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

#include <SurgSim/Physics/Fem2DElementTriangle.h>

Inheritance diagram for SurgSim::Physics::Fem2DElementTriangle:

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

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

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

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

virtual double	getVolume (const SurgSim::Math::OdeState &state) const override
	Gets the element's volume based on the input state. More...

virtual void	addForce (const SurgSim::Math::OdeState &state, SurgSim::Math::Vector *F, double scale=1.0) override
	Adds the element's force (computed for a given state) to a complete system force vector F (assembly) More...

virtual void	addMass (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *M, double scale=1.0) override
	Adds the element's mass matrix M (computed for a given state) to a complete system mass matrix M (assembly) More...

virtual void	addDamping (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *D, double scale=1.0) override
	Adds the element's damping matrix D (= -df/dv) (computed for a given state) to a complete system damping matrix D (assembly) More...

virtual void	addStiffness (const SurgSim::Math::OdeState &state, SurgSim::Math::Matrix *K, double scale=1.0) override
	Adds the element's stiffness matrix K (= -df/dx) (computed for a given state) to a complete system stiffness matrix K (assembly) More...

virtual void	addFMDK (const SurgSim::Math::OdeState &state, SurgSim::Math::Vector F, SurgSim::Math::Matrix M, SurgSim::Math::Matrix D, SurgSim::Math::Matrix K) override
	Adds the element's 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 (const SurgSim::Math::OdeState &state, double alphaM, double alphaD, double alphaK, const SurgSim::Math::Vector &x, SurgSim::Math::Vector *F)
	Adds the element's 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...

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

virtual 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. More...

virtual	~FemElement ()
	Virtual destructor. More...

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 bool	update (const SurgSim::Math::OdeState &state)
	Update the element based on a given state. More...

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

Protected Member Functions
void	computeInitialRotation (const SurgSim::Math::OdeState &state)
	Computes the triangle element's initial rotation. More...

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

void	computeMass (const SurgSim::Math::OdeState &state, Eigen::Matrix< double, 18, 18 > *m)
	Computes the triangle's mass matrix. 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 neta) const
	Batoz derivative dHx/dxi. More...

std::array< double, 9 >	batozDhxDneta (double xi, double neta) const
	Batoz derivative dHx/dneta. More...

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

std::array< double, 9 >	batozDhyDneta (double xi, double neta) const
	Batoz derivative dHy/dneta. More...

Matrix39Type	batozStrainDisplacement (double xi, double neta) 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...

Protected Attributes
Eigen::Matrix< double, 18, 1 >	m_x0
	The element's rest state. More...

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

Eigen::Matrix< double, 18, 18 >	m_M
	Mass matrix (in global coordinate frame) More...

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

Eigen::Matrix< double, 18, 18 >	m_K
	Stiffness matrix (in global coordinate frame) More...

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

double	m_restArea
	The triangle rest area. More...

double	m_thickness
	Thickness of the element. More...

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

SurgSim::Math::Vector3d	m_xij
	Thin-plate (bending/twisting) specific data structure DOF simulated: (z, thetaX, thetaY) "A Study Of Three-Node Triangular Plate Bending Elements", Jean-Louis Batoz Numerical Methods in Engineering, vol 15, 1771-1812 (1980) Indices are as follow: 0 1 2 denotes triangle's points ABC: 4 (mid-edge 12) 5 (mid-edge 20) 6 (mid-edge 01) denotes mid-edge points Data structures having only mid-edge information are 0 based (0->4 (mid-egde 12) ; 1->5 ; 2->6) More...

SurgSim::Math::Vector3d	m_yij

SurgSim::Math::Vector3d	m_lij_sqr

SurgSim::Math::Vector3d	m_ak

SurgSim::Math::Vector3d	m_bk

SurgSim::Math::Vector3d	m_ck

SurgSim::Math::Vector3d	m_dk

SurgSim::Math::Vector3d	m_ek

SurgSim::Math::Vector3d	m_Pk

SurgSim::Math::Vector3d	m_qk

SurgSim::Math::Vector3d	m_tk

SurgSim::Math::Vector3d	m_rk

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

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

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

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

double	m_nu
	Poisson ratio (unitless) More...

Private Types
typedef Eigen::Matrix< double, 3, 3 >	Matrix33Type

typedef Eigen::Matrix< double, 3, 6 >	Matrix36Type

typedef Eigen::Matrix< double, 6, 6 >	Matrix66Type

typedef Eigen::Matrix< double, 3, 9 >	Matrix39Type

typedef Eigen::Matrix< double, 9, 9 >	Matrix99Type

Detailed Description

2D FemElement based on a triangle with a constant thickness

The triangle is modelled as a shell (6DOF) which is decomposed into a membrane (in-plane 2DOF (X,Y)) and a plate (bending/twisting 3DOF (Z, ThetaX,ThetaY)). The thin-plate assumption does not consider the drilling dof (ThetaZ). The system includes the 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)

Note: The plate mass matrix is not detailed in the above paper, but the analytical equations; have been derived from it.; The element is considered to have a constant thickness.

Member Typedef Documentation

typedef Eigen::Matrix<double, 3, 3> SurgSim::Physics::Fem2DElementTriangle::Matrix33Type

private

typedef Eigen::Matrix<double, 3, 6> SurgSim::Physics::Fem2DElementTriangle::Matrix36Type

private

typedef Eigen::Matrix<double, 3, 9> SurgSim::Physics::Fem2DElementTriangle::Matrix39Type

private

typedef Eigen::Matrix<double, 6, 6> SurgSim::Physics::Fem2DElementTriangle::Matrix66Type

private

typedef Eigen::Matrix<double, 9, 9> SurgSim::Physics::Fem2DElementTriangle::Matrix99Type

private

Constructor & Destructor Documentation

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

explicit

Constructor.

Parameters

nodeIds An array of 3 node ids (A, B, C) defining this triangle element with respect to a DeformableRepresentaitonState which is passed to the initialize method.

Member Function Documentation

void SurgSim::Physics::Fem2DElementTriangle::addDamping	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	D,
		double	scale = `1.0`
	)

overridevirtual

Adds the element's damping matrix D (= -df/dv) (computed for a given state) to a complete system damping matrix D (assembly)

Parameters

	state	The state to compute the damping matrix with
[in,out]	D	The complete system damping matrix to add the element damping matrix into
	scale	A factor to scale the added damping matrix with

Note: The element's damping matrix is a square matrix of size getNumDofPerNode() x getNumNodes().; This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode().; The beam uses linear elasticity (not visco-elasticity), so it does not have any damping.

Implements SurgSim::Physics::FemElement.

void SurgSim::Physics::Fem2DElementTriangle::addFMDK	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Vector *	F,
		SurgSim::Math::Matrix *	M,
		SurgSim::Math::Matrix *	D,
		SurgSim::Math::Matrix *	K
	)

overridevirtual

Adds the element's force vector, mass, stiffness and damping matrices (computed for a given state) into a complete system data structure F, M, D, K (assembly)

Parameters

	state	The state to compute everything with
[in,out]	F	The complete system force vector to add the element force into
[in,out]	M	The complete system mass matrix to add the element mass matrix into
[in,out]	D	The complete system damping matrix to add the element damping matrix into
[in,out]	K	The complete system stiffness matrix to add the element stiffness matrix into

Note: This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode().

Implements SurgSim::Physics::FemElement.

void SurgSim::Physics::Fem2DElementTriangle::addForce	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Vector *	F,
		double	scale = `1.0`
	)

overridevirtual

Adds the element's force (computed for a given state) to a complete system force vector F (assembly)

Parameters

	state	The state to compute the force with
[in,out]	F	The complete system force vector to add the element's force into
	scale	A factor to scale the added force with

Note: The element's force is of size (getNumDofPerNode() x getNumNodes()).; This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode().

Implements SurgSim::Physics::FemElement.

void SurgSim::Physics::Fem2DElementTriangle::addMass	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	M,
		double	scale = `1.0`
	)

overridevirtual

Adds the element's mass matrix M (computed for a given state) to a complete system mass matrix M (assembly)

Parameters

	state	The state to compute the mass matrix with
[in,out]	M	The complete system mass matrix to add the element's mass-matrix into
	scale	A factor to scale the added mass matrix with

Note: The element's mass matrix is a square matrix of size getNumDofPerNode() x getNumNodes().; This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode()

Implements SurgSim::Physics::FemElement.

void SurgSim::Physics::Fem2DElementTriangle::addMatVec	(	const SurgSim::Math::OdeState &	state,
		double	alphaM,
		double	alphaD,
		double	alphaK,
		const SurgSim::Math::Vector &	x,
		SurgSim::Math::Vector *	F
	)

virtual

Adds the element's matrix-vector contribution F += (alphaM.M + alphaD.D + alphaK.K).x (computed for a given state) into a complete system data structure F (assembly)

Parameters

	state	The state to compute everything with
	alphaM	The scaling factor for the mass contribution
	alphaD	The scaling factor for the damping contribution
	alphaK	The scaling factor for the stiffness contribution
	x	A complete system vector to use as the vector in the matrix-vector multiplication
[in,out]	F	The complete system force vector to add the element matrix-vector contribution into

Note: This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode().

Implements SurgSim::Physics::FemElement.

void SurgSim::Physics::Fem2DElementTriangle::addStiffness	(	const SurgSim::Math::OdeState &	state,
		SurgSim::Math::Matrix *	K,
		double	scale = `1.0`
	)

overridevirtual

Adds the element's stiffness matrix K (= -df/dx) (computed for a given state) to a complete system stiffness matrix K (assembly)

Parameters

	state	The state to compute the stiffness matrix with
[in,out]	K	The complete system stiffness matrix to add the element stiffness matrix into
	scale	A factor to scale the added stiffness matrix with

Note: The element stiffness matrix is square of size getNumDofPerNode() x getNumNodes().; This method supposes that the incoming state contains information with the same number of dof per node as getNumDofPerNode()

Implements SurgSim::Physics::FemElement.

std::array< double, 9 > SurgSim::Physics::Fem2DElementTriangle::batozDhxDneta	(	double	xi,
		double	neta
	)		const

protected

Batoz derivative dHx/dneta.

Parameters