Function Operators
StandardFunctionOperators
StandardFunctionOperators are abstract types that represent fundamental (linear) operators, such as the identity, gradient, divergence, and others. These operators provide a unified interface for evaluating finite element basis functions in various ways.
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) |
The transformation from the reference domain to the physical domain differs for each finite element class. As a result, the evaluation of each function operator must be implemented specifically for every finite element class. Not all function operators are currently available for every dimension or element type, but new implementations are added as needed or upon request.
Additionally, function operators can be combined with user-defined kernels to postprocess/construct more advanced operators from the available primitives (for example, the deviatoric part of a tensor).
ExtendableFEMBase.AbstractFunctionOperator — Typeabstract type AbstractFunctionOperatorroot type for FunctionOperators.
ExtendableFEMBase.Curl2D — Typeabstract type Curl2D <: ??evaluates the curl of some two-dimensional vector field, i.e. Curl2D((u1,u2)) = du2/dx1 - du1/dx2
ExtendableFEMBase.Curl3D — Typeabstract type Curl3D <: ??evaluates the curl of some three-dimensional vector field, i.e. Curl3D(u) = abla imes u
ExtendableFEMBase.CurlScalar — Typeabstract type CurlScalar <: ??evaluates the curl of some scalar function in 2D, i.e. the rotated gradient.
ExtendableFEMBase.Deviator — Typeabstract type Deviator <: ??evaluates the deviator of a matrix-valued function
ExtendableFEMBase.Divergence — Typeabstract type Divergence <: ??evaluates the divergence of the finite element function.
ExtendableFEMBase.Gradient — Typeabstract type Gradient <: ??evaluates the gradient of the finite element function.
ExtendableFEMBase.Hessian — Typeabstract type Hessian <: ??evaluates the full Hessian of some (possibly vector-valued) finite element function.
ExtendableFEMBase.Identity — Typeabstract type Identity <: ??identity operator: evaluates finite element function.
ExtendableFEMBase.IdentityComponent — Typeabstract type IdentityComponent{c} <: ??identity operator: evaluates only the c-th component of the finite element function.
ExtendableFEMBase.Laplacian — Typeabstract type Laplacian <: ??evaluates the Laplacian of some (possibly vector-valued) finite element function.
ExtendableFEMBase.NormalFlux — Typeabstract type NormalFlux <: ??evaluates 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.StandardFunctionOperator — Typeabstract type StandardFunctionOperator <: ??root type for StandardFunctionOperators
ExtendableFEMBase.SymmetricGradient — Typeabstract type SymmetricGradient{offdiagval} <: ??evaluates 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} <: ??evaluates 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 <: ??evaluates the tangent-flux of the finite element function.
ExtendableFEMBase.TangentialGradient — Typeabstract type TangentialGradient <: ??evaluates the gradient of the tangential part of some vector-valued finite element function.
ExtendableFEMBase.Trace — Typeabstract type Trace <: ??evaluates the trace of a matrix-valued function.
ExtendableFEMBase.DefaultName4Operator — MethodDefaultName4Operator(_::Type{<:??}) -> String
default name of the operator for print-outs
ExtendableFEMBase.Length4Operator — MethodLength4Operator(
_::Type{<:id},
xdim::Int64,
ncomponents::Int64
) -> Int64
number of expected components for the operator in xdim dimensions and a function value dimension of ncomponents
ExtendableFEMBase.NeededDerivative4Operator — MethodNeededDerivative4Operator(_::Type{<:id}) -> Int64
number of derivatives that are needed to evaluate the operator
ExtendableFEMBase.QuadratureOrderShift4Operator — MethodQuadratureOrderShift4Operator(operator) -> Any
default quadrature order shift for an operator (basically equals minus the number of derivatives)
ReconstructionOperators
Special operators are provided to evaluate a primitive operator on a reconstructed version of a test function. These are useful for advanced discretizations and post-processing.
ExtendableFEMBase.Reconstruct — Typeabstract type Reconstruct{FETypeR, O} <: ??reconstruction 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 discretizations 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 H(div) function. Currently, such operators are implemented for the vector-valued Crouzeix-Raviart (H1CR) and Bernardi–Raugel (H1BR) finite element types, as well as for the P2-bubble (H1P2B) element in two dimensions.
Example: Reconst{HDIVRT0{d}, Identity} reconstructs 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 combined into an OperatorPair, allowing you to provide two operators in each argument of an assembly pattern. Both operators must be well-defined on the relevant element geometries and finite element spaces, and their actions must be compatible with the input and result fields. This feature is experimental and may have limitations in some cases. An OperatorTriple is also available for combining three operators.