|
KASKADE 7 development version
|
A class for representing finite element functions. More...
#include <functionspace.hh>
A class for representing finite element functions.
A class for representing finite element function space elements (also known as finite element functions). Finite element functions are piecewisely smooth functions (on each codim 0 entity, i.e. cell, of an associated grid) that may or may not be globally continuous or differentiable. FE functions support pointwise evaluation and differentiation, but on sets of measure zero (more precisely, on the codim 1 entities, i.e. faces, of the grid), these values are undefined, and calling the corresponding methods may yield arbitrary values.
| FunctionSpace | Each FunctionSpaceElement is associated with an FEFunctionSpace (it "lives" in this space). This function space is defined in the first template parameter |
| m | FunctionSpaceElements may have multiple components. The number of components attached to each shape function is given by this template parameter. Note that even with m=1, the function can be vector-valued if the shape functions themselves are vector-valued. |
Usually, FSElements are created as part of a VariableSet, which in turn is created via a VariableSetDescription, containing information about variables and their spaces (VariableDescription s). FSElements contain the data for the variables.
If a GridManager is used, FunctionSpaceElements are automatically prolongated, if the grid is refined, and approximated, if the grid is coarsened.
If a VariableSet vs is at hand, the FunctionSpaceElements can be accessed as public data-members in a boost::fusion::vector vs.data. This means: boost::fusion::at_c<N>(vs.data) gives a reference to the N'th FSElement in vs.
Simple assignments are performed by the assignment operator. For more complex specification of FE function values see spaceTransfer, interpolateGlobally, and interpolateGloballyWeak.
Definition at line 314 of file functionspace.hh.
Public Types | |
| using | Space = FunctionSpace |
| FEFunctionSpace, this function is an element of. More... | |
| using | GridView = typename Space::GridView |
| the grid view this function is defined on More... | |
| using | ctype = typename GridView::ctype |
| the scalar type used for spatial coordinates More... | |
| typedef FunctionSpaceElement< Space, m > | Self |
| own type More... | |
| typedef Space::Scalar | Scalar |
| scalar field type More... | |
| using | field_type = typename Space::field_type |
| scalar field type More... | |
| typedef Dune::FieldVector< Scalar, m > | StorageValueType |
| type of the elements of the data vector More... | |
| using | SfValueType = typename Space::SfValueType |
| type of shape function values More... | |
| using | block_type = StorageValueType |
| type of coefficient vector entries More... | |
| typedef Dune::BlockVector< StorageValueType > | StorageType |
| type of the data vector More... | |
| typedef Dune::FieldVector< Scalar, components > | ValueType |
| return type of the function value(...) More... | |
| typedef Dune::FieldMatrix< Scalar, components, Space::dimworld > | DerivativeType |
| return type of the function derivative (...) More... | |
| using | HessianType = Tensor< Scalar, components, Space::dimworld, Space::dimworld > |
| return type of the function hessian(...) More... | |
Public Member Functions | |
Constructors & Destructor | |
| FunctionSpaceElement (Space const &fs) | |
| Constructor. More... | |
| FunctionSpaceElement (Self const &fse) | |
| Copy constructor. More... | |
| ~FunctionSpaceElement () | |
Assignment | |
| Self & | operator= (Self const &fse) |
| Copy assignment. More... | |
| Self & | operator= (Self &&fse) |
| Move assignment. More... | |
| template<class OtherSpace > | |
| Self & | operator= (FunctionSpaceElement< OtherSpace, m > const &f) |
| Assignment from functions belonging to other function spaces. More... | |
| template<template< class, class > class DomainMapper> | |
| Self & | operator= (FunctionSpaceElement< FEFunctionSpace< BoundaryMapper< DomainMapper, Scalar, typename Space::GridView > >, m > const &bf) |
| Assignment from an FE function which is restricted to the boundary to a suitable FE function. More... | |
| template<class Mapper > | |
| std::enable_if_t< std::is_same< Mapper, FunctionSpace_Detail::ChooseDomainMapper< typename Space::Mapper > >::value, Self & > | operator= (FunctionSpaceElement< FEFunctionSpace< Mapper >, m > const &f) |
| Assignment from a suitable FE function to an FE function which is restricted to the boundary. More... | |
| template<class Functor > | |
| Self & | operator= (WeakFunctionViewAdaptor< Functor > const &f) |
| Assignment from a weak function view adaptor. More... | |
| template<class Space , class Functor > | |
| Self & | operator= (FunctionViewAdaptor< Space, Functor > const &f) |
| Assignment from a function view adaptor. More... | |
| Self & | operator= (StorageType const &v) |
| Assignment from vector of coefficients. More... | |
| Self & | operator= (StorageValueType const &a) |
| Sets all coefficients to the given value. As StorageValueType is a vector one value for each component can be specified. More... | |
| Self & | operator= (Scalar val) |
| Sets all coefficients to the given value. More... | |
Basic linear algebra | |
| Self & | operator+= (Self const &f) |
| Addition of a function from the same (type of) space. Note that despite the same type the function spaces may actually be different (e.g. different polynomial order). In this case, interpolation is employed. More... | |
| template<class OtherSpace > | |
| Self & | operator+= (FunctionSpaceElement< OtherSpace, m > const &f) |
| Addition of a function from a different function space, but with the same number of components. More... | |
| Self & | operator+= (StorageType const &v) |
| Addition of coefficient vectors. More... | |
| Self & | operator+= (Scalar val) |
| Addition of a constant value to the coefficient vector. Note that this does in general not add a constant function, except for the notable case of Lagrange finite elements. More... | |
| Self & | operator-= (Self const &f) |
| Subtraction of a function from the same (type of) space. More... | |
| template<class OtherSpace > | |
| Self & | operator-= (FunctionSpaceElement< OtherSpace, m > const &f) |
| Subtraction of a function from a different function space, but with the same number of components. More... | |
| Self & | operator-= (StorageType const &v) |
| Subtraction of coefficient vectors. More... | |
| Self & | axpy (Scalar a, Self const &f) |
| Computes \( y \leftarrow ax + y \) for a function x from the same (type of) space. Note that despite the same type the function spaces may actually be different (e.g. different polynomial order). In this case, interpolation is employed. More... | |
| template<class OtherSpace > | |
| Self & | axpy (Scalar a, FunctionSpaceElement< OtherSpace, m > const &f) |
| Computes \( y \leftarrow ax + y \) for a function x from a different function space, but with the same number of components. More... | |
| Self & | axpy (Scalar a, StorageType const &v) |
| Computes \( y \leftarrow av + y \) for coefficient vectors. More... | |
| Self & | operator*= (Scalar a) |
| Scaling. More... | |
Function and derivative evaluation | |
| ValueType | value (typename Space::Evaluator const &evaluator) const |
| Evaluates the FE function at the position used to construct the evaluator. More... | |
| ValueType | value (Cell< GridView > const &cell, LocalPosition< GridView > const &local) const |
| Evaluates the FE function at the specified local position in the given cell. More... | |
| ValueType | value (GlobalPosition< GridView > const &x) const |
| Evaluates the FE function at the specified global position. More... | |
| DerivativeType | derivative (typename FunctionSpace::Evaluator const &evaluator) const |
| Evaluates the derivative of the FE function wrt global coordinates at the position used to construct the evaluator. More... | |
| DerivativeType | derivative (Cell< GridView > const &cell, LocalPosition< GridView > const &local) const |
| Evaluates the derivative of the FE function at the given local position inside the given cell. More... | |
| DerivativeType | derivative (GlobalPosition< GridView > const &global) const |
| Evaluates the FE function's gradient at the specified global position. More... | |
| HessianType | hessian (typename FunctionSpace::Evaluator const &evaluator) const |
| Evaluates the Hessian of the FE function at the position used to construct the evaluator. More... | |
| HessianType | hessian (Cell< GridView > const &cell, LocalPosition< GridView > const &local) const |
| Evaluates the Hessian of the FE function at the given local position inside the given cell. More... | |
| HessianType | hessian (GlobalPosition< GridView > const &global) const |
| Evaluates the FE function's Hessian at the specified global position. More... | |
Access to coefficient vector | |
| StorageType const & | coefficients () const |
| Provides read-only access to the coefficients of the finite element basis functions. More... | |
| StorageType & | coefficients () |
| Provides access to the coefficients of the finite element basis functions. More... | |
| StorageValueType & | operator[] (size_t i) |
| EXPERIMENTAL Provides access to the coefficients of the finite element basis functions. More... | |
| StorageValueType const & | operator[] (size_t i) const |
| EXPERIMENTAL Provides read-only access to the coefficients of the finite element basis functions. More... | |
Miscellaneous | |
| int | dim () const |
| Number of scalar degrees of freedom. More... | |
| int | size () const |
| Number of coefficient blocks. More... | |
| Space const & | space () const |
| Provides access to the FEFunctionSpace to which this function belongs. More... | |
| int | order (Cell< GridView > const &cell) const |
| The polynomial order of the function on the provided cell. More... | |
| template<class SFS > | |
| int | order (SFS const &sfs) const |
| void | transfer (typename TransferData< Space >::TransferMatrix const &transferMatrix) |
Static Public Attributes | |
| static int const | components = Space::sfComponents * m |
| components at each point of evaluation More... | |
Protected Attributes | |
| Dune::BlockVector< StorageValueType > | data |
| Space const * | sp |
Related Functions | |
(Note that these are not member functions.) | |
| template<class Space , int m, class OutIter > | |
| OutIter | vectorToSequence (FunctionSpaceElement< Space, m > const &v, OutIter iter) |
| Writes the coefficients into a flat sequence. <Space,m> More... | |
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::block_type = StorageValueType |
type of coefficient vector entries
This is provided for compatibility with Dune::BlockVector.
Definition at line 351 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::ctype = typename GridView::ctype |
the scalar type used for spatial coordinates
Definition at line 324 of file functionspace.hh.
| typedef Dune::FieldMatrix<Scalar,components,Space::dimworld> Kaskade::FunctionSpaceElement< FunctionSpace, m >::DerivativeType |
return type of the function derivative (...)
The derivative is always computed wrt the global world coordinate system, even for functions defined on a lower dimensional manifold (e.g. in shell or plate problems). In that case, the function is extended to a neighborhood of the manifold in the canonical way - constant extension normal to the manifold. The derivative of that extension wrt the global world coordinate system is then returned. Hence, the number of entries in the derivative is always the world dimension.
Design rationale: The alternative would be to consider the derivative wrt a coordinate system of the tangent space. The choice of such a coordinate system is, however, not unique.
Definition at line 374 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::field_type = typename Space::field_type |
scalar field type
This is provided for consistency with Dune::BlockVector.
Definition at line 336 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::GridView = typename Space::GridView |
the grid view this function is defined on
Definition at line 321 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::HessianType = Tensor<Scalar,components,Space::dimworld,Space::dimworld> |
return type of the function hessian(...)
Definition at line 377 of file functionspace.hh.
| typedef Space::Scalar Kaskade::FunctionSpaceElement< FunctionSpace, m >::Scalar |
scalar field type
Definition at line 329 of file functionspace.hh.
| typedef FunctionSpaceElement<Space, m> Kaskade::FunctionSpaceElement< FunctionSpace, m >::Self |
own type
Definition at line 327 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::SfValueType = typename Space::SfValueType |
type of shape function values
Definition at line 344 of file functionspace.hh.
| using Kaskade::FunctionSpaceElement< FunctionSpace, m >::Space = FunctionSpace |
FEFunctionSpace, this function is an element of.
Definition at line 318 of file functionspace.hh.
| typedef Dune::BlockVector<StorageValueType> Kaskade::FunctionSpaceElement< FunctionSpace, m >::StorageType |
type of the data vector
Definition at line 354 of file functionspace.hh.
| typedef Dune::FieldVector<Scalar,m> Kaskade::FunctionSpaceElement< FunctionSpace, m >::StorageValueType |
type of the elements of the data vector
Definition at line 339 of file functionspace.hh.
| typedef Dune::FieldVector<Scalar,components> Kaskade::FunctionSpaceElement< FunctionSpace, m >::ValueType |
return type of the function value(...)
Definition at line 360 of file functionspace.hh.
|
inline |
Constructor.
The function is initialized to zero. Space is a FEFunctionSpace.
Definition at line 389 of file functionspace.hh.
|
inline |
Copy constructor.
Definition at line 397 of file functionspace.hh.
|
inline |
Definition at line 403 of file functionspace.hh.
|
inline |
Computes \( y \leftarrow ax + y \) for a function x from a different function space, but with the same number of components.
Here, interpolation is used.
Definition at line 720 of file functionspace.hh.
|
inline |
Computes \( y \leftarrow ax + y \) for a function x from the same (type of) space. Note that despite the same type the function spaces may actually be different (e.g. different polynomial order). In this case, interpolation is employed.
Definition at line 704 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::axpy().
|
inline |
Computes \( y \leftarrow av + y \) for coefficient vectors.
Definition at line 729 of file functionspace.hh.
|
inline |
Provides access to the coefficients of the finite element basis functions.
Definition at line 924 of file functionspace.hh.
|
inline |
Provides read-only access to the coefficients of the finite element basis functions.
Definition at line 919 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::axpy(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator+=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator-=(), Kaskade::LinearProductSpace< Scalar_, Seq >::operator=(), Kaskade::CheckPoint::readData(), Kaskade::AmiraMeshReader::readData(), Kaskade::AmiraMeshReader::readDeformation(), Kaskade::AmiraMeshReader::readDeformation2(), Kaskade::vectorFromSequence(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::vectorToSequence(), and Kaskade::CheckPoint::writeData().
|
inline |
Evaluates the derivative of the FE function at the given local position inside the given cell.
On codimension 1 entitites, the value is undefined if Space::continuity<1. The components are sorted as described for value().
Definition at line 830 of file functionspace.hh.
|
inline |
Evaluates the FE function's gradient at the specified global position.
Definition at line 845 of file functionspace.hh.
|
inline |
Evaluates the derivative of the FE function wrt global coordinates at the position used to construct the evaluator.
On codimension 1 entitites, the value is undefined if Space::continuity<1. The components are sorted as described for value().
Definition at line 807 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::derivative().
|
inline |
Number of scalar degrees of freedom.
Definition at line 946 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::size().
|
inline |
Evaluates the Hessian of the FE function at the given local position inside the given cell.
On codimension 1 entitites, the value is undefined if Space::continuity<2. The components are sorted as described for value().
Definition at line 886 of file functionspace.hh.
|
inline |
Evaluates the FE function's Hessian at the specified global position.
Definition at line 901 of file functionspace.hh.
|
inline |
Evaluates the Hessian of the FE function at the position used to construct the evaluator.
On codimension 1 entitites, the value is undefined if Space::continuity<2. The components are sorted as described for value().
Definition at line 859 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::hessian().
|
inline |
Definition at line 732 of file functionspace.hh.
|
inline |
Addition of a function from a different function space, but with the same number of components.
Here, interpolation is used.
Definition at line 638 of file functionspace.hh.
|
inline |
Addition of a constant value to the coefficient vector. Note that this does in general not add a constant function, except for the notable case of Lagrange finite elements.
Definition at line 654 of file functionspace.hh.
|
inline |
Addition of a function from the same (type of) space. Note that despite the same type the function spaces may actually be different (e.g. different polynomial order). In this case, interpolation is employed.
Definition at line 622 of file functionspace.hh.
|
inline |
Addition of coefficient vectors.
Definition at line 647 of file functionspace.hh.
|
inline |
Subtraction of a function from a different function space, but with the same number of components.
Here, interpolation of the given function to our function space is used.
Definition at line 683 of file functionspace.hh.
|
inline |
Subtraction of a function from the same (type of) space.
Note that despite the same type the function spaces may actually be different (e.g. different polynomial order). In this case, interpolation is employed.
Definition at line 667 of file functionspace.hh.
|
inline |
Subtraction of coefficient vectors.
Definition at line 692 of file functionspace.hh.
|
inline |
Assignment from an FE function which is restricted to the boundary to a suitable FE function.
If the underlying mappers are the same for both FE function, assignment can be done fast by simply copying the coefficients. Otherwise interpolation will be used. (TODO: Currently interpolateGlobally with a boundary FE function as second parameter does not work as it should. Therefore, before doing the interpolation the boundary FE function is assigned to a suitable FE function, which is then used for interpolation.)
Definition at line 484 of file functionspace.hh.
|
inline |
Assignment from a suitable FE function to an FE function which is restricted to the boundary.
If the underlying mappers are the same for both FE function, assignment can be done fast by simply copying the coefficients. Otherwise interpolation will be used. This overload will only be available if this FE function belongs to a boundary space (otherwise ChooseDomainMapper yields void which is not the same as Mapper and type substitution will fail).
Definition at line 519 of file functionspace.hh.
|
inline |
Assignment from functions belonging to other function spaces.
This assignment is defined by interpolation and hence results in pointwise equality only if the right hand side f lies inside the function space represented by our space. In cases where the coefficients of the involved spaces are just subsets (either way round), the interpolation approach is probably of suboptimal performance.
Definition at line 467 of file functionspace.hh.
|
inline |
Assignment from a function view adaptor.
| Functor | the actual callable defining the function view |
Usage example doubling an existing FE function:
Definition at line 575 of file functionspace.hh.
|
inline |
Sets all coefficients to the given value.
Note that this does in general not result in a constant function (a notable exception being Lagrange elements, where this property indeed holds).
Definition at line 607 of file functionspace.hh.
|
inline |
Move assignment.
Note that even if the type of function (and hence of the function space) is the same, the actual function spaces may be different (e.g., by different polynomial order). If different spaces are involved, assignment is just interpolation.
Definition at line 444 of file functionspace.hh.
|
inline |
Copy assignment.
Note that even if the type of function (and hence of the function space) is the same, the actual function spaces may be different (e.g., by different polynomial order). If different spaces are involved, assignment is just interpolation.
Definition at line 425 of file functionspace.hh.
|
inline |
Assignment from vector of coefficients.
The size of the provided coefficient vector has to match the size of the current coefficient vector.
Definition at line 586 of file functionspace.hh.
|
inline |
Sets all coefficients to the given value. As StorageValueType is a vector one value for each component can be specified.
Note that this does in general not result in a constant function (a notable exception being Lagrange elements, where this property indeed holds).
Definition at line 599 of file functionspace.hh.
|
inline |
Assignment from a weak function view adaptor.
| Functor | the actual callable defining the function view |
Usage example creating a conic function:
Definition at line 555 of file functionspace.hh.
|
inline |
EXPERIMENTAL Provides access to the coefficients of the finite element basis functions.
Definition at line 929 of file functionspace.hh.
|
inline |
EXPERIMENTAL Provides read-only access to the coefficients of the finite element basis functions.
Definition at line 934 of file functionspace.hh.
|
inline |
The polynomial order of the function on the provided cell.
Definition at line 964 of file functionspace.hh.
|
inline |
Definition at line 972 of file functionspace.hh.
|
inline |
Number of coefficient blocks.
This is dim()/m. Method is added for interface consistency with Dune::BlockVector.
Definition at line 954 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=().
|
inline |
Provides access to the FEFunctionSpace to which this function belongs.
Definition at line 959 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::axpy(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::derivative(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::hessian(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator+=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator-=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::order(), and Kaskade::FunctionSpaceElement< FunctionSpace, m >::value().
|
inline |
Group of members that are necessary for automatic grid transfer after refinement. The member transferMe is connected to a boost::signal in function space automatically at construction
Definition at line 982 of file functionspace.hh.
|
inline |
Evaluates the FE function at the specified local position in the given cell.
If several values on the same cell are to be calculated and performance is important, consider creating an evaluator and compute the function's values using the evaluator.
Definition at line 776 of file functionspace.hh.
|
inline |
Evaluates the FE function at the specified global position.
\WARNING This method is comparatively DEAD SLOW.
The components are sorted as described for value().
Definition at line 792 of file functionspace.hh.
|
inline |
Evaluates the FE function at the position used to construct the evaluator.
On codimension >0 entitites, the value is undefined if Space::continuity<0. For vectorial shape functions and m>1, the components are grouped by shape function component first. Example with shape functions with 3 components and m==2: [sf0m0,sf1m0,sf2m0,sf0m1,sf1m1,sf2m1]
Definition at line 752 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=(), and Kaskade::FunctionSpaceElement< FunctionSpace, m >::value().
|
static |
components at each point of evaluation
Definition at line 357 of file functionspace.hh.
|
protected |
Definition at line 1025 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::axpy(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::coefficients(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::derivative(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::dim(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::FunctionSpaceElement(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::hessian(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator*=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator+=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator-=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator[](), Kaskade::FunctionSpaceElement< FunctionSpace, m >::size(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::transfer(), and Kaskade::FunctionSpaceElement< FunctionSpace, m >::value().
|
protected |
Definition at line 1026 of file functionspace.hh.
Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::axpy(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::FunctionSpaceElement(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator+=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator-=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=(), Kaskade::FunctionSpaceElement< FunctionSpace, m >::space(), and Kaskade::FunctionSpaceElement< FunctionSpace, m >::~FunctionSpaceElement().
1.9.4