Detailed Description

template<typename T>
class gismo::gsFunctionSet< T >

Interface for the set of functions defined on a domain (the total number of functions in the set equals to $S$ )

All G+Smo objects that evaluate function[s] derive from this object. Examples are gsMatrix, gsFunction and gsGeometry.

The available evaluation procedures are:

Name of procedure	Evaluate what
`eval` (points)	value
`deriv` (points)	first derivative(s)
`deriv2` (points)	second derivative(s)

There are two classes of functions for evaluation that differ in the arguments and in the returned values. The argument can either be

1) a matrix;
2) a gsTransform object representing the parametrization of the domain.

In the first case the the matrix specifies a set of points of a size $N$ (one point per column) and it is the caller responsibility to distinguish between functions defined on the physical and on the parametric domain. In the second case the evaluator will fetch the appropriate points depending on whether it is defined on the parametric or in the physical space. In complex situations: in particular when the physical domain of a function is the parametric domain of another the automatic mechanism is not guaranteed to work.

The result is a matrix in which the i-th column contains the requested values or derivatives at the i-th point, i.e. the point whose coordinate are the i-th column of the input matrix or of the matrix supplied by the gsTransform object. On each column the data is grouped in blocks corresponding to different functions, so that that if the requested evaluation contains s values $ v_1, \ldots, v_s $ for each pair $ (f_i, p_j) $ , $ i = 1, \ldots, S $, $ j = 1, \ldots, N $ , of (function, point) the output matrix looks like

\[ \left[ \begin{array}{ccccc} v_1(f_1,p_1) & v_1(f_1,p_2) & \ldots & v_1(f_1,p_N)\\ v_2(f_1,p_1) & v_2(f_1,p_2) & \ldots & v_2(f_1,p_N)\\ \vdots & \vdots & & \vdots\\ v_s(f_1,p_1) & v_s(f_1,p_2) & \ldots & v_s(f_1,p_N)\\ v_1(f_2,p_1) & v_1(f_2,p_2) & \ldots & v_1(f_2,p_N)\\ \vdots & \vdots & & \vdots\\ v_s(f_S,p_1) & v_s(f_S,p_2) & \ldots & v_s(f_S,p_N) \end{array} \right] \]

Any implementation of the gsFunctionSet interface must:

1) overload the virtual function int size() that returns the number of functions in the set;

2) overload the needed evaluation functions with gsMatrix argument: eval_into, deriv_into, deriv2_into; the one that are not implemented will fail at runtime printing, which function need implementing to the console

and possibly:

3) write optimized versions of div, curl, laplacian, and ect. evaluation. By default they are computed by evaluating the derivatives and applying the definition.

Template Parameters

T	type for real numbers

Inheritance diagram for gsFunctionSet< T >:

Public Types
typedef memory::shared_ptr < gsFunctionSet >	Ptr
	Shared pointer for gsFunctionSet.

typedef memory::unique_ptr < gsFunctionSet >	uPtr
	Unique pointer for gsFunctionSet.

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...

virtual void	active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
	Indices of active (non-zero) function(s) for each point. More...

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.

uPtr	clone ()
	Clone methode. Produceds a deep copy inside a uPtr.

virtual void	compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
	Computes function data. More...

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) basis at points u. More...

virtual void	deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Second derivatives. More...

virtual void	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	First derivatives. More...

virtual short_t	domainDim () const =0
	Dimension of the (source) domain. More...

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 the function(s). More...

std::vector< gsMatrix< T > >	evalAllDers (const gsMatrix< T > &u, int n) const
	Evaluate all derivatives upto order n,. More...

virtual void	evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result) const
	Evaluate the nonzero functions and their derivatives up to order n. If n is -1 then no computation is performed.

const gsFunction< T > &	function (const index_t k) const
	Helper which casts and returns the k-th piece of this function set as a gsFunction.

virtual index_t	nPieces () const
	Number of pieces in the domain of definition.

virtual const gsFunctionSet &	piece (const index_t) const
	Returns the piece(s) of the function(s) at subdomain k.

virtual std::ostream &	print (std::ostream &os) const
	Prints the object as a string.

virtual index_t	size () const
	size More...

virtual short_t	targetDim () const
	Dimension of the target space. More...

Member Function Documentation

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

inline

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

See Also: active_into()

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

virtual

Indices of active (non-zero) function(s) for each point.

The columns are sorted in increasing order, if on a point there are less active then the number of rows in the result matrix (some other point has more actives) then the rest of the column is filled with 0s.

Parameters

u
result

Reimplemented in gsHTensorBasis< d, T >, gsBasis< T >, gsBasis< Scalar >, gsBasis< real_t >, gsTensorBSplineBasis< 1, T >, gsTensorBSplineBasis< d, T >, gsTensorBSplineBasis< domainDim+1, T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsTHBSplineBasis< d, T >, gsLagrangeBasis< T >, gsMappedSingleBasis< d, T >, gsFunction< T >, gsConstantBasis< T >, gsLegendreBasis< T >, gsMonomialBasis< T >, and gsMvLegendreBasis< T >.

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

virtual

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 >.

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

Evaluate the derivatives,.

See Also: deriv_into()

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

Evaluates the second derivatives of active (i.e., non-zero) basis 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

Second derivatives.

For scalar valued functions $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}$ format is:

\[ \left[ \begin{array}{ccccc} \partial_{1}\partial_{1}f_1(p_1) & \partial_{1}\partial_{1}f_1(p_2) & \ldots & \partial_{1}\partial_{1}f_1(p_N)\\ \partial_{2}\partial_{2}f_1(p_1) & \partial_{2}\partial_{2}f_1(p_2) & \ldots & \partial_{2}\partial_{2}f_1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}\partial_{k}f_1(p_1) & \partial_{k}\partial_{k}f_1(p_2) & \ldots & \partial_{k}\partial_{k}f_1(p_N)\\ \partial_{1}\partial_{2}f_1(p_1) & \partial_{1}\partial_{2}f_1(p_2) & \ldots & \partial_{1}\partial_{2}f_1(p_N)\\ \partial_{1}\partial_{3}f_1(p_1) & \partial_{1}\partial_{3}f_1(p_2) & \ldots & \partial_{1}\partial_{3}f_1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{1}\partial_{k}f_1(p_1) & \partial_{1}\partial_{k}f_1(p_2) & \ldots & \partial_{1}\partial_{k}f_1(p_N)\\ \partial_{2}\partial_{3}f_1(p_1) & \partial_{2}\partial_{3}f_1(p_2) & \ldots & \partial_{2}\partial_{3}f_1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{2}\partial_{k}f_1(p_1) & \partial_{2}\partial_{k}f_1(p_2) & \ldots & \partial_{2}\partial_{k}f_1(p_N)\\ \partial_{3}\partial_{4}f_1(p_1) & \partial_{3}\partial_{4}f_1(p_2) & \ldots & \partial_{3}\partial_{4}f_1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k-1}\partial_{k}f_1(p_1) & \partial_{k-1}\partial_{k}f_1(p_2) & \ldots & \partial_{k-1}\partial_{k}f_1(p_N)\\ \partial_{1}\partial_{1}f_2(p_1) & \partial_{1}\partial_{1}f_2(p_2) & \ldots & \partial_{1}\partial_{1}f_2(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k-1}\partial_{k}f_S(p_1) & \partial_{k-1}\partial_{k}f_S(p_2) & \ldots & \partial_{k-1}\partial_{k}f_S(p_N)\\ \end{array} \right] \]

For vector valued functions function $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}^{m}$ the format is:

\[ \left[ \begin{array}{ccccc} \partial_{1}\partial_{1}f_1^{(1)}(p_1) & \partial_{1}\partial_{1}f_1^{(1)}(p_2) & \ldots & \partial_{1}\partial_{1}f_1^{(1)}(p_N)\\ \partial_{2}\partial_{2}f_1^{(1)}(p_1) & \partial_{2}\partial_{2}f_1^{(1)}(p_2) & \ldots & \partial_{2}\partial_{2}f_1^{(1)}(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}\partial_{k}f_1^{(1)}(p_1) & \partial_{k}\partial_{k}f_1^{(1)}(p_2) & \ldots & \partial_{k}\partial_{k}f_1^{(1)}(p_N)\\ \partial_{1}\partial_{2}f_1^{(1)}(p_1) & \partial_{1}\partial_{2}f_1^{(1)}(p_2) & \ldots & \partial_{1}\partial_{2}f_1^{(1)}(p_N)\\ \partial_{1}\partial_{3}f_1^{(1)}(p_1) & \partial_{1}\partial_{3}f_1^{(1)}(p_2) & \ldots & \partial_{1}\partial_{3}f_1^{(1)}(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k-1}\partial_{k}f_1{(1)}(p_1) & \partial_{k-1}\partial_{k}f_1^{(1)}(p_2) & \ldots & \partial_{k-1}\partial_{k}f_1^{(1)}(p_N)\\ \partial_{1}\partial_{1}f_1^{(2)}(p_1) & \partial_{1}\partial_{1}f_1^{(2)}(p_2) & \ldots & \partial_{1}\partial_{1}f_1^{(2)}(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k-1}\partial_{k}f_1^{(m)}(p_1) & \partial_{k-1}\partial_{k}f_1^{(m)}(p_2) & \ldots & \partial_{k-1}\partial_{k}f_1^{(m)}(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k-1}\partial_{k}f_S^{(m)}(p_1) & \partial_{k-1}\partial_{k}f_S^{(m)}(p_2) & \ldots & \partial_{k-1}\partial_{k}f_S^{(m)}(p_N)\\ \end{array} \right] \]

where $ f^{(i)}_j$ is the $i$-th component of function $ f_j$ of the set.

Parameters

u
result

Reimplemented in gsBasis< T >, gsBasis< Scalar >, gsBasis< real_t >, gsTensorBSplineBasis< 1, T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsGeometry< T >, gsFunctionExpr< T >, gsFunction< T >, gsLagrangeBasis< T >, gsMappedSingleBasis< d, T >, gsConstantFunction< T >, gsTHBSplineBasis< d, T >, gsPreCICEFunction< T >, gsLegendreBasis< T >, gsAffineFunction< T >, gsHBSplineBasis< d, T >, gsMappedSingleSpline< d, T >, gsMonomialBasis< T >, gsMvLegendreBasis< T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

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

virtual

First derivatives.

For scalar valued functions $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}$ format is:

\[ \left[ \begin{array}{ccccc} \partial_{1}f_1(p_1) & \partial_{1}f_1(p_2) & \ldots & \partial_{1}f_1(p_N)\\ \partial_{2}f_1(p_1) & \partial_{2}f_1(p_2) & \ldots & \partial_{2}f_1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}f_1(p_1) & \partial_{k}f_1(p_2) & \ldots & \partial_{k}f_1(p_N)\\ \partial_{1}f_2(p_1) & \partial_{1}f_2(p_2) & \ldots & \partial_{1}f_2(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}f_S(p_1) & \partial_{k}f_S(p_2) & \ldots & \partial_{k}f_S(p_N)\\ \end{array} \right] \]

For vector valued functions function $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}^{m}$ the format is:

\[ \left[ \begin{array}{ccccc} \partial_{1}f_1^1(p_1) & \partial_{1}f_1^1(p_2) & \ldots & \partial_{1}f_1^1(p_N)\\ \partial_{2}f_1^1(p_1) & \partial_{1}f_2^1(p_2) & \ldots & \partial_{1}f_2^1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}f_1^1(p_1) & \partial_{k}f_1^1(p_2) & \ldots & \partial_{k}f_1^1(p_N)\\ \partial_{1}f_1^2(p_1) & \partial_{1}f_1^2(p_2) & \ldots & \partial_{1}f_1^2(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}f_1^2(p_1) & \partial_{k}f_1^2(p_2) & \ldots & \partial_{k}f_1^2(p_N)\\ \partial_{1}f_2^1(p_1) & \partial_{1}f_2^1(p_2) & \ldots & \partial_{1}f_2^1(p_N)\\ \vdots & \vdots & & \vdots\\ \partial_{k}f_S^{(m)}(p_1) & \partial_{k}f_S^{(m)}(p_2) & \ldots & \partial_{k}f_S^{(m)}(p_N) \end{array} \right] \]

where $f^{(i)}_j$ is the $i$-th component of function $f_j$ of the set.

Parameters

u
result

Reimplemented in gsBasis< T >, gsBasis< Scalar >, gsBasis< real_t >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsTensorBSplineBasis< 1, T >, gsGeometry< T >, gsFunctionExpr< T >, gsLagrangeBasis< T >, gsMappedSingleBasis< d, T >, gsFunction< T >, gsConstantFunction< T >, gsTHBSplineBasis< d, T >, gsPreCICEFunction< T >, gsAffineFunction< T >, gsHBSplineBasis< d, T >, gsLegendreBasis< T >, gsMappedSingleSpline< d, T >, gsMvLegendreBasis< T >, gsMonomialBasis< T >, gsConstantBasis< T >, gsGeometrySlice< T >, gsSquaredDistance< T >, gsBasisFun< T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

virtual short_t domainDim ( ) const

pure virtual

Dimension of the (source) domain.

Returns: For $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$, returns $n$.

Implemented in gsTHBSplineBasis< d, T >, gsPatchIdField< T >, gsGeometry< T >, gsParamField< T >, gsNormalField< T >, gsMultiBasis< T >, gsMultiBasis< real_t >, gsMultiPatch< T >, gsMultiPatch< real_t >, gsFsiLoad< T >, gsMaterialMatrixEvalSingle< T, out >, gsFunctionExpr< T >, gsMaterialMatrixIntegrateSingle< T, out >, gsTensorBSplineBasis< 1, T >, gsJacDetField< T >, gsDetFunction< T >, gsConstantFunction< T >, gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, gsRationalBasis< gsBSplineBasis< T > >, gsRemapInterface< T >, gsGradientField< T >, gsShellStressFunction< T >, gsPiecewiseFunction< T >, gsLagrangeBasis< T >, gsCPPInterface< T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsSquaredDistance< T >, gsHBSplineBasis< d, T >, gsAffineFunction< T >, gsPreCICEFunction< T >, gsElementErrorPlotter< T >, gsMappedSingleBasis< d, T >, gsMappedSingleSpline< d, T >, gsConstantBasis< T >, gsSurface< T >, gsLegendreBasis< T >, gsBasisFun< T >, gsCurve< T >, gsAbsError< T >, gsVolume< T >, gsBulk< T >, gsGeometryTransform< T >, gsFuncCoordinate< T >, gsGeometrySlice< T >, gsCauchyStressFunction< T >, and gsMonomialBasis< T >.

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

Evaluate the function,.

See Also: eval_into()

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

virtual

Evaluates the function(s).

For scalar valued functions $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}$ format is:

\[ \left[ \begin{array}{ccccc} f_1(p_1) & f_1(p_2) & \ldots & f_1(p_N)\\ f_2(p_1) & f_2(p_2) & \ldots & f_2(p_N)\\ \vdots & \vdots & & \vdots\\ f_S(p_1) & f_S(p_2) & \ldots & f_S(p_N) \end{array} \right] \]

For vector valued functions function $f_1, \ldots, f_S$ from $\mathbb{R}^n\rightarrow\mathbb{R}^m$ the format is:

\[ \left[ \begin{array}{ccccc} f_1^1(p_1) & f_1^{(1)}(p_2) & \ldots & f_1^{(1)}(p_N)\\ f_1^2(p_1) & f_1^{(2)}(p_2) & \ldots & f_1^{(2)}(p_N)\\ \vdots & \vdots & & \vdots\\ f_1^{(m)}(p_1) & f_1^{(m)}(p_2) & \ldots & f_1^{(m)}(p_N)\\ f_2^{(1)}(p_1) & f_2^{(1)}(p_2) & \ldots & f_2^{(1)}(p_N)\\ \vdots & \vdots & & \vdots\\ f_S^{(m)}(p_1) & f_S^{(m)}(p_2) & \ldots & f_S^{(m)}(p_N) \end{array} \right] \]

where $f^{(i)}_j$ is the $i$-th component of function $f_j$ of the set.

Parameters

u
result

Reimplemented in gsBasis< T >, gsBasis< Scalar >, gsBasis< real_t >, gsTHBSplineBasis< d, T >, gsPatchIdField< T >, gsMaterialMatrixEvalSingle< T, out >, gsParamField< T >, gsTensorBSplineBasis< 1, T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsShellStressFunction< T >, gsMaterialMatrixIntegrateSingle< T, out >, gsNormalField< T >, gsFsiLoad< T >, gsFunctionExpr< T >, gsGeometry< T >, gsLagrangeBasis< T >, gsMappedSingleBasis< d, T >, gsPiecewiseFunction< T >, gsConstantFunction< T >, gsJacDetField< T >, gsDetFunction< T >, gsRemapInterface< T >, gsFunction< T >, gsGradientField< T >, gsCPPInterface< T >, gsHBSplineBasis< d, T >, gsAffineFunction< T >, gsPreCICEFunction< T >, gsLegendreBasis< T >, gsMvLegendreBasis< T >, gsMappedSingleSpline< d, T >, gsConstantBasis< T >, gsMonomialBasis< T >, gsCauchyStressFunction< T >, gsBasisFun< T >, gsGeometrySlice< T >, gsGeometryTransform< T >, gsFuncCoordinate< T >, gsAbsError< T >, gsElementErrorPlotter< T >, and gsSquaredDistance< T >.

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

Evaluate all derivatives upto order n,.

See Also: evalAllDers_into

virtual index_t size ( ) const

inlinevirtual

size

Warning: gsFunction and gsGeometry have size() == 1. This should not be confused with the size eg. of gsGeometry::basis(), which is the number of basis functions in the basis

Returns: the size of the function set: the total number of functions

Reimplemented in gsHTensorBasis< d, T >, gsFunction< T >, gsMappedSingleBasis< d, T >, gsMultiBasis< T >, gsLagrangeBasis< T >, gsMultiBasis< real_t >, gsTensorBSplineBasis< 1, T >, gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, gsRationalBasis< gsBSplineBasis< T > >, gsPiecewiseFunction< T >, gsMultiPatch< T >, gsMultiPatch< real_t >, gsLegendreBasis< T >, gsMvLegendreBasis< T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsConstantBasis< T >, gsMonomialBasis< T >, and gsPatchwiseFunction< T >.

virtual short_t targetDim ( ) const

inlinevirtual

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 >.

Detailed Description

template<typename T> class gismo::gsFunctionSet< T >

Public Types

Public Member Functions

Member Function Documentation

template<typename T>
class gismo::gsFunctionSet< T >