Class representing a mixed finite element. More...

#include <FiniteElement.h>

Inheritance diagram for MxFiniteElement:

Collaboration diagram for MxFiniteElement:

[legend]

Public Member Functions
	MxFiniteElement (const std::vector< size_t > &n, size_t ip=0)
	The constructor initializes the size of each basis.

size_t	getNoBasis () const override
	Returns the number of bases.

const Vector &	basis (char b) const override
	Returns a const reference to the basis function values.

Vector &	basis (char b) override
	Returns a reference to the basis function values.

const Matrix &	grad (char b) const override
	Returns a const reference to the basis function derivatives.

const Matrix3D &	hess (char b) const override
	Returns a const reference to the basis function 2nd-derivatives.

const Matrix4D &	hess2 (char b) const override
	Returns a const reference to the basis function 3rd-derivatives.

bool	Jacobian (Matrix &Jac, const Matrix &Xnod, unsigned short int gBasis, const BasisValuesPtrs &bfs)
	Sets up the Jacobian matrix of the coordinate mapping. More...

bool	Jacobian (Matrix &Jac, Vec3 &n, const Matrix &Xnod, unsigned short int gBasis, const BasisValuesPtrs &bfs, size_t t1, size_t t2, size_t nBasis=0, const Matrix *Xnod2=nullptr)
	Sets up the Jacobian matrix of the coordinate mapping on a boundary. More...

bool	Hessian (Matrix3D &Hess, const Matrix &Jac, const Matrix &Xnod, unsigned short int gBasis, const BasisValuesPtrs &bfs)
	Sets up the Hessian matrix of the coordinate mapping. More...

void	piolaMapping (const double detJ, const Matrix &Ji, const Matrix &Xnod, const BasisValuesPtrs &bfs)
	Calculates the Piola basis functions and their derivatives. More...

void	piolaBasis (const double detJ, const Matrix &J)
	Calculates the Piola basis functions. More...

Public Member Functions inherited from FiniteElement
	FiniteElement (size_t n=0, size_t i=0)
	Default constructor.

Public Member Functions inherited from ItgPoint
	ItgPoint (size_t i=0)
	Default constructor.

	ItgPoint (double a, double b=0.0, double c=0.0, size_t i=0)
	Constructor initializing the spline domain parameters.

	ItgPoint (const double *par, size_t i=0)
	Alternative constructor initializing the spline domain parameters.

virtual	~ItgPoint ()
	Empty destructor.

Protected Member Functions
void	piolaGradient (const double detJ, const Matrix &J, const Matrix &Ji, const Matrix &Xnod, const BasisValuesPtrs &bfs)
	Calculates the Piola derivatives. More...

Matrix &	grad (char b) override
	Returns a reference to the basis function derivatives.

Matrix3D &	hess (char b) override
	Returns a reference to the basis function 2nd-derivatives.

std::ostream &	write (std::ostream &os) const override
	Writes the finite element object to the given output stream.

Private Attributes
std::vector< Vector >	M
	Basis function values.

std::vector< Matrix >	dMdX
	First derivatives of the basis functions.

std::vector< Matrix3D >	d2MdX2
	Second derivatives of the basis functions.

std::vector< Matrix4D >	d3MdX3
	Third derivatives of the basis functions.

Additional Inherited Members
Public Attributes inherited from FiniteElement
double	detJxW
	Weighted determinant of the coordinate mapping.

Vector	N
	Basis function values.

Matrix	dNdX
	First derivatives (gradient) of the basis functions.

Matrix3D	d2NdX2
	Second derivatives of the basis functions.

Matrix4D	d3NdX3
	Third derivatives of the basis functions.

Matrix	G
	Covariant basis / Matrix used for stabilized methods.

Matrix	H
	Hessian.

Matrix	P
	Matrix holding Piola-mapped basis function values. More...

Matrix	dPdX
	Matrix holding Piola-mapped basis derivatives. More...

short int	p
	Polynomial order of the basis in u-direction.

short int	q
	Polynomial order of the basis in v-direction.

short int	r
	Polynomial order of the basis in r-direction.

double	age
	Time since birth of this element.

double	h
	Characteristic element size/diameter.

Vec3Vec	XC
	Array with element corner coordinate vectors.

Vector	Navg
	Volume-averaged basis function values.

Matrix	Xn
	Matrix of element nodal coordinates.

Tensor	Te
	Local-to-global element transformation matrix.

std::vector< Tensor >	Tn
	Array of element nodal rotation matrices.

Vec3Vec	En
	Array of nodal eccentricity vectors.

Public Attributes inherited from ItgPoint
size_t	iGP
	Global integration point counter.

double	u
	First spline parameter of the point.

double	v
	Second spline parameter of the point.

double	w
	Third spline parameter of the point.

int	iel
	Identifier of the element containing this point.

size_t	idx
	Global index (0-based) of the element containing this point.

double	xi
	First local coordinate within current element.

double	eta
	Second local coordinate within current element.

double	zeta
	Third local coordinate within current element.

Detailed Description

Class representing a mixed finite element.

Member Function Documentation

◆ Hessian()

bool MxFiniteElement::Hessian	(	Matrix3D &	Hess,
		const Matrix &	Jac,
		const Matrix &	Xnod,
		unsigned short int	gBasis,
		const BasisValuesPtrs &	bfs
	)

Sets up the Hessian matrix of the coordinate mapping.

Parameters

[out]	Hess	The Hessian matrix
[in]	Jac	The inverse of the Jacobian matrix
[in]	Xnod	Matrix of element nodal coordinates
[in]	gBasis	1-based index of basis representing the geometry
[in]	bfs	Basis function derivatives

This method also calculates the second-derivatives of the basis functions with respect to the Cartesian coordinates, using the same geometry mapping for all bases.

References getNoBasis(), grad(), hess(), utl::Hessian(), and utl::matrix3d< T >::multiply().

Referenced by ASMs2Dmx::integrate(), ASMs3Dmx::integrate(), ASMu2Dmx::integrate(), and ASMu3Dmx::integrate().

◆ Jacobian() [1/2]

bool MxFiniteElement::Jacobian	(	Matrix &	Jac,
		const Matrix &	Xnod,
		unsigned short int	gBasis,
		const BasisValuesPtrs &	bfs
	)

Sets up the Jacobian matrix of the coordinate mapping.

Parameters

[out]	Jac	The inverse of the Jacobian matrix
[in]	Xnod	Matrix of element nodal coordinates
[in]	gBasis	1-based index of basis representing the geometry
[in]	bfs	Basis function values and derivatives

This method also calculates the first-derivatives of the basis functions with respect to the Cartesian coordinates, using the same geometry mapping for all bases.

References FiniteElement::detJxW, getNoBasis(), grad(), utl::Jacobian(), and utl::matrix< T >::multiply().

Referenced by ASMs2DmxLag::evalSolution(), ASMs3DmxLag::evalSolution(), ASMs2Dmx::evalSolution(), ASMs3Dmx::evalSolution(), ASMs2Dmx::integrate(), ASMs2DmxLag::integrate(), ASMs3Dmx::integrate(), ASMs3DmxLag::integrate(), ASMu2Dmx::integrate(), and ASMu3Dmx::integrate().

◆ Jacobian() [2/2]

bool MxFiniteElement::Jacobian	(	Matrix &	Jac,
		Vec3 &	n,
		const Matrix &	Xnod,
		unsigned short int	gBasis,
		const BasisValuesPtrs &	bfs,
		size_t	t1,
		size_t	t2,
		size_t	nBasis = `0`,
		const Matrix *	Xnod2 = `nullptr`
	)

Sets up the Jacobian matrix of the coordinate mapping on a boundary.

Parameters

[out]	Jac	The inverse of the Jacobian matrix
[out]	n	Outward-directed unit normal vector on the boundary
[in]	Xnod	Matrix of element nodal coordinates
[in]	gBasis	1-based index of basis representing the geometry
[in]	bfs	Derivatives of basis functions
[in]	t1	First parametric tangent direction of the boundary
[in]	t2	Second parametric tangent direction of the boundary
[in]	nBasis	Number of basis functions
[in]	Xnod2	Matrix of element nodal coordinates for neighbor element

If the input argument nBasis is half (or less than half) of the size of the internal basis function value array, it is used to flag that we are doing element interface terms, and the basis function values of both elements sharing the interface are stored internally. If nBasis is zero (default), it is reset to the size of the dNxdu argument, which will fit normal cases.

References FiniteElement::detJxW, getNoBasis(), grad(), utl::Jacobian(), and utl::matrix< T >::multiply().

◆ piolaBasis()

void MxFiniteElement::piolaBasis	(	const double	detJ,
		const Matrix &	J
	)

Calculates the Piola basis functions.

Parameters

[in]	detJ	Determinant of Jacobian of the geometry mapping
[in]	J	The Jacobian of the geometry mapping

References basis(), utl::matrix< T >::multiply(), FiniteElement::P, utl::matrix< T >::rows(), and utl::vector< T >::size().

Referenced by ASMs2Dmx::evalSolutionPiola(), ASMu2Dmx::evalSolutionPiola(), and piolaMapping().

◆ piolaGradient()

void MxFiniteElement::piolaGradient	(	const double	detJ,
		const Matrix &	J,
		const Matrix &	Ji,
		const Matrix &	Xnod,
		const BasisValuesPtrs &	bfs
	)

protected

Calculates the Piola derivatives.

Parameters

[in]	detJ	Determinant of Jacobian of the geometry mapping
[in]	J	Jacobian of the geometry mapping
[in]	Ji	Inverse jacobian of the geometry mapping
[in]	Xnod	Matrix of element nodal coordinates
[in]	bfs	Derivatives of basis functions

References basis(), utl::detJacGradient(), FiniteElement::dPdX, utl::matrix< T >::getColumn(), FiniteElement::H, utl::JacobianGradient(), utl::matrix< T >::multiply(), utl::matrix< T >::resize(), utl::matrix< T >::rows(), and utl::vector< T >::size().

Referenced by piolaMapping().

◆ piolaMapping()

void MxFiniteElement::piolaMapping	(	const double	detJ,
		const Matrix &	Ji,
		const Matrix &	Xnod,
		const BasisValuesPtrs &	bfs
	)

Calculates the Piola basis functions and their derivatives.

Parameters

[in]	detJ	Determinant of Jacobian of the geometry mapping
[in]	Ji	Inverse jacobian of the geometry mapping
[in]	Xnod	Matrix of element nodal coordinates
[in]	bfs	Derivatives of basis functions

References utl::matrix< T >::multiply(), piolaBasis(), and piolaGradient().

Referenced by ASMs2Dmx::evalSolution(), ASMs2Dmx::integrate(), and ASMu2Dmx::integrate().

The documentation for this class was generated from the following files:

src/ASM/FiniteElement.h
src/ASM/FiniteElement.C

Public Member Functions

Protected Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ Hessian()

◆ Jacobian() [1/2]

◆ Jacobian() [2/2]

◆ piolaBasis()

◆ piolaGradient()

◆ piolaMapping()