coressponding input and result fields). This feature is still experimental and might have issues in some cases. OperatorTriple for a combination of three operators is also available.