Topologically Rectangular Domain - a domain dexcribing a topologically rectangular problem - primarily contains functions to access the position of the global boundary relative to the macro element boundary, as well as first and second derivates of the global boundary wrt the macro element boundary NOTE : suitable for HermiteElementQuadMesh. More...

#include <topologically_rectangular_domain.h>

Inheritance diagram for oomph::TopologicallyRectangularDomain:

Public Types
typedef void(*	BoundaryFctPt) (const double &s, Vector< double > &r)
	boundary function pointer - for a given boundary takes the macro element coordinate position on that boundary and for that position returns the global coordinates (x) coordinates, or derivatives - dx_i/dm_t or second derivatives d2x_i/dm_t^2 More...

Public Member Functions
	TopologicallyRectangularDomain (BoundaryFctPt north_pt, BoundaryFctPt east_pt, BoundaryFctPt south_pt, BoundaryFctPt west_pt)
	Constructor - domain boundaries are described with four boundary function pointers describing the topology of the north, east, south, and west boundaries. More...

	TopologicallyRectangularDomain (const double &l_x, const double &l_y)
	Constructor - takes length of domain in x and y direction as arguements. Assumes domain is rectangular, and the south west (lower left) corner is at 0,0. More...

	TopologicallyRectangularDomain (const double &x_min, const double &x_max, const double &y_min, const double &y_max)
	Constructor - takes the minimum and maximum coordinates of the of an assumed rectanguler domain in the x and y direction. More...

	TopologicallyRectangularDomain (const TopologicallyRectangularDomain &)=delete
	Broken copy constructor. More...

void	operator= (const TopologicallyRectangularDomain &)=delete
	Broken assignment operator. More...

	~TopologicallyRectangularDomain ()
	Destructor - empty; clean up done in base class. More...

void	set_boundary_derivative_functions (BoundaryFctPt d_north_pt, BoundaryFctPt d_east_pt, BoundaryFctPt d_south_pt, BoundaryFctPt d_west_pt)
	allows the boundary derivate function pointers to be set. To compute the derivatives of the problem domain global coordinates (x_i) wrt the macro element coordinates (m_i), dx_i/dm_t is required along the domain boundaries (where dm_t is the macro element coordinate tangential to the domain boundary). The derivatives dx_i/dm_t can either be prescribed with function pointers, or if the function pointers are not provided then dx_i/dm_t is computed with finite differencing. Note - these functions are only required for domains contructed with boundary function pointers More...

void	set_boundary_second_derivative_functions (BoundaryFctPt d2_north_pt, BoundaryFctPt d2_east_pt, BoundaryFctPt d2_south_pt, BoundaryFctPt d2_west_pt)
	allows the boundary second derivate function pointers to be set. To compute the second derivatives of the problem domain global coordinates (x_i) wrt the macro element coordinates (m_i), d2x_i/dm_t^2 is required along the domain boundaries (where dm_t is the macro element coordinate tangential to the domain boundary). The derivatives d2x_i/dm_t^2 can either be prescribed with function pointers, or if the function pointers are not provided then dx_i/dm_t is computed with finite differencing. Note - these functions are only required for domains contructed with boundary function pointers More...

void	macro_element_boundary (const unsigned &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	returns the global coordinate position (f) of macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence) More...

void	dmacro_element_boundary (const unsigned &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	returns the derivates of the global coordinate position (f) wrt to the macro element coordinate at macro macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence) More...

void	d2macro_element_boundary (const unsigned &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	returns the second derivates of the global coordinate position (f) wrt to the macro element coordinate at macro macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence) More...

Public Member Functions inherited from oomph::Domain
	Domain ()
	Constructor. More...

	Domain (const Domain &)=delete
	Broken copy constructor. More...

void	operator= (const Domain &)=delete
	Broken assignment operator. More...

virtual	~Domain ()
	Destructor: Strictly speaking, whoever creates an object dynamically should be responsible for the cleanup of said object but it makes sense here for the Domain to generically kill any MacroElements left over in the MacroElement container (if it hasn't already been done in the derived class) to avoid memory leaks. More...

MacroElement *	macro_element_pt (const unsigned &i)
	Access to i-th macro element. More...

unsigned	nmacro_element ()
	Number of macro elements in domain. More...

void	output (const std::string &filename, const unsigned &nplot)
	Output macro elements. More...

void	output (std::ostream &outfile, const unsigned &nplot)
	Output macro elements. More...

virtual void	macro_element_boundary (const double &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary i_direct (e.g. N/S/W/E in 2D) at continuous time, t. More...

void	macro_element_boundary (const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary i_direct (e.g. N/S/W/E in 2D) at current time: f(s). More...

void	output_macro_element_boundaries (const std::string &filename, const unsigned &nplot)
	Output all macro element boundaries as tecplot zones. More...

void	output_macro_element_boundaries (std::ostream &outfile, const unsigned &nplot)
	Output all macro element boundaries as tecplot zones. More...

virtual void	dmacro_element_boundary (const double &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary derivatives i_direct (e.g. N/S/W/E in 2D) at continuous time level t. Broken virtual. More...

void	dmacro_element_boundary (const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary derivatives i_direct (e.g. N/S/W/E in 2D) at current time: f(s). More...

virtual void	d2macro_element_boundary (const double &t, const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary seocond derivatives i_direct (e.g. N/S/W/E in 2D) at continuous time level t. Broken virtual. More...

void	d2macro_element_boundary (const unsigned &i_macro, const unsigned &i_direct, const Vector< double > &s, Vector< double > &f)
	Vector representation of the i_macro-th macro element boundary second derivatives i_direct (e.g. N/S/W/E in 2D) at current time: f(s). More...

Private Member Functions
void	r_N (const Vector< double > &s, Vector< double > &f)
	takes the macro element coordinate position along the north boundary and returns the global coordinate position along that boundary More...

void	r_E (const Vector< double > &s, Vector< double > &f)
	takes the macro element coordinate position along the east boundary and returns the global coordinate position along that boundary More...

void	r_S (const Vector< double > &s, Vector< double > &f)
	takes the macro element coordinate position along the south boundary and returns the global coordinate position along that boundary More...

void	r_W (const Vector< double > &s, Vector< double > &f)
	takes the macro element coordinate position along the west boundary and returns the global coordinate position along that boundary access down boundary function pointer More...

void	dr_N (const Vector< double > &s, Vector< double > &dr)
	takes the macro element coordinate position along the north boundary and returns the derivates of the global coordinates with respect to the boundary More...

void	dr_E (const Vector< double > &s, Vector< double > &dr)
	takes the macro element coordinate position along the E boundary and returns the derivates of the global coordinates with respect to the boundary More...

void	dr_S (const Vector< double > &s, Vector< double > &dr)
	takes the macro element coordinate position along the south boundary and returns the derivates of the global coordinates with respect to the boundary More...

void	dr_W (const Vector< double > &s, Vector< double > &dr)
	takes the macro element coordinate position along the W boundary and returns the derivates of the global coordinates with respect to the boundary More...

void	d2r_N (const Vector< double > &s, Vector< double > &d2r)
	takes the macro element coordinate position along the north boundary and returns the second derivates of the global coordinates with respect to the boundary More...

void	d2r_E (const Vector< double > &s, Vector< double > &d2r)
	takes the macro element coordinate position along the east boundary and returns the second derivates of the global coordinates with respect to the boundary More...

void	d2r_S (const Vector< double > &s, Vector< double > &d2r)
	takes the macro element coordinate position along the south boundary and returns the second derivates of the global coordinates with respect to the boundary More...

void	d2r_W (const Vector< double > &s, Vector< double > &d2r)
	takes the macro element coordinate position along the west boundary and returns the second derivates of the global coordinates with respect to the boundary More...

Private Attributes
BoundaryFctPt	North_boundary_fn_pt
	Function pointer to prescribe the north boundary of this topologically rectangular domain. More...

BoundaryFctPt	East_boundary_fn_pt
	Function pointer to prescribe the east boundary of this topologically rectangular domain. More...

BoundaryFctPt	South_boundary_fn_pt
	Function pointer to prescribe the north boundary of this topologically rectangular domain. More...

BoundaryFctPt	West_boundary_fn_pt
	Function pointer to prescribe the west boundary of this topologically rectangular domain. More...

BoundaryFctPt	dNorth_boundary_fn_pt
	Function pointer to prescribe the derivates of global coordinates wrt to the macro element coordinate tangential to the north boundary. More...

BoundaryFctPt	dEast_boundary_fn_pt
	Function pointer to prescribe the derivates of global coordinates wrt to the macro element coordinate tangential to the east boundary. More...

BoundaryFctPt	dSouth_boundary_fn_pt
	Function pointer to prescribe the derivates of global coordinates wrt to the macro element coordinate tangential to the south boundary. More...

BoundaryFctPt	dWest_boundary_fn_pt
	Function pointer to prescribe the derivates of global coordinates wrt to the macro element coordinate tangential to the west boundary. More...

BoundaryFctPt	d2North_boundary_fn_pt
	Function pointer to prescribe the second derivates of global coordinates wrt to the macro element coordinate tangential to the north boundary. More...

BoundaryFctPt	d2East_boundary_fn_pt
	Function pointer to prescribe the second derivates of global coordinates wrt to the macro element coordinate tangential to the east boundary. More...

BoundaryFctPt	d2South_boundary_fn_pt
	Function pointer to prescribe the second derivates of global coordinates wrt to the macro element coordinate tangential to the south boundary. More...

BoundaryFctPt	d2West_boundary_fn_pt
	Function pointer to prescribe the second derivates of global coordinates wrt to the macro element coordinate tangential to the west boundary. More...

Vector< double >	x_south_west
	coordinate position of south west corner of domain (only used if boundary functions are not used) More...

Vector< double >	x_north_east
	coordinate position of north east corner of domain (only used if boundary functions are not used) More...

Additional Inherited Members
Protected Attributes inherited from oomph::Domain
Vector< MacroElement * >	Macro_element_pt
	Vector of pointers to macro elements. More...

Detailed Description

Topologically Rectangular Domain - a domain dexcribing a topologically rectangular problem - primarily contains functions to access the position of the global boundary relative to the macro element boundary, as well as first and second derivates of the global boundary wrt the macro element boundary NOTE : suitable for HermiteElementQuadMesh.

Definition at line 50 of file topologically_rectangular_domain.h.

Member Typedef Documentation

◆ BoundaryFctPt

typedef void(* oomph::TopologicallyRectangularDomain::BoundaryFctPt) (const double &s, Vector< double > &r)

boundary function pointer - for a given boundary takes the macro element coordinate position on that boundary and for that position returns the global coordinates (x) coordinates, or derivatives - dx_i/dm_t or second derivatives d2x_i/dm_t^2

Definition at line 57 of file topologically_rectangular_domain.h.

Constructor & Destructor Documentation

◆ TopologicallyRectangularDomain() [1/4]

oomph::TopologicallyRectangularDomain::TopologicallyRectangularDomain	(	BoundaryFctPt	north_pt,
		BoundaryFctPt	east_pt,
		BoundaryFctPt	south_pt,
		BoundaryFctPt	west_pt
	)

Constructor - domain boundaries are described with four boundary function pointers describing the topology of the north, east, south, and west boundaries.

Definition at line 36 of file topologically_rectangular_domain.cc.

References d2East_boundary_fn_pt, d2North_boundary_fn_pt, d2South_boundary_fn_pt, d2West_boundary_fn_pt, dEast_boundary_fn_pt, dNorth_boundary_fn_pt, dSouth_boundary_fn_pt, dWest_boundary_fn_pt, East_boundary_fn_pt, oomph::Domain::Macro_element_pt, North_boundary_fn_pt, South_boundary_fn_pt, and West_boundary_fn_pt.

◆ TopologicallyRectangularDomain() [2/4]

oomph::TopologicallyRectangularDomain::TopologicallyRectangularDomain	(	const double &	l_x,
		const double &	l_y
	)

Constructor - takes length of domain in x and y direction as arguements. Assumes domain is rectangular, and the south west (lower left) corner is at 0,0.

Definition at line 152 of file topologically_rectangular_domain.cc.

References d2East_boundary_fn_pt, d2North_boundary_fn_pt, d2South_boundary_fn_pt, d2West_boundary_fn_pt, dEast_boundary_fn_pt, dNorth_boundary_fn_pt, dSouth_boundary_fn_pt, dWest_boundary_fn_pt, East_boundary_fn_pt, oomph::Domain::Macro_element_pt, North_boundary_fn_pt, South_boundary_fn_pt, West_boundary_fn_pt, x_north_east, and x_south_west.

◆ TopologicallyRectangularDomain() [3/4]

oomph::TopologicallyRectangularDomain::TopologicallyRectangularDomain	(	const double &	x_min,
		const double &	x_max,
		const double &	y_min,
		const double &	y_max
	)

Constructor - takes the minimum and maximum coordinates of the of an assumed rectanguler domain in the x and y direction.

Definition at line 196 of file topologically_rectangular_domain.cc.

References d2East_boundary_fn_pt, d2North_boundary_fn_pt, d2South_boundary_fn_pt, d2West_boundary_fn_pt, dEast_boundary_fn_pt, dNorth_boundary_fn_pt, dSouth_boundary_fn_pt, dWest_boundary_fn_pt, East_boundary_fn_pt, oomph::Domain::Macro_element_pt, North_boundary_fn_pt, South_boundary_fn_pt, West_boundary_fn_pt, x_north_east, and x_south_west.

◆ TopologicallyRectangularDomain() [4/4]

oomph::TopologicallyRectangularDomain::TopologicallyRectangularDomain ( const TopologicallyRectangularDomain & )

delete

Broken copy constructor.

◆ ~TopologicallyRectangularDomain()

oomph::TopologicallyRectangularDomain::~TopologicallyRectangularDomain ( )

inline

Destructor - empty; clean up done in base class.

Definition at line 91 of file topologically_rectangular_domain.h.

Member Function Documentation

◆ d2macro_element_boundary()

void oomph::TopologicallyRectangularDomain::d2macro_element_boundary	(	const unsigned &	t,
		const unsigned &	i_macro,
		const unsigned &	i_direct,
		const Vector< double > &	s,
		Vector< double > &	f
	)

virtual

returns the second derivates of the global coordinate position (f) wrt to the macro element coordinate at macro macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence)

Reimplemented from oomph::Domain.

Definition at line 364 of file topologically_rectangular_domain.cc.

References d2r_E(), d2r_N(), d2r_S(), d2r_W(), oomph::QuadTreeNames::E, oomph::QuadTreeNames::N, s, oomph::QuadTreeNames::S, and oomph::QuadTreeNames::W.

◆ d2r_E()

void oomph::TopologicallyRectangularDomain::d2r_E	(	const Vector< double > &	s,
		Vector< double > &	d2r
	)

private

takes the macro element coordinate position along the east boundary and returns the second derivates of the global coordinates with respect to the boundary

Definition at line 675 of file topologically_rectangular_domain.cc.

References d2East_boundary_fn_pt, dEast_boundary_fn_pt, e, East_boundary_fn_pt, and s.

Referenced by d2macro_element_boundary().

◆ d2r_N()

void oomph::TopologicallyRectangularDomain::d2r_N	(	const Vector< double > &	s,
		Vector< double > &	d2r
	)

private

takes the macro element coordinate position along the north boundary and returns the second derivates of the global coordinates with respect to the boundary

Definition at line 622 of file topologically_rectangular_domain.cc.

References d2North_boundary_fn_pt, dNorth_boundary_fn_pt, e, North_boundary_fn_pt, and s.

Referenced by d2macro_element_boundary().

◆ d2r_S()

void oomph::TopologicallyRectangularDomain::d2r_S	(	const Vector< double > &	s,
		Vector< double > &	d2r
	)

private

takes the macro element coordinate position along the south boundary and returns the second derivates of the global coordinates with respect to the boundary

Definition at line 728 of file topologically_rectangular_domain.cc.

References d2South_boundary_fn_pt, dSouth_boundary_fn_pt, e, s, and South_boundary_fn_pt.

Referenced by d2macro_element_boundary().

◆ d2r_W()

void oomph::TopologicallyRectangularDomain::d2r_W	(	const Vector< double > &	s,
		Vector< double > &	d2r
	)

private

takes the macro element coordinate position along the west boundary and returns the second derivates of the global coordinates with respect to the boundary

Definition at line 781 of file topologically_rectangular_domain.cc.

References d2West_boundary_fn_pt, dWest_boundary_fn_pt, e, s, and West_boundary_fn_pt.

Referenced by d2macro_element_boundary().

◆ dmacro_element_boundary()

void oomph::TopologicallyRectangularDomain::dmacro_element_boundary	(	const unsigned &	t,
		const unsigned &	i_macro,
		const unsigned &	i_direct,
		const Vector< double > &	s,
		Vector< double > &	f
	)

virtual

returns the derivates of the global coordinate position (f) wrt to the macro element coordinate at macro macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence)

Reimplemented from oomph::Domain.

Definition at line 328 of file topologically_rectangular_domain.cc.

References dr_E(), dr_N(), dr_S(), dr_W(), oomph::QuadTreeNames::E, oomph::QuadTreeNames::N, s, oomph::QuadTreeNames::S, and oomph::QuadTreeNames::W.

◆ dr_E()

void oomph::TopologicallyRectangularDomain::dr_E	(	const Vector< double > &	s,
		Vector< double > &	dr
	)

private

takes the macro element coordinate position along the E boundary and returns the derivates of the global coordinates with respect to the boundary

Definition at line 514 of file topologically_rectangular_domain.cc.

References dEast_boundary_fn_pt, e, East_boundary_fn_pt, s, x_north_east, and x_south_west.

Referenced by dmacro_element_boundary().

◆ dr_N()

void oomph::TopologicallyRectangularDomain::dr_N	(	const Vector< double > &	s,
		Vector< double > &	dr
	)

private

takes the macro element coordinate position along the north boundary and returns the derivates of the global coordinates with respect to the boundary

Definition at line 478 of file topologically_rectangular_domain.cc.

References dNorth_boundary_fn_pt, e, North_boundary_fn_pt, s, x_north_east, and x_south_west.

Referenced by dmacro_element_boundary().

◆ dr_S()

void oomph::TopologicallyRectangularDomain::dr_S	(	const Vector< double > &	s,
		Vector< double > &	dr
	)

private

takes the macro element coordinate position along the south boundary and returns the derivates of the global coordinates with respect to the boundary

Definition at line 550 of file topologically_rectangular_domain.cc.

References dSouth_boundary_fn_pt, e, s, South_boundary_fn_pt, x_north_east, and x_south_west.

Referenced by dmacro_element_boundary().

◆ dr_W()

void oomph::TopologicallyRectangularDomain::dr_W	(	const Vector< double > &	s,
		Vector< double > &	dr
	)

private

takes the macro element coordinate position along the W boundary and returns the derivates of the global coordinates with respect to the boundary

Definition at line 586 of file topologically_rectangular_domain.cc.

References dWest_boundary_fn_pt, e, s, West_boundary_fn_pt, x_north_east, and x_south_west.

Referenced by dmacro_element_boundary().

◆ macro_element_boundary()

void oomph::TopologicallyRectangularDomain::macro_element_boundary	(	const unsigned &	t,
		const unsigned &	i_macro,
		const unsigned &	i_direct,
		const Vector< double > &	s,
		Vector< double > &	f
	)

virtual

returns the global coordinate position (f) of macro element position s on boundary i_direct (e.g. N/S/W/E in 2D) at time t (no time dependence)

Implements oomph::Domain.

Definition at line 292 of file topologically_rectangular_domain.cc.

References oomph::QuadTreeNames::E, oomph::QuadTreeNames::N, r_E(), r_N(), r_S(), r_W(), s, oomph::QuadTreeNames::S, and oomph::QuadTreeNames::W.

◆ operator=()

void oomph::TopologicallyRectangularDomain::operator= ( const TopologicallyRectangularDomain & )

delete

Broken assignment operator.

◆ r_E()

void oomph::TopologicallyRectangularDomain::r_E	(	const Vector< double > &	s,
		Vector< double > &	f
	)

private

takes the macro element coordinate position along the east boundary and returns the global coordinate position along that boundary

Definition at line 419 of file topologically_rectangular_domain.cc.

References East_boundary_fn_pt, s, x_north_east, and x_south_west.

Referenced by macro_element_boundary().

◆ r_N()

void oomph::TopologicallyRectangularDomain::r_N	(	const Vector< double > &	s,
		Vector< double > &	f
	)

private

takes the macro element coordinate position along the north boundary and returns the global coordinate position along that boundary

Definition at line 400 of file topologically_rectangular_domain.cc.

References North_boundary_fn_pt, s, x_north_east, and x_south_west.

Referenced by macro_element_boundary().

◆ r_S()

void oomph::TopologicallyRectangularDomain::r_S	(	const Vector< double > &	s,
		Vector< double > &	f
	)

private

takes the macro element coordinate position along the south boundary and returns the global coordinate position along that boundary

Definition at line 438 of file topologically_rectangular_domain.cc.

References s, South_boundary_fn_pt, x_north_east, and x_south_west.

Referenced by macro_element_boundary().

◆ r_W()

void oomph::TopologicallyRectangularDomain::r_W	(	const Vector< double > &	s,
		Vector< double > &	f
	)

private

takes the macro element coordinate position along the west boundary and returns the global coordinate position along that boundary access down boundary function pointer

Definition at line 458 of file topologically_rectangular_domain.cc.

References s, West_boundary_fn_pt, x_north_east, and x_south_west.

Referenced by macro_element_boundary().

◆ set_boundary_derivative_functions()

void oomph::TopologicallyRectangularDomain::set_boundary_derivative_functions	(	BoundaryFctPt	d_north_pt,
		BoundaryFctPt	d_east_pt,
		BoundaryFctPt	d_south_pt,
		BoundaryFctPt	d_west_pt
	)

allows the boundary derivate function pointers to be set. To compute the derivatives of the problem domain global coordinates (x_i) wrt the macro element coordinates (m_i), dx_i/dm_t is required along the domain boundaries (where dm_t is the macro element coordinate tangential to the domain boundary). The derivatives dx_i/dm_t can either be prescribed with function pointers, or if the function pointers are not provided then dx_i/dm_t is computed with finite differencing. Note - these functions are only required for domains contructed with boundary function pointers

Definition at line 248 of file topologically_rectangular_domain.cc.

References dEast_boundary_fn_pt, dNorth_boundary_fn_pt, dSouth_boundary_fn_pt, and dWest_boundary_fn_pt.

◆ set_boundary_second_derivative_functions()

void oomph::TopologicallyRectangularDomain::set_boundary_second_derivative_functions	(	BoundaryFctPt	d2_north_pt,
		BoundaryFctPt	d2_east_pt,
		BoundaryFctPt	d2_south_pt,
		BoundaryFctPt	d2_west_pt
	)

allows the boundary second derivate function pointers to be set. To compute the second derivatives of the problem domain global coordinates (x_i) wrt the macro element coordinates (m_i), d2x_i/dm_t^2 is required along the domain boundaries (where dm_t is the macro element coordinate tangential to the domain boundary). The derivatives d2x_i/dm_t^2 can either be prescribed with function pointers, or if the function pointers are not provided then dx_i/dm_t is computed with finite differencing. Note - these functions are only required for domains contructed with boundary function pointers

Definition at line 274 of file topologically_rectangular_domain.cc.

References d2East_boundary_fn_pt, d2North_boundary_fn_pt, d2South_boundary_fn_pt, and d2West_boundary_fn_pt.