Evaluates the one-dimensional basis functions used by the solver. More...

#include "bspline.hpp"

Static Public Member Functions
static void	beval3 (Real ai, int i, int n, Real Bik, int pos)
	Evaluate the bounded quadratic basis in cell `i`.

static void	beval3D (Real ai, int i, int n, Real Bik, int pos)
	Evaluate bounded quadratic derivatives with respect to \(\xi\).

static void	beval4 (Real ai, int i, int n, Real Bik, int pos)
	Evaluate the bounded cubic basis in cell `i`.

static void	beval4D (Real ai, int i, int n, Real Bik, int pos)
	Evaluate bounded cubic derivatives with respect to \(\xi\).

static void	eval (Real ai, int degree, Real Bik, int pos)
	Dispatch to the cardinal evaluator selected by ORB5 degree.

static void	eval1 (Real ai, Real Bik, int pos)
	Evaluate the constant cardinal basis.

static void	eval1D (Real ai, Real Bik, int pos)
	Evaluate the \(\xi\) derivative of the constant cardinal basis.

static void	eval2 (Real ai, Real Bik, int pos)
	Evaluate the two linear cardinal basis functions.

static void	eval2D (Real ai, Real Bik, int pos)
	Evaluate the \(\xi\) derivatives of the linear cardinal basis.

static void	eval3 (Real ai, Real Bik, int pos)
	Evaluate the three quadratic cardinal basis functions.

static void	eval3D (Real ai, Real Bik, int pos)
	Evaluate the \(\xi\) derivatives of the quadratic cardinal basis.

static void	eval4 (Real ai, Real Bik, int pos)
	Evaluate the four cubic cardinal basis functions.

static void	eval4D (Real ai, Real Bik, int pos)
	Evaluate the \(\xi\) derivatives of the cubic cardinal basis.

static void	evald (Real ai, int degree, Real Bik, int pos)
	Dispatch to the cardinal derivative evaluator selected by degree.

static int	find_cell (Real x, Real x0, Real dx_inv)
	Convert a coordinate to its zero-based uniform-mesh cell index.

static Real	normalize_ksi (Real x, Real x_cell, Real dx)
	Normalize a coordinate to its local cell coordinate.

Detailed Description

Evaluates the one-dimensional basis functions used by the solver.

The cardinal routines operate on a uniform mesh. The bounded variants replace the cardinal coefficients in the first and last cells so that the physical end points are represented correctly. In this API degree follows the historical ORB5 convention: it is the number of nonzero basis functions (1–4), so values 1, 2, 3, and 4 denote constant, linear, quadratic, and cubic polynomials, respectively.

All derivative routines return derivatives with respect to the dimensionless cell coordinate \(\xi\). A derivative with respect to the physical coordinate is obtained by multiplying by the inverse cell width.

Fortran reference: solver/BSplines.F90, modules BSplinesCardinalDegree1 through BSplinesCardinalDegree4, BSplinesBoundedDegree3, and BSplinesBoundedDegree4.

Member Function Documentation

◆ beval3()

static void orb5x::BSpline::beval3	(	Real	ai,
		int	i,
		int	n,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the bounded quadratic basis in cell i.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[in]	i	Zero-based cell index; `i == n` denotes the final grid point.
[in]	n	Number of mesh intervals.
[out]	Bik	Array of at least three elements receiving basis values.
[out]	pos	Array of at least three elements receiving offsets {-2,-1,0}.

◆ beval3D()

static void orb5x::BSpline::beval3D	(	Real	ai,
		int	i,
		int	n,
		Real *	Bik,
		int *	pos
	)

static

Evaluate bounded quadratic derivatives with respect to \(\xi\).

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[in]	i	Zero-based cell index; `i == n` denotes the final grid point.
[in]	n	Number of mesh intervals.
[out]	Bik	Array of at least three elements receiving derivative values.
[out]	pos	Array of at least three elements receiving offsets {-2,-1,0}.

◆ beval4()

static void orb5x::BSpline::beval4	(	Real	ai,
		int	i,
		int	n,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the bounded cubic basis in cell i.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[in]	i	Zero-based cell index; `i == n` denotes the final grid point.
[in]	n	Number of mesh intervals.
[out]	Bik	Array of at least four elements receiving basis values.
[out]	pos	Array of at least four elements receiving offsets {-3,-2,-1,0}.

◆ beval4D()

static void orb5x::BSpline::beval4D	(	Real	ai,
		int	i,
		int	n,
		Real *	Bik,
		int *	pos
	)

static

Evaluate bounded cubic derivatives with respect to \(\xi\).

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[in]	i	Zero-based cell index; `i == n` denotes the final grid point.
[in]	n	Number of mesh intervals.
[out]	Bik	Array of at least four elements receiving derivative values.
[out]	pos	Array of at least four elements receiving offsets {-3,-2,-1,0}.

◆ eval()

static void orb5x::BSpline::eval	(	Real	ai,
		int	degree,
		Real *	Bik,
		int *	pos
	)

static

Dispatch to the cardinal evaluator selected by ORB5 degree.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\) within the cell.
[in]	degree	Number of active basis functions, in the range 1–4.
[out]	Bik	Array of at least `degree` elements receiving basis values.
[out]	pos	Array of at least `degree` elements receiving index offsets.

Note: An unsupported degree writes zero to the first output elements.

◆ eval1()

static void orb5x::BSpline::eval1	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the constant cardinal basis.

Parameters

[in]	ai	Dimensionless coordinate within the cell; unused for this degree.
[out]	Bik	Array of at least one element receiving the basis value.
[out]	pos	Array of at least one element receiving the basis-index offset.

◆ eval1D()

static void orb5x::BSpline::eval1D	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the \(\xi\) derivative of the constant cardinal basis.

Parameters

[in]	ai	Dimensionless coordinate within the cell; unused for this degree.
[out]	Bik	Array of at least one element receiving the derivative value.
[out]	pos	Array of at least one element receiving the basis-index offset.

◆ eval2()

static void orb5x::BSpline::eval2	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the two linear cardinal basis functions.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[out]	Bik	Array of at least two elements receiving the basis values.
[out]	pos	Array of at least two elements receiving offsets {-1,0}.

◆ eval2D()

static void orb5x::BSpline::eval2D	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the \(\xi\) derivatives of the linear cardinal basis.

Parameters

[in]	ai	Dimensionless coordinate within the cell; derivatives are constant.
[out]	Bik	Array of at least two elements receiving derivative values.
[out]	pos	Array of at least two elements receiving offsets {-1,0}.

◆ eval3()

static void orb5x::BSpline::eval3	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the three quadratic cardinal basis functions.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[out]	Bik	Array of at least three elements receiving the basis values.
[out]	pos	Array of at least three elements receiving offsets {-2,-1,0}.

◆ eval3D()

static void orb5x::BSpline::eval3D	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the \(\xi\) derivatives of the quadratic cardinal basis.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[out]	Bik	Array of at least three elements receiving derivative values.
[out]	pos	Array of at least three elements receiving offsets {-2,-1,0}.

◆ eval4()

static void orb5x::BSpline::eval4	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the four cubic cardinal basis functions.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[out]	Bik	Array of at least four elements receiving the basis values.
[out]	pos	Array of at least four elements receiving offsets {-3,-2,-1,0}.

◆ eval4D()

static void orb5x::BSpline::eval4D	(	Real	ai,
		Real *	Bik,
		int *	pos
	)

static

Evaluate the \(\xi\) derivatives of the cubic cardinal basis.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\in[0,1]\) within the cell.
[out]	Bik	Array of at least four elements receiving derivative values.
[out]	pos	Array of at least four elements receiving offsets {-3,-2,-1,0}.

◆ evald()

static void orb5x::BSpline::evald	(	Real	ai,
		int	degree,
		Real *	Bik,
		int *	pos
	)

static

Dispatch to the cardinal derivative evaluator selected by degree.

Parameters

[in]	ai	Dimensionless coordinate \(\xi\) within the cell.
[in]	degree	Number of active basis functions, in the range 1–4.
[out]	Bik	Array of at least `degree` elements receiving \(dB/d\xi\).
[out]	pos	Array of at least `degree` elements receiving index offsets.

Note: An unsupported degree writes zero to the first output elements.

◆ find_cell()

static int orb5x::BSpline::find_cell	(	Real	x,
		Real	x0,
		Real	dx_inv
	)

static

Convert a coordinate to its zero-based uniform-mesh cell index.

Parameters

[in]	x	Coordinate to locate, in the same units as `x0`.
[in]	x0	Coordinate of grid point zero.
[in]	dx_inv	Inverse mesh spacing, in inverse units of `x`.

Returns: Truncation of \((x-x_0)/\Delta x\) toward zero.

Note: This helper does not clamp the result to a mesh extent.

◆ normalize_ksi()

static Real orb5x::BSpline::normalize_ksi	(	Real	x,
		Real	x_cell,
		Real	dx
	)

static

Normalize a coordinate to its local cell coordinate.

Parameters

[in]	x	Coordinate to normalize.
[in]	x_cell	Coordinate of the left cell boundary, in units of `x`.
[in]	dx	Cell width, in units of `x`.

Returns: Dimensionless value \((x-x_{cell})/\Delta x\).

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

bspline.hpp

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ beval3()

◆ beval3D()

◆ beval4()

◆ beval4D()

◆ eval()

◆ eval1()

◆ eval1D()

◆ eval2()

◆ eval2D()

◆ eval3()

◆ eval3D()

◆ eval4()

◆ eval4D()

◆ evald()

◆ find_cell()

◆ normalize_ksi()