Linear Lagrange finite element on triangular reference element. More...
#include <lf/uscalfe/uscalfe.h>
Public Member Functions | |
| FeLagrangeO1Tria (const FeLagrangeO1Tria &)=default | |
| FeLagrangeO1Tria (FeLagrangeO1Tria &&) noexcept=default | |
| FeLagrangeO1Tria & | operator= (const FeLagrangeO1Tria &)=default |
| FeLagrangeO1Tria & | operator= (FeLagrangeO1Tria &&) noexcept=default |
| FeLagrangeO1Tria ()=default | |
| ~FeLagrangeO1Tria () override=default | |
| 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. | |
| size_type | NumRefShapeFunctions () const override |
| The local shape functions: barycentric coordinate functions. | |
| size_type | NumRefShapeFunctions (dim_t codim) const override |
| Exactly one shape function attached to each node, none to other sub-entities. | |
| size_type | NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const override |
| Exactly one shape function attached to each node, none to other sub-entities. | |
| 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 |
| Evalutation nodes are just the vertices of the triangle. | |
| size_type | NumEvaluationNodes () const override |
| Three 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. | |
| virtual Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > | NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodvals) const |
| Computes the linear combination of reference shape functions matching function values at evaluation nodes. | |
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::uscalfe::FeLagrangeO1Tria< SCALAR >
Linear Lagrange finite element on triangular reference element.
This is a specialization of fe::ScalarReferenceFiniteElement. Refer to its documentation.
The reference shape functions are
\[ \widehat{b}^1(\widehat{\mathbf{x}}) = 1-\widehat{x}_1-\widehat{x}_2\quad,\quad \widehat{b}^2(\widehat{\mathbf{x}}) = \widehat{x}_1\quad,\quad \widehat{b}^3(\widehat{\mathbf{x}}) = \widehat{x}_2 \;. \]
The basis function \(\widehat{b}^i\) is associated with vertex \(i\) of the triangle.
- See also
- fe::ScalarReferenceFiniteElement
Constructor & Destructor Documentation
◆ FeLagrangeO1Tria() [1/3]
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default |
◆ FeLagrangeO1Tria() [2/3]
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defaultnoexcept |
◆ FeLagrangeO1Tria() [3/3]
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default |
◆ ~FeLagrangeO1Tria()
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overridedefault |
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 >.
◆ 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 >.
◆ EvaluationNodes()
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inlineoverridevirtual |
Evalutation nodes are just the vertices of the triangle.
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 135 of file lagr_fe.h.
References lf::base::RefEl::NodeCoords(), and lf::uscalfe::FeLagrangeO1Tria< SCALAR >::RefEl().
◆ 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 >.
◆ NumEvaluationNodes()
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inlineoverridevirtual |
Three evaluation nodes.
Tell the number of evaluation (interpolation) nodes.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 142 of file lagr_fe.h.
References lf::base::RefEl::NumNodes(), and lf::uscalfe::FeLagrangeO1Tria< SCALAR >::RefEl().
◆ NumRefShapeFunctions() [1/3]
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inlineoverridevirtual |
The local shape functions: barycentric coordinate 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 >.
◆ NumRefShapeFunctions() [2/3]
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inlineoverridevirtual |
Exactly one shape function attached to each node, none to other sub-entities.
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 >.
◆ NumRefShapeFunctions() [3/3]
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inlineoverridevirtual |
Exactly one shape function attached to each node, none to other sub-entities.
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 >.
◆ 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 67 of file lagr_fe.h.
References lf::base::RefEl::kTria().
Referenced by lf::uscalfe::FeLagrangeO1Tria< SCALAR >::EvaluationNodes(), and lf::uscalfe::FeLagrangeO1Tria< SCALAR >::NumEvaluationNodes().
The documentation for this class was generated from the following file:
- /__w/lehrfempp/lehrfempp/lib/lf/uscalfe/lagr_fe.h
