This class provides an encapsulated access to all static public member functions of finite element classes. More...

#include <fe_interface.h>

Public Types
typedef unsigned int(*	n_dofs_at_node_ptr) (const ElemType, const Order, const unsigned int)

Public Member Functions
virtual	~FEInterface ()
	Destructor. More...

Static Public Member Functions
static unsigned int	n_shape_functions (const unsigned int dim, const FEType &fe_t, const ElemType t)

static unsigned int	n_dofs (const unsigned int dim, const FEType &fe_t, const ElemType t)

static unsigned int	n_dofs (const unsigned int dim, const FEType &fe_t, const Elem *elem)
	Similar to the function above but takes an Elem * and accounts for p-refinement internally, if any. More...

static unsigned int	n_dofs_at_node (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int n)

static n_dofs_at_node_ptr	n_dofs_at_node_function (const unsigned int dim, const FEType &fe_t)

static unsigned int	n_dofs_per_elem (const unsigned int dim, const FEType &fe_t, const ElemType t)

static void	dofs_on_side (const Elem *const elem, const unsigned int dim, const FEType &fe_t, unsigned int s, std::vector< unsigned int > &di)
	Fills the vector di with the local degree of freedom indices associated with side `s` of element `elem` Automatically decides which finite element class to use. More...

static void	dofs_on_edge (const Elem *const elem, const unsigned int dim, const FEType &fe_t, unsigned int e, std::vector< unsigned int > &di)
	Fills the vector di with the local degree of freedom indices associated with edge `e` of element `elem` Automatically decides which finite element class to use. More...

static void	nodal_soln (const unsigned int dim, const FEType &fe_t, const Elem *elem, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
	Build the nodal soln from the element soln. More...

static Point	map (unsigned int dim, const FEType &fe_t, const Elem *elem, const Point &p)

static Point	inverse_map (const unsigned int dim, const FEType &fe_t, const Elem *elem, const Point &p, const Real tolerance=TOLERANCE, const bool secure=true)

static void	inverse_map (const unsigned int dim, const FEType &fe_t, const Elem *elem, const std::vector< Point > &physical_points, std::vector< Point > &reference_points, const Real tolerance=TOLERANCE, const bool secure=true)

static bool	on_reference_element (const Point &p, const ElemType t, const Real eps=TOLERANCE)

static Real	shape (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int i, const Point &p)

static Real	shape (const unsigned int dim, const FEType &fe_t, const Elem *elem, const unsigned int i, const Point &p)

template<typename OutputType >
static void	shape (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int i, const Point &p, OutputType &phi)

template<typename OutputType >
static void	shape (const unsigned int dim, const FEType &fe_t, const Elem *elem, const unsigned int i, const Point &p, OutputType &phi)

static Real	shape_deriv (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int i, const unsigned int j, const Point &p)

static Real	shape_deriv (const unsigned int dim, const FEType &fe_t, const Elem *elem, const unsigned int i, const unsigned int j, const Point &p)

static void	compute_data (const unsigned int dim, const FEType &fe_t, const Elem *elem, FEComputeData &data)
	Lets the appropriate child of `FEBase` compute the requested data for the input specified in `data`, and sets the values in `data`. More...

static void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
	Computes the constraint matrix contributions (for non-conforming adapted meshes) corresponding to variable number `var_number`. More...

static void	compute_periodic_constraints (DofConstraints &constraints, DofMap &dof_map, const PeriodicBoundaries &boundaries, const MeshBase &mesh, const PointLocatorBase point_locator, const unsigned int variable_number, const Elem elem)
	Computes the constraint matrix contributions (for periodic boundary conditions) corresponding to variable number `var_number`. More...

static unsigned int	max_order (const FEType &fe_t, const ElemType &el_t)

static bool	extra_hanging_dofs (const FEType &fe_t)

static FEFieldType	field_type (const FEType &fe_type)

static FEFieldType	field_type (const FEFamily &fe_family)

static unsigned int	n_vec_dim (const MeshBase &mesh, const FEType &fe_type)

static FEContinuity	get_continuity (const FEType &fe_type)
	Returns the input FEType's FEContinuity based on the underlying FEFamily and potentially the Order, although we do not currently support FEs with order-dependent continuity. More...

Private Member Functions
	FEInterface ()
	Empty constructor. More...

Static Private Member Functions
static bool	is_InfFE_elem (const ElemType et)

static unsigned int	ifem_n_shape_functions (const unsigned int dim, const FEType &fe_t, const ElemType t)

static unsigned int	ifem_n_dofs (const unsigned int dim, const FEType &fe_t, const ElemType t)

static unsigned int	ifem_n_dofs_at_node (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int n)

static unsigned int	ifem_n_dofs_per_elem (const unsigned int dim, const FEType &fe_t, const ElemType t)

static void	ifem_nodal_soln (const unsigned int dim, const FEType &fe_t, const Elem *elem, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)

static Point	ifem_map (const unsigned int dim, const FEType &fe_t, const Elem *elem, const Point &p)

static Point	ifem_inverse_map (const unsigned int dim, const FEType &fe_t, const Elem *elem, const Point &p, const Real tolerance=TOLERANCE, const bool secure=true)

static void	ifem_inverse_map (const unsigned int dim, const FEType &fe_t, const Elem *elem, const std::vector< Point > &physical_points, std::vector< Point > &reference_points, const Real tolerance=TOLERANCE, const bool secure=true)

static bool	ifem_on_reference_element (const Point &p, const ElemType t, const Real eps)

static Real	ifem_shape (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int i, const Point &p)

static Real	ifem_shape (const unsigned int dim, const FEType &fe_t, const Elem *elem, const unsigned int i, const Point &p)

static Real	ifem_shape_deriv (const unsigned int dim, const FEType &fe_t, const ElemType t, const unsigned int i, const unsigned int j, const Point &p)

static Real	ifem_shape_deriv (const unsigned int dim, const FEType &fe_t, const Elem *elem, const unsigned int i, const unsigned int j, const Point &p)

static void	ifem_compute_data (const unsigned int dim, const FEType &fe_t, const Elem *elem, FEComputeData &data)

Detailed Description

This class provides an encapsulated access to all static public member functions of finite element classes.

Using this class, one need not worry about the correct finite element class.

Author: Daniel Dreyer

Date: 2002-2007 Interface class which provides access to FE functions.

Definition at line 65 of file fe_interface.h.

Member Typedef Documentation

◆ n_dofs_at_node_ptr

typedef unsigned int(* libMesh::FEInterface::n_dofs_at_node_ptr) (const ElemType, const Order, const unsigned int)

Definition at line 126 of file fe_interface.h.

Constructor & Destructor Documentation

◆ FEInterface()

libMesh::FEInterface::FEInterface ( )

private

Empty constructor.

Do not create an object of this type.

◆ ~FEInterface()

virtual libMesh::FEInterface::~FEInterface ( )

virtual

Destructor.

Definition at line 79 of file fe_interface.h.

79 {}

Member Function Documentation

◆ compute_constraints()

static void libMesh::FEInterface::compute_constraints	(	DofConstraints &	constraints,
		DofMap &	dof_map,
		const unsigned int	variable_number,
		const Elem *	elem
	)

static

Computes the constraint matrix contributions (for non-conforming adapted meshes) corresponding to variable number var_number.

◆ compute_data()

static void libMesh::FEInterface::compute_data	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		FEComputeData &	data
	)

static

Lets the appropriate child of FEBase compute the requested data for the input specified in data, and sets the values in data.

See this as a generalization of shape(). With infinite elements disabled, computes values for all shape functions of elem evaluated at p.

Note: On a p-refined element, fe_t.order should be the base order of the element.

◆ compute_periodic_constraints()

static void libMesh::FEInterface::compute_periodic_constraints	(	DofConstraints &	constraints,
		DofMap &	dof_map,
		const PeriodicBoundaries &	boundaries,
		const MeshBase &	mesh,
		const PointLocatorBase *	point_locator,
		const unsigned int	variable_number,
		const Elem *	elem
	)

static

Computes the constraint matrix contributions (for periodic boundary conditions) corresponding to variable number var_number.

◆ dofs_on_edge()

static void libMesh::FEInterface::dofs_on_edge	(	const Elem *const	elem,
		const unsigned int	dim,
		const FEType &	fe_t,
		unsigned int	e,
		std::vector< unsigned int > &	di
	)

static

Fills the vector di with the local degree of freedom indices associated with edge e of element elem Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the base order of the element.

Referenced by libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()().

◆ dofs_on_side()

static void libMesh::FEInterface::dofs_on_side	(	const Elem *const	elem,
		const unsigned int	dim,
		const FEType &	fe_t,
		unsigned int	s,
		std::vector< unsigned int > &	di
	)

static

Fills the vector di with the local degree of freedom indices associated with side s of element elem Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the base order of the element.

Referenced by libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()().

◆ extra_hanging_dofs()

static bool libMesh::FEInterface::extra_hanging_dofs ( const FEType & fe_t )

static

Returns: true if separate degrees of freedom must be allocated for vertex DoFs and edge/face DoFs at a hanging node.

◆ field_type() [1/2]

static FEFieldType libMesh::FEInterface::field_type ( const FEType & fe_type )

static

Returns: The number of components of a vector-valued element. Scalar-valued elements return 1.

◆ field_type() [2/2]

static FEFieldType libMesh::FEInterface::field_type ( const FEFamily & fe_family )

static

Returns: The number of components of a vector-valued element. Scalar-valued elements return 1.

◆ get_continuity()

static FEContinuity libMesh::FEInterface::get_continuity ( const FEType & fe_type )

static

Returns the input FEType's FEContinuity based on the underlying FEFamily and potentially the Order, although we do not currently support FEs with order-dependent continuity.

These should exactly match the FEBase::get_continuity() specializations/overrides for the different FE types.

◆ ifem_compute_data()

static void libMesh::FEInterface::ifem_compute_data	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		FEComputeData &	data
	)

staticprivate

◆ ifem_inverse_map() [1/2]

static Point libMesh::FEInterface::ifem_inverse_map	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const Point &	p,
		const Real	tolerance = `TOLERANCE`,
		const bool	secure = `true`
	)

staticprivate

◆ ifem_inverse_map() [2/2]

static void libMesh::FEInterface::ifem_inverse_map	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const std::vector< Point > &	physical_points,
		std::vector< Point > &	reference_points,
		const Real	tolerance = `TOLERANCE`,
		const bool	secure = `true`
	)

staticprivate

◆ ifem_map()

static Point libMesh::FEInterface::ifem_map	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const Point &	p
	)

staticprivate

◆ ifem_n_dofs()

static unsigned int libMesh::FEInterface::ifem_n_dofs	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

staticprivate

◆ ifem_n_dofs_at_node()

static unsigned int libMesh::FEInterface::ifem_n_dofs_at_node	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	n
	)

staticprivate

◆ ifem_n_dofs_per_elem()

static unsigned int libMesh::FEInterface::ifem_n_dofs_per_elem	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

staticprivate

◆ ifem_n_shape_functions()

static unsigned int libMesh::FEInterface::ifem_n_shape_functions	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

staticprivate

◆ ifem_nodal_soln()

static void libMesh::FEInterface::ifem_nodal_soln	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const std::vector< Number > &	elem_soln,
		std::vector< Number > &	nodal_soln
	)

staticprivate

◆ ifem_on_reference_element()

static bool libMesh::FEInterface::ifem_on_reference_element	(	const Point &	p,
		const ElemType	t,
		const Real	eps
	)

staticprivate

◆ ifem_shape() [1/2]

static Real libMesh::FEInterface::ifem_shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	i,
		const Point &	p
	)

staticprivate

◆ ifem_shape() [2/2]

static Real libMesh::FEInterface::ifem_shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const unsigned int	i,
		const Point &	p
	)

staticprivate

◆ ifem_shape_deriv() [1/2]

static Real libMesh::FEInterface::ifem_shape_deriv	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	i,
		const unsigned int	j,
		const Point &	p
	)

staticprivate

◆ ifem_shape_deriv() [2/2]

static Real libMesh::FEInterface::ifem_shape_deriv	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const unsigned int	i,
		const unsigned int	j,
		const Point &	p
	)

staticprivate

◆ inverse_map() [1/2]

static Point libMesh::FEInterface::inverse_map	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const Point &	p,
		const Real	tolerance = `TOLERANCE`,
		const bool	secure = `true`
	)

static

Returns: The location (on the reference element) of the point p located in physical space. This function requires inverting the (probably nonlinear) transformation map, so it is not trivial. The optional parameter tolerance defines how close is "good enough." The map inversion iteration computes the sequence $\{ p_n \}$ , and the iteration is terminated when $\|p - p_n\| < \mbox{\texttt{tolerance}}$

Referenced by libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()(), FETest< order, family, elem_type >::testGradU(), FETest< order, family, elem_type >::testGradUComp(), and FETest< order, family, elem_type >::testU().

◆ inverse_map() [2/2]

static void libMesh::FEInterface::inverse_map	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const std::vector< Point > &	physical_points,
		std::vector< Point > &	reference_points,
		const Real	tolerance = `TOLERANCE`,
		const bool	secure = `true`
	)

static

Returns: The location (on the reference element) of the points physical_points located in physical space. This function requires inverting the (probably nonlinear) transformation map, so it is not trivial. The location of each point on the reference element is returned in the vector reference_points. The optional parameter tolerance defines how close is "good enough." The map inversion iteration computes the sequence $\{ p_n \}$ , and the iteration is terminated when $\|p - p_n\| < \mbox{\texttt{tolerance}}$

◆ is_InfFE_elem()

static bool libMesh::FEInterface::is_InfFE_elem ( const ElemType et )

staticprivate

Returns: true if et is an element to be processed by class InfFE, false otherwise or when !LIBMESH_ENABLE_INFINITE_ELEMENTS.

◆ map()

static Point libMesh::FEInterface::map	(	unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const Point &	p
	)

static

Returns: The point in physical space corresponding to the reference point p which is passed in.

◆ max_order()

static unsigned int libMesh::FEInterface::max_order	(	const FEType &	fe_t,
		const ElemType &	el_t
	)

static

Returns: The maximum polynomial degree that the given finite element family can support on the given geometric element.

◆ n_dofs() [1/2]

static unsigned int libMesh::FEInterface::n_dofs	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

static

Returns: The number of shape functions associated with this finite element. Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the total order of the element.

◆ n_dofs() [2/2]

static unsigned int libMesh::FEInterface::n_dofs	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem
	)

static

Similar to the function above but takes an Elem * and accounts for p-refinement internally, if any.

This function is designed to prevent users from needing to trick FEInterface::n_dofs() into giving them the right number of dofs when working with p-refined elements. See, e.g. FEInterface::compute_data().

◆ n_dofs_at_node()

static unsigned int libMesh::FEInterface::n_dofs_at_node	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	n
	)

static

Returns: The number of dofs at node n for a finite element of type fe_t. Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the total order of the element.

Referenced by libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()().

◆ n_dofs_at_node_function()

static n_dofs_at_node_ptr libMesh::FEInterface::n_dofs_at_node_function	(	const unsigned int	dim,
		const FEType &	fe_t
	)

static

Returns: A function which evaluates n_dofs_at_node for the requested FE type and dimension.

◆ n_dofs_per_elem()

static unsigned int libMesh::FEInterface::n_dofs_per_elem	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

static

Returns: The number of dofs interior to the element, not associated with any interior nodes. Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the total order of the element.

◆ n_shape_functions()

static unsigned int libMesh::FEInterface::n_shape_functions	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t
	)

static

Returns: The number of shape functions associated with this finite element of type fe_t. Automatically decides which finite element class to use.

On a p-refined element, fe_t.order should be the total order of the element.

◆ n_vec_dim()

static unsigned int libMesh::FEInterface::n_vec_dim	(	const MeshBase &	mesh,
		const FEType &	fe_type
	)

static

Returns: The number of components of a vector-valued element. Scalar-valued elements return 1.

◆ nodal_soln()

static void libMesh::FEInterface::nodal_soln	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const std::vector< Number > &	elem_soln,
		std::vector< Number > &	nodal_soln
	)

static

Build the nodal soln from the element soln.

This is the solution that will be plotted. Automatically passes the request to the appropriate finite element class member. To indicate that results from this specific implementation of nodal_soln should not be used, the vector nodal_soln is returned empty.

On a p-refined element, fe_t.order should be the base order of the element.

◆ on_reference_element()

static bool libMesh::FEInterface::on_reference_element	(	const Point &	p,
		const ElemType	t,
		const Real	eps = `TOLERANCE`
	)

static

Returns: true if the point p is located on the reference element for element type t, false otherwise.

Since we are doing floating point comparisons, the parameter eps can be specified to indicate a tolerance. For example, $\xi \le 1$ becomes $\xi \le 1 + \epsilon$ .

◆ shape() [1/4]

static Real libMesh::FEInterface::shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	i,
		const Point &	p
	)

static

Returns: The value of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate finite element class member.

Note: On a p-refined element, fe_t.order should be the total order of the element.

◆ shape() [2/4]

static Real libMesh::FEInterface::shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const unsigned int	i,
		const Point &	p
	)

static

Returns: The value of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate finite element class member.

Note: On a p-refined element, fe_t.order should be the base order of the element.

◆ shape() [3/4]

template<typename OutputType >

static void libMesh::FEInterface::shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	i,
		const Point &	p,
		OutputType &	phi
	)

static

Returns: The value of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate scalar finite element class member.

Note: On a p-refined element, fe_t.order should be the total order of the element.

◆ shape() [4/4]

template<typename OutputType >

static void libMesh::FEInterface::shape	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const unsigned int	i,
		const Point &	p,
		OutputType &	phi
	)

static

Returns: The value of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate scalar finite element class member.

Note: On a p-refined element, fe_t.order should be the total order of the element.

◆ shape_deriv() [1/2]

static Real libMesh::FEInterface::shape_deriv	(	const unsigned int	dim,
		const FEType &	fe_t,
		const ElemType	t,
		const unsigned int	i,
		const unsigned int	j,
		const Point &	p
	)

static

Returns: The $j^{th}$ coordinate of the gradient of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate scalar finite element class member.

Note: On a p-refined element, fe_t.order should be the total order of the element.

◆ shape_deriv() [2/2]

static Real libMesh::FEInterface::shape_deriv	(	const unsigned int	dim,
		const FEType &	fe_t,
		const Elem *	elem,
		const unsigned int	i,
		const unsigned int	j,
		const Point &	p
	)

static

Returns: The $j^{th}$ coordinate of the gradient of the $i^{th}$ shape function at point p. This method allows you to specify the dimension, element type, and order directly. Automatically passes the request to the appropriate scalar finite element class member.

Note: On a p-refined element, fe_t.order should be the total order of the element.

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

include/fe/fe_interface.h

Public Types

Public Member Functions

Static Public Member Functions

Private Member Functions

Static Private Member Functions

Detailed Description

Member Typedef Documentation

◆ n_dofs_at_node_ptr

Constructor & Destructor Documentation

◆ FEInterface()

◆ ~FEInterface()

Member Function Documentation

◆ compute_constraints()

◆ compute_data()

◆ compute_periodic_constraints()

◆ dofs_on_edge()

◆ dofs_on_side()

◆ extra_hanging_dofs()

◆ field_type() [1/2]

◆ field_type() [2/2]

◆ get_continuity()

◆ ifem_compute_data()

◆ ifem_inverse_map() [1/2]

◆ ifem_inverse_map() [2/2]

◆ ifem_map()

◆ ifem_n_dofs()

◆ ifem_n_dofs_at_node()

◆ ifem_n_dofs_per_elem()

◆ ifem_n_shape_functions()

◆ ifem_nodal_soln()

◆ ifem_on_reference_element()

◆ ifem_shape() [1/2]

◆ ifem_shape() [2/2]

◆ ifem_shape_deriv() [1/2]

◆ ifem_shape_deriv() [2/2]

◆ inverse_map() [1/2]

◆ inverse_map() [2/2]

◆ is_InfFE_elem()

◆ map()

◆ max_order()

◆ n_dofs() [1/2]

◆ n_dofs() [2/2]

◆ n_dofs_at_node()

◆ n_dofs_at_node_function()

◆ n_dofs_per_elem()

◆ n_shape_functions()

◆ n_vec_dim()

◆ nodal_soln()

◆ on_reference_element()

◆ shape() [1/4]

◆ shape() [2/4]

◆ shape() [3/4]

◆ shape() [4/4]

◆ shape_deriv() [1/2]

◆ shape_deriv() [2/2]