HybridCore

Classes providing cell and face quadrature rules, and values of basis functions at the nodes. More...

Classes
class	HArDCore3D::ElementQuad
class	HArDCore3D::UVector
class	HArDCore3D::HybridCore

Typedefs
typedef Family< MonomialScalarBasisCell >	HArDCore3D::HybridCore::PolyCellBasisType
	type for cell basis
typedef Family< MonomialScalarBasisFace >	HArDCore3D::HybridCore::PolyFaceBasisType
	type for face basis
typedef Family< MonomialScalarBasisEdge >	HArDCore3D::HybridCore::PolyEdgeBasisType
	type for edge basis

Functions
	HArDCore3D::ElementQuad::ElementQuad (const HybridCore &hho, const size_t iT, const size_t doeT, const size_t doeF)
	Class constructor: loads the quadrature rules and values of basis functions/gradients at these points.
const QuadratureRule &	HArDCore3D::ElementQuad::get_quadT () const
	Returns quadrature rules in cell.
const QuadratureRule &	HArDCore3D::ElementQuad::get_quadF (size_t ilF) const
	Returns quadrature rules on face with local number ilF.
const boost::multi_array< double, 2 > &	HArDCore3D::ElementQuad::get_phiT_quadT () const
	Returns values of cell basis functions at cell quadrature nodes.
const boost::multi_array< VectorRd, 2 > &	HArDCore3D::ElementQuad::get_dphiT_quadT () const
	Returns values of gradients of cell basis functions at cell quadrature nodes.
const boost::multi_array< double, 2 > &	HArDCore3D::ElementQuad::get_phiT_quadF (size_t ilF) const
	Returns values of cell basis functions at face quadrature nodes, for face with local number ilF.
const boost::multi_array< double, 2 >	HArDCore3D::ElementQuad::get_phiF_quadF (size_t ilF) const
	Returns values of face basis functions at face quadrature nodes, for face with local number ilF.
const boost::multi_array< VectorRd, 2 > &	HArDCore3D::ElementQuad::get_dphiT_quadF (size_t ilF) const
	Returns values of gradients of cell basis functions at face quadrature nodes, for face with local number ilF.
boost::multi_array< VectorRd, 2 >	HArDCore3D::ElementQuad::get_vec_phiT_quadT (size_t degree) const
	Builds on the fly the values of vector cell basis functions at cell quadrature nodes. The vector basis is obtained by tensorization of the scalar one: \((\phi_1,0,0), (\phi_2,0,0), ..., (\phi_N,0,0), (0,\phi_1,0), (0,\phi_2,0) ... (0,\phi_N,0), (0,0,\phi_1) ... (0,0,\phi_N)\).
boost::multi_array< VectorRd, 2 >	HArDCore3D::ElementQuad::get_vec_phiT_quadF (size_t ilF, size_t degree) const
	Builds on the fly the values of vector cell basis functions at face quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.
boost::multi_array< VectorRd, 2 >	HArDCore3D::ElementQuad::get_vec_phiF_quadF (size_t ilF, size_t degree) const
	Builds on the fly the values of vector face basis functions at face quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.
template<typename GeometricSupport >
size_t const	HArDCore3D::DimPoly (int m)
template<>
const size_t	HArDCore3D::DimPoly< Cell > (const int m)
	Compute the size of the basis of 3-variate polynomials up to degree m.
template<>
const size_t	HArDCore3D::DimPoly< Face > (const int m)
	Compute the size of the basis of 2-variate polynomials up to degree m.
template<>
const size_t	HArDCore3D::DimPoly< Edge > (const int m)
	Compute the size of the basis of 1-variate polynomials up to degree m.
	HArDCore3D::UVector::UVector (const Eigen::VectorXd values, const Mesh &mesh, const int cell_deg, const size_t face_deg)
Eigen::VectorXd &	HArDCore3D::UVector::asVectorXd () const
	Return the values as an Eigen vector.
const int	HArDCore3D::UVector::get_cell_deg () const
	Return the cell degree.
const size_t	HArDCore3D::UVector::get_face_deg () const
	Return the face degree.
const size_t	HArDCore3D::UVector::n_cell_dofs () const
	Number of dofs in each cell.
const size_t	HArDCore3D::UVector::n_face_dofs () const
	Number of dofs on each face.
const size_t	HArDCore3D::UVector::n_total_cell_dofs () const
	Total number of cell dofs (in the vector, this is where the face dofs start)
Eigen::VectorXd	HArDCore3D::UVector::restr (size_t iT) const
	Extract the restriction of the unknowns corresponding to cell iT and its faces.
UVector	HArDCore3D::UVector::operator+ (const UVector &b)
	Overloads the addition: adds the coefficients.
UVector	HArDCore3D::UVector::operator- (const UVector &b)
	Overloads the subtraction: subtracts the coefficients.
double	HArDCore3D::UVector::operator() (size_t index) const
	Overloads the (): returns the corresponding coefficient.
	HArDCore3D::HybridCore::HybridCore (const Mesh *mesh_ptr, const int cell_deg, const size_t face_deg, const int edge_deg, const bool use_threads=true, std::ostream &output=std::cout, const bool ortho=true)
	Class constructor: initialises the data structure with the given mesh, and desired polynomial degrees of the basis functions.
const Mesh *	HArDCore3D::HybridCore::get_mesh () const
	Returns a pointer to the mesh.
const int	HArDCore3D::HybridCore::CellDegree () const
	Return the degree of cell polynomials.
const int	HArDCore3D::HybridCore::CellDegreePos () const
const size_t	HArDCore3D::HybridCore::FaceDegree () const
	Return the degree of face polynomials.
const PolyCellBasisType &	HArDCore3D::HybridCore::CellBasis (size_t iT) const
	Return cell basis for element with global index iT.
const PolyFaceBasisType &	HArDCore3D::HybridCore::FaceBasis (size_t iF) const
	Return face basis for face with global index iF.
const PolyEdgeBasisType &	HArDCore3D::HybridCore::EdgeBasis (size_t iE) const
	Return edge basis for edge with global index iE.
double	HArDCore3D::HybridCore::L2norm (const UVector &Xh) const
	Compute L2 norm of a discrete function (using cell values)
double	HArDCore3D::HybridCore::H1norm (const UVector &Xh) const
	Compute discrete H1 norm of a discrete function.
template<typename ContinuousFunction >
UVector	HArDCore3D::HybridCore::interpolate (const ContinuousFunction &f, const int deg_cell, const size_t deg_face, size_t doe) const
	Compute the interpolant in the discrete space of a continuous function.
Eigen::VectorXd	HArDCore3D::HybridCore::compute_weights (size_t iT) const
	Computes the weights to get cell values from face values when l=-1.
double	HArDCore3D::HybridCore::evaluate_in_cell (const UVector Xh, size_t iT, VectorRd x) const
	Evaluates a discrete function in the cell iT at point x.
double	HArDCore3D::HybridCore::evaluate_in_face (const UVector Xh, size_t iF, VectorRd x) const
	Evaluates a discrete function on the face iF at point x.
Eigen::VectorXd	HArDCore3D::HybridCore::VertexValues (const UVector Xh, const std::string from_dofs)
	From a hybrid function, computes a vector of values at the vertices of the mesh.

Detailed Description

Classes providing cell and face quadrature rules, and values of basis functions at the nodes.

Classes providing tools to implement schemes having polynomial unknowns in the cells, on the faces, and on the edges.

Typedef Documentation

PolyCellBasisType

typedef Family<MonomialScalarBasisCell> HArDCore3D::HybridCore::PolyCellBasisType

type for cell basis

PolyEdgeBasisType

typedef Family<MonomialScalarBasisEdge> HArDCore3D::HybridCore::PolyEdgeBasisType

type for edge basis

PolyFaceBasisType

typedef Family<MonomialScalarBasisFace> HArDCore3D::HybridCore::PolyFaceBasisType

type for face basis

Function Documentation

asVectorXd()

Eigen::VectorXd & HArDCore3D::UVector::asVectorXd ( ) const

inline

Return the values as an Eigen vector.

CellBasis()

const PolyCellBasisType & HArDCore3D::HybridCore::CellBasis ( size_t  iT ) const

inline

Return cell basis for element with global index iT.

CellDegree()

const int HArDCore3D::HybridCore::CellDegree ( ) const

inline

Return the degree of cell polynomials.

CellDegreePos()

const int HArDCore3D::HybridCore::CellDegreePos ( ) const

inline

compute_weights()

Eigen::VectorXd HybridCore::compute_weights

(

size_t

iT

)

const

Computes the weights to get cell values from face values when l=-1.

DimPoly()

template<typename GeometricSupport >

size_t const HArDCore3D::DimPoly ( int  m )

inline

DimPoly< Cell >()

template<>

const size_t HArDCore3D::DimPoly< Cell > ( const int  m )

inline

Compute the size of the basis of 3-variate polynomials up to degree m.

Parameters

m	Polynomial degree

DimPoly< Edge >()

template<>

const size_t HArDCore3D::DimPoly< Edge > ( const int  m )

inline

Compute the size of the basis of 1-variate polynomials up to degree m.

Parameters

m	Polynomial degree

DimPoly< Face >()

template<>

const size_t HArDCore3D::DimPoly< Face > ( const int  m )

inline

Compute the size of the basis of 2-variate polynomials up to degree m.

Parameters

m	Polynomial degree

EdgeBasis()

const PolyEdgeBasisType & HArDCore3D::HybridCore::EdgeBasis ( size_t  iE ) const

inline

Return edge basis for edge with global index iE.

ElementQuad()

ElementQuad::ElementQuad	(	const HybridCore &	hho,
		const size_t	iT,
		const size_t	doeT,
		const size_t	doeF
	)

Class constructor: loads the quadrature rules and values of basis functions/gradients at these points.

Parameters

hho	A reference to the hybridcore instance
iT	Global index of cell
doeT	The degree of exactness for cell quadratures
doeF	The degree of exactness of face quadratures

evaluate_in_cell()

double HybridCore::evaluate_in_cell	(	const UVector	Xh,
		size_t	iT,
		VectorRd	x
	)		const

Evaluates a discrete function in the cell iT at point x.

evaluate_in_face()

double HybridCore::evaluate_in_face	(	const UVector	Xh,
		size_t	iF,
		VectorRd	x
	)		const

Evaluates a discrete function on the face iF at point x.

FaceBasis()

const PolyFaceBasisType & HArDCore3D::HybridCore::FaceBasis ( size_t  iF ) const

inline

Return face basis for face with global index iF.

FaceDegree()

const size_t HArDCore3D::HybridCore::FaceDegree ( ) const

inline

Return the degree of face polynomials.

get_cell_deg()

const int HArDCore3D::UVector::get_cell_deg ( ) const

inline

Return the cell degree.

get_dphiT_quadF()

const boost::multi_array< VectorRd, 2 > & HArDCore3D::ElementQuad::get_dphiT_quadF ( size_t  ilF ) const

inline

Returns values of gradients of cell basis functions at face quadrature nodes, for face with local number ilF.

get_dphiT_quadT()

const boost::multi_array< VectorRd, 2 > & HArDCore3D::ElementQuad::get_dphiT_quadT ( ) const

inline

Returns values of gradients of cell basis functions at cell quadrature nodes.

get_face_deg()

const size_t HArDCore3D::UVector::get_face_deg ( ) const

inline

Return the face degree.

get_mesh()

const Mesh * HArDCore3D::HybridCore::get_mesh ( ) const

inline

Returns a pointer to the mesh.

get_phiF_quadF()

const boost::multi_array< double, 2 > HArDCore3D::ElementQuad::get_phiF_quadF ( size_t  ilF ) const

inline

Returns values of face basis functions at face quadrature nodes, for face with local number ilF.

get_phiT_quadF()

const boost::multi_array< double, 2 > & HArDCore3D::ElementQuad::get_phiT_quadF ( size_t  ilF ) const

inline

Returns values of cell basis functions at face quadrature nodes, for face with local number ilF.

get_phiT_quadT()

const boost::multi_array< double, 2 > & HArDCore3D::ElementQuad::get_phiT_quadT ( ) const

inline

Returns values of cell basis functions at cell quadrature nodes.

get_quadF()

const QuadratureRule & HArDCore3D::ElementQuad::get_quadF ( size_t  ilF ) const

inline

Returns quadrature rules on face with local number ilF.

get_quadT()

const QuadratureRule & HArDCore3D::ElementQuad::get_quadT ( ) const

inline

Returns quadrature rules in cell.

get_vec_phiF_quadF()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiF_quadF	(	size_t	ilF,
		size_t	degree
	)		const

Builds on the fly the values of vector face basis functions at face quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

local number of face required degree of basis function

get_vec_phiT_quadF()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiT_quadF	(	size_t	ilF,
		size_t	degree
	)		const

Builds on the fly the values of vector cell basis functions at face quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

local number of face maximum degree of basis functions required

get_vec_phiT_quadT()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiT_quadT

(

size_t

degree

)

const

Builds on the fly the values of vector cell basis functions at cell quadrature nodes. The vector basis is obtained by tensorization of the scalar one: \((\phi_1,0,0), (\phi_2,0,0), ..., (\phi_N,0,0), (0,\phi_1,0), (0,\phi_2,0) ... (0,\phi_N,0), (0,0,\phi_1) ... (0,0,\phi_N)\).

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

maximal degree of basis functions that is required

H1norm()

double HybridCore::H1norm

(

const UVector &

Xh

)

const

Compute discrete H1 norm of a discrete function.

Parameters

Xh	Vector of unknowns

HybridCore()

HybridCore::HybridCore	(	const Mesh *	mesh_ptr,
		const int	cell_deg,
		const size_t	face_deg,
		const int	edge_deg,
		const bool	use_threads = true,
		std::ostream &	output = std::cout,
		const bool	ortho = true
	)

Class constructor: initialises the data structure with the given mesh, and desired polynomial degrees of the basis functions.

The orthonormalisation comes at a cost in terms of manipulation of the basis functions. This should only be used when the polynomial degree is large and/or the cell is distorted. However, in these cases, it can make a huge difference on the observed convergence rate.

Parameters

mesh_ptr	A pointer to the loaded mesh
cell_deg	The degree of the cell polynomials
face_deg	The degree of the face polynomials
edge_deg	The degree of the edge polynomials
use_threads	Optional argument to indicate if threads should be used
output	Optional argument for specifying outputs of messages
ortho	Optional argument for choosing to orthonormalise or not the basis functions

interpolate()

template<typename ContinuousFunction >

UVector HArDCore3D::HybridCore::interpolate	(	const ContinuousFunction &	f,
		const int	deg_cell,
		const size_t	deg_face,
		size_t	doe
	)		const

Compute the interpolant in the discrete space of a continuous function.

Returns

vector Xh of coefficients on the basis functions, as described in class UVector.

Parameters

f	function to interpolate
deg_cell	degree of cell polynomials for interpolation
deg_face	degree of face polynomials for interpolation
doe	degree of exactness of the quadrature rules to compute the interpolate

L2norm()

double HybridCore::L2norm

(

const UVector &

Xh

)

const

Compute L2 norm of a discrete function (using cell values)

Parameters

Xh	Vector of unknowns

n_cell_dofs()

const size_t HArDCore3D::UVector::n_cell_dofs ( ) const

inline

Number of dofs in each cell.

n_face_dofs()

const size_t HArDCore3D::UVector::n_face_dofs ( ) const

inline

Number of dofs on each face.

n_total_cell_dofs()

const size_t HArDCore3D::UVector::n_total_cell_dofs ( ) const

inline

Total number of cell dofs (in the vector, this is where the face dofs start)

operator()()

double HArDCore3D::UVector::operator() ( size_t  index ) const

inline

Overloads the (): returns the corresponding coefficient.

operator+()

UVector HArDCore3D::UVector::operator+ ( const UVector &  b )

inline

Overloads the addition: adds the coefficients.

operator-()

UVector HArDCore3D::UVector::operator- ( const UVector &  b )

inline

Overloads the subtraction: subtracts the coefficients.

restr()

Eigen::VectorXd UVector::restr

(

size_t

iT

)

const

Extract the restriction of the unknowns corresponding to cell iT and its faces.

UVector()

UVector::UVector	(	const Eigen::VectorXd	values,
		const Mesh &	mesh,
		const int	cell_deg,
		const size_t	face_deg
	)

Parameters

values	values of the vector
mesh	reference to the mesh
cell_deg	polynomial degrees in cell
face_deg	polynomial degrees on faces

VertexValues()

Eigen::VectorXd HybridCore::VertexValues	(	const UVector	Xh,
		const std::string	from_dofs
	)

From a hybrid function, computes a vector of values at the vertices of the mesh.

Parameters

Xh	vector of discrete unknowns on cell and face polynomials
from_dofs	Type of unknowns to use: "cell" or "face"