sll_m_fekete_integration Module Reference

Description

Fekete quadrature rules for a triangle.

Author

Laura S. Mendoza

This module contains the Fekete quadrature rule adapted to triangles. The main functions are taken from the following site but have been modified to respect Selalib's structure: http://people.sc.fsu.edu/~jburkardt/f_src/triangle_fekete_rule/

Derived types and interfaces
interface	function_2D
	2d real function More...

Functions/Subroutines
subroutine	fekete_degree (rule, degree)
	returns the degree of a Fekete rule for the triangle. More...
subroutine, public	sll_s_fekete_order_num (rule, order_num)
	returns the order of a Fekete rule for the triangle. More...
subroutine	fekete_rule (rule, order_num, xy, w)
	returns the points and weights of a Fekete rule. More...
subroutine	rearrange_fekete_rule (n, xy, w)
	Re arranges the order of the quadrature point to go from lower-left to top right. More...
subroutine	fekete_rule_num (rule_num)
	returns the number of Fekete rules available. More...
subroutine	fekete_suborder (rule, suborder_num, suborder)
	returns the suborders for a Fekete rule. More...
subroutine	fekete_suborder_num (rule, suborder_num)
	returns the number of suborders for a Fekete rule. More...
subroutine	fekete_subrule (rule, suborder_num, suborder_xyz, suborder_w)
	returns a compressed Fekete rule More...
integer(kind=i32) function	wrapping (ival, ilo, ihi)
	forces an I4 to lie between given limits by wrapping. More...
subroutine	reference_to_physical_t3 (node_xy, n, ref, phy)
	maps T3 reference points to physical points. More...
real(kind=f64) function, dimension(:, :), allocatable, public	sll_f_fekete_points_and_weights (node_xy2, rule)
	Gives the fekete points coordinates and associated weights for a certain rule in a given triangle. More...
real(kind=f64) function, public	sll_f_fekete_integral (f, pxy)
	Fekete quadrature rule over a triangle. More...
subroutine	triangle_area (node_xy, area)
subroutine	write_quadrature (rule)
	Writes fekete points coordinates of a hex-mesh reference triangle. More...
subroutine	write_basis_values (deg, rule)
	Writes on a file values of boxsplines on fekete points. More...
subroutine, public	sll_s_write_all_django_files (num_cells, deg, rule, transf)
	This function is supposed to write all django input files needed for a Django/Jorek simulation. More...

Function/Subroutine Documentation

fekete_degree()

subroutine sll_m_fekete_integration::fekete_degree	(	integer(kind=i32), intent(in)	rule,
		integer(kind=i32), intent(out)	degree
	)

returns the degree of a Fekete rule for the triangle.

This code was first written by John Burkardt and is available online under the GNU LGPL license. Reference: Mark Taylor, Beth Wingate, Rachel Vincent, An Algorithm for Computing Fekete Points in the Triangle, SIAM Journal on Numerical Analysis, Volume 38, Number 5, 2000, pages 1707-1720.

Parameters

[IN]	rule integer rule of quadrature for the fekete quadrature
[OUT]	degree integer the polynomial degree of exactness of

Definition at line 61 of file sll_m_fekete_integration.F90.

fekete_rule()

subroutine sll_m_fekete_integration::fekete_rule ( integer(kind=i32), intent(in)  rule, integer(kind=i32), intent(in)  order_num, real(kind=f64), dimension(2, order_num), intent(out)  xy, real(kind=f64), dimension(order_num), intent(out)  w  )

private

returns the points and weights of a Fekete rule.

This code was first written by John Burkardt and is available online under the GNU LGPL license.

Parameters

[IN]	rule integer the index of the rule
[IN]	order_num integer the order (number of points of the rule)
[OUT]	xy real table the points of the rule
[OUT]	w real table the weigths of the rule

Definition at line 125 of file sll_m_fekete_integration.F90.

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fekete_rule_num()

subroutine sll_m_fekete_integration::fekete_rule_num ( integer(kind=i32), intent(out)  rule_num )

private

returns the number of Fekete rules available.

This code was first written by John Burkardt and is available online under the GNU LGPL license.

Parameters

[OUT]

rule_num integer the number of rules available

Definition at line 237 of file sll_m_fekete_integration.F90.

fekete_suborder()

subroutine sll_m_fekete_integration::fekete_suborder ( integer(kind=i32), intent(in)  rule, integer(kind=i32), intent(in)  suborder_num, integer(kind=i32), dimension(suborder_num), intent(out)  suborder  )

private

returns the suborders for a Fekete rule.

This code was first written by John Burkardt and is available online under the GNU LGPL license.

Parameters

[IN]	rule integer the index of the rule
[IN]	suborder_num integer the number of suborder of the rule
[OUT]	suborder integer array the suborders of the rule

Definition at line 250 of file sll_m_fekete_integration.F90.

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fekete_suborder_num()

subroutine sll_m_fekete_integration::fekete_suborder_num ( integer(kind=i32), intent(in)  rule, integer(kind=i32), intent(out)  suborder_num  )

private

returns the number of suborders for a Fekete rule.

This code was first written by John Burkardt and is available online under the GNU LGPL license.

Parameters

[IN]	rule integer the index of the rule
[OUTPUT]	suborder_num the number of suborder of the rule

Definition at line 299 of file sll_m_fekete_integration.F90.

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fekete_subrule()

subroutine sll_m_fekete_integration::fekete_subrule ( integer(kind=i32), intent(in)  rule, integer(kind=i32), intent(in)  suborder_num, real(kind=f64), dimension(3, suborder_num), intent(out)  suborder_xyz, real(kind=f64), dimension(suborder_num), intent(out)  suborder_w  )

private

returns a compressed Fekete rule

This code was first written by John Burkardt and is available online under the GNU LGPL license. Discussion: The listed weights are twice what we want...since we want them to sum to 1/2, reflecting the area of a unit triangle. So we simple halve the values before exiting this routine.

Parameters

[IN]	rule integer the index of the rule
[IN]	suborder_num the number of suborders
[OUT]	suborder_xyz(3, suborder_num) real the number of suborders of the rule
[OUT]	suborder_w(suborder_num) real the suborder weights

Definition at line 339 of file sll_m_fekete_integration.F90.

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rearrange_fekete_rule()

subroutine sll_m_fekete_integration::rearrange_fekete_rule ( integer(kind=i32), intent(in)  n, real(kind=f64), dimension(2, n), intent(out)  xy, real(kind=f64), dimension(n), intent(out)  w  )

private

Re arranges the order of the quadrature point to go from lower-left to top right.

Parameters

[IN]	n integer number of quadrature points
[INOUT]	xy real table the points of the rule
[INOUT]	w real table the weigths of the rule

Definition at line 196 of file sll_m_fekete_integration.F90.

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reference_to_physical_t3()

subroutine sll_m_fekete_integration::reference_to_physical_t3 ( real(kind=f64), dimension(2, 3), intent(in)  node_xy, integer(kind=i32), intent(in)  n, real(kind=f64), dimension(2, n), intent(in)  ref, real(kind=f64), dimension(2, n), intent(out)  phy  )

private

maps T3 reference points to physical points.

This code was first written by John Burkardt and is available online under the GNU LGPL license. Given the vertices of an order 3 physical triangle and a point (XSI,ETA) in the reference triangle, the routine computes the value of the corresponding image point (X,Y) in physical space.

This routine is also appropriate for an order 4 triangle, as long as the fourth node is the centroid of the triangle.

This routine may also be appropriate for an order 6 triangle, if the mapping between reference and physical space is linear. This implies, in particular, that the sides of the image triangle are straight and that the "midside" nodes in the physical triangle are literally halfway along the sides of the physical triangle.

Reference Element T3:

| 1 3 | |\ | | \ S | \ | | \ | | \ 0 1--—2 | +–0–R–1-->

Parameters

[IN]	node_xy(2,3) integer the coordinates of the vertices. The vertices are assumed to be the images of (0,0), (1,0) and (0,1).
[IN]	n integer the number of objects to transform
[IN]	ref(2,n) points in the reference triangle
[OUT]	phy(2,n) corresponding points in the physical triangle

Definition at line 748 of file sll_m_fekete_integration.F90.

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sll_f_fekete_integral()

real(kind=f64) function, public sll_m_fekete_integration::sll_f_fekete_integral	(	procedure(function_2d)	f,
		real(kind=f64), dimension(2, 3), intent(in)	pxy
	)

Fekete quadrature rule over a triangle.

To integrate the function \( f(x) \) (real-valued and over a triangle) we use the Fekete formula

\[ \int_{\Omega} f(x)dx \approx \sum_{k=1}^{N} w_k f(x_k) \]

The only quadrature rule possible for now is 1 (10 points)

Parameters

[in]	f	the function to be integrated
[in]	pxy	array of dimesion (2,3) containg the coordinates of the edges of the triangle

Returns

The value of the integral

Definition at line 816 of file sll_m_fekete_integration.F90.

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sll_f_fekete_points_and_weights()

real(kind=f64) function, dimension(:, :), allocatable, public sll_m_fekete_integration::sll_f_fekete_points_and_weights	(	real(kind=f64), dimension(2, 3), intent(in)	node_xy2,
		integer(kind=i32), intent(in)	rule
	)

Gives the fekete points coordinates and associated weights for a certain rule in a given triangle.

This code was first written by John Burkardt and is available online under the GNU LGPL license.

Parameters

[IN]	node_xy2 array of dimesion (2,3) containg the coordinates of the edges of the triangle
[IN]	rule integer quadrature rule

Returns

xyw array of dimesion (3,n) containg the fekete points and weights using the rule number given in parameter. xyw(1,:) contains the x coordinates of the fekete points, xyw(2, :) the y coordinates and xyw(3, :) contains the associated weights.

Definition at line 775 of file sll_m_fekete_integration.F90.

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sll_s_fekete_order_num()

subroutine, public sll_m_fekete_integration::sll_s_fekete_order_num	(	integer(kind=i32), intent(in)	rule,
		integer(kind=i32), intent(out)	order_num
	)

returns the order of a Fekete rule for the triangle.

This code was first written by John Burkardt and is available online under the GNU LGPL license. Reference: Mark Taylor, Beth Wingate, Rachel Vincent, An Algorithm for Computing Fekete Points in the Triangle, SIAM Journal on Numerical Analysis, Volume 38, Number 5, 2000, pages 1707-1720.

Parameters

[IN]	rule integer the index of the rule
[OUT]	order_num integer the order (number of points) of the rule

Definition at line 99 of file sll_m_fekete_integration.F90.

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sll_s_write_all_django_files()

subroutine, public sll_m_fekete_integration::sll_s_write_all_django_files	(	integer(kind=i32), intent(in)	num_cells,
		integer(kind=i32), intent(in)	deg,
		integer(kind=i32), intent(in)	rule,
		character(len=*), intent(in)	transf
	)

This function is supposed to write all django input files needed for a Django/Jorek simulation.

Parameters

[in]	num_cells	integer number of cells in a radius of the hexagonal mesh
[in]	deg	integer degree of the splines that will be used for the interpolation

Definition at line 1032 of file sll_m_fekete_integration.F90.

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triangle_area()

subroutine sll_m_fekete_integration::triangle_area ( real(kind=f64), dimension(2, 3)  node_xy, real(kind=f64)  area  )

private

Definition at line 863 of file sll_m_fekete_integration.F90.

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wrapping()

integer(kind=i32) function sll_m_fekete_integration::wrapping ( integer(kind=i32), intent(in)  ival, integer(kind=i32), intent(in)  ilo, integer(kind=i32), intent(in)  ihi  )

private

forces an I4 to lie between given limits by wrapping.

This code was first written by John Burkardt and is available online under the GNU LGPL license. ILO = 4, IHI = 8

I Value

-2 8 -1 4 0 5 1 6 2 7 3 8 4 4 5 5 6 6 7 7 8 8 9 4 10 5 11 6 12 7 13 8 14 4

Parameters

[IN]	ival an integer value
[IN]	ilo the desired lower bound
[IN]	ihi the desired upper bound
[OUT]	res a wrapped version of ival

Definition at line 691 of file sll_m_fekete_integration.F90.

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write_basis_values()

subroutine sll_m_fekete_integration::write_basis_values ( integer(kind=i32), intent(in)  deg, integer(kind=i32), intent(in)  rule  )

private

Writes on a file values of boxsplines on fekete points.

Following CAID structure, we write a file with the values of the basis function (box splines) on a reference element (triangle) fekete points. Output for DJANGO. Output file : boxsplines_basis_values.txt

Parameters

[in]

deg

integer with degree of splines

Number of non null box splines on a cell

Number of derivatives to be computed

The displament vector correspond to the translation done to obtain the other non null basis functions

Definition at line 945 of file sll_m_fekete_integration.F90.

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write_quadrature()

subroutine sll_m_fekete_integration::write_quadrature ( integer(kind=i32), intent(in)  rule )

private

Writes fekete points coordinates of a hex-mesh reference triangle.

Takes the reference triangle of a hexmesh and computes the fekete points on it. Then it writes the results in a file following CAID/Django nomenclature. Output file : boxsplines_quadrature.txt

Parameters

[in]

rule

integer for the fekete quadrature rule

Definition at line 884 of file sll_m_fekete_integration.F90.

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