Class for Fem Element 3D based on a cube volume discretization. More...

#include <Fem3DElementCube.h>

Inheritance diagram for SurgSim::Physics::Fem3DElementCube:

Public Member Functions
	Fem3DElementCube ()
	Constructor.

	Fem3DElementCube (std::array< size_t, 8 > nodeIds)
	Constructor. More...

	Fem3DElementCube (std::shared_ptr< FemElementStructs::FemElementParameter > elementData)
	Constructor for FemElement object factory. More...

void	initialize (const SurgSim::Math::OdeState &state) override
	Initializes the element once everything has been set. More...

double	getVolume (const SurgSim::Math::OdeState &state) const override
	Gets the element volume based on the input state (in m-3) More...

SurgSim::Math::Vector	computeCartesianCoordinate (const SurgSim::Math::OdeState &state, const SurgSim::Math::Vector &naturalCoordinate) const override
	Computes a given natural coordinate in cartesian coordinates. More...

SurgSim::Math::Vector	computeNaturalCoordinate (const SurgSim::Math::OdeState &state, const SurgSim::Math::Vector &cartesianCoordinate) const override
	Computes a natural coordinate given a global coordinate. More...

Public Member Functions inherited from SurgSim::Physics::FemElement
	FemElement ()
	Constructor.

virtual	~FemElement ()
	Virtual destructor.

size_t	getNumDofPerNode () const
	Gets the number of degree of freedom per node. More...

size_t	getNumNodes () const
	Gets the number of nodes connected by this element. More...

size_t	getNodeId (size_t elementNodeId) const
	Gets the elementNodeId-th node id. More...

const std::vector< size_t > &	getNodeIds () const
	Gets the node ids for this element. More...

void	setYoungModulus (double E)
	Sets the Young modulus (in N.m-2) More...

double	getYoungModulus () const
	Gets the Young modulus (in N.m-2) More...

void	setPoissonRatio (double nu)
	Sets the Poisson ratio (unitless) More...

double	getPoissonRatio () const
	Gets the Poisson ratio (unitless) More...

void	setMassDensity (double rho)
	Sets the mass density (in Kg.m-3) More...

double	getMassDensity () const
	Gets the mass density (in Kg.m-3) More...

double	getMass (const SurgSim::Math::OdeState &state) const
	Gets the element mass based on the input state (in Kg) More...

virtual void	addForce (SurgSim::Math::Vector *F, double scale) const
	Adds the element force (computed for a given state) to a complete system force vector F (assembly) More...

virtual void	addForce (SurgSim::Math::Vector *F) const

virtual void	addMass (SurgSim::Math::SparseMatrix *M, double scale) const
	Adds the element mass matrix M (computed for a given state) to a complete system mass matrix M (assembly) More...

virtual void	addMass (SurgSim::Math::SparseMatrix *M) const

virtual void	addDamping (SurgSim::Math::SparseMatrix *D, double scale) const
	Adds the element damping matrix D (= -df/dv) (comuted for a given state) to a complete system damping matrix D (assembly) More...

virtual void	addDamping (SurgSim::Math::SparseMatrix *D) const

virtual void	addStiffness (SurgSim::Math::SparseMatrix *K, double scale) const
	Adds the element stiffness matrix K (= -df/dx) (computed for a given state) to a complete system stiffness matrix K (assembly) More...

virtual void	addStiffness (SurgSim::Math::SparseMatrix *K) const

virtual void	addFMDK (SurgSim::Math::Vector F, SurgSim::Math::SparseMatrix M, SurgSim::Math::SparseMatrix D, SurgSim::Math::SparseMatrix K) const
	Adds the element force vector, mass, stiffness and damping matrices (computed for a given state) into a complete system data structure F, M, D, K (assembly) More...

virtual void	addMatVec (double alphaM, double alphaD, double alphaK, const SurgSim::Math::Vector &x, SurgSim::Math::Vector F, SurgSim::Math::Vector extractedX, SurgSim::Math::Vector *acumulator) const
	Adds the element matrix-vector contribution F += (alphaM.M + alphaD.D + alphaK.K).x (computed for a given state) into a complete system data structure F (assembly) More...

bool	isValidCoordinate (const SurgSim::Math::Vector &naturalCoordinate) const
	Determines whether a given natural coordinate is valid. More...

template<typename T , int Opt, typename StorageIndex >
void	assembleMatrixBlocks (const Eigen::Ref< const Math::Matrix > &subMatrix, const std::vector< size_t > &blockIds, size_t blockSize, Eigen::SparseMatrix< T, Opt, StorageIndex > *matrix) const
	Add a sub-matrix made of squared-blocks into a matrix that could be un-initialized. More...

template<typename T , int Opt, typename StorageIndex >
void	assembleMatrixBlocksNoInitialize (const Eigen::Ref< const Math::Matrix > &subMatrix, const std::vector< size_t > &blockIds, size_t blockSize, Eigen::SparseMatrix< T, Opt, StorageIndex > *matrix) const
	Add a sub-matrix made of squared-blocks into a matrix that is already initialized. More...

void	updateFMDK (const Math::OdeState &state, int options)
	Update the FemElement based on the given state. More...

Protected Member Functions
void	initializeMembers ()
	Initializes variables needed before Initialize() is called.

void	buildConstitutiveMaterialMatrix (Eigen::Matrix< double, 6, 6 > *constitutiveMatrix)
	Build the constitutive material 6x6 matrix. More...

void	computeStiffness (const SurgSim::Math::OdeState &state, Eigen::Matrix< double, 6, 24 > strain, Eigen::Matrix< double, 6, 24 > stress, SurgSim::Math::Matrix *k)
	Computes the cube stiffness matrix along with the strain and stress matrices. More...

void	computeMass (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *m)
	Computes the cube mass matrix. More...

void	doUpdateFMDK (const Math::OdeState &state, int options) override
	Update the FemElement based on the given state. More...

void	addStrainStressStiffnessAtPoint (const SurgSim::Math::OdeState &state, const SurgSim::Math::gaussQuadraturePoint &epsilon, const SurgSim::Math::gaussQuadraturePoint &eta, const SurgSim::Math::gaussQuadraturePoint &mu, Eigen::Matrix< double, 6, 24 > strain, Eigen::Matrix< double, 6, 24 > stress, SurgSim::Math::Matrix *k)
	Helper method to evaluate strain-stress and stiffness integral terms with a discrete sum using a Gauss quadrature rule. More...

void	addMassMatrixAtPoint (const SurgSim::Math::OdeState &state, const SurgSim::Math::gaussQuadraturePoint &epsilon, const SurgSim::Math::gaussQuadraturePoint &eta, const SurgSim::Math::gaussQuadraturePoint &mu, SurgSim::Math::Matrix *m)
	Helper method to evaluate mass integral terms with a discrete sum using a Gauss quadrature rule. More...

void	evaluateJ (const SurgSim::Math::OdeState &state, double epsilon, double eta, double mu, SurgSim::Math::Matrix33d J, SurgSim::Math::Matrix33d Jinv, double *detJ) const
	Helper method to evaluate matrix J = d(x,y,z)/d(epsilon,eta,mu) at a given 3D parametric location J expresses the 3D space coordinate frames variation w.r.t. More...

void	evaluateStrainDisplacement (double epsilon, double eta, double mu, const SurgSim::Math::Matrix33d &Jinv, Eigen::Matrix< double, 6, 24 > *B) const
	Helper method to evaluate the strain-displacement matrix at a given 3D parametric location c.f. More...

double	shapeFunction (size_t i, double epsilon, double eta, double mu) const
	Shape functions \(N_i(\epsilon, \eta, \mu) = (1\pm\epsilon)(1\pm\eta)(1\pm\mu)/8\). More...

double	dShapeFunctiondepsilon (size_t i, double epsilon, double eta, double mu) const
	Shape functions derivative \(dN_i/d\epsilon(\epsilon, \eta, \mu) = \pm(1\pm\eta)(1\pm\mu)/8\). More...

double	dShapeFunctiondeta (size_t i, double epsilon, double eta, double mu) const
	Shape functions derivative \(dN_i/d\eta(\epsilon, \eta, \mu) = \pm(1\pm\epsilon)(1\pm\mu)/8\). More...

double	dShapeFunctiondmu (size_t i, double epsilon, double eta, double mu) const
	Shape functions derivative \(dN_i/d\mu(\epsilon, \eta, \mu) = \pm(1\pm\epsilon)(1\pm\eta)/8\). More...

Protected Member Functions inherited from SurgSim::Physics::FemElement
void	setNumDofPerNode (size_t numDofPerNode)
	Sets the number of degrees of freedom per node. More...

void	initializeFMDK ()
	Initialize f, M, D, K variables.

virtual void	doInitializeFMDK ()
	Function to be overridden by the derived classes to initialize the f, M, D, K variables.

Protected Attributes
double	m_restVolume
	Cube rest volume.

Eigen::Matrix< double, 24, 1 >	m_elementRestPosition
	The cube rest state (nodes ordered by m_nodeIds)

Eigen::Matrix< double, 6, 24 >	m_strain
	Strain matrix (usually noted \(\epsilon\))

Eigen::Matrix< double, 6, 24 >	m_stress
	Stress matrix (usually noted \(\sigma\))

Eigen::Matrix< double, 6, 6 >	m_constitutiveMaterial
	Constitutive material matrix (Hooke's law in this case) defines the relationship between stress and strain.


std::array< double, 8 >	m_shapeFunctionsEpsilonSign

std::array< double, 8 >	m_shapeFunctionsEtaSign

std::array< double, 8 >	m_shapeFunctionsMuSign

Protected Attributes inherited from SurgSim::Physics::FemElement
size_t	m_numDofPerNode
	Number of degree of freedom per node for this element.

std::vector< size_t >	m_nodeIds
	Node ids connected by this element.

SurgSim::Math::Vector	m_f
	The force vector.

SurgSim::Math::Matrix	m_K
	The stiffness matrix.

SurgSim::Math::Matrix	m_M
	The mass matrix.

SurgSim::Math::Matrix	m_D
	The damping matrix.

bool	m_useDamping
	Flag to specify of the damping is used.

double	m_rho
	Mass density (in Kg.m-3)

double	m_E
	Young modulus (in N.m-2)

double	m_nu
	Poisson ratio (unitless)

Additional Inherited Members
Public Types inherited from SurgSim::Physics::FemElement
typedef SurgSim::Framework::ObjectFactory1< FemElement, std::shared_ptr< FemElementStructs::FemElementParameter > >	FactoryType

Static Public Member Functions inherited from SurgSim::Physics::FemElement
static FactoryType &	getFactory ()

Detailed Description

Class for Fem Element 3D based on a cube volume discretization.

Note: The stiffness property of the cube is derived from; http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch11.d/AFEM.Ch11.pdf; The mass property of the cube is derived from the kinetic energy computed on the cube's volume; (c.f. internal documentation on cube mass matrix computation for details).; Integral terms over the volume are evaluated using the Gauss-Legendre 2-points quadrature.; http://en.wikipedia.org/wiki/Gaussian_quadrature; Note that this technique is accurate for any polynomial evaluation up to degree 3.; In our case, the shape functions \(N_i\) are linear (of degree 1). So for exmaple,; in the mass matrix we have integral terms like \(\int_V N_i.N_j dV\), which are of degree 2.; Fem3DElementCube uses linear elasticity (not visco-elasticity), so it does not give any damping.

Constructor & Destructor Documentation

§ Fem3DElementCube() [1/2]

SurgSim::Physics::Fem3DElementCube::Fem3DElementCube ( std::array< size_t, 8 > nodeIds )

explicit

Constructor.

Parameters

nodeIds An array of 8 node ids defining this cube element in an overall mesh

Note: It is required that getVolume() is positive, to do so, it needs (looking at the cube from; the exterior, face normal 'n' pointing outward):; the 1st 4 nodeIds (ABCD) should define any face CW i.e. (AB^AC or AB^AD or AC^AD).n < 0; the last 4 nodeIds (EFGH) should define the opposite face CCW i.e. (EF^EG or EF^EH or EG^EH).n > 0; A warning will be logged when the initialize function is called if this condition is not met, but the; simulation will keep running. Behavior will be undefined because of possible negative volume terms.

§ Fem3DElementCube() [2/2]

SurgSim::Physics::Fem3DElementCube::Fem3DElementCube ( std::shared_ptr< FemElementStructs::FemElementParameter > elementData )

explicit

Constructor for FemElement object factory.

Parameters

elementData A FemElement3D struct defining this cube element in an overall mesh

Note: It is required that getVolume() is positive, to do so, it needs (looking at the cube from; the exterior, face normal 'n' pointing outward):; the 1st 4 nodeIds (ABCD) should define any face CW i.e. (AB^AC or AB^AD or AC^AD).n < 0; the last 4 nodeIds (EFGH) should define the opposite face CCW i.e. (EF^EG or EF^EH or EG^EH).n > 0; A warning will be logged when the initialize function is called if this condition is not met, but the; simulation will keep running. Behavior will be undefined because of possible negative volume terms.

Exceptions

SurgSim::Framework::AssertionFailure if nodeIds has a size different than 8

Member Function Documentation

§ addMassMatrixAtPoint()

void SurgSim::Physics::Fem3DElementCube::addMassMatrixAtPoint	(	const SurgSim::Math::OdeState &	state,
		const SurgSim::Math::gaussQuadraturePoint &	epsilon,
		const SurgSim::Math::gaussQuadraturePoint &	eta,
		const SurgSim::Math::gaussQuadraturePoint &	mu,
		SurgSim::Math::Matrix *	m
	)

protected

Helper method to evaluate mass integral terms with a discrete sum using a Gauss quadrature rule.

Parameters

	state	The state to compute the evaluation with
	epsilon,eta,mu	The Gauss quadrature points to evaluate the data at
[out]	m	The matrix in which to add the evaluations

§ addStrainStressStiffnessAtPoint()

void SurgSim::Physics::Fem3DElementCube::addStrainStressStiffnessAtPoint	(	const SurgSim::Math::OdeState &	state,
		const SurgSim::Math::gaussQuadraturePoint &	epsilon,
		const SurgSim::Math::gaussQuadraturePoint &	eta,
		const SurgSim::Math::gaussQuadraturePoint &	mu,
		Eigen::Matrix< double, 6, 24 > *	strain,
		Eigen::Matrix< double, 6, 24 > *	stress,
		SurgSim::Math::Matrix *	k
	)

protected

Helper method to evaluate strain-stress and stiffness integral terms with a discrete sum using a Gauss quadrature rule.

Parameters

	state	The state to compute the evaluation with
	epsilon,eta,mu	The Gauss quadrature points to evaluate the data at
[out]	strain,stress,k	The matrices in which to add the evaluations

§ buildConstitutiveMaterialMatrix()

void SurgSim::Physics::Fem3DElementCube::buildConstitutiveMaterialMatrix ( Eigen::Matrix< double, 6, 6 > * constitutiveMatrix )

protected

Build the constitutive material 6x6 matrix.

Parameters

[out] constitutiveMatrix The 6x6 constitutive material matrix

§ computeCartesianCoordinate()

SurgSim::Math::Vector SurgSim::Physics::Fem3DElementCube::computeCartesianCoordinate	(	const SurgSim::Math::OdeState &	state,
		const SurgSim::Math::Vector &	naturalCoordinate
	)		const

overridevirtual

Computes a given natural coordinate in cartesian coordinates.

Parameters

state	The state at which to transform coordinates
naturalCoordinate	The coordinates to transform

Returns: The resultant cartesian coordinates

Implements SurgSim::Physics::FemElement.

§ computeMass()

void SurgSim::Physics::Fem3DElementCube::computeMass	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	m
	)

protected

Computes the cube mass matrix.

Parameters

	state	The state to compute the mass matrix from
[out]	m	The mass matrix to store the result into

§ computeNaturalCoordinate()

SurgSim::Math::Vector SurgSim::Physics::Fem3DElementCube::computeNaturalCoordinate	(	const SurgSim::Math::OdeState &	state,
		const SurgSim::Math::Vector &	cartesianCoordinate
	)		const

overridevirtual

Computes a natural coordinate given a global coordinate.

Parameters

state	The state at which to transform coordinates
cartesianCoordinate	The coordinates to transform

Returns: The resultant natural coordinates

Implements SurgSim::Physics::FemElement.

§ computeStiffness()

void SurgSim::Physics::Fem3DElementCube::computeStiffness	(	const SurgSim::Math::OdeState &	state,
		Eigen::Matrix< double, 6, 24 > *	strain,
		Eigen::Matrix< double, 6, 24 > *	stress,
		SurgSim::Math::Matrix *	k
	)

protected

Computes the cube stiffness matrix along with the strain and stress matrices.

Parameters

	state	The state to compute the stiffness matrix from
[out]	strain,stress,k	The strain, stress and stiffness matrices to store the result into

§ doUpdateFMDK()

void SurgSim::Physics::Fem3DElementCube::doUpdateFMDK	(	const Math::OdeState &	state,
		int	options
	)

overrideprotectedvirtual

Update the FemElement based on the given state.

Parameters

state	\((x, v)\) the current position and velocity to evaluate the various terms with
options	Flag to specify which of the F, M, D, K needs to be updated

Implements SurgSim::Physics::FemElement.

§ dShapeFunctiondepsilon()

double SurgSim::Physics::Fem3DElementCube::dShapeFunctiondepsilon	(	size_t	i,
		double	epsilon,
		double	eta,
		double	mu
	)		const

protected

Shape functions derivative \(dN_i/d\epsilon(\epsilon, \eta, \mu) = \pm(1\pm\eta)(1\pm\mu)/8\).

Parameters

i	The node id (w.r.t. local element) to evaluate at
epsilon,eta,mu	The 3D parametric coordinates each within \([-1 +1]\)

Returns: dNi/depsilon(epsilon, eta, mu)

Note: A check is performed on the nodeId i but not on the 3D parametric coordinates range

See also: N; m_shapeFunctionsEpsilonSign, m_shapeFunctionsEtaSign, m_shapeFunctionsMuSign

§ dShapeFunctiondeta()

double SurgSim::Physics::Fem3DElementCube::dShapeFunctiondeta	(	size_t	i,
		double	epsilon,
		double	eta,
		double	mu
	)		const

protected

Shape functions derivative \(dN_i/d\eta(\epsilon, \eta, \mu) = \pm(1\pm\epsilon)(1\pm\mu)/8\).

Parameters

i	The node id (w.r.t. local element) to evaluate at
epsilon,eta,mu	The 3D parametric coordinates each within \([-1 +1]\)

Returns: dNi/depsilon(epsilon, eta, mu)

Note: A check is performed on the nodeId i but not on the 3D parametric coordinates range

See also: N; m_shapeFunctionsEpsilonSign, m_shapeFunctionsEtaSign, m_shapeFunctionsMuSign

§ dShapeFunctiondmu()

double SurgSim::Physics::Fem3DElementCube::dShapeFunctiondmu	(	size_t	i,
		double	epsilon,
		double	eta,
		double	mu
	)		const

protected

Shape functions derivative \(dN_i/d\mu(\epsilon, \eta, \mu) = \pm(1\pm\epsilon)(1\pm\eta)/8\).

Parameters

i	The node id (w.r.t. local element) to evaluate at
epsilon,eta,mu	The 3D parametric coordinates each within \([-1 +1]\)

Returns: dNi/depsilon(epsilon, eta, mu)

Note: A check is performed on the nodeId i but not on the 3D parametric coordinates range

See also: N; m_shapeFunctionsEpsilonSign, m_shapeFunctionsEtaSign, m_shapeFunctionsMuSign

§ evaluateJ()

void SurgSim::Physics::Fem3DElementCube::evaluateJ	(	const SurgSim::Math::OdeState &	state,
		double	epsilon,
		double	eta,
		double	mu,
		SurgSim::Math::Matrix33d *	J,
		SurgSim::Math::Matrix33d *	Jinv,
		double *	detJ
	)		const

protected

Helper method to evaluate matrix J = d(x,y,z)/d(epsilon,eta,mu) at a given 3D parametric location J expresses the 3D space coordinate frames variation w.r.t.

parametric coordinates

Parameters

	state	The state to compute the evaluation with
	epsilon,eta,mu	The 3D parametric coordinates to evaluate the data at (within \([-1 +1]\))
[out]	J,Jinv,detJ	The J matrix with its inverse and determinant evaluated at (epsilon, eta, mu)

§ evaluateStrainDisplacement()

void SurgSim::Physics::Fem3DElementCube::evaluateStrainDisplacement	(	double	epsilon,
		double	eta,
		double	mu,
		const SurgSim::Math::Matrix33d &	Jinv,
		Eigen::Matrix< double, 6, 24 > *	B
	)		const

protected

Helper method to evaluate the strain-displacement matrix at a given 3D parametric location c.f.

http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch11.d/AFEM.Ch11.pdf for more details

Parameters

	epsilon,eta,mu	The 3D parametric coordinates to evaluate the data at (within \([-1 +1]\))
	Jinv	The inverse of matrix J (3D global coords to 3D parametric coords)
[out]	B	The strain-displacement matrix

§ getVolume()

double SurgSim::Physics::Fem3DElementCube::getVolume ( const SurgSim::Math::OdeState & state ) const

overridevirtual

Gets the element volume based on the input state (in m-3)

Parameters

state The state to compute the volume with

Returns: The volume of this element (in m-3)

Implements SurgSim::Physics::FemElement.

§ initialize()

void SurgSim::Physics::Fem3DElementCube::initialize ( const SurgSim::Math::OdeState & state )

overridevirtual

Initializes the element once everything has been set.

Parameters

state The state to initialize the FemElement with

Note: We use the theory of linear elasticity, so this method precomputes the stiffness and mass matrices; The 8 node ids must be valid in the given state, if they aren't an ASSERT will be raised; The 8 node ids must define a cube with positive volume, if they don't an ASSERT will be raised; In order to do so (looking at the cube from the exterior, face normal 'n' pointing outward),; the 1st 4 nodeIds (ABCD) should define any face CW i.e. (AB^AC or AB^AD or AC^AD).n < 0; the last 4 nodeIds (EFGH) should define the opposite face CCW i.e. (EF^EG or EF^EH or EG^EH).n > 0; A warning will be logged in if this condition is not met, but the simulation will keep running.; Behavior will be undefined because of possible negative volume terms.

Reimplemented from SurgSim::Physics::FemElement.

§ shapeFunction()

double SurgSim::Physics::Fem3DElementCube::shapeFunction	(	size_t	i,
		double	epsilon,
		double	eta,
		double	mu
	)		const

protected

Shape functions \(N_i(\epsilon, \eta, \mu) = (1\pm\epsilon)(1\pm\eta)(1\pm\mu)/8\).

\( \begin{array}{r | r r r} i & sign(\epsilon) & sign(\eta) & sign(\mu) \\ \hline 0 & -1 & -1 & -1 \\ 1 & +1 & -1 & -1 \\ 2 & +1 & +1 & -1 \\ 3 & -1 & +1 & -1 \\ 4 & -1 & -1 & +1 \\ 5 & +1 & -1 & +1 \\ 6 & +1 & +1 & +1 \\ 7 & -1 & +1 & +1 \end{array} \)

Parameters

i	The node id (w.r.t. local element) to evaluate at
epsilon,eta,mu	The 3D parametric coordinates each within \([-1 +1]\)

Returns: Ni(epsilon, eta, mu)

Note: A check is performed on the nodeId i but not on the 3D parametric coordinates range

See also: m_shapeFunctionsEpsilonSign, m_shapeFunctionsEtaSign, m_shapeFunctionsMuSign; dNdepsilon, dNdeta, dNdmu

Member Data Documentation

§ m_shapeFunctionsEpsilonSign

std::array<double, 8> SurgSim::Physics::Fem3DElementCube::m_shapeFunctionsEpsilonSign

protected

Shape functions parameters \(N_i(\epsilon, \eta, \mu) = (1\pm\epsilon)(1\pm\eta)(1\pm\mu)/8 = (1+\epsilon.sign(\epsilon_i))(1+\eta.sign(\eta_i))(1+\mu.sign(\mu_i))/8 \textbf{ with } (\epsilon, \eta, \mu) \in [-1 +1]^3\)

We choose to only store the sign of epsilon, eta and mu for each shape functions.

See also: N

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

SurgSim/Physics/Fem3DElementCube.h
SurgSim/Physics/Fem3DElementCube.cpp

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

§ Fem3DElementCube() [1/2]

§ Fem3DElementCube() [2/2]

Member Function Documentation

§ addMassMatrixAtPoint()

§ addStrainStressStiffnessAtPoint()

§ buildConstitutiveMaterialMatrix()

§ computeCartesianCoordinate()

§ computeMass()

§ computeNaturalCoordinate()

§ computeStiffness()

§ doUpdateFMDK()

§ dShapeFunctiondepsilon()

§ dShapeFunctiondeta()

§ dShapeFunctiondmu()

§ evaluateJ()

§ evaluateStrainDisplacement()

§ getVolume()

§ initialize()

§ shapeFunction()

Member Data Documentation

§ m_shapeFunctionsEpsilonSign