Decorator class around a ScalarReferenceFiniteElement to represent discontinuous shape functions. More...
Public Member Functions | |
| DiscontinuousScalarReferenceFiniteElement ()=default | |
| DiscontinuousScalarReferenceFiniteElement (const DiscontinuousScalarReferenceFiniteElement &)=delete | |
| DiscontinuousScalarReferenceFiniteElement (DiscontinuousScalarReferenceFiniteElement &&) noexcept=default | |
| DiscontinuousScalarReferenceFiniteElement & | operator= (const DiscontinuousScalarReferenceFiniteElement &)=delete |
| DiscontinuousScalarReferenceFiniteElement & | operator= (DiscontinuousScalarReferenceFiniteElement &&) noexcept=default |
| DiscontinuousScalarReferenceFiniteElement (std::shared_ptr< const lf::fe::ScalarReferenceFiniteElement< SCALAR > > cfe) | |
| bool | isInitialized () const |
| Reports initialization status of the object. More... | |
| lf::base::RefEl | RefEl () const override |
| Tells the type of reference cell underlying the parametric finite element. More... | |
| unsigned | Degree () const override |
| Request the maximal polynomial degree of the basis functions in this finite element. More... | |
| size_type | NumRefShapeFunctions () const override |
| Total number of reference shape functions associated with the reference cell. More... | |
| size_type | NumRefShapeFunctions (dim_t codim) const override |
| size_type | NumRefShapeFunctions (dim_t codim, sub_idx_t) const override |
| Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | EvalReferenceShapeFunctions (const Eigen::MatrixXd &local) const override |
| Evaluation of all reference shape functions in a number of points. More... | |
| Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | GradientsReferenceShapeFunctions (const Eigen::MatrixXd &local) const override |
| Computation of the gradients of all reference shape functions in a number of points. More... | |
| Eigen::MatrixXd | EvaluationNodes () const override |
| Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case. More... | |
| size_type | NumEvaluationNodes () const override |
| Tell the number of evaluation (interpolation) nodes. More... | |
| Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > | NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodvals) const override |
| Computes the linear combination of reference shape functions matching function values at evaluation nodes. More... | |
| ~DiscontinuousScalarReferenceFiniteElement () override=default | |
 Public Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| virtual | ~ScalarReferenceFiniteElement ()=default |
| virtual base::RefEl | RefEl () const =0 |
| Tells the type of reference cell underlying the parametric finite element. More... | |
| virtual unsigned int | Degree () const =0 |
| Request the maximal polynomial degree of the basis functions in this finite element. More... | |
| dim_t | Dimension () const |
| Returns the spatial dimension of the reference cell. More... | |
| virtual size_type | NumRefShapeFunctions () const |
| Total number of reference shape functions associated with the reference cell. More... | |
| virtual size_type | NumRefShapeFunctions (dim_t codim) const |
| The number of interior reference shape functions for sub-entities of a particular co-dimension. More... | |
| virtual size_type | NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const =0 |
| The number of interior reference shape functions for every sub-entity. More... | |
| virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0 |
| Evaluation of all reference shape functions in a number of points. More... | |
| virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0 |
| Computation of the gradients of all reference shape functions in a number of points. More... | |
| virtual Eigen::MatrixXd | EvaluationNodes () const =0 |
| Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case. More... | |
| virtual size_type | NumEvaluationNodes () const =0 |
| Tell the number of evaluation (interpolation) nodes. More... | |
| 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. More... | |
Private Attributes | |
| std::shared_ptr< const lf::fe::ScalarReferenceFiniteElement< SCALAR > > | cfe_ |
Additional Inherited Members | |
| Public Types inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR > | |
| using | Scalar = SCALAR |
| The underlying scalar type. More... | |
| 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 Functions 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. More... | |
| template<typename SCALAR > | |
| std::ostream & | operator<< (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &fe_desc) |
| Stream output operator: just calls the ScalarReferenceFiniteElement::print() method. More... | |
Detailed Description
class projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >
Decorator class around a ScalarReferenceFiniteElement to represent discontinuous shape functions.
- Template Parameters
-
SCALAR The scalar type of the shape functions e.g. 'double'
The class decorates any lf::fe::ScalarReferenceFiniteElement and forwards most calls to the decorated instance. The exception are methods requesting the number of shape functions associated with certain codimensions or subentities. Here the class changes the underlying implementation and associates all shape functions to the underlying entity of codimension 0.
In particular this class is used to represent \(L^2(\Omega) \) conforming finite elements. Standard \( H^1(\Omega) \) Lagrangian finite elements fullfill some continuity constraints, since certain shape functions are associated with vertices or edges of the cells. These continuity constraints are broken, by considering all shape functions as interior.
Definition at line 42 of file discontinuous_scalar_reference_finite_element.h.
Constructor & Destructor Documentation
◆ DiscontinuousScalarReferenceFiniteElement() [1/4]
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default |
Default constructor, does not initialize this class (invalid state). If any method is called upon it, an error is thrown.
◆ DiscontinuousScalarReferenceFiniteElement() [2/4]
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delete |
◆ DiscontinuousScalarReferenceFiniteElement() [3/4]
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defaultnoexcept |
◆ DiscontinuousScalarReferenceFiniteElement() [4/4]
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inlineexplicit |
Definition at line 60 of file discontinuous_scalar_reference_finite_element.h.
◆ ~DiscontinuousScalarReferenceFiniteElement()
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overridedefault |
virtual destructor
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 76 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ 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 102 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ EvaluationNodes()
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inlineoverridevirtual |
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 114 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ 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 108 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ isInitialized()
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inline |
Reports initialization status of the object.
Objects built by the default constructor are undefined
Definition at line 69 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_.
Referenced by projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::Degree(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::EvalReferenceShapeFunctions(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::EvaluationNodes(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::GradientsReferenceShapeFunctions(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NodalValuesToDofs(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NumEvaluationNodes(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions(), and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::RefEl().
◆ 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 124 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ NumEvaluationNodes()
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inlineoverridevirtual |
Tell the number of evaluation (interpolation) nodes.
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 119 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ NumRefShapeFunctions() [1/3]
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inlineoverridevirtual |
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 81 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ NumRefShapeFunctions() [2/3]
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inlineoverridevirtual |
Associates all shape functions to the underlying entity of codim 0
Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 88 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ NumRefShapeFunctions() [3/3]
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inlineoverridevirtual |
Associates all shape functions to the underlying entity of codim 0
Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.
Definition at line 95 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
◆ operator=() [1/2]
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delete |
◆ 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 71 of file discontinuous_scalar_reference_finite_element.h.
References projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::cfe_, and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized().
Member Data Documentation
◆ cfe_
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private |
The underlying (continuous) scalar-valued paramteric finite element
Definition at line 135 of file discontinuous_scalar_reference_finite_element.h.
Referenced by projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::Degree(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::EvalReferenceShapeFunctions(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::EvaluationNodes(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::GradientsReferenceShapeFunctions(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::isInitialized(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NodalValuesToDofs(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NumEvaluationNodes(), projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions(), and projects::dpg::DiscontinuousScalarReferenceFiniteElement< SCALAR >::RefEl().
The documentation for this class was generated from the following file:
- /home/nico/bildung/SemVI/thesis/lehrfempp/projects/dpg/discontinuous_scalar_reference_finite_element.h
