2D FemElement based on a triangle with a constant thickness More...

#include <Fem2DElementTriangle.h>

Inheritance diagram for SurgSim::Physics::Fem2DElementTriangle:

Public Member Functions
	Fem2DElementTriangle ()
	Constructor.

	Fem2DElementTriangle (std::array< size_t, 3 > nodeIds)
	Constructor. More...

	Fem2DElementTriangle (std::shared_ptr< FemElementStructs::FemElementParameter > elementData)
	Constructor for FemElement object factory. More...

void	setThickness (double thickness)
	Sets the triangle's thickness. More...

double	getThickness () const
	Gets the triangle's thickness. More...

void	initialize (const SurgSim::Math::OdeState &state) override
	Initialize the FemElement 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.

SurgSim::Math::Matrix33d	computeRotation (const SurgSim::Math::OdeState &state)
	Computes the triangle element's rotation given a state. More...

virtual void	computeLocalStiffness (const SurgSim::Math::OdeState &state, Eigen::Matrix< double, 18, 18 > *localStiffnessMatrix)
	Computes the triangle's local stiffness matrix. More...

void	computeStiffness (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *stiffnessMatrix)
	Computes the triangle's stiffness matrix. More...

virtual void	computeLocalMass (const SurgSim::Math::OdeState &state, Eigen::Matrix< double, 18, 18 > *localMassMatrix)
	Computes the triangle's local mass matrix. More...

void	computeMass (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *massMatrix)
	Computes the triangle's mass matrix. More...

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

void	computeShapeFunctionsParameters (const SurgSim::Math::OdeState &restState)
	Compute the various shape functions (membrane and plate deformations) parameters. More...

std::array< double, 9 >	batozDhxDxi (double xi, double eta) const
	Batoz derivative dHx/dxi. More...

std::array< double, 9 >	batozDhxDeta (double xi, double eta) const
	Batoz derivative dHx/deta. More...

std::array< double, 9 >	batozDhyDxi (double xi, double eta) const
	Batoz derivative dHy/dxi. More...

std::array< double, 9 >	batozDhyDeta (double xi, double eta) const
	Batoz derivative dHy/deta. More...

Matrix39Type	batozStrainDisplacement (double xi, double eta) const
	Batoz strain displacement matrix evaluated at a given point. 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
Eigen::Matrix< double, 18, 1 >	m_x0
	The element's rest state.

SurgSim::Math::Matrix33d	m_initialRotation
	Initial rotation matrix for the element.

Eigen::Matrix< double, 18, 18 >	m_MLocal
	Stiffness matrix (in local coordinate frame)

Eigen::Matrix< double, 18, 18 >	m_KLocal
	Stiffness matrix (in local coordinate frame)

double	m_restArea
	The triangle rest area.

double	m_thickness
	Thickness of the element.

SurgSim::Math::Matrix33d	m_membraneShapeFunctionsParameters
	Membrane (in-plane) deformation. More...


SurgSim::Math::Vector3d	m_xij
	Batoz variable \(x_{ij} = x_i - x_j\).

SurgSim::Math::Vector3d	m_yij
	Batoz variable \(y_{ij} = y_i - y_j\).

SurgSim::Math::Vector3d	m_lij_sqr
	Batoz variable \(l_{ij}^2 = x_{ij}^2 + y_{ij}^2\).

SurgSim::Math::Vector3d	m_ak
	Batoz variable \(a_{k} = -x_{ij}/l_i^2\).

SurgSim::Math::Vector3d	m_bk
	Batoz variable \(b_{k} = 3/4 x_{ij} y_{ij}/l_{ij}2\).

SurgSim::Math::Vector3d	m_ck
	Batoz variable \(c_{k} = (1/4 x_{ij}^2 - 1/2 y_{ij}^2)/l_{ij}^2\).

SurgSim::Math::Vector3d	m_dk
	Batoz variable \(d_{k} = -y_{ij}/l_{ij}^2\).

SurgSim::Math::Vector3d	m_ek
	Batoz variable \(e_{k} = (1/4 y_{ij}^2 - 1/2 x_{ij}^2)/l_{ij}^2\).


SurgSim::Math::Vector3d	m_Pk
	Batoz variable \(P_{k} = -6x_{ij}/l_{ij}^2 = 6.\textrm{m_a}_k\).

SurgSim::Math::Vector3d	m_qk
	Batoz variable \(q_{k} = 3x_{ij}y_{ij}/l_{ij}^2 = 4.\textrm{m_b}_k\).

SurgSim::Math::Vector3d	m_tk
	Batoz variable \(t_{k} = -6y_{ij}/l_{ij}^2 = 6.\textrm{m_d}_k\).

SurgSim::Math::Vector3d	m_rk
	Batoz variable \(r_{k} = 3y_{ij}^2/l_{ij}^2\).


SurgSim::Math::Matrix	m_integral_dT_d
	Plate mass matrix: integral terms related to the dof \((z)\).

SurgSim::Math::Matrix	m_integralHyiHyj
	Plate mass matrix: integral terms related to the dof \((\theta_x)\).

SurgSim::Math::Matrix	m_integralHxiHxj
	Plate mass matrix: integral terms related to the dof \((\theta_y)\).

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

2D FemElement based on a triangle with a constant thickness

The triangle is modeled as a shell (6DOF) which is decomposed into a membrane (in-plane 2DOF \((x, y)\)) and a plate (bending/twisting 3DOF \((z, \theta_x, \theta_y)\)). The thin-plate assumption does not consider the drilling dof \(\theta_z\). The system includes the drilling DOF for completeness but does not assign any mass or stiffness to it.

The membrane (in-plane) equations (mass and stiffness) are following "Theory of Matrix Structural Analysis" from J.S. Przemieniecki.

The thin-plate (bending) equations (mass and stiffness) are following "A Study Of Three-Node Triangular Plate Bending Elements", Jean-Louis Batoz Numerical Methods in Engineering, vol 15, 1771-1812 (1980).
The plate mass matrix is not detailed in the above paper, but the analytical equations have been derived from it. Moreover, to account for contribution of the displacement along z to the plate mass matrix, we used a cubic expression of this displacement given in: "Shell elements: modelizations DKT, DST, DKTG and Q4g", Code_Aster, 2013, Thomas De Soza.

Note: The element is considered to have a constant thickness.; The element uses linear elasticity (not visco-elasticity), so it does not have any damping.

Constructor & Destructor Documentation

§ Fem2DElementTriangle() [1/2]

SurgSim::Physics::Fem2DElementTriangle::Fem2DElementTriangle ( std::array< size_t, 3 > nodeIds )

explicit

Constructor.

Parameters

nodeIds An array of 3 node ids defining this triangle element with respect to a DeformableRepresentationState which is passed to the initialize method.

§ Fem2DElementTriangle() [2/2]

SurgSim::Physics::Fem2DElementTriangle::Fem2DElementTriangle ( std::shared_ptr< FemElementStructs::FemElementParameter > elementData )

explicit

Constructor for FemElement object factory.

Parameters

elementData A FemElement struct defining this triangle element with respect to a DeformableRepresentationState which is passed to the initialize method.

Exceptions

SurgSim::Framework::AssertionFailure if nodeIds has a size different than 3

Member Function Documentation

§ batozDhxDeta()

std::array< double, 9 > SurgSim::Physics::Fem2DElementTriangle::batozDhxDeta	(	double	xi,
		double	eta
	)		const

protected

Batoz derivative dHx/deta.

Parameters

xi,eta The parametric coordinate (in [0 1] and xi+eta<1.0)

Returns: The vector dHx/deta evaluated at (xi, eta)

§ batozDhxDxi()

std::array< double, 9 > SurgSim::Physics::Fem2DElementTriangle::batozDhxDxi	(	double	xi,
		double	eta
	)		const

protected

Batoz derivative dHx/dxi.

Parameters

xi,eta The parametric coordinate (in [0 1] and xi+eta<1.0)

Returns: The vector dHx/dxi evaluated at (xi, eta)

§ batozDhyDeta()

std::array< double, 9 > SurgSim::Physics::Fem2DElementTriangle::batozDhyDeta	(	double	xi,
		double	eta
	)		const

protected

Batoz derivative dHy/deta.

Parameters

xi,eta The parametric coordinate (in [0 1] and xi+eta<1.0)

Returns: The vector dHy/deta evaluated at (xi, eta)

§ batozDhyDxi()

std::array< double, 9 > SurgSim::Physics::Fem2DElementTriangle::batozDhyDxi	(	double	xi,
		double	eta
	)		const

protected

Batoz derivative dHy/dxi.

Parameters

xi,eta The parametric coordinate (in [0 1] and xi+eta<1.0)

Returns: The vector dHy/dxi evaluated at (xi, eta)

§ batozStrainDisplacement()

Fem2DElementTriangle::Matrix39Type SurgSim::Physics::Fem2DElementTriangle::batozStrainDisplacement	(	double	xi,
		double	eta
	)		const

protected

Batoz strain displacement matrix evaluated at a given point.

Parameters

xi,eta The parametric coordinate (in [0 1] and xi+eta<1.0)

Returns: The 3x9 strain displacement matrix evaluated at (xi, eta)

§ computeCartesianCoordinate()

SurgSim::Math::Vector SurgSim::Physics::Fem2DElementTriangle::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.

§ computeLocalMass()

void SurgSim::Physics::Fem2DElementTriangle::computeLocalMass	(	const SurgSim::Math::OdeState &	state,
		Eigen::Matrix< double, 18, 18 > *	localMassMatrix
	)

protectedvirtual

Computes the triangle's local mass matrix.

Parameters

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

§ computeLocalStiffness()

void SurgSim::Physics::Fem2DElementTriangle::computeLocalStiffness	(	const SurgSim::Math::OdeState &	state,
		Eigen::Matrix< double, 18, 18 > *	localStiffnessMatrix
	)

protectedvirtual

Computes the triangle's local stiffness matrix.

Parameters

	state	The state to compute the local stiffness matrix from
[out]	localStiffnessMatrix	The local stiffness matrix to store the result into

§ computeMass()

void SurgSim::Physics::Fem2DElementTriangle::computeMass	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	massMatrix
	)

protected

Computes the triangle's mass matrix.

Parameters

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

§ computeNaturalCoordinate()

SurgSim::Math::Vector SurgSim::Physics::Fem2DElementTriangle::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.

§ computeRotation()

SurgSim::Math::Matrix33d SurgSim::Physics::Fem2DElementTriangle::computeRotation ( const SurgSim::Math::OdeState & state )

protected

Computes the triangle element's rotation given a state.

Parameters

state The state to compute the rotation from

Returns: The rotation matrix of the element in the given state

§ computeShapeFunctionsParameters()

void SurgSim::Physics::Fem2DElementTriangle::computeShapeFunctionsParameters ( const SurgSim::Math::OdeState & restState )

protected

Compute the various shape functions (membrane and plate deformations) parameters.

Parameters

restState the rest state to compute the shape functions parameters from

§ computeStiffness()

void SurgSim::Physics::Fem2DElementTriangle::computeStiffness	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	stiffnessMatrix
	)

protected

Computes the triangle's stiffness matrix.

Parameters

	state	The state to compute the stiffness matrix from
[out]	stiffnessMatrix	The stiffness matrix to store the result into

§ doUpdateFMDK()

void SurgSim::Physics::Fem2DElementTriangle::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.

§ getThickness()

double SurgSim::Physics::Fem2DElementTriangle::getThickness ( ) const

Gets the triangle's thickness.

Returns: The thickness of the triangle

§ getVolume()

double SurgSim::Physics::Fem2DElementTriangle::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::Fem2DElementTriangle::initialize ( const SurgSim::Math::OdeState & state )

overridevirtual

Initialize the FemElement once everything has been set.

Parameters

state The state to initialize the FemElement with

Reimplemented from SurgSim::Physics::FemElement.

§ setThickness()

void SurgSim::Physics::Fem2DElementTriangle::setThickness ( double thickness )

Sets the triangle's thickness.

Parameters

thickness The thickness of the triangle

Member Data Documentation

§ m_membraneShapeFunctionsParameters

SurgSim::Math::Matrix33d SurgSim::Physics::Fem2DElementTriangle::m_membraneShapeFunctionsParameters

protected

Membrane (in-plane) deformation.

DOF simulated: (x, y). "Theory of Matrix Structural Analysis" from J.S. Przemieniecki. Shape functions are \(f_i(x, y) = a_i + b_i.x + c_i.y\), here we store \((a_i, b_i, c_i)\) on each row.

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

SurgSim/Physics/Fem2DElementTriangle.h
SurgSim/Physics/Fem2DElementTriangle.cpp
SurgSim/Physics/Fem2DElementTriangle_computeLocalMembraneMass.dox
SurgSim/Physics/Fem2DElementTriangle_computeLocalPlateMass.dox

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

§ Fem2DElementTriangle() [1/2]

§ Fem2DElementTriangle() [2/2]

Member Function Documentation

§ batozDhxDeta()

§ batozDhxDxi()

§ batozDhyDeta()

§ batozDhyDxi()

§ batozStrainDisplacement()

§ computeCartesianCoordinate()

§ computeLocalMass()

§ computeLocalStiffness()

§ computeMass()

§ computeNaturalCoordinate()

§ computeRotation()

§ computeShapeFunctionsParameters()

§ computeStiffness()

§ doUpdateFMDK()

§ getThickness()

§ getVolume()

§ initialize()

§ setThickness()

Member Data Documentation

§ m_membraneShapeFunctionsParameters