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KASKADE 7 development version
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Management of degrees of freedom. More...
#include <concepts.hh>
Management of degrees of freedom.
LocalToGlobalMappers define how a global FE coefficient vector is to be interpreted as a FE function in terms of shape functions on each cell of the grid.
Finite element functions are given by piecewise linear combinations of shape functions on cells. On each cell \( T \), a finite element function \( u \) is given as
\[ u|_T(x) = \sum_{j=1}^n (K a_{I})_j \psi(x) \phi_{j}(x) = \psi(x) (\phi^T(x) K a). \]
Here, \( a \) denotes the global coefficient vector, \( I \in {\bf N}^m, \, m\le n, \) is the set of \( m \) global degrees of freedom that is associated to this cell, and \( K\in {\bf R}^{n\times m} \) is a matrix (called combiner) describing which linear combination of the \( n \) shape functions yields which global ansatz function (FE basis function) restricted to the cell:
\[ (\varphi|_T)_i = \sum_{j=1}^n K_{ji} (\psi\phi)_j \]
Here, the index \( i \) of \( (\varphi|_T)_i \) is local to the cell \( T \), and thus applies to the restriction \( \varphi|_T \) and not only to \( \varphi \).
Often, \( K \) is simply the identity, or at least diagonal (e.g. for edge elements, where the global orientation of edges has to be taken into account), but it may be non-square and arbitrarily dense (e.g. for hp-methods or hanging nodes). \( \psi(x) \) is a linear geometric transformation of shape function values (and gradients). Often this is the identity, but for vectorial shape functions it may depend on the Jacobian at the evaluation point inside the cell (e.g. for edge elements). Even in the most trivial case, \( \psi \) transforms at least the shape function gradients due to the chain rule. \(\phi\) are the shape functions themselves. They may be scalar or vectorial. Note that vectorial finite element functions can perfectly well be defined in terms of scalar shape functions: Each component is defined in terms of the scalar shape functions. Often the set of shape functions is the same on each cell, but this need not be the case (e.g. for hp-methods).
Working with sets of evaluation points \( x_i \), the representation formula may conveniently be written as
\[ u|_T = \Psi \Phi K a_I, \]
with \( u \) being the column vector of values \( u(x_i)\), \(\Psi = \mathrm{diag}(\psi(x_i))\), and \( \Phi_{ij} = \phi_j(x_i) \).
LocalToGlobalMappers define the ingredients for the above representation. Their interface is described here.
Definition at line 70 of file concepts.hh.
Public Types | |
| typedef unspecified | Scalar |
| The type of scalars used for shape function values. More... | |
| using | GridView = unspecified |
| The grid view on which the finite element functions are defined. More... | |
| typedef unspecified | Grid |
| The grid type of the grid view. More... | |
| typedef unspecified | IndexSet |
| The entity indexing of the grid view. More... | |
| typedef unspecified | ShapeFunctionSet |
| The type of shape function sets living on the cells. More... | |
| typedef unspecified | GlobalIndexRange |
| A standard sequence with value type IndexSet::IndexType. More... | |
| typedef unspecified | SortedIndexRange |
| A standard sequence with value type std::pair<IndexSet::IndexType,int>. More... | |
| typedef ConverterConcept | Converter |
| The type of the geometrical transformation \( \psi \). More... | |
| typedef CombinerConcept | Combiner |
| The type of the algebraic combiner \( K \) associated with this cell. More... | |
| using | Cell = Kaskade::Cell< Grid > |
| The type of codimension 0 entities in the grid. More... | |
Public Member Functions | |
| GridView const & | gridView () const |
| Returns the grid view used. More... | |
| ShapeFunctionSet const & | shapefunctions (Cell const &cell) const |
| Returns the shape function set living on the given cell. More... | |
| ShapeFunctionSet const & | shapefunctions (size_t index) const |
| Returns the shape function set living on the cell with given index. More... | |
| size_t | size () const |
| Returns the number of global degrees of freedom managed. More... | |
| int | maxOrder () const |
| Returns the maximal polynomial order of shape functions encountered in any cell. More... | |
| SortedIndexRange | sortedIndices (Cell const &cell) const |
| Returns an immutable sequence of (global index, local index) pairs sorted in ascending global index order. More... | |
| SortedIndexRange | sortedIndices (size_t n) const |
| Returns an immutable sequence of (global index, local index) pairs sorted in ascending global index order. More... | |
| GlobalIndexRange | globalIndices (Cell const &cell) const |
| The global indices \( I \) associated with this cell. Returns a const sequence containing the global indices of the ansatz functions with nonvanishing support on this cell. More... | |
| GlobalIndexRange | globalIndices (size_t index) const |
| The global indices \( I \) associated with the cell with given index. Returns a const sequence containing the global indices of the ansatz functions with nonvanishing support on this cell. More... | |
| Combiner | combiner (Cell const &cell, size_t index) const |
| Returns the algebraic combinator. More... | |
| void | update () |
| Recomputes the node management, e.g. after grid modifications. More... | |
The type of codimension 0 entities in the grid.
Definition at line 109 of file concepts.hh.
The type of the algebraic combiner \( K \) associated with this cell.
This type represents a matrix of small to moderate size. For performance reasons, it is a class of its own with special interface.
Definition at line 106 of file concepts.hh.
The type of the geometrical transformation \( \psi \).
Definition at line 99 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::GlobalIndexRange |
A standard sequence with value type IndexSet::IndexType.
Definition at line 91 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::Grid |
The grid type of the grid view.
Definition at line 80 of file concepts.hh.
| using LocalToGlobalMapperConcept::GridView = unspecified |
The grid view on which the finite element functions are defined.
Definition at line 77 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::IndexSet |
The entity indexing of the grid view.
Definition at line 83 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::Scalar |
The type of scalars used for shape function values.
Definition at line 74 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::ShapeFunctionSet |
The type of shape function sets living on the cells.
Definition at line 86 of file concepts.hh.
| typedef unspecified LocalToGlobalMapperConcept::SortedIndexRange |
A standard sequence with value type std::pair<IndexSet::IndexType,int>.
Definition at line 96 of file concepts.hh.
Returns the algebraic combinator.
| cell | the grid cell for which the combiner is requested |
| index | the index of the cell |
| GlobalIndexRange LocalToGlobalMapperConcept::globalIndices | ( | Cell const & | cell | ) | const |
The global indices \( I \) associated with this cell. Returns a const sequence containing the global indices of the ansatz functions with nonvanishing support on this cell.
| GlobalIndexRange LocalToGlobalMapperConcept::globalIndices | ( | size_t | index | ) | const |
The global indices \( I \) associated with the cell with given index. Returns a const sequence containing the global indices of the ansatz functions with nonvanishing support on this cell.
| GridView const & LocalToGlobalMapperConcept::gridView | ( | ) | const |
Returns the grid view used.
| int LocalToGlobalMapperConcept::maxOrder | ( | ) | const |
Returns the maximal polynomial order of shape functions encountered in any cell.
| ShapeFunctionSet const & LocalToGlobalMapperConcept::shapefunctions | ( | Cell const & | cell | ) | const |
Returns the shape function set living on the given cell.
The shape function set is persistent, i.e. it exists and is valid even after mesh refinement.
| ShapeFunctionSet const & LocalToGlobalMapperConcept::shapefunctions | ( | size_t | index | ) | const |
Returns the shape function set living on the cell with given index.
| size_t LocalToGlobalMapperConcept::size | ( | ) | const |
Returns the number of global degrees of freedom managed.
| SortedIndexRange LocalToGlobalMapperConcept::sortedIndices | ( | Cell const & | cell | ) | const |
Returns an immutable sequence of (global index, local index) pairs sorted in ascending global index order.
| SortedIndexRange LocalToGlobalMapperConcept::sortedIndices | ( | size_t | n | ) | const |
Returns an immutable sequence of (global index, local index) pairs sorted in ascending global index order.
| void LocalToGlobalMapperConcept::update | ( | ) |
Recomputes the node management, e.g. after grid modifications.
1.9.4