gsTensorBasis< d, T > Class Template Reference

Detailed Description

template<short_t d, class T>
class gismo::gsTensorBasis< d, T >

Abstract base class for tensor product bases.

Parameters

d	dimension of the parameter domain
Basis_t	type of the coordinate-wise bases

Inheritance diagram for gsTensorBasis< d, T >:

Collaboration diagram for gsTensorBasis< d, T >:

Public Types
typedef Basis_t **	iterator
	Iterators on coordinate bases.
typedef T	Scalar_t
	Coefficient type.

Public Member Functions
gsMatrix< index_t >	active (const gsMatrix< T > &u) const
	Returns the indices of active (nonzero) functions at points u, as a list of indices. More...
void	active_cwise (const gsMatrix< T > &u, gsVector< index_t, d > &low, gsVector< index_t, d > &upp) const
virtual void	active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
	Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support. More...
gsMatrix< index_t >	allBoundary () const
void	anchor_into (index_t i, gsMatrix< T > &result) const
	Returns the anchors (graville absissae) that represent the members of the basis.
gsMatrix< T >	anchors () const
	Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function. More...
void	anchors_into (gsMatrix< T > &result) const
	Returns the anchors (graville absissae) that represent the members of the basis.
const gsBasis< T > &	basis (const index_t k) const
	Helper which casts and returns the k-th piece of this function set as a gsBasis.
const_iterator	begin () const
iterator	begin ()
gsMatrix< index_t >	boundary (boxSide const &s) const
	Returns the indices of the basis functions that are nonzero at the domain boundary as single-column-matrix.
gsBasis< T >::uPtr	boundaryBasis (boxSide const &s)
	Returns the boundary basis for side s.
gsMatrix< index_t >	boundaryOffset (boxSide const &s, index_t offset) const
uPtr	clone ()
	Clone methode. Produceds a deep copy inside a uPtr.
gsMatrix< index_t >	coefSlice (short_t dir, index_t k) const
	Returns all the basis functions with tensor-numbering. More...
gsSparseMatrix< T >	collocationMatrix (gsMatrix< T > const &u) const
	Computes the collocation matrix w.r.t. points u. More...
Basis_t &	component (short_t dir)
	For a tensor product basis, return the 1-d basis for the i-th parameter component.
const Basis_t &	component (short_t dir) const
	For a tensor product basis, return the (const) 1-d basis for the i-th parameter component.
virtual uPtr	componentBasis (boxComponent b) const
	Returns the basis that corresponds to the component.
virtual uPtr	componentBasis_withIndices (boxComponent b, gsMatrix< index_t > &indices, bool noBoundary=true) const
	Returns the basis that corresponds to the component. More...
virtual void	compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
	Computes function data. More...
virtual void	connectivity (const gsMatrix< T > &nodes, gsMesh< T > &mesh) const
virtual void	connectivityAtAnchors (gsMesh< T > &mesh) const
virtual gsBasis::uPtr	create () const
	Create an empty basis of the derived type and return a pointer to it.
short_t	degree (short_t i) const
	Returns the degree of the basis wrt variable i.
virtual void	degreeDecrease (short_t const &i=1, short_t const dir=-1)
	Lower the degree of the basis by the given amount, preserving knots multiplicity.
virtual void	degreeElevate (short_t const &i=1, short_t const dir=-1)
	Elevate the degree of the basis by the given amount, preserve smoothness.
virtual void	degreeIncrease (short_t const &i=1, short_t const dir=-1)
	Elevate the degree of the basis by the given amount, preserve knots multiplicity.
virtual void	degreeReduce (short_t const &i=1, short_t const dir=-1)
	Reduce the degree of the basis by the given amount, preserve smoothness.
gsMatrix< T >	deriv (const gsMatrix< T > &u) const
	Evaluate the derivatives,. More...
gsMatrix< T >	deriv2 (const gsMatrix< T > &u) const
	Evaluates the second derivatives of active (i.e., non-zero) functions at points u. More...
virtual void	deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate the second derivatives of all active basis function at points u. More...
virtual void	deriv2Single_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate the (partial) derivatives of the i-th basis function at points u into result.
virtual void	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluates the first partial derivatives of the nonzero basis function. More...
virtual void	derivSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluates the (partial) derivatives of the i-th basis function at points u into result. More...
virtual std::string	detail () const
	Prints the object as a string with extended details.
virtual gsDomain< T > *	domain () const
short_t	domainDim () const
	Dimension of the (source) domain. More...
size_t	elementIndex (const gsVector< T > &u) const
	Returns an index for the element which contains point u.
gsMatrix< T >	elementInSupportOf (index_t j) const
	Returns (the coordinates of) an element in the support of basis function j.
virtual void	elevateContinuity (int const &i=1)
	Elevates the continuity of the basis along element boundaries.
const_iterator	end () const
iterator	end ()
gsMatrix< T >	eval (const gsMatrix< T > &u) const
	Evaluate the function,. More...
virtual void	eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluates nonzero basis functions at point u into result. More...
virtual void	eval_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Evaluate an element of the space given by coefs at points u.
std::vector< gsMatrix< T > >	evalAllDers (const gsMatrix< T > &u, int n, bool sameElement=false) const
	Evaluate all derivatives upto order n,. More...
virtual void	evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
	Evaluate the nonzero functions and their derivatives up to order n at points u into result. More...
void	evalSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate the i-th basis function at points u into result.
gsBasisFun< T >	function (index_t i) const
	Returns the i-th basis function as a gsFunction. More...
void	getComponentsForSide (boxSide const &s, std::vector< Basis_t * > &rr) const
	Returns the components for a basis on the face s.
virtual T	getMaxCellLength () const
	Get the maximum mesh size, as expected for approximation error estimates.
virtual T	getMinCellLength () const
	Get the minimum mesh size, as expected for inverse inequalities.
	gsTensorBasis (const gsTensorBasis &o)
	Copy Constructor.
	gsTensorBasis (Basis_t *x, Basis_t *y)
	Constructor 2D (takes ownership of the passed bases)
	gsTensorBasis (Basis_t *x, Basis_t *y, Basis_t *z)
	Constructor 3D (takes ownership of the passed bases)
	gsTensorBasis (Basis_t *x, Basis_t *y, Basis_t *z, Basis_t *w)
	Constructor 4D (takes ownership of the passed bases)
	gsTensorBasis (iterator it)
	Constructor nD (takes ownership of the passed bases)
unsigned	index (unsigned i, unsigned j, unsigned k=0) const
index_t	index (gsVector< index_t, d > const &v) const
bool	indexOnBoundary (const gsVector< index_t, d > &ind) const
	Returns true iff the basis function with multi-index ind is on the boundary.
bool	indexOnBoundary (const index_t m) const
	Returns true iff the basis function indexed m is on the boundary.
virtual gsGeometry< T >::uPtr	interpolateAtAnchors (gsMatrix< T > const &vals) const
	Applies interpolation of values pts using the anchors as parameter points. May be reimplemented in derived classes with more efficient algorithms. (by default uses interpolateData(pts,vals)
memory::unique_ptr< gsGeometry < T > >	interpolateData (gsMatrix< T > const &vals, gsMatrix< T > const &pts) const
	Applies interpolation given the parameter values pts and values vals.
gsGeometry< T >::uPtr	interpolateGrid (gsMatrix< T > const &vals, std::vector< gsMatrix< T > >const &grid) const
bool	isActive (const index_t i, const gsVector< T > &u) const
	Returns true if there the point u with non-zero value or derivatives when evaluated at the basis function i.
virtual bool	isRational () const
	Returns false, since all bases that inherit from gsBasis are not rational.
gsBasis< T >::domainIter	makeDomainIterator () const
	Create a domain iterator for the computational mesh of this basis, that points to the first element of the domain.
gsBasis< T >::domainIter	makeDomainIterator (const boxSide &s) const
	Create a boundary domain iterator for the computational mesh this basis, that points to the first element on the boundary of the domain.
virtual memory::unique_ptr < gsGeometry< T > >	makeGeometry (gsMatrix< T > coefs) const =0
	Create a gsGeometry of proper type for this basis with the given coefficient matrix.
virtual memory::unique_ptr < gsBasis< T > >	makeNonRational () const
void	matchWith (const boundaryInterface &bi, const gsBasis< T > &other, gsMatrix< index_t > &bndThis, gsMatrix< index_t > &bndOther, index_t offset) const
	Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi. More...
short_t	maxDegree () const
	If the basis is of polynomial or piecewise polynomial type, then this function returns the maximum polynomial degree.
short_t	minDegree () const
	If the basis is of polynomial or piecewise polynomial type, then this function returns the minimum polynomial degree.
virtual index_t	nPieces () const
	Number of pieces in the domain of definition.
size_t	numElements (boxSide const &s=boundary::none) const
	The number of elements on side s.
void	numElements_cwise (gsVector< unsigned > &result) const
	Returns the number of elements (component wise)
const gsBasis< T > &	piece (const index_t k) const
	Returns the piece(s) of the function(s) at subdomain k.
virtual std::ostream &	print (std::ostream &os) const =0
	Prints the object as a string, pure virtual function of gsTensorBasis.
virtual void	reduceContinuity (int const &i=1)
	Reduces the continuity of the basis along element boundaries.
virtual void	refine (gsMatrix< T > const &boxes, int refExt=0)
	Refine the basis on the area defined by the matrix boxes. More...
void	refineElements (std::vector< index_t > const &elements)
	Refine elements defined by their tensor-index. More...
virtual void	refineElements_withCoefs (gsMatrix< T > &coefs, std::vector< index_t > const &boxes)
	Refine basis and geometry coefficients to levels. More...
virtual void	reverse ()
	Reverse the basis.
void	setDegree (short_t const &i)
	Set the degree of the basis (either elevate or reduce) in order to have degree equal to i wrt to each variable.
void	setDegreePreservingMultiplicity (short_t const &i)
	Set the degree of the basis (either increase or decrecee) in order to have degree equal to i.
index_t	size () const
	Returns the number of elements in the basis.
index_t	size (short_t k) const
	The number of basis functions in the direction of the k-th parameter component.
template<int s>
void	size_cwise (gsVector< index_t, s > &result) const
	The number of basis functions in the direction of the k-th parameter component.
virtual const gsBasis &	source () const
virtual gsBasis &	source ()
unsigned	stride (short_t dir) const
	Returns the stride for dimension dir.
void	stride_cwise (gsVector< index_t, d > &result) const
	Returns the strides for all dimensions.
gsMatrix< T >	support () const
	Returns (a bounding box for) the domain of the whole basis. More...
gsMatrix< T >	support (const index_t &i) const
	Returns (a bounding box for) the support of the i-th basis function. More...
gsMatrix< T >	supportInterval (index_t dir) const
	Returns an interval that contains the parameter values in direction dir. More...
virtual short_t	targetDim () const
	Dimension of the target space. More...
gsVector< index_t, d >	tensorIndex (const index_t &m) const
	Returns the tensor index of the basis function with global index m.
virtual gsBasis::uPtr	tensorize (const gsBasis &other) const
	Return a tensor basis of this and other.
short_t	totalDegree () const
	If the basis is of polynomial or piecewise polynomial type, then this function returns the total polynomial degree.
virtual void	uniformCoarsen (int numKnots=1)
	Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times. More...
virtual void	uniformCoarsen_withCoefs (gsMatrix< T > &coefs, int numKnots=1)
	Coarsen the basis uniformly. More...
void	uniformCoarsen_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1)
	Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones. More...
virtual void	uniformRefine (int numKnots=1, int mul=1, int dir=-1)
	Refine the basis uniformly by inserting numKnots new knots with multiplicity mul on each knot span.
void	uniformRefine_withCoefs (gsMatrix< T > &coefs, int numKnots=1, int mul=1, int dir=-1)
void	uniformRefine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1, int mul=1)
virtual const gsMatrix< T > &	weights () const
	Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
virtual gsMatrix< T > &	weights ()
	Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
Evaluation functions
gsMatrix< T >	evalSingle (index_t i, const gsMatrix< T > &u) const
	Evaluate a single basis function i at points u.
gsMatrix< T >	derivSingle (index_t i, const gsMatrix< T > &u) const
	Evaluate a single basis function i derivative at points u.
gsMatrix< T >	deriv2Single (index_t i, const gsMatrix< T > &u) const
	Evaluate the second derivative of a single basis function i at points u.
gsVector< index_t >	numActive (const gsMatrix< T > &u) const
	Number of active basis functions at an arbitrary parameter value. More...
virtual void	numActive_into (const gsMatrix< T > &u, gsVector< index_t > &result) const
	Returns the number of active (nonzero) basis functions at points u in result.
virtual void	activeCoefs_into (const gsVector< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions. More...
virtual void	evalAllDersSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
	Evaluate the basis function i and its derivatives up to order n at points u into result.
virtual void	evalDerSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
	Evaluate the (partial) derivative(s) of order n the i-th basis function at points u into result.
virtual gsMatrix< T >	laplacian (const gsMatrix< T > &u) const
	Compute the Laplacian of all nonzero basis functions at points u.

Static Public Attributes
static const short_t	Dim
	Dimension of the parameter domain.

Geometry evaluation functions
These functions evaluate not the individual basis functions of the basis, but a geometry object which is represented by a coefficient matrix w.r.t. this basis object. For the format of the coefficient matrix, see gsGeometry. These functions have default implementations which simply compute the basis function values and perform linear combination, but they may be overridden in derived classes if a higher-performance implementation is possible.
gsMatrix< T >	evalFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
	Evaluate the function described by coefs at points u. More...
virtual void	evalFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result. More...
gsMatrix< T >	derivFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
	Evaluate the derivatives of the function described by coefs at points u. More...
virtual void	derivFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Evaluate the derivatives of the function described by coefs at points u. More...
virtual void	jacobianFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Evaluate the Jacobian of the function described by coefs at points u. Jacobian matrices are stacked in blocks.
gsMatrix< T >	deriv2Func (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
	Evaluates the second derivatives of the function described by coefs at points u. More...
virtual void	deriv2Func_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
	Evaluates the second derivatives of the function described by coefs at points u. More...
virtual void	evalAllDersFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, const unsigned n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
	Evaluates all derivatives up to order n of the function described by coefs at points u. More...
static void	linearCombination_into (const gsMatrix< T > &coefs, const gsMatrix< index_t > &actives, const gsMatrix< T > &values, gsMatrix< T > &result, bool sameElement=false)
	Computes the linear combination coefs * values( actives ) More...

Member Function Documentation

gsMatrix<index_t> active ( const gsMatrix< T > &  u ) const

inlineinherited

Returns the indices of active (nonzero) functions at points u, as a list of indices.

See Also

active_into()

void active_cwise	(	const gsMatrix< T > &	u,
		gsVector< index_t, d > &	low,
		gsVector< index_t, d > &	upp
	)		const

Returns a box with the coordinate-wise active functions

Parameters

u	evaluation points
low	lower left corner of the box
upp	upper right corner of the box

void active_into ( const gsMatrix< T > &  u, gsMatrix< index_t > &  result  ) const

virtual

Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support.

Tensor indexing in result

Assume that the parameter domain is three dimensional. Let n1, n2, and n3 denote the number of univariate basis functions in the first, second and third coordinate direction, respectively.
Let the trivariate tensor product basis function B_I be defined by
B_I(x,y,z) = B_i(x) * B_j(y) * B_k(z).
Then, the index I, which is returned in result, is computed as
I = i + j * n1 + k * n1*n2.
Examples:
I <-> (i,j,k)
0 <-> (0,0,0)
1 <-> (1,0,0)
2 <-> (2,0,0)
...
(n1-1) <-> (n1-1,0,0)
n1 <-> (0,1,0)
n1+1 <-> (1,1,0)
n1+2 <-> (2,1,0)
...
n1*n2-1 <-> (n1,n2,0)
n1*n2 <-> (0,0,1)
n1*n2+1 <-> (1,0,1)
...
n1*n2*n3-1 <-> (n1,n2,n3)

Parameters

[in]	u	gsMatrix containing evaluation points. Each column represents one evaluation point.
[out]	result	For every column i of u, a column containing the active basis functions at evaluation point u.col(i)

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.

void activeCoefs_into ( const gsVector< T > &  u, const gsMatrix< T > &  coefs, gsMatrix< T > &  result  ) const

virtualinherited

Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions.

Parameters

[in]	u	gsVector containing an evaluation point.
[in]	coefs	gsMatrix is a coefficient matrix with as many rows as the size of the basis
[out]	result	For every column i of u, a column containing the indices of the active basis functions at evaluation point u.col(i).

gsMatrix< index_t > allBoundary ( ) const

virtual

Returns the indices of the basis functions that touch the domain boundary

Reimplemented from gsBasis< T >.

gsMatrix<T> anchors ( ) const

inlineinherited

Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function.

The exact definition of the anchor points depends on the particular basis. For instance, for a Bspline basis these are the Greville abscissae. In general, evaluating a function at the anchor points should provide enough information to interpolate that function using this basis.

const_iterator begin ( ) const

inline

Get a const-iterator to the beginning of the bases vector

Returns

an iterator to the beginning of the bases vector

iterator begin ( )

inline

Get an iterator to the beginning of the bases vector

Returns

an iterator to the beginning of the bases vector

gsMatrix< index_t > boundaryOffset ( boxSide const &  s, index_t  offset  ) const

virtual

Returns the indices of the basis functions that touch the domain boundary

Reimplemented from gsBasis< T >.

gsMatrix< index_t > coefSlice	(	short_t	dir,
		index_t	k
	)		const

Returns all the basis functions with tensor-numbering.

Parameters

k	in direction
dir	Detailed explanation:

Tensor-numbering in N-variate tensor-product basis means that each basis function is assigned an identifier (i_0, i_1, ..., i_{N-1}). This function returns indices of basis functions with i_dir = k and the returned indices are numbering of the basis functions in the basis (i.e., 0,1, ..., basis.size() ).

Example:

Bivariate tensor-product basis functions have tensor numbering (a,b). Calling dir=0, k=1 gives all functions with tensor-numbering (1,b). Calling dir=1, k=3 gives all functions with tensor-numbering (a,3).

gsSparseMatrix< T > collocationMatrix ( gsMatrix< T > const &  u ) const

inlineinherited

Computes the collocation matrix w.r.t. points u.

The collocation matrix is a sparse matrix with u.cols rows and size() columns. The entry (i,j) is the value of basis function j at evaluation point i.

gsBasis< T >::uPtr componentBasis_withIndices ( boxComponent  b, gsMatrix< index_t > &  indices, bool  noBoundary = true  ) const

virtualinherited

Returns the basis that corresponds to the component.

Parameters

b	The component
indices	The row vector where the indices are stored to
noBoundary	If true, the transfer matrix does not include parts belonging to lower-order components (i.e., edges without corners or faces without corners and edges)

void compute ( const gsMatrix< T > &  in, gsFuncData< T > &  out  ) const

virtualinherited

Computes function data.

This function evaluates the functions and their derivatives at the points in and writes them in the corresponding fields of out. Which field to write (and what to compute) is controlled by the out.flags (see also gsFuncData).

The input points in are expected to be compatible with the implementation/representation of the function, i.e. they should be points inside the domain of definitition of the function

Parameters

[in]	in
[out]	out

Reimplemented in gsGeometry< T >, and gsConstantFunction< T >.

void connectivity ( const gsMatrix< T > &  nodes, gsMesh< T > &  mesh  ) const

virtual

Returns the connectivity structure of the basis The returned mesh has vertices the rows of matrix nodes

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void connectivityAtAnchors ( gsMesh< T > &  mesh ) const

inlinevirtualinherited

Returns the connectivity structure of the basis The returned mesh has the anchor points as vertices

gsMatrix< T > deriv ( const gsMatrix< T > &  u ) const

inherited

Evaluate the derivatives,.

See Also

deriv_into()

gsMatrix< T > deriv2 ( const gsMatrix< T > &  u ) const

inherited

Evaluates the second derivatives of active (i.e., non-zero) functions at points u.

See documentation for deriv2_into() (the one without input parameter coefs) for details.

See Also

deriv2_into()

Parameters

[in]

u

Evaluation points in columns.

Returns

For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one without input parameter coefs) for details.

void deriv2_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluate the second derivatives of all active basis function at points u.

Input parameter u is a gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

result is a gsMatrix of size (K * d) x N, where
K is the number of active basis functions at the evaluation point.
Each column of result corresponds to a column of u. It contains the "pure" and the mixed derivatives for each active basis function, "above" each other.

Example (bivariate): Let \(B_i(x,y)\), d = 2 be bivariate basis functions, and let the functions with indices 3,4,7, and 8 (K = 4) be active at an evaluation point u. Then, the corresponding column of result represents:
\( ( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xx}\, B_4(u), \partial_{yy}\, B_4(u), \partial_{xy}\, B_4(u), \partial_{xx}\, B_7(u), ... , \partial_{xy}\, B_8(u) )^T \)

Example (trivariate): Let \(B_i(x,y,z)\), d = 3 be trivariate basis functions, and let the functions with indices 3,4,7, and 8 be active at an evaluation point u. Then, the corresponding column of result represents:
\(( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{zz}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xz}\, B_3(u), \partial_{yz}\, B_3(u), \partial_{xx}\, B_4(u), ... , \partial_{yz}\, B_8(u) )^T \)

Parameters

[in]	u	Evaluation points in columns (see above for format).
[in,out]	result	For every column of u, a column containing the second derivatives as described above.

See also deriv2() (the one without input parameter coefs).

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsMatrix<T> deriv2Func ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs  ) const

inlineinherited

Evaluates the second derivatives of the function described by coefs at points u.

See documentation for deriv2_into() (the one with input parameter coefs) for details.

Parameters

[in]	u	Evaluation points in columns.
[in]	coefs	Coefficient matrix describing the geometry in this basis.

Returns

For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one with input parameter coefs) for details.

void deriv2Func_into ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs, gsMatrix< T > &  result  ) const

virtualinherited

Evaluates the second derivatives of the function described by coefs at points u.

...i.e., evaluates a linear combination of coefs * (2nd derivatives of basis functions), into result.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size N x n, where
N is the number of points = number of basis functions and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

Let the function \( f: \mathbb R^3 \to \mathbb R^3\) be given by

\[ f = ( f_1, f_2, f_3)^T = \sum_{i=1}^N c_i B_i(x,y,z), \]

where \( B_i(x,y,z)\) are scalar basis functions and \(c_i\) are the corresponding (m-dimensional) control points. Then, for each column in u, the corresponding column in result represents

\[ ( \partial_{xx}\ f_1, \partial_{yy}\ f_1, \partial_{zz}\ f_1, \partial_{xy}\ f_1, \partial_{xz}\ f_1, \partial_{yz}\ f_1, \partial_{xx}\ f_2, \partial_{yy}\ f_2, \ldots , \partial_{xz}\ f_3, \partial_{yz}\ f_3)^T. \]

at the respective evaluation point.

Parameters

[in]	u	Evaluation points in columns (see above for format).
[in]	coefs	Coefficient matrix describing the geometry in this basis.
[out]	result	For every column of u, a column containing the second derivatives at the respective point in the format described above.

This function has a default implementation that may be overridden in derived classes for higher performance.
See also deriv2() (the one with input parameter coefs).

void deriv_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluates the first partial derivatives of the nonzero basis function.

Let
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()).
N denote the number of evaluation points.
The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed derivatives in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the gradients of all active functions "above" each other.
For example, for scalar basis functions \(B_i : (x,y,z)-> R\), a column represents
\((dx B_1, dy B_1, dz B_1, dx B_2, dy B_2, dz B_2, ... , dx B_n, dy B_N, dz B_N)^T\),
where the order the basis functions \(B_i\) is as returned by active() and active_into().

Parameters

[in]	u	Evaluation points given as gsMatrix of size d x N. See above for details.
[in,out]	result	gsMatrix of size (K*d) x N. See above for details.

Todo:

Rename to _ grad_into

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsMatrix<T> derivFunc ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs  ) const

inlineinherited

Evaluate the derivatives of the function described by coefs at points u.

Parameters

u	evaluation points as N column vectors
coefs	coefficient matrix describing the geometry in this basis, n columns

Returns

For every column of u, the result matrix will contain one Jacobian matrix of size d * n, such that the total size of the result is n x (d * n) x N

void derivFunc_into ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs, gsMatrix< T > &  result  ) const

virtualinherited

Evaluate the derivatives of the function described by coefs at points u.

Evaluates a linear combination of coefs*BasisFunctionDerivatives, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Let the function \(f: \mathbb R^d \to \mathbb R^m \) be described by the coefficients coefs, i.e.,
each evaluation point is in \(\mathbb R^d\), and
each coefficient is a point in \(\mathbb R^m\).

The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The K coefficients coefs are given as a gsMatrix of size K x m. Each row of coefs represents one coefficient in \(\mathbb R^m\).

The gsMatrix result contains the following data:
For every column of u, the corresponding column in the matrix result contains the gradients of the m components of the function above each other. Hence, the size of result is (d*m) x N.

Example 1:
Let \(f(s,t)\) be a bivariate scalar function, \(f:\mathbb R^2 \to \mathbb R\) (i.e., d=2, m=1), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{cccc} \partial_s f(u_1) & \partial_s f(u_2) & \ldots & \partial_t f(u_{N}) \\ \partial_t f(u_1) & \partial_t f(u_2) & \ldots & \partial_t f(u_{N}) \end{array} \right) \]

Example 2:
Let \(f(s,t) = ( f_1(s,t), f_2(s,t), f_3(s,t) )\) represent a surface in space, \(f:\mathbb R^2 \to \mathbb R^3\) (i.e., d=2, m=3), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{ccccccc} \partial_s f_1(u_1) & \partial_s f_1(u_2) & \ldots & \partial_s f_1(u_N) \\ \partial_t f_1(u_1) & \partial_t f_1(u_2) & \ldots & \partial_t f_1(u_N) \\ \partial_s f_2(u_1) & \partial_s f_2(u_2) & \ldots & \partial_s f_2(u_N) \\ \partial_t f_2(u_1) & \partial_t f_2(u_2) & \ldots & \partial_t f_2(u_N) \\ \partial_s f_3(u_1) & \partial_s f_3(u_2) & \ldots & \partial_s f_3(u_N) \\ \partial_t f_3(u_1) & \partial_t f_3(u_2) & \ldots & \partial_t f_3(u_N) \\ \end{array} \right) \]

Parameters

[in]	u	Evaluation points as d x N-matrix.
[in]	coefs	Coefficient matrix describing the geometry in this basis as K x m-matrix. K should equal the size() of the basis, i.e., the number basis functions.
[in,out]	result	gsMatrix of size d*m x N, see above for format.

where
d is the dimension of the parameter domain
m is the dimension of the physical domain
N is the number of evaluation points
K is the number of coefficients

void derivSingle_into ( index_t  i, const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluates the (partial) derivatives of the i-th basis function at points u into result.

See deriv_into() for detailed documentation.

Todo:

rename grad_into

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsDomain< T > * domain ( ) const

virtualinherited

Return the gsDomain which represents the parameter domain of this basis. Currently unused.

Reimplemented in gsTensorBSplineBasis< 1, T >, gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, gsRationalBasis< gsBSplineBasis< T > >, gsLagrangeBasis< T >, gsLegendreBasis< T >, and gsMvLegendreBasis< T >.

short_t domainDim ( ) const

inlinevirtual

Dimension of the (source) domain.

Returns

For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(n\).

Implements gsFunctionSet< T >.

const_iterator end ( ) const

inline

Get a const-iterator to the end of the bases vector

Returns

an iterator to the end of the bases vector

iterator end ( )

inline

Get an iterator to the end of the bases vector

Returns

an iterator to the end of the bases vector

gsMatrix< T > eval ( const gsMatrix< T > &  u ) const

inherited

Evaluate the function,.

See Also

eval_into()

void eval_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluates nonzero basis functions at point u into result.

Let...
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()). N denote the number of evaluation points.
The n evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed function values in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the values of all active functions "above" each other.
For example, for scalar basis functions Bi : (x,y,z)-> R, a column represents
(B1, B2, ... , BN)^T,
where the order the basis functions Bi is as returned by active() and active_into().

Parameters

[in]	u	Evaluation points given as gsMatrix of size d x N. See above for details.
[in,out]	result	gsMatrix of size K x N. See above for details.

Todo:

more efficient ?

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

std::vector< gsMatrix< T > > evalAllDers ( const gsMatrix< T > &  u, int  n, bool  sameElement = false  ) const

inherited

Evaluate all derivatives upto order n,.

See Also

evalAllDers_into

void evalAllDers_into ( const gsMatrix< T > &  u, int  n, std::vector< gsMatrix< T > > &  result, bool  sameElement = false  ) const

virtual

Evaluate the nonzero functions and their derivatives up to order n at points u into result.

The derivatives (the 0-th derivative is the function value) are stored in a result. result is a std::vector, where result[i] is a gsMatrix which contains the i-th derivatives.

The entries in result[0], result[1], and result[2] are ordered as in eval_into(), deriv_into(), and deriv2_into(), respectively. For i > 2, the derivatives are stored in lexicographical order, e.g. for order i = 3 and dimension 2 the derivatives are stored as follows: \( \partial_{xxx}, \, \partial_{xxy}, \, \partial_{xyy}, \, \partial_{yyy}.\, \)

Parameters

[in]	u	Evaluation points, each column corresponds to one evaluation point.
[in]	n	All derivatives up to order n are computed and stored in result.
[in,out]	result	See above for format.

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void evalAllDersFunc_into ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs, const unsigned  n, std::vector< gsMatrix< T > > &  result, bool  sameElement = false  ) const

virtualinherited

Evaluates all derivatives up to order n of the function described by coefs at points u.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size K x n, where
K is the number of (active) basis functions (=size()) and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

result is a std::vector, where the entry result[i] contains the gsMatrix corresponding to the i-th derivatives. The format of the respective entry is as in
evalFunc_into()
derivFunc_into()
deriv2Func_into()

Todo:

finish documentation

Parameters

[in]	u
[in]	coefs
[in]	n
[out]	result

gsMatrix<T> evalFunc ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs  ) const

inlineinherited

Evaluate the function described by coefs at points u.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters

u	evaluation points as m column vectors
coefs	coefficient matrix describing the geometry in this basis, n columns

Returns

a matrix of size n x m with one function value as a column vector per evaluation point

void evalFunc_into ( const gsMatrix< T > &  u, const gsMatrix< T > &  coefs, gsMatrix< T > &  result  ) const

virtualinherited

Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters

	u	evaluation points as N column vectors
	coefs	coefficient matrix describing the geometry in this basis, n columns
[out]	result	a matrix of size n x N with one function value as a column vector per evaluation point

Reimplemented in gsRationalBasis< SrcT >, gsTensorBSplineBasis< 1, T >, and gsMonomialBasis< T >.

gsBasisFun< T > function ( index_t  i ) const

inherited

Returns the i-th basis function as a gsFunction.

Note that the gsBasisFun object only holds a reference to the current basis, so it is invalidated when the basis is destroyed.

unsigned index ( unsigned  i, unsigned  j, unsigned  k = 0  ) const

inline

Returns the global index of the basis function created by components of indices i,j,k (for 2d or 3d only)

index_t index ( gsVector< index_t, d > const &  v ) const

inline

Returns the global index of the basis function created by components of indices given in the vector v

gsGeometry< T >::uPtr interpolateGrid	(	gsMatrix< T > const &	vals,
		std::vector< gsMatrix< T > >const &	grid
	)		const

Interpolates values on a tensor-grid of points, given in tensor form (d coordinate-wise vectors). Samples vals should be ordered as the tensor-basis coefficients

void linearCombination_into ( const gsMatrix< T > &  coefs, const gsMatrix< index_t > &  actives, const gsMatrix< T > &  values, gsMatrix< T > &  result, bool  sameElement = false  )

staticinherited

Computes the linear combination coefs * values( actives )

Todo:

documentation

Parameters

[in]	coefs	gsMatrix of size K x m, where K should equal size() of the basis (i.e., the number of basis functions).
[in]	actives	gsMatrix of size numAct x numPts
[in]	values	gsMatrix of size stride*numAct x numPts
[out]	result	gsMatrix of size stride x numPts

virtual memory::unique_ptr<gsBasis<T> > makeNonRational ( ) const

inlinevirtualinherited

Clone the source of this basis in case of rational basis, same as clone() otherwise

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

void matchWith ( const boundaryInterface &  bi, const gsBasis< T > &  other, gsMatrix< index_t > &  bndThis, gsMatrix< index_t > &  bndOther, index_t  offset  ) const

virtual

Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi.

NOTE: bndThis will have offset but bndOther will NOT have an offset (hence offset 0)

Reimplemented from gsBasis< T >.

gsVector<index_t> numActive ( const gsMatrix< T > &  u ) const

inlineinherited

Number of active basis functions at an arbitrary parameter value.

Usually, this is used for getting the active functions on one element, assuming that this number doesn't change for different parameters inside the element.

void refine ( gsMatrix< T > const &  boxes, int  refExt = 0  )

virtualinherited

Refine the basis on the area defined by the matrix boxes.

boxes is a d x n-matrix (n even), where d is the dimension of the parameter domain.
n must be even, and every 2 successive columns in the matrix define a box in the parameter domain (the first column represents the coordinates of the lower corner, the second column the coordinates of the upper corner).

Example: The input of the matrix

\[ \left[\begin{array}{cccc} 0 & 0.2 & 0.8 & 1 \\ 0.4 & 0.6 & 0.2 & 0.4 \end{array} \right] \]

results in refinement of the two boxes \([0,0.2]\times[0.4,0.6]\) and \([0.8,1]\times[0.2,0.4]\).

Parameters

[in]	boxes	gsMatrix of size d x n, see above for description of size and meaning.
[in]	refExt	Extension to be applied to the refinement boxes

Reimplemented in gsHTensorBasis< d, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.

void refineElements ( std::vector< index_t > const &  elements )

virtual

Refine elements defined by their tensor-index.

In a tensor mesh, each element has a unique index computed as follows:

Let \(n_i\) denote the number of basis functions in the i-th component, and let \(k_i\) denote the index of an element in the (1-dimensional) mesh of the i-th component. The global index of element \((a,b,c)\) is given by \(a + b \cdot n_1 + c\cdot n_1 \cdot n_2\).

Parameters

[in]

elements

vector of unsigned containing the indices of the elements that should be refined (see above).

Reimplemented from gsBasis< T >.

void refineElements_withCoefs ( gsMatrix< T > &  coefs, std::vector< index_t > const &  boxes  )

virtualinherited

Refine basis and geometry coefficients to levels.

Refines the basis as well as the coefficients. The refinement and the format of the input depend on the implementation of refineElements().

Reimplemented in gsHTensorBasis< d, T >, and gsRationalBasis< SrcT >.

virtual const gsBasis& source ( ) const

inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

virtual gsBasis& source ( )

inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

gsMatrix< T > support ( ) const

virtual

Returns (a bounding box for) the domain of the whole basis.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > support ( const index_t &  i ) const

virtual

Returns (a bounding box for) the support of the i-th basis function.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > supportInterval ( index_t  dir ) const

inherited

Returns an interval that contains the parameter values in direction dir.

Returns a 1x2 matrix, containing the two endpoints of the interval.

virtual short_t targetDim ( ) const

inlinevirtualinherited

Dimension of the target space.

Returns

For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(m\).

Reimplemented in gsPatchIdField< T >, gsGeometry< T >, gsParamField< T >, gsNormalField< T >, gsMultiBasis< T >, gsMultiBasis< real_t >, gsMultiPatch< T >, gsMultiPatch< real_t >, gsFsiLoad< T >, gsMaterialMatrixEvalSingle< T, out >, gsMaterialMatrixIntegrateSingle< T, out >, gsFunctionExpr< T >, gsJacDetField< T >, gsDetFunction< T >, gsConstantFunction< T >, gsShellStressFunction< T >, gsGradientField< T >, gsPiecewiseFunction< T >, gsSquaredDistance< T >, gsAffineFunction< T >, gsPreCICEFunction< T >, gsMappedSingleSpline< d, T >, gsBasisFun< T >, gsAbsError< T >, gsGeometrySlice< T >, gsGeometryTransform< T >, gsFuncCoordinate< T >, and gsCauchyStressFunction< T >.

virtual void uniformCoarsen ( int  numKnots = 1 )

inlinevirtual

Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times.

This function is the oposite of gsBasis::uniformRefine

The execution of

basis->uniformRefine (nKnots, mul)
basis->uniformCoarsen(nKnots);

results in no overall change in "basis". However,

basis->uniformCoarsen(nKnots);
basis->uniformRefine (nKnots, mul)

is not guaranteed to keep "basis" unchanged.

See Also

gsBasis::uniformRefine

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void uniformCoarsen_withCoefs ( gsMatrix< T > &  coefs, int  numKnots = 1  )

virtualinherited

Coarsen the basis uniformly.

The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.

This function is equivalent to

gsSparseMatrix<T,RowMajor> transfer;
basis->uniformCoarsen_withTransfer(transfer, numKnots);
coefs = transfer * coefs;

See Also

gsBasis::uniformRefine

Reimplemented in gsHTensorBasis< d, T >.

void uniformCoarsen_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer, int  numKnots = 1  )

virtual

Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones.

The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions

See Also

gsBasis::uniformCoarsen

Reimplemented from gsBasis< T >.

void uniformRefine_withCoefs ( gsMatrix< T > &  coefs, int  numKnots = 1, int  mul = 1, int  dir = -1  )

virtual

Refine the basis uniformly and perform knot refinement for the given coefficient vector

Reimplemented from gsBasis< T >.

void uniformRefine_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer, int  numKnots = 1, int  mul = 1  )

virtual

Refine the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones

Reimplemented from gsBasis< T >.