BasisFunctions – B-spline Basis Function Library

The module defining spline basis objects and their arguments

OBJECT CONSTRUCTOR
- Basis (Int)
=>

METHODS OF BASIS TYPE
- PlotBasisFunc (Sub)
- FindSpan (Fn)
- FindMult (Fn)
- FindSpanMult (Sub)
- BasisFuns (Sub)
- OneBasisFun (Fn)
- AllBasisFuns (Sub)
- DersBasisFuns (Sub)
- DersOneBasisFun (Sub)
- InsertKnot (Sub)
- UpdateBasis (Sub)
- BSegments (Sub)
- ParLen (Fn)
- GrevilleAbscissae (Fn)
- get_nquad (Fn)
=>

TYPE CONSTRUCTORS
- basis_constructor (Fn)
=>

PROCEDURES
- bin (Fn)
- sort (Fn)
=>

Recursive quicksort using binary tree pivot.

Uses

- ogpf
- misc
- Parameters
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Interfaces

public interface basis

Spline basis constructor

private pure function basisConstructor(nk, p, k) result(me)

Arguments

Type	Intent		Name
integer,	intent(in)	::	nk	Number of the knots
integer,	intent(in)	::	p	Polynomial degree $p$
real(kind=wp),	intent(in)	::	k(nk)	Open knot vector `kv(0:m)` $= \left\{u_{0},..., u_{m}\right\}$

Return Value type(basis)

Spline basis object

Derived Types

type, public :: basis

Spline basis function class

Components

Type	Visibility	Attributes		Name
integer,	public		::	p	Polynomial degree $p$
integer,	public		::	nk	Number of the knots
integer,	public		::	m	The highest index of the knots `( nk-1 )`
integer,	public		::	nb	Number of basis functions
integer,	public		::	n	The highest index of the basis functions `( nb-1 )`
real(kind=wp),	public,	allocatable	::	kv(:)	Open knot vector `kv(0:m)` $= \left\{u_{0},..., u_{m}\right\}$
real(kind=wp),	public,	allocatable	::	unik(:)	Unique knot values
integer,	public,	allocatable	::	mult(:)	Multiplicity of each unique knot value
integer,	public		::	minreg	Minimum reqularity of the basis functions,`= me%p-max(me%mult(2:ubound(me%mult)-1)`
integer,	public		::	maxreg	Maximum reqularity of the basis functions,`= me%p-min(me%mult(2:ubound(me%mult)-1)`
integer,	public		::	nbez	Number of elements with nonzero measure, i.e., number of Bezier segments

Constructor

Spline basis constructor

private pure function basisConstructor(nk, p, k)

Type-Bound Procedures

procedure, public :: FindSpan	Knot span index search
procedure, public :: FindMult	Knot multiplicity computation
procedure, public :: FindSpanMult	Knot span and multiplicity computation
procedure, public :: BasisFuns	Computation of the non-vanishing basis functions
procedure, public :: OneBasisFun	Computation of a single basis function
procedure, public :: AllBasisFuns	Computation of a all non-zero basis function of all degrees from $0$ up to $p$
procedure, public :: DersBasisFuns	Computation of the derivatives of the non-vanishing basis functions
procedure, public :: DersOneBasisFun	Computation of the the derivatives of a single basis function
procedure, public :: InsertKnot	Knot insertion of a given knot into the knot vector
procedure, public :: UpdateBasis	Updating the basis (the same contents with constructor)
procedure, public :: BSegments	Computation of the segments' data of the basis
procedure, public :: ParLen	Get the length of parameter space
procedure, public :: GrevilleAbscissae	Computation of the Greville abscissae
procedure, public :: Plot => PlotBasisFunc	Plot basis functions
procedure, public :: get_nquad	Get the number of quadrature points

Functions

private pure elemental function FindSpan(me, u) result(span)

This procedure is an elemental function and does a binary search and return the integer :: span value, where $u$ lies on. Note that, indexing starts with zero. If $u=u_{m}$ , the index is $n (= m - p - 1)$ , i.e., $u \in {(u_{n}, u_{nb}]}$ , where "nb $(= n+1)$ " is number of control points of the univariate spline basis object.

Arguments

Type	Intent	Optional	Attributes		Name
class(basis),	intent(in)			::	me
real(kind=wp),	intent(in)			::	u	Given $u$ value

Return Value integer

Knot span index of $u$ value

private pure elemental function FindMult(me, i, u) result(mult)

Computes the multiplicity of a knot

Arguments

Type	Intent		Name
class(basis),	intent(in)	::	me
integer,	intent(in)	::	i	Starting index for search
real(kind=wp),	intent(in)	::	u	Given $u$ value

Return Value integer

Multiplicity of $u$ value

private pure elemental function OneBasisFun(me, i, u) result(Nip)

Computes a single basis function, $N_{i,p}(u)$ .

Arguments

Type	Intent		Name
class(basis),	intent(in)	::	me
integer,	intent(in)	::	i	Basis function index
real(kind=wp),	intent(in)	::	u	Given $u$ value

Return Value real(kind=wp)

$N_{i,p}(u)$

private pure function ParLen(me) result(len)

Return the length of the parameter space, i.e., $u_{m}-u_{0}$ , where $m$ is the highest index of the knot vector.

Arguments

Type	Intent	Optional	Attributes		Name
class(basis),	intent(in)			::	me

Return Value real(kind=wp)

Length of the parameter space

private pure function GrevilleAbscissae(me, dsc) result(pt)

Return the Greville abscissae, $\{u_i\}$ , of the knot vector,

Arguments

Type	Intent	Optional	Attributes		Name
class(basis),	intent(in)			::	me
logical,	intent(in),	optional		::	dsc	If true, then the Greville abscissae are modified by shifting the boundary points into the domain

Return Value real(kind=wp) (0:me%n)

The array containing Greville abscissae

private pure elemental function get_nquad(me, key, reduced_int, pdeq) result(ngp)

Return the number of quadrature points for a successive integration

Arguments

Type	Intent	Optional		Name
class(basis),	intent(in)		::	me
character(len=*),	intent(in),	optional	::	key	The key defining whether the quadrature is over the Bezier segments or over whole patch Valid inputs: - `"elementwise"` - `"patchwise"`
logical,	intent(in),	optional	::	reduced_int	Reduced integration flag, if true, the routine will return the number of quadrature points for reduced integration
character(len=*),	intent(in),	optional	::	pdeq	Internal name of the relevant partial differential equation Valid inputs: - `"Timoshenko"` - `"Mindlin"` - `"Shell"`

Return Value integer

private pure function basisConstructor(nk, p, k) result(me)

Arguments

Type	Intent		Name
integer,	intent(in)	::	nk	Number of the knots
integer,	intent(in)	::	p	Polynomial degree $p$
real(kind=wp),	intent(in)	::	k(nk)	Open knot vector `kv(0:m)` $= \left\{u_{0},..., u_{m}\right\}$

Return Value type(basis)

Spline basis object

private pure function bin(n, r)

Arguments

Type	Intent	Optional	Attributes		Name
integer,	intent(in)			::	n
integer,	intent(in)			::	r

Return Value integer(kind=i8p)

private pure recursive function sort(x) result(res)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:)	::	x	Input array

Return Value real(kind=wp), dimension(size(x))

Subroutines

private subroutine PlotBasisFunc(me, first, last, d, plotRes, title, work, fname, terminal, showPlot)

This subroutine plots the basis functions or their $d$ th order derivatives having index from first to last by computing at interpolation points defined by plotRes, and print the image file into "|cwd|/data/work/fname.ext" if the workspace work is presented, "|cwd|/data/img/fname.ext" if else. Plotting requires GNUPlot installed on your computer. The input parameter terminal set the output terminal of GNUPlot.

Arguments

Type	Intent	Optional		Name
class(basis),	intent(in)		::	me
integer,	intent(in),	optional	::	first	First index to be plotted
integer,	intent(in),	optional	::	last	Last index to be plotted
integer,	intent(in),	optional	::	d	Derivatives degree
integer,	intent(in),	optional	::	plotRes	Resolution of the calculation, Default = 100
character(len=*),	intent(in),	optional	::	title	Title of the graph, Default = "Basis functions"
character(len=*),	intent(in),	optional	::	work	Current work name to create folder
character(len=*),	intent(in),	optional	::	fname	Filename of the .png output, if GNUPlot* terminal is `png`. Default = "BasisFunctions"
character(len=*),	intent(in),	optional	::	terminal	GNUPlot terminal, Default = `wxt`
logical,	intent(in),	optional	::	showPlot	Key to open created `png` files

private pure elemental subroutine FindSpanMult(me, u, k, s)

Computes the span and multiplicity of a knot

Arguments

Type	Intent		Name
class(basis),	intent(in)	::	me
real(kind=wp),	intent(in)	::	u	Given $u$ value
integer,	intent(out)	::	k	Knot span index of $u$ value
integer,	intent(out)	::	s	Multiplicity of $u$ value

private pure subroutine BasisFuns(me, u, N, span)

Computes the nonvanishing basis functions based on Algorithm A2.2 in the NURBS Book ¹.

Arguments

Type	Intent	Optional		Name
class(basis),	intent(in)		::	me
real(kind=wp),	intent(in)		::	u	Given $u$ value
real(kind=wp),	intent(out)		::	N(0:me%p)	$= \{ N_{i-p,p}(u) \ldots N_{i,p}(u)\}$
integer,	intent(in),	optional	::	span	Knot span where $u$ lies on

private pure subroutine AllBasisFuns(me, u, N, span)

Computes all non-zero basis function of all degrees from $0$ up to $p$ .

Arguments

Type	Intent	Optional		Name
class(basis),	intent(in)		::	me
real(kind=wp),	intent(in)		::	u	Given $u$ value
real(kind=wp),	intent(out)		::	N(0:me%p,0:me%p)	$= \begin{bmatrix} {N_{i - p,0}(u)} & \cdots &{N_{i - p,p}(u)}\\ \vdots & \ddots & \vdots \\ {N_{i,0}(u)} & \cdots &{N_{i,p}(u)} \end{bmatrix}$
integer,	intent(in),	optional	::	span	Knot span where $u$ lies on

private pure subroutine DersBasisFuns(me, u, n, ders, span)

Computes all nonzero basis functions and their derivatives based on Algorithm A2.3 in the NURBS Book [^1].

Arguments

Type	Intent	Optional		Name
class(basis),	intent(in)		::	me
real(kind=wp),	intent(in)		::	u	Given $u$ value
integer,	intent(in)		::	n	Number of derivatives ( $n \leq p$ )
real(kind=wp),	intent(out)		::	ders(0:me%p,0:n)	$= \begin{bmatrix} {N_{i - p,p}^{(0)}(u)} & \cdots &{N_{i - p,p}^{(k)}(u)}\\ \vdots & \ddots & \vdots \\ {N_{i,p}^{(0)}(u)} & \cdots &{N_{i,p}^{(k)}(u)} \end{bmatrix}$
integer,	intent(in),	optional	::	span	Knot span where $u$ lies on

private pure subroutine DersOneBasisFun(me, i, u, n, ders)

Computes a single basis function and its derivatives based on Algorithm A2.5 in the NURBS Book [^1].

Arguments

Type	Intent		Name
class(basis),	intent(in)	::	me
integer,	intent(in)	::	i	Basis function index
real(kind=wp),	intent(in)	::	u	Given $u$ value
integer,	intent(in)	::	n	Number of derivatives ( $n \leq p$ )
real(kind=wp),	intent(out)	::	ders(0:n)	$= \{ N_{i,p}^{(0)}(u), \ldots, N_{i,p}^{(k)}(u) \}$

private pure subroutine InsertKnot(me, u, r, span)

Insert knot $\bar{u}$ into ${[u_{k}, u_{k+1})}$ $r$ times.

Arguments

Type	Intent	Optional		Name
class(basis),	intent(inout)		::	me
real(kind=wp),	intent(in)		::	u	knot value $\bar{u}$
integer,	intent(in)		::	r	multiplicity of new knot value
integer,	intent(in),	optional	::	span	knot span

private pure subroutine UpdateBasis(me, nk, k, p)

Updates all component of the basis according to input knot vector.

Arguments

Type	Intent	Optional		Name
class(basis),	intent(inout)		::	me
integer,	intent(in)		::	nk	Number of the knots
real(kind=wp),	intent(in)		::	k(nk)	New knot vector
integer,	intent(in),	optional	::	p	New basis degree

private pure subroutine BSegments(me, nb, knots)

Returns the number of Bezier segments, integer :: nb, of the basis and the real(wp) :: knots(:) array containing the unique values of the knot vector.

Arguments

Type	Intent	Optional	Attributes		Name
class(basis),	intent(in)			::	me
integer,	intent(out)			::	nb	Number of Bezier segments
real(kind=wp),	intent(out),	optional	allocatable	::	knots(:)	The unique knot values

BasisFunctions Module

Contents

Uses

Contents

Interfaces

public interface basis

private pure function basisConstructor(nk, p, k) result(me)

Arguments

Return Value type(basis)

Derived Types

type, public :: basis

Components

Constructor

Type-Bound Procedures

Functions

private pure elemental function FindSpan(me, u) result(span)

Arguments

Return Value integer

private pure elemental function FindMult(me, i, u) result(mult)

Arguments

Return Value integer

private pure elemental function OneBasisFun(me, i, u) result(Nip)

Arguments

Return Value real(kind=wp)

private pure function ParLen(me) result(len)

Arguments

Return Value real(kind=wp)

private pure function GrevilleAbscissae(me, dsc) result(pt)

Arguments

Return Value real(kind=wp) (0:me%n)

private pure elemental function get_nquad(me, key, reduced_int, pdeq) result(ngp)

Arguments

Return Value integer

private pure function basisConstructor(nk, p, k) result(me)

Arguments

Return Value type(basis)

private pure function bin(n, r)

Arguments

Return Value integer(kind=i8p)

private pure recursive function sort(x) result(res)

Arguments

Return Value real(kind=wp), dimension(size(x))

Subroutines

private subroutine PlotBasisFunc(me, first, last, d, plotRes, title, work, fname, terminal, showPlot)

Arguments

private pure elemental subroutine FindSpanMult(me, u, k, s)

Arguments

private pure subroutine BasisFuns(me, u, N, span)

Arguments

private pure subroutine AllBasisFuns(me, u, N, span)

Arguments

private pure subroutine DersBasisFuns(me, u, n, ders, span)

Arguments

private pure subroutine DersOneBasisFun(me, i, u, n, ders)

Arguments

private pure subroutine InsertKnot(me, u, r, span)

Arguments

private pure subroutine UpdateBasis(me, nk, k, p)

Arguments

private pure subroutine BSegments(me, nb, knots)

Arguments