beams Namespace Reference

Kynema API: kynema::beams Namespace Reference

Kynema API

A flexible multibody structural dynamics code for wind turbines

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Classes
struct	CalculateForceFC

struct	CalculateForceFD

struct	CalculateInertialQuadraturePointValues

struct	CalculateJacobian
	Functor to calculate Jacobians and unit tangent vectors at quadrature points for beam elements. More...

struct	CalculateOuu

struct	CalculatePuu

struct	CalculateQPPosition
	Functor to calculate current position and orientation at quadrature points. More...

struct	CalculateQuadraturePointValues

struct	CalculateQuu

struct	CalculateStiffnessQuadraturePointValues

struct	CalculateStrain

struct	CalculateSystemMatrix

struct	CalculateTemporaryVariables

struct	HollowCircleProperties
	Struct containing geometric properties for a hollow circular cross-section. More...

struct	IntegrateInertiaMatrixElement

struct	IntegrateResidualVectorElement

struct	IntegrateStiffnessMatrixElement

struct	InterpolateQPPosition
	Interpolates quadrature point positions from nodal positions using shape functions. More...

struct	InterpolateQPRotation
	A Kernel which interpolates a rotation quaternion on a given element from its nodes to all of it quadrature points. More...

struct	InterpolateQPState_r
	Interpolates the rotation (r) part of the state at a quadrature point. More...

struct	InterpolateQPState_rprime
	Interpolates the rotation derivative (r') part of the state at a quadrature point. More...

struct	InterpolateQPState_u
	Interpolates the displacement (u) part of the state at a quadrature point. More...

struct	InterpolateQPState_uprime
	Interpolates the displacement derivative (u') part of the state at a quadrature point. More...

struct	InterpolateQPVector
	A Kernel which interpolates a vector quantity from nodes on a given element to a quadrature point given a basis function. More...

struct	InterpolateToQuadraturePointForInertia

struct	InterpolateToQuadraturePointForStiffness

struct	InterpolateToQuadraturePoints
	Interpolates various quantities from nodes to quadrature points for beam elements. More...

struct	UpdateNodeState

struct	UpdateNodeStateElement

Functions
std::vector< std::array< double, 2 > >	CreateTrapezoidalQuadrature (std::span< const double > grid)
	Creates a trapezoidal quadrature rule based on a given grid.

std::vector< std::array< double, 2 > >	CreateGaussLegendreLobattoQuadrature (std::span< const double > grid, size_t order)

std::vector< std::array< double, 2 > >	CreateGaussLegendreQuadrature (const size_t order)
	Creates Gauss-Legendre (GL) quadrature points and weights on [-1, 1].

static std::array< std::array< double, 6 >, 6 >	GenerateStiffnessMatrix (double EA, double EI_x, double EI_y, double GKt, double GA, double kxs, double kys, double x_C, double y_C, double theta_p, double x_S, double y_S, double theta_s)
	Generates a 6x6 cross-sectional stiffness matrix for use in beam elements.

static std::array< std::array< double, 6 >, 6 >	GenerateMassMatrix (double m, double I_x, double I_y, double I_p, double x_G=0., double y_G=0., double theta_i=0.)
	Generates a 6x6 cross-sectional mass matrix for use in beam elements.

static HollowCircleProperties	CalculateHollowCircleProperties (double outer_diameter, double wall_thickness, double nu=0.33)
	Calculates geometric properties for a hollow circular cross-section.

static BeamSection	GenerateHollowCircleSection (double s, double E, double G, double rho, double outer_diameter, double wall_thickness, double nu, double x_C=0., double y_C=0., double theta_p=0., double x_S=0., double y_S=0., double theta_s=0., double x_G=0., double y_G=0., double theta_i=0)
	Generates a BeamSection with 6x6 mass and stiffness matrices for a hollow circular cross-section.

template<typename shape_matrix_type , typename node_type , typename qp_type >
KOKKOS_INLINE_FUNCTION void	InterpVector3 (const shape_matrix_type &shape_matrix, const node_type &node_v, const qp_type &qp_v)

template<typename shape_matrix_type , typename node_type , typename qp_type >
KOKKOS_INLINE_FUNCTION void	InterpVector4 (const shape_matrix_type &shape_matrix, const node_type &node_v, const qp_type &qp_v)

template<typename shape_matrix_type , typename node_type , typename qp_type >
KOKKOS_INLINE_FUNCTION void	InterpQuaternion (const shape_matrix_type &shape_matrix, const node_type &node_v, const qp_type &qp_v)

template<typename shape_matrix_type , typename jacobian_type , typename node_type , typename qp_type >
KOKKOS_INLINE_FUNCTION void	InterpVector3Deriv (const shape_matrix_type &shape_matrix_deriv, const jacobian_type &jacobian, const node_type &node_v, const qp_type &qp_v)

template<typename shape_matrix_type , typename jacobian_type , typename node_type , typename qp_type >
KOKKOS_INLINE_FUNCTION void	InterpVector4Deriv (const shape_matrix_type &shape_matrix_deriv, const jacobian_type &jacobian, const node_type &node_v, const qp_type &qp_v)

void	PopulateNodeX0 (const BeamElement &elem, std::span< const Node > nodes, const Kokkos::View< double *[7], Kokkos::LayoutStride, Kokkos::HostSpace > &node_x0)
	Populate the node initial position and orientation.

void	PopulateQPWeight (const BeamElement &elem, const Kokkos::View< double *, Kokkos::LayoutStride, Kokkos::HostSpace > &qp_weight)
	Populate the integration weights at each quadrature point.

std::vector< double >	MapNodePositions (const BeamElement &elem, std::span< const Node > nodes)
	Map node positions from [0,1] to [-1,1].

void	PopulateShapeFunctionValues (const BeamElement &elem, std::span< const Node > nodes, const Kokkos::View< double **, Kokkos::LayoutStride, Kokkos::HostSpace > &shape_interp)
	Populate shape function values at each quadrature point.

void	PopulateShapeFunctionDerivatives (const BeamElement &elem, std::span< const Node > nodes, const Kokkos::View< double **, Kokkos::LayoutStride, Kokkos::HostSpace > &shape_deriv)
	Populate shape function derivatives at each quadrature point.

std::vector< double >	MapSectionPositions (const BeamElement &elem)
	Map section positions from [0,1] to [-1,1].

void	PopulateQPMstar (const BeamElement &elem, const Kokkos::View< double *[6][6], Kokkos::LayoutStride, Kokkos::HostSpace > &qp_Mstar)
	Populate mass matrix values at each quadrature point.

void	PopulateQPCstar (const BeamElement &elem, const Kokkos::View< double *[6][6], Kokkos::LayoutStride, Kokkos::HostSpace > &qp_Cstar)
	Populate stiffness matrix values at each quadrature point.

Function Documentation

◆ CalculateHollowCircleProperties()

static HollowCircleProperties kynema::beams::CalculateHollowCircleProperties	(	double	outer_diameter,
		double	wall_thickness,
		double	nu = `0.33`
	)

static

Calculates geometric properties for a hollow circular cross-section.

Parameters

outer_diameter	Outer diameter of the hollow circle [Length]
wall_thickness	Wall thickness [Length]
nu	Poisson's ratio for shear correction factor calculation [dimensionless]

Returns: HollowCircleProperties struct containing geometric properties

◆ CreateGaussLegendreLobattoQuadrature()

std::vector< std::array< double, 2 > > kynema::beams::CreateGaussLegendreLobattoQuadrature	(	std::span< const double >	grid,
		size_t	order
	)

inline

◆ CreateGaussLegendreQuadrature()

std::vector< std::array< double, 2 > > kynema::beams::CreateGaussLegendreQuadrature ( const size_t order )

inline

Creates Gauss-Legendre (GL) quadrature points and weights on [-1, 1].

GL quadrature provides optimal accuracy for polynomial integration. The points are roots of P_n(x) where P_n is the nth Legendre polynomial. GL quadrature does NOT include the endpoints (-1, 1).

Parameters

order Number of quadrature points (n >= 1). Returns n quadrature points.

Returns: Vector of {point, weight} pairs for GL quadrature, sorted by point value

◆ CreateTrapezoidalQuadrature()

std::vector< std::array< double, 2 > > kynema::beams::CreateTrapezoidalQuadrature ( std::span< const double > grid )

inline

Creates a trapezoidal quadrature rule based on a given grid.

This function generates a set of quadrature points and weights for trapezoidal integration over a specified grid. The quadrature points are mapped to the interval [-1, 1].

Parameters

grid	A span of grid points defining the integration domain.

Returns: A vector of arrays, where each array contains a quadrature point and its corresponding weight.

◆ GenerateHollowCircleSection()

static BeamSection kynema::beams::GenerateHollowCircleSection	(	double	s,
		double	E,
		double	G,
		double	rho,
		double	outer_diameter,
		double	wall_thickness,
		double	nu,
		double	x_C = `0.`,
		double	y_C = `0.`,
		double	theta_p = `0.`,
		double	x_S = `0.`,
		double	y_S = `0.`,
		double	theta_s = `0.`,
		double	x_G = `0.`,
		double	y_G = `0.`,
		double	theta_i = `0`
	)

static

Generates a BeamSection with 6x6 mass and stiffness matrices for a hollow circular cross-section.

This is a convenience function specifically for hollow circular sections commonly used in wind turbine towers. It calculates the geometric properties and then calls the general GenerateMassMatrix and GenerateStiffnessMatrix functions.

Parameters

s	Normalized position along beam element [dimensionless]
E	Young's modulus [Force/Length²]
G	Shear modulus [Force/Length²]
rho	Material density [Mass/Length³]
outer_diameter	Outer diameter of the hollow circle [Length]
wall_thickness	Wall thickness [Length]
nu	Poisson's ratio [dimensionless]
x_C	x-coordinate of elastic centroid relative to reference point [Length]
y_C	y-coordinate of elastic centroid relative to reference point [Length]
theta_p	Rotation angle from reference axes to principal bending axes [radians]
x_S	x-coordinate of shear center relative to reference point [Length]
y_S	y-coordinate of shear center relative to reference point [Length]
theta_s	Rotation angle from reference axes to principal shear axes [radians]
x_G	x-coordinate of center of gravity relative to reference point [Length]
y_G	y-coordinate of center of gravity relative to reference point [Length]
theta_i	Rotation angle from reference axes to principal inertia axes [radians]

Returns: BeamSection object containing stiffness matrix, mass matrix, and position

Note: For hollow circular sections, the elastic centroid, shear center, and center of gravity all coincide at the geometric center, so offset parameters are typically zero.; Principal axes align with the reference axes due to circular symmetry.

◆ GenerateMassMatrix()

static std::array< std::array< double, 6 >, 6 > kynema::beams::GenerateMassMatrix	(	double	m,
		double	I_x,
		double	I_y,
		double	I_p,
		double	x_G = `0.`,
		double	y_G = `0.`,
		double	theta_i = `0.`
	)

static

Generates a 6x6 cross-sectional mass matrix for use in beam elements.

This function constructs the cross-sectional mass matrix that relates the generalized accelerations to the generalized inertial forces in a beam cross-section. Returns the mass matrix at a given point 'O' and w.r.t. given orientation axes based on the values at the center of gravity 'G' and in the inertial axis frame.

Parameters

m	Mass per unit length [Mass/Length]
I_x	Mass moment of inertia about local x-axis in principal/inertial frame and at center of gravity G [Mass×Length]
I_y	Mass moment of inertia about local y-axis in principal/inertial frame and at center of gravity G [Mass×Length]
I_p	Polar mass moment of inertia in principal/inertial frame and at center of gravity G (I_x + I_y) [Mass×Length]
x_G	x-coordinate of center of gravity relative to reference point i.e. distance FROM O TO G [Length]
y_G	y-coordinate of center of gravity relative to reference point i.e. distance FROM O TO G [Length]
theta_i	Rotation angle (around z) FROM reference axes TO principal inertial axes [radians]

Returns: 6x6 cross-sectional mass matrix

◆ GenerateStiffnessMatrix()

static std::array< std::array< double, 6 >, 6 > kynema::beams::GenerateStiffnessMatrix	(	double	EA,
		double	EI_x,
		double	EI_y,
		double	GKt,
		double	GA,
		double	kxs,
		double	kys,
		double	x_C,
		double	y_C,
		double	theta_p,
		double	x_S,
		double	y_S,
		double	theta_s
	)

static

Generates a 6x6 cross-sectional stiffness matrix for use in beam elements.

This function constructs the cross-sectional stiffness matrix that relates the generalized strains to the generalized forces/moments in a beam cross-section. The matrix accounts for coupling between axial, bending, shear, and torsional deformations due to offset between the elastic centroid and shear center locations. Returns a 6x6 stiffness matrix at the cross section origin 'O', based on inputs at the centroid 'C' and shear center 'S' locations.

Parameters

EA	Axial stiffness (E×A) [Force]
EI_x	Bending stiffness around local x-axis in principal frame and at the centroid 'C' [Force × Length²]
EI_y	Bending stiffness around local y-axis in principal frame and at the centroid 'C' [Force × Length²]
GKt	Torsional stiffness around principal frame and at shear center 'S' (G×J) [Force × Length²]
GA	Shear stiffness around principal frame and at shear center 'S' (G×A) [Force]
kxs	Shear correction factor in x-direction [dimensionless]
kys	Shear correction factor in y-direction [dimensionless]
x_C	x-coordinate of elastic centroid relative to reference point i.e. distance FROM 'O' TO 'C' [Length]
y_C	y-coordinate of elastic centroid relative to reference point i.e. distance FROM 'O' TO 'C' [Length]
theta_p	Rotation angle (around z) FROM reference axes TO principal bending axes [radians]
x_S	x-coordinate of shear center relative to reference point i.e. distance FROM 'O' TO 'S' [Length]
y_S	y-coordinate of shear center relative to reference point i.e. distance FROM 'O' TO 'S' [Length]
theta_s	Rotation angle (around z) FROM reference axes TO principal shear axes [radians]

Returns: 6x6 cross-sectional stiffness matrix

◆ InterpQuaternion()

template<typename shape_matrix_type , typename node_type , typename qp_type >

KOKKOS_INLINE_FUNCTION void kynema::beams::InterpQuaternion	(	const shape_matrix_type &	shape_matrix,
		const node_type &	node_v,
		const qp_type &	qp_v
	)

◆ InterpVector3()

template<typename shape_matrix_type , typename node_type , typename qp_type >

KOKKOS_INLINE_FUNCTION void kynema::beams::InterpVector3	(	const shape_matrix_type &	shape_matrix,
		const node_type &	node_v,
		const qp_type &	qp_v
	)

◆ InterpVector3Deriv()

template<typename shape_matrix_type , typename jacobian_type , typename node_type , typename qp_type >

KOKKOS_INLINE_FUNCTION void kynema::beams::InterpVector3Deriv	(	const shape_matrix_type &	shape_matrix_deriv,
		const jacobian_type &	jacobian,
		const node_type &	node_v,
		const qp_type &	qp_v
	)

◆ InterpVector4()

template<typename shape_matrix_type , typename node_type , typename qp_type >

KOKKOS_INLINE_FUNCTION void kynema::beams::InterpVector4	(	const shape_matrix_type &	shape_matrix,
		const node_type &	node_v,
		const qp_type &	qp_v
	)

◆ InterpVector4Deriv()

template<typename shape_matrix_type , typename jacobian_type , typename node_type , typename qp_type >

KOKKOS_INLINE_FUNCTION void kynema::beams::InterpVector4Deriv	(	const shape_matrix_type &	shape_matrix_deriv,
		const jacobian_type &	jacobian,
		const node_type &	node_v,
		const qp_type &	qp_v
	)

◆ MapNodePositions()

std::vector< double > kynema::beams::MapNodePositions	(	const BeamElement &	elem,
		std::span< const Node >	nodes
	)

inline

Map node positions from [0,1] to [-1,1].

◆ MapSectionPositions()

std::vector< double > kynema::beams::MapSectionPositions ( const BeamElement & elem )

inline

Map section positions from [0,1] to [-1,1].

◆ PopulateNodeX0()

void kynema::beams::PopulateNodeX0	(	const BeamElement &	elem,
		std::span< const Node >	nodes,
		const Kokkos::View< double *[7], Kokkos::LayoutStride, Kokkos::HostSpace > &	node_x0
	)

inline

Populate the node initial position and orientation.

◆ PopulateQPCstar()

void kynema::beams::PopulateQPCstar	(	const BeamElement &	elem,
		const Kokkos::View< double *[6][6], Kokkos::LayoutStride, Kokkos::HostSpace > &	qp_Cstar
	)

inline

Populate stiffness matrix values at each quadrature point.

◆ PopulateQPMstar()

void kynema::beams::PopulateQPMstar	(	const BeamElement &	elem,
		const Kokkos::View< double *[6][6], Kokkos::LayoutStride, Kokkos::HostSpace > &	qp_Mstar
	)

inline

Populate mass matrix values at each quadrature point.

◆ PopulateQPWeight()

void kynema::beams::PopulateQPWeight	(	const BeamElement &	elem,
		const Kokkos::View< double *, Kokkos::LayoutStride, Kokkos::HostSpace > &	qp_weight
	)

inline

Populate the integration weights at each quadrature point.

◆ PopulateShapeFunctionDerivatives()

void kynema::beams::PopulateShapeFunctionDerivatives	(	const BeamElement &	elem,
		std::span< const Node >	nodes,
		const Kokkos::View< double **, Kokkos::LayoutStride, Kokkos::HostSpace > &	shape_deriv
	)

inline

Populate shape function derivatives at each quadrature point.

◆ PopulateShapeFunctionValues()

void kynema::beams::PopulateShapeFunctionValues	(	const BeamElement &	elem,
		std::span< const Node >	nodes,
		const Kokkos::View< double **, Kokkos::LayoutStride, Kokkos::HostSpace > &	shape_interp
	)

inline

Populate shape function values at each quadrature point.

beams Namespace Reference

Classes

Functions

Function Documentation

◆ CalculateHollowCircleProperties()

◆ CreateGaussLegendreLobattoQuadrature()

◆ CreateGaussLegendreQuadrature()

◆ CreateTrapezoidalQuadrature()

◆ GenerateHollowCircleSection()

◆ GenerateMassMatrix()

◆ GenerateStiffnessMatrix()

◆ InterpQuaternion()

◆ InterpVector3()

◆ InterpVector3Deriv()

◆ InterpVector4()

◆ InterpVector4Deriv()

◆ MapNodePositions()

◆ MapSectionPositions()

◆ PopulateNodeX0()

◆ PopulateQPCstar()

◆ PopulateQPMstar()

◆ PopulateQPWeight()

◆ PopulateShapeFunctionDerivatives()

◆ PopulateShapeFunctionValues()