sll_m_box_splines Module Reference

Description

Provides capabilities for values and derivatives interpolation with box splines on hexagonal meshes.

Author

Laura S. Mendoza

This module contains the computation of box splines of arbitrary degree. There is a special optimized algorithm for degree 2. The box splines here are defined only on hexagonal meshes (regular equilateral triangles elements). The only boundary condition supported right now is dirichlet. Reference: http://ieeexplore.ieee.org/document/1642713/

Derived types and interfaces
type	sll_t_box_spline_2d
	Basic type for 2 dimensional box splines. More...
interface	sll_o_delete
	Generic sub-routine defined for 2D box spline types. Deallocates the memory associated with the given box spline object. More...

Functions/Subroutines
type(sll_t_box_spline_2d) function, pointer, public	sll_f_new_box_spline_2d (mesh, bc_type)
	Creates a box spline element. More...
subroutine	compute_coeff_box_spline_2d (spline, data, deg)
	Computes box splines coefficients. More...
subroutine	compute_coeff_box_spline_2d_diri (spline, data, deg)
	Computes box splines coefficients for dirichlet BC. More...
subroutine	compute_coeff_box_spline_2d_prdc (spline, data, deg)
	Computes box splines coefficients with periodic BC. More...
subroutine	compute_coeff_box_spline_2d_neum (spline, data, deg)
	Computes box splines coefficients with neumann BC. More...
function	choose (n, k)
	Computes the binomial coefficient (n, k) More...
function	chi_gen_val (x1_in, x2_in, deg)
	Computes the box spline of degree deg on (x1, x2) More...
real(kind=f64) function, public	sll_f_compute_box_spline (spline, x1, x2, deg)
	Computes the value of a box spline. More...
real(kind=f64) function	change_basis_x1 (spline, x1, x2)
	1st coo. of (x1, x2) in reference hex-mesh coo. More...
real(kind=f64) function	change_basis_x2 (spline, x1, x2)
	2nd coo. of (x1, x2) in reference hex-mesh coo. More...
real(kind=f64) function, public	sll_f_hex_interpolate_value (mesh_geom, x1, x2, spline, deg)
	Interpolates on point of cartesian coordinates (x1, x2) More...
integer(kind=i32) function, dimension(3 *deg *deg)	non_zeros_splines (mesh, cell_index, deg)
	Computes indices of non null splines on a given cell. More...
real(kind=f64) function, public	sll_f_boxspline_x1_derivative (x1, x2, deg)
	Computes x-derivative on (x,y) More...
real(kind=f64) function, public	sll_f_boxspline_x2_derivative (x1, x2, deg)
	Computes y-derivative on (x,y) More...
real(kind=f64) function, public	sll_f_boxspline_val_der (x1, x2, deg, nderiv1, nderiv2)
	Computes the values or derivatives of box splines. More...
subroutine, public	sll_s_write_connectivity (mesh, deg)
	Writes connectivity for CAID / DJANGO. More...
subroutine	delete_box_spline_2d (spline)
	Generic sub-routine defined for 2D box spline types. Deallocates the memory associated with the given box spline object. More...

Function/Subroutine Documentation

change_basis_x1()

real(kind=f64) function sll_m_box_splines::change_basis_x1 ( type(sll_t_box_spline_2d), pointer  spline, real(kind=f64), intent(in)  x1, real(kind=f64), intent(in)  x2  )

private

1st coo. of (x1, x2) in reference hex-mesh coo.

This function allows to change a point of coordinates (x1, x2) on the spline basis to the mesh basis. Gives 1st coordinate.

Parameters

[in]	spline	box spline who contains the reference hexagonal mesh
[in]	x1	real containing first coordinate of point
[in]	x2	real containing second coordinate of point

Returns

the first coordinate after change of coordinate sys.

Definition at line 452 of file sll_m_box_splines.F90.

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change_basis_x2()

real(kind=f64) function sll_m_box_splines::change_basis_x2 ( type(sll_t_box_spline_2d), pointer  spline, real(kind=f64), intent(in)  x1, real(kind=f64), intent(in)  x2  )

private

2nd coo. of (x1, x2) in reference hex-mesh coo.

This function allows to change a point of coordinates (x1, x2) on the spline basis to the mesh basis. Gives 2nd coordinate.

Parameters

[in]	spline	box spline who contains the reference hexagonal mesh
[in]	x1	real containing first coordinate of point
[in]	x2	real containing second coordinate of point

Returns

the second coordinate after change of coordinate sys.

Definition at line 494 of file sll_m_box_splines.F90.

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chi_gen_val()

function sll_m_box_splines::chi_gen_val ( intent(in)  x1_in, intent(in)  x2_in, intent(in)  deg  )

private

Computes the box spline of degree deg on (x1, x2)

Computes the value of the box spline (chi) of degree deg on the point of cartesian coordiantes (x1, x2). The algorithm is specific to the hexagonal mesh and has been optimized for degree 2 splines. Reference : @Condat and Van De Ville (2006) "Three directional Box Splines: Characterization and Efficient Evaluation."

Parameters

[in]	x1_in	real containing first coordinate of point where spline is to be evaluated
[in]	x2_in	real containing second coordinate of point where spline is to be evaluated
[in]	deg	integer containing the degree of the spline

Returns

chi_gen_val value of the box spline

Definition at line 298 of file sll_m_box_splines.F90.

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choose()

function sll_m_box_splines::choose ( intent(in)  n, intent(in)  k  )

private

Computes the binomial coefficient (n, k)

Computes the binomial coefficient (n, k)

Parameters

[in]	n	integer
[in]	k	integer

Returns

res binomial coefficient

\[ \dfrac{n!}{(k!(n-k)!} \]

Definition at line 270 of file sll_m_box_splines.F90.

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compute_coeff_box_spline_2d()

subroutine sll_m_box_splines::compute_coeff_box_spline_2d ( class(sll_t_box_spline_2d), intent(inout)  spline, real(kind=f64), dimension(:), intent(in), target  data, integer(kind=i32), intent(in)  deg  )

private

Computes box splines coefficients.

Computes box splines coefficients for a box spline of degree deg and fitted to the data vector

Parameters

[in]	data	vector containing the data to be fit
[in]	deg	integer representing the box spline degree
[in]	spline	box spline type element, containting the mesh, bc, ...

Definition at line 129 of file sll_m_box_splines.F90.

compute_coeff_box_spline_2d_diri()

subroutine sll_m_box_splines::compute_coeff_box_spline_2d_diri ( class(sll_t_box_spline_2d), intent(inout)  spline, real(kind=f64), dimension(:), intent(in), target  data, integer(kind=i32), intent(in)  deg  )

private

Computes box splines coefficients for dirichlet BC.

Computes box splines coefficients for a box spline of degree deg and fitted to the data vector on a hexmesh with dirichlet BC

Parameters

[in]	data	vector containing the data to be fit
[in]	deg	integer representing the box spline degree
[in]	spline	box spline type element, containting the mesh, bc, ...

Definition at line 161 of file sll_m_box_splines.F90.

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compute_coeff_box_spline_2d_neum()

subroutine sll_m_box_splines::compute_coeff_box_spline_2d_neum ( class(sll_t_box_spline_2d), intent(inout)  spline, target  data, intent(in)  deg  )

private

Computes box splines coefficients with neumann BC.

NOT YET IMPLEMENTED. Computes neuman box splines coefficients for a box spline of degree deg and fitted to the data vector

Parameters

[in]	data	vector containing the data to be fit
[in]	deg	integer representing the box spline degree
[in]	spline	box spline type element, containting the mesh, bc, ...

Definition at line 247 of file sll_m_box_splines.F90.

compute_coeff_box_spline_2d_prdc()

subroutine sll_m_box_splines::compute_coeff_box_spline_2d_prdc ( class(sll_t_box_spline_2d), intent(inout)  spline, target  data, intent(in)  deg  )

private

Computes box splines coefficients with periodic BC.

NOT YET IMPLEMENTED. Computes periodic box splines coefficients for a box spline of degree deg and fitted to the data vector

Parameters

[in]	data	vector containing the data to be fit
[in]	deg	integer representing the box spline degree
[in]	spline	box spline type element, containting the mesh, bc, ...

Definition at line 223 of file sll_m_box_splines.F90.

delete_box_spline_2d()

subroutine sll_m_box_splines::delete_box_spline_2d ( type(sll_t_box_spline_2d), intent(inout), pointer  spline )

private

Generic sub-routine defined for 2D box spline types. Deallocates the memory associated with the given box spline object.

Parameters

[in,out]

spline_object.

Definition at line 866 of file sll_m_box_splines.F90.

non_zeros_splines()

integer(kind=i32) function, dimension(3*deg*deg) sll_m_box_splines::non_zeros_splines ( type(sll_t_hex_mesh_2d), intent(in), pointer  mesh, integer(kind=i32), intent(in)  cell_index, integer(kind=i32), intent(in)  deg  )

private

Computes indices of non null splines on a given cell.

The function returns for a given cell and a certain degree the indices of the splines of that degree that are different to 0 on that cell.

Parameters

[IN]	mesh hexagonal mesh, the domain
[IN]	cell_index index of the cell where we wish to know the indices of the non vanishing splines
[IN]	deg integer of the degree of the splines
[OUT]	index_nZ vector of size 3*deg*deg containing the global indices of all non-zero splines

Definition at line 632 of file sll_m_box_splines.F90.

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sll_f_boxspline_val_der()

real(kind=f64) function, public sll_m_box_splines::sll_f_boxspline_val_der	(	real(kind=f64), intent(in)	x1,
		real(kind=f64), intent(in)	x2,
		integer(kind=i32), intent(in)	deg,
		integer(kind=i32), intent(in)	nderiv1,
		integer(kind=i32), intent(in)	nderiv2
	)

Computes the values or derivatives of box splines.

Depending on nderiv1 and nderiv2 will compute box spline or derivative with respect to x (nderiv1 > 0) or/and to y (nderiv2 > 0)

Parameters

[in]	x1	real containing second coordinate of point
[in]	x2	real containing second coordinate of point
[in]	deg	integer with degree of splines
[in]	nderiv1	integer number of times to derive on the x direction
[in]	nderiv2	integer number of times to derive on the y direction

Returns

real nderiv-derivatives of boxspline

no derivative to compute

derivative with respect to the second coo

Definition at line 768 of file sll_m_box_splines.F90.

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sll_f_boxspline_x1_derivative()

real(kind=f64) function, public sll_m_box_splines::sll_f_boxspline_x1_derivative	(	real(kind=f64), intent(in)	x1,
		real(kind=f64), intent(in)	x2,
		integer(kind=i32), intent(in)	deg
	)

Computes x-derivative on (x,y)

Using the 5 point stencil computes the x-derivative on (x,y)

Parameters

[in]	x1	real containing second coordinate of point
[in]	x2	real containing second coordinate of point
[in]	deg	integer with degree of splines

Returns

real derivative on x1 of box spline

Definition at line 703 of file sll_m_box_splines.F90.

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sll_f_boxspline_x2_derivative()

real(kind=f64) function, public sll_m_box_splines::sll_f_boxspline_x2_derivative	(	real(kind=f64), intent(in)	x1,
		real(kind=f64), intent(in)	x2,
		integer(kind=i32), intent(in)	deg
	)

Computes y-derivative on (x,y)

Using the 5 point stencil computes the y-derivative on (x,y)

Parameters

[in]	x1	real containing second coordinate of point
[in]	x2	real containing second coordinate of point
[in]	deg	integer with degree of splines

Returns

real derivative on x2 of box spline

Definition at line 734 of file sll_m_box_splines.F90.

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sll_f_compute_box_spline()

real(kind=f64) function, public sll_m_box_splines::sll_f_compute_box_spline	(	type(sll_t_box_spline_2d), intent(in), pointer	spline,
		real(kind=f64), intent(in)	x1,
		real(kind=f64), intent(in)	x2,
		integer(kind=i32), intent(in)	deg
	)

Computes the value of a box spline.

This function computes the value of a box spline of degree deg at the point (x1,x2)

Parameters

[in]	spline	box spline which contains the reference hexagonal mesh
[in]	x1	real containing first coordinate of point
[in]	x2	real containing second coordinate of point
[in]	deg	real containing the degree of the spline to be computed

Returns

the value of the box spline at (x1,x2)

Definition at line 428 of file sll_m_box_splines.F90.

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sll_f_hex_interpolate_value()

real(kind=f64) function, public sll_m_box_splines::sll_f_hex_interpolate_value	(	type(sll_t_hex_mesh_2d), intent(in), pointer	mesh_geom,
		real(kind=f64), intent(in)	x1,
		real(kind=f64), intent(in)	x2,
		type(sll_t_box_spline_2d), pointer	spline,
		integer(kind=i32), intent(in)	deg
	)

Interpolates on point of cartesian coordinates (x1, x2)

Interpolation with box splines of degree deg on the point of cartesian coordinates (x1, x2) on the mesh mesh_geom

Parameters

[in]	mesh_geom	geometric mesh
[in]	x1	real containing 1st coordinate of point where to interpolate
[in]	x2	real containing 2nd coordinate of point where to interpolate
[in]	spline	box spline with coefficients already pre computed
[in]	deg	integer with degree of splines

Returns

real Interpolated value

Definition at line 542 of file sll_m_box_splines.F90.

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sll_f_new_box_spline_2d()

type(sll_t_box_spline_2d) function, pointer, public sll_m_box_splines::sll_f_new_box_spline_2d	(	type(sll_t_hex_mesh_2d), pointer	mesh,
		integer(kind=i32), intent(in)	bc_type
	)

Creates a box spline element.

Creates a box spline element to the associated mesh and with the given boundary conditions. Also allocates memory for the coefficients

Parameters

[in]	mesh	Hexagonal mesh on which the box spline will be defined
[in]	bc_type	Integer defining the boundary condition, usual values are sll_p_periodic, sll_p_dirichlet, ...

Returns

box spline element

Definition at line 103 of file sll_m_box_splines.F90.

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sll_s_write_connectivity()

subroutine, public sll_m_box_splines::sll_s_write_connectivity	(	type(sll_t_hex_mesh_2d), pointer	mesh,
		integer(kind=i32), intent(in)	deg
	)

Writes connectivity for CAID / DJANGO.

write connectivity info for CAID/DJANGO. This function was intented to couple Pigasus poisson solver to the hex-mesh. Output file : boxsplines_connectivity.txt

Parameters

[in]	mesh	pointer to the hexagonal mesh
[in]	deg	integer designing box splines degree

Definition at line 819 of file sll_m_box_splines.F90.

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