Hierarchic Finite Elements of arbitrary degree on quadrilaterals. More...
#include <lf/fe/fe.h>
Public Member Functions | |
| FeHierarchicQuad (const FeHierarchicQuad &)=default | |
| FeHierarchicQuad (FeHierarchicQuad &&) noexcept=default | |
| FeHierarchicQuad & | operator= (const FeHierarchicQuad &)=default |
| FeHierarchicQuad & | operator= (FeHierarchicQuad &&) noexcept=default |
| ~FeHierarchicQuad () override=default | |
| FeHierarchicQuad (unsigned interior_degree, std::array< unsigned, 4 > edge_degrees, const quad::QuadRuleCache &qr_cache, std::span< const lf::mesh::Orientation > rel_orient) | |
| lf::base::RefEl | RefEl () const override |
| Tells the type of reference cell underlying the parametric finite element. | |
| unsigned | Degree () const override |
| Request the maximal polynomial degree of the basis functions in this finite element. | |
| lf::base::size_type | NumRefShapeFunctions () const override |
| The local shape functions. | |
| lf::base::size_type | NumRefShapeFunctions (dim_t codim) const override |
| One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral. | |
| lf::base::size_type | NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const override |
| One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral. | |
| Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override |
| Evaluation of all reference shape functions in a number of points. | |
| Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override |
| Computation of the gradients of all reference shape functions in a number of points. | |
| Eigen::MatrixXd | EvaluationNodes () const override |
| Evaluation nodes are the vertices, the points of a quadrature rule on each edge and the points of a quadrature rule on the interior of the quadrilateral. | |
| lf::base::size_type | NumEvaluationNodes () const override |
| Tell the number of evaluation (interpolation) nodes. | |
| Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > | NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodevals) const override |
| Computes the linear combination of reference shape functions matching function values at evaluation nodes. | |
 Public Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| virtual | ~ScalarReferenceFiniteElement ()=default |
| dim_t | Dimension () const |
| Returns the spatial dimension of the reference cell. | |
Private Attributes | |
| unsigned | interior_degree_ |
| std::array< unsigned, 4 > | edge_degrees_ |
| std::array< const quad::QuadRule *, 4 > | qr_dual_edge_ |
| FeHierarchicSegment< SCALAR > | fe1d_ |
| std::span< const lf::mesh::Orientation > | rel_orient_ |
Additional Inherited Members | |
| Public Types inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| using | Scalar = SCALAR |
| The underlying scalar type. | |
| Protected Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| ScalarReferenceFiniteElement ()=default | |
| ScalarReferenceFiniteElement (const ScalarReferenceFiniteElement &)=default | |
| ScalarReferenceFiniteElement (ScalarReferenceFiniteElement &&) noexcept=default | |
| ScalarReferenceFiniteElement & | operator= (const ScalarReferenceFiniteElement &)=default |
| ScalarReferenceFiniteElement & | operator= (ScalarReferenceFiniteElement &&) noexcept=default |
| Related Symbols inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| template<class SCALAR > | |
| void | PrintInfo (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &srfe, unsigned int ctrl=0) |
| Print information about a ScalarReferenceFiniteElement to the given stream. | |
| template<typename SCALAR > | |
| std::ostream & | operator<< (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &fe_desc) |
| Stream output operator: just calls the ScalarReferenceFiniteElement::print() method. | |
Detailed Description
class lf::fe::FeHierarchicQuad< SCALAR >
Hierarchic Finite Elements of arbitrary degree on quadrilaterals.
The basis functions on the quadrilateral has a tensor product structure and can thus be represented by products of basis functions on segments. The vertex basis functions on the reference quad are therefore given by
\[ \begin{align*} \widehat{b^{\cdot}}^0(x, y) &:= (1 - x)(1 - y) \\ \widehat{b^{\cdot}}^1(x, y) &:= x(1 - y) \\ \widehat{b^{\cdot}}^2(x, y) &:= xy \\ \widehat{b^{\cdot}}^3(x, y) &:= (1 - x)y. \end{align*} \]
The edge basis functions can be written as
\[ \begin{align*} \widehat{b^{-}}^{0,n} &:= (1-y)L_n(x) \\ \widehat{b^{-}}^{1,n} &:= xL_n(y) \\ \widehat{b^{-}}^{2,n} &:= yL_n(1-x) \\ \widehat{b^{-}}^{3,n} &:= (1-x)L_n(1-y) \end{align*} \]
where \( n \geq 2 \) is the degree of the basis function. Finally, the face bubbles are given by
\[ \widehat{b^{\square}}^{n,m}(x, y) := L_n(x)L_m(y) \]
where \( n \geq 2 \), \( m \geq 2 \).
The dual basis is therefore also quite simple, as we can recycle the one from the segments by first applying the dual basis along the \(x\)-axis and then apply the dual basis to the resulting 1d function.
- Attention
- Note that for the basis functions associated with the edges, depending on the
lf::mesh::Orientationof the according edge, the local coordinate may be flipped to ensure continuity of the function space over the cell interfaces of the mesh. The basis functions and the dual basis must be adjusted accordingly in this case.
A complete description of the basis functions and dual basis can be found here.
- See also
- ScalarReferenceFiniteElement
Definition at line 1141 of file hierarchic_fe.h.
Constructor & Destructor Documentation
◆ FeHierarchicQuad() [1/3]
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default |
◆ FeHierarchicQuad() [2/3]
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defaultnoexcept |
◆ ~FeHierarchicQuad()
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overridedefault |
◆ FeHierarchicQuad() [3/3]
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inline |
Definition at line 1149 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::base::RefEl::kSegment().
Member Function Documentation
◆ Degree()
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inlineoverridevirtual |
Request the maximal polynomial degree of the basis functions in this finite element.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1179 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs().
◆ EvalReferenceShapeFunctions()
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inlineoverridevirtual |
Evaluation of all reference shape functions in a number of points.
- Parameters
-
refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N, where dim is the dimension of the reference element, that is =0 for points, =1 for edges, =2 for triangles and quadrilaterals
- Returns
- An Eigen Matrix of size
NumRefShapeFunctions() x refcoords.cols()which contains the shape functions evaluated at every quadrature point.
Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler or the documentation of the class.
- Note
- shape functions are assumed to be real-valued.
Example: Triangular Linear Lagrangian finite elements
There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x2 matrix:
\[ \begin{pmatrix}\hat{b}^1(\mathbf{x}_1) & \hat{b}^1(\mathbf{x}_2) \\ \hat{b}^2(\mathbf{x}_1) & \hat{b}^2(\mathbf{x}_2) \\ \hat{b}^3(\mathbf{x}_1)\ & \hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1248 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.
◆ EvaluationNodes()
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inlineoverridevirtual |
Evaluation nodes are the vertices, the points of a quadrature rule on each edge and the points of a quadrature rule on the interior of the quadrilateral.
Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case.
- Returns
- A d x N matrix, where d is the dimension of the reference cell, and N the number of interpolation nodes. The columns of this matrix contain their reference coordinates.
Every parametric scalar finite element implicitly defines a local interpolation operator by duality with the reference shape functions. This interpolation operator can be realized through evaluations at certain evaluation nodes, which are provided by this method.
Unisolvence
The evaluation points must satisfy the following requirement: If the values of a function belonging to the span of the reference shape functions are known in the evaluation nodes, then this function is uniquely determined. This entails that the number of evaluation nodes must be at least as big as the number of reference shape functions.
- Note
- It is not required that any vector a values at evaluation nodes can be produced by a suitable linear combination of reference shape functions. For instance, this will not be possible, if there are more evaluation points than reference shape functions. If both sets have the same size, however, the interpolation problem has a solution for any vector of values at the evluation points.
Example: Principal lattice
For triangular Lagrangian finite elements of degree p the evaluation nodes, usually called "interpolation nodes" in this context, can be chosen as \(\left(\frac{j}{p},\frac{k}{p}\right),\; 0\leq j,k \leq p, j+k\leq p\).
Moment-based degrees of freedom
For some finite element spaces the interpolation functional may be defined based on integrals over edges. In this case the evaluation nodes will be quadrature nodes for the approximate evaluation of these integrals.
The quadrature rule must be exact for the polynomials contained in the local finite element spaces.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1452 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::mesh::positive, lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.
◆ GradientsReferenceShapeFunctions()
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inlineoverridevirtual |
Computation of the gradients of all reference shape functions in a number of points.
- Parameters
-
refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N.
- Returns
- An Eigen Matrix of size
NumRefShapeFunctions() x (dim * refcoords.cols())wheredimis the dimension of the reference finite element. The gradients are returned in chunks of rows of this matrix.
Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler.
Example: Triangular Linear Lagrangian finite elements
There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x4 matrix:
\[ \begin{pmatrix} \grad^T\hat{b}^1(\mathbf{x}_1) & \grad^T\hat{b}^1(\mathbf{x}_2) \\ \grad^T\hat{b}^2(\mathbf{x}_1) & \grad^T\hat{b}^2(\mathbf{x}_2) \\ \grad^T\hat{b}^3(\mathbf{x}_1) & \grad^T\hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1335 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.
◆ NodalValuesToDofs()
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inlineoverridevirtual |
Computes the linear combination of reference shape functions matching function values at evaluation nodes.
- Parameters
-
nodvals row vector of function values at evaluation nodes The length of this vector must agree with NumEvaluationNodes().
- Returns
- The coefficients of the local "nodal interpolant" with respect to the reference shape functions. This is a row vector of length NumRefShapeFunctions().
If the evaluation nodes are interpolation nodes, that is, if the set of reference shape functions forms a cardinal basis with respect to these nodes, then we have NumEvaluationNodes() == NumRefShapeFunctions() and the linear mapping realized by NodalValuesToDofs() is the identity mapping.
- Note
- default implementation is the identity mapping
Requirement: reproduction of local finite element functions
If the vector of values at the evaluation nodes agrees with a vector of function values of a linear combination of reference shape functions at the evaluation nodes, then this method must return the very coefficients of the linear combination.
Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1545 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::ILegendreDx(), lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::LegendreDx(), lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.
◆ NumEvaluationNodes()
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inlineoverridevirtual |
Tell the number of evaluation (interpolation) nodes.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1530 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, and lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_.
◆ NumRefShapeFunctions() [1/3]
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inlineoverridevirtual |
The local shape functions.
Total number of reference shape functions associated with the reference cell.
- Note
- the total number of shape functions is the sum of the number of interior shape functions for all sub-entities and the entity itself.
Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1188 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().
◆ NumRefShapeFunctions() [2/3]
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inlineoverridevirtual |
One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral.
The number of interior reference shape functions for sub-entities of a particular co-dimension.
- Parameters
-
codim co-dimension of the subentity
- Returns
- number of interior reference shape function belonging to the specified sub-entity.
- Note
- this method will throw an exception when different numbers of reference shape functions are assigned to different sub-entities of the same co-dimension
Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1199 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_.
◆ NumRefShapeFunctions() [3/3]
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inlineoverridevirtual |
One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral.
The number of interior reference shape functions for every sub-entity.
- Parameters
-
codim do-dimension of the subentity subidx local index of the sub-entity
- Returns
- number of interior reference shape function belonging to the specified sub-entity.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1228 of file hierarchic_fe.h.
References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().
◆ operator=() [1/2]
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default |
◆ operator=() [2/2]
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defaultnoexcept |
◆ RefEl()
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inlineoverridevirtual |
Tells the type of reference cell underlying the parametric finite element.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 1175 of file hierarchic_fe.h.
References lf::base::RefEl::kQuad().
Member Data Documentation
◆ edge_degrees_
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private |
Definition at line 1672 of file hierarchic_fe.h.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::FeHierarchicQuad(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().
◆ fe1d_
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private |
Definition at line 1675 of file hierarchic_fe.h.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), and lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes().
◆ interior_degree_
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private |
Definition at line 1671 of file hierarchic_fe.h.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().
◆ qr_dual_edge_
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private |
Definition at line 1673 of file hierarchic_fe.h.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), and lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes().
◆ rel_orient_
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private |
Definition at line 1676 of file hierarchic_fe.h.
Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), and lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs().
The documentation for this class was generated from the following file:
- /__w/lehrfempp/lehrfempp/lib/lf/fe/hierarchic_fe.h
