gsHBSpline< d, T > Class Template Reference

Detailed Description

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

A hierarchical B-Spline function, in d dimensions.

This is the geometry type associated with gsHBSplineBasis.

Template Parameters

T	is the coefficient type

Inheritance diagram for gsHBSpline< d, T >:

Collaboration diagram for gsHBSpline< d, T >:

Public Types
typedef memory::shared_ptr < gsHBSpline >	Ptr
	Shared pointer for gsHBSpline.
typedef memory::unique_ptr < gsHBSpline >	uPtr
	Unique pointer for gsHBSpline.

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_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.
T	closestPointTo (const gsVector< T > &pt, gsVector< T > &result, const T accuracy=1e-6, const bool useInitialPoint=false) const
virtual void	computeMap (gsMapData< T > &InOut) const
	Computes map 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...
T	directedHausdorffDistance (const gsGeometry &other, const index_t nsamples=1000, const T accuracy=1e-6) const
virtual T	distanceL2 (gsFunction< T > const &) const
	Computes the L2-distance between this function and the field and a function func.
gsMatrix< T >	eval (const gsMatrix< T > &u) const
	Evaluate the function,. More...
std::vector< gsMatrix< T > >	evalAllDers (const gsMatrix< T > &u, int n) const
	Evaluate all derivatives upto order n,. More...
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.
gsGeometrySlice< T >	getIsoParametricSlice (index_t dir_fixed, T par) const
	gsHBSpline ()
	Default empty constructor.
	gsHBSpline (const Basis *basis, const gsMatrix< T > *coefs)
	Construct HB-Spline by basis functions and coefficient matrix.
	gsHBSpline (const Basis &basis, const gsMatrix< T > &coefs)
	Construct HB-Spline by basis functions and coefficient matrix.
	gsHBSpline (const gsTensorBSpline< d, T > &tbsp)
	Construct B-Spline from a Tensor B-Spline.
T	HausdorffDistance (const gsGeometry &other, const index_t nsamples=1000, const T accuracy=1e-6, const bool directed=false) const
	Computes the Hausdorff distance between *this to other.
size_t	id () const
	Returns the patch index for this patch.
virtual void	invertPoints (const gsMatrix< T > &points, gsMatrix< T > &result, const T accuracy=1e-6, const bool useInitialPoint=false) const
int	newtonRaphson (const gsVector< T > &value, gsVector< T > &arg, bool withSupport=true, const T accuracy=1e-6, int max_loop=100, T damping_factor=1) const
virtual index_t	nPieces () const
	Number of pieces in the domain of definition.
int	orientation () const
	Evaluates if the geometry orientation coincide with the ambient orientation. This is computed in the center of the parametrization and will fail to be meaningful if the geometry is singular. returns one if codimension is not zero.
virtual gsMatrix< T >	parameterCenter () const
	Returns a "central" point inside inside the parameter domain.
gsMatrix< T >	parameterCenter (const boxCorner &bc) const
	Get coordinates of the boxCorner bc in the parameter domain.
gsMatrix< T >	parameterCenter (const boxSide &bs) const
	Get coordinates of the midpoint of the boxSide bs in the parameter domain.
virtual const gsFunction &	piece (const index_t k) const
	Returns the piece(s) of the function(s) at subdomain k.
void	recoverPoints (gsMatrix< T > &xyz, gsMatrix< T > &uv, index_t k, const T accuracy=1e-6) const
void	setId (const size_t i)
	Sets the patch index for this patch.
index_t	size () const
	size More...
void	slice (index_t dir_fixed, T par, BoundaryGeometryType &result) const
Evaluation functions
re-implemented from gsFunction, see also Evaluation members
void	eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate the function at points u into result. More...
virtual void	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate derivatives of the function \(f:\mathbb{R}^d\rightarrow\mathbb{R}^n\) at points u into result. More...
virtual void	deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate second derivatives of the function at points u into result. 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.
virtual void	compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
	Computes function data. More...
Accessors
virtual const gsBasis< T > &	basis () const =0
	Returns a const reference to the basis of the geometry. More...
virtual gsBasis< T > &	basis ()=0
	Returns a reference to the basis of the geometry. More...
short_t	targetDim () const
	Dimension of the ambient physical space (overriding gsFunction::targetDim())
short_t	coefDim () const
	Dimension n of the coefficients (control points)
short_t	geoDim () const
	Dimension n of the absent physical space.
virtual short_t	domainDim () const
	Dimension d of the parameter domain (overriding gsFunction::domainDim()).
short_t	parDim () const
	Dimension d of the parameter domain (same as domainDim()).
short_t	coDim () const
	Co-dimension of the geometric object.
gsMatrix< T >	support () const
	Returns the range of parameters (same as parameterRange())
gsMatrix< T >	parameterRange () const
	Returns the range of parameters as a matrix with two columns, [lower upper].
std::vector< boxSide >	locateOn (const gsMatrix< T > &u, gsVector< bool > &onG2, gsMatrix< T > &preIm, bool lookForBoundary=false, real_t tol=1.e-6) const
	Get back the side of point u. More...
Coefficient access functions
These functions allow direct access to the coefficient matrix of the geometry.
gsMatrix< T > &	coefs ()
const gsMatrix< T > &	coefs () const
	Returns the coefficient matrix of the geometry.
gsMatrix< T >::RowXpr	coef (index_t i)
	Returns the i-th coefficient of the geometry as a row expression.
gsMatrix< T >::ConstRowXpr	coef (index_t i) const
	Returns the i-th coefficient of the geometry as a row expression.
T &	coef (index_t i, index_t j)
	Returns the j-th coordinate of the i-th coefficient of the geometry.
const T	coef (index_t i, index_t j) const
	Returns the j-th coordinate of the i-th coefficient of the geometry.
void	setCoefs (gsMatrix< T > cc)
	Set the coefficient matrix of the geometry, taking ownership of the matrix.
unsigned	coefsSize () const
	Return the number of coefficients (control points)
Transformation functions
These functions apply various linear and affine transformations to the coefficients.
void	linearTransform (const gsMatrix< T > &mat)
	Apply the given square matrix to every control point.
void	rotate (T angle, const gsVector< T, 3 > &axis)
	Apply 3D Rotation by angle radians around axis axis.
void	rotate (T angle)
	Apply 2D Rotation by angle radians.
void	scale (T s, int coord=-1)
	Apply Scaling by factor s.
void	scale (gsVector< T > const &v)
	Apply Scaling coord-wise by a vector v.
void	translate (gsVector< T > const &v)
	Apply translation by vector v.
gsMatrix< T >::RowXpr	coefAtCorner (boxCorner const &c)
	Returns the control point at corner c.
gsMatrix< T >::ConstRowXpr	coefAtCorner (boxCorner const &c) const
	Returns the control point at corner c.
Other miscellaneous functions
virtual void	uniformRefine (int numKnots=1, int mul=1, int dir=-1)
	Refine the geometry uniformly, inserting numKnots new knots into each knot span.
virtual void	uniformCoarsen (int numKnots=1)
	Coarsen the geometry uniformly, removing numKnots new knots into each knot span.
void	refineElements (std::vector< index_t > const &boxes)
	Refines the basis and adjusts the coefficients to keep the geometry the same. More...
void	unrefineElements (std::vector< index_t > const &boxes)
gsGeometry::uPtr	coord (const index_t c) const
void	embed3d ()
	Embeds coefficients in 3D.
void	embed (index_t N, bool pad_right=true)
	Embeds coefficients in N dimensions For the new dimensions zeros are added (or removed) on the right (if pad_right is true) or on the left (if pad_right is false)
short_t	degree (const short_t &i) const
	Returns the degree wrt direction i.
virtual void	insertKnot (T knot, index_t dir, index_t i=1)
	Inserts knot knot at direction dir, i times.
virtual void	degreeElevate (short_t const i=1, short_t const dir=-1)
	Elevate the degree by the given amount i for the direction dir. If dir is -1 then degree elevation is done for all directions. Uses gsBasis<T>::degreeElevate.
virtual void	degreeIncrease (short_t const i=1, short_t const dir=-1)
	Elevate the degree by the given amount i for the direction dir. If dir is -1 then degree elevation is done for all directions. Uses gsBasis<T>::degreeIncrease.
virtual void	degreeReduce (short_t const i=1, short_t const dir=-1)
	Reduces the degree by the given amount i for the direction dir. If dir is -1 then degree reduction is done for all directions. Uses gsBasis<T>::degreeReduce.
virtual void	degreeDecrease (short_t const i=1, short_t const dir=-1)
	Reduces the degree by the given amount i for the direction dir. If dir is -1 then degree reduction is done for all directions. Uses gsBasis<T>::degreeDecrease.
virtual void	hessian_into (const gsMatrix< T > &u, gsMatrix< T > &result, index_t coord) const
void	controlNet (gsMesh< T > &mesh) const
	Return the control net of the geometry.
void	outerNormal_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes the outer normals at parametric points u. More...
std::vector< gsGeometry * >	boundary () const
	Get boundary of this geometry as a vector of new gsGeometry instances.
virtual gsGeometry::uPtr	boundary (boxSide const &s) const
	Get parametrization of boundary side s as a new gsGeometry uPtr.
virtual gsGeometry::uPtr	iface (const boundaryInterface &bi, const gsGeometry &other) const
	Computes and returns the interface with other as a new geometry.
gsGeometry::uPtr	component (boxComponent const &bc) const
	Get parametrization of box component bc as a new gsGeometry uPtr.
virtual std::ostream &	print (std::ostream &os) const
	Prints the object as a string.
virtual void	merge (gsGeometry *other)
	Merge the given other geometry into this one.
virtual void	toMesh (gsMesh< T > &msh, int npoints) const
gsGeometry::uPtr	approximateLinearly () const
void	evaluateMesh (gsMesh< T > &mesh) const
virtual std::vector < gsGeometry< T > * >	uniformSplit (index_t dir=-1) const
Evaluation functions
These functions allow one to evaluate the function as well as its derivatives at one or more points in the parameter space. See also Evaluation members.
virtual void	eval_component_into (const gsMatrix< T > &u, const index_t comp, gsMatrix< T > &result) const
	Evaluate the function for component comp in the target dimension at points u into result.
virtual void	jacobian_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the Jacobian matrix.
void	div_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the divergence matrix.
gsMatrix< T >	jacobian (const gsMatrix< T > &u) const
virtual gsMatrix< T >	hessian (const gsMatrix< T > &u, index_t coord=0) const
virtual gsMatrix< T >	laplacian (const gsMatrix< T > &u) const
	Evaluate the Laplacian at points u. More...

Protected Attributes
gsBasis< T > *	m_basis
	Pointer to the basis of this geometry.
gsMatrix< T >	m_coefs
	Coefficient matrix of size coefsSize() x geoDim()
size_t	m_id

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

inlinevirtualinherited

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 from gsFunctionSet< T >.

virtual const gsBasis<T>& basis ( ) const

pure virtualinherited

Returns a const reference to the basis of the geometry.

Note

This function will return the derived concrete type of the basis.

Implemented in gsConstantFunction< T >.

virtual gsBasis<T>& basis ( )

pure virtualinherited

Returns a reference to the basis of the geometry.

Note

This function will return the derived concrete type of the basis.

Implemented in gsConstantFunction< T >.

T closestPointTo ( const gsVector< T > &  pt, gsVector< T > &  result, const T  accuracy = 1e-6, const bool  useInitialPoint = false  ) const

inherited

Returns the parameters of closest point to pt as an argument, and the Euclidean distance as a return value

gsMatrix<T>& coefs ( )

inlineinherited

Returns the coefficient matrix of the geometry Coefficient matrix of size coefsSize() x geoDim()

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 from gsFunctionSet< T >.

Reimplemented in gsConstantFunction< T >.

void computeMap ( gsMapData< T > &  InOut ) const

virtualinherited

Computes map function data.

This function evaluates the functions and their derivatives at the points InOut.points and writes them in the corresponding fields of InOut. Which field to write (and what to compute) is controlled by the InOut.flags (see also gsMapData). This is intended for parametrizations only and it works on functions sets of cardinality 1 only.

Parameters

[in,out]

InOut

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) 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

virtualinherited

Evaluate second derivatives of the function at points u into result.

Let d be the dimension of the source space ( d = domainDim() ).
Let n be the dimension of the image/target space ( n = targetDim() ).
Let N denote the number of evaluation points.

Parameters

[in]	u	gsMatrix of size d x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (S*n) x N, where S=d*(d+1)/2. Each column in result corresponds to one point (i.e., one column in u) and contains the following values (for d=3, n=3): \( (\partial_{xx} f^{(1)}, \partial_{yy} f^{(1)}, \partial_{zz} f^{(1)}, \partial_{xy} f^{(1)}, \partial_{xz} f^{(1)}, \partial_{yz} f^{(1)}, \partial_{xx} f^{(2)},\ldots,\partial_{yz} f^{(3)} )^T\)

Warning

By default uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunction< T >.

Reimplemented in gsConstantFunction< T >.

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

virtualinherited

Evaluate derivatives of the function \(f:\mathbb{R}^d\rightarrow\mathbb{R}^n\) at points u into result.

Let d be the dimension of the source space ( d = domainDim() ).
Let n be the dimension of the image/target space ( n = targetDim() ).
Let N denote the number of evaluation points.

Let \( f:\mathbb R^2 \rightarrow \mathbb R^3 \), i.e., \( f(x,y) = ( f^{(1)}(x,y), f^{(2)}(x,y), f^{(3)}(x,y) )^T\),
and let \( u = ( u_1, \ldots, u_N) = ( (x_1,y_1)^T, \ldots, (x_N, y_N)^T )\).
Then, result is of the form

\[ \left[ \begin{array}{cccc} \partial_x f^{(1)}(u_1) & \partial_x f^{(1)}(u_2) & \ldots & \partial_x f^{(1)}(u_N) \\ \partial_y f^{(1)}(u_1) & \partial_y f^{(1)}(u_2) & \ldots & \partial_y f^{(1)}(u_N) \\ \partial_x f^{(2)}(u_1) & \partial_x f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_y f^{(2)}(u_1) & \partial_y f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_x f^{(3)}(u_1) & \partial_x f^{(3)}(u_2) & \ldots & \partial_x f^{(3)}(u_N)\\ \partial_y f^{(3)}(u_1) & \partial_y f^{(3)}(u_2) & \ldots & \partial_y f^{(3)}(u_N) \end{array} \right] \]

Parameters

[in]	u	gsMatrix of size d x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (d * n) x N. Each row of result corresponds to one component in the target space and contains the gradients for each evaluation point, as row vectors, one after the other (see above for details on the format).

Warning

By default, gsFunction uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunction< T >.

Reimplemented in gsConstantFunction< T >.

T directedHausdorffDistance ( const gsGeometry< T > &  other, const index_t  nsamples = 1000, const T  accuracy = 1e-6  ) const

inherited

Computes the Hausdorff distance in a single direction from *this to other. The Hausdorff distance is computed by taking the maximum of the shortest distances between points of this and other.

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

virtualinherited

Evaluate the function at points u into result.

Let d be the dimension of the source space ( d = domainDim() ).
Let n be the dimension of the image/target space ( n = targetDim() ).
Let N denote the number of evaluation points.

Parameters

[in]	u	gsMatrix of size d x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size n x N, where each column of u represents the result of the function at the respective valuation point.

Implements gsFunction< T >.

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

inherited

Evaluate all derivatives upto order n,.

See Also

evalAllDers_into

void evaluateMesh ( gsMesh< T > &  mesh ) const

inherited

Updates the vertices of input mesh by evaluating the geometry at vertices. Vertices of the new mesh are

{ geom(v) | v vertex of input mesh }

gsGeometrySlice< T > getIsoParametricSlice ( index_t  dir_fixed, T  par  ) const

inherited

Gives back an isoParametric slice of the geometry with fixed par in direction dim_fixed as an gsGeometrySlice object.

virtual gsMatrix<T> hessian ( const gsMatrix< T > &  u, index_t  coord = 0  ) const

inlinevirtualinherited

Evaluates the Hessian (matrix of second partial derivatives) of coordinate coord at points u.

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

virtualinherited

Compute the Hessian matrix of the coordinate coord evaluated at points u

Reimplemented from gsFunction< T >.

void invertPoints ( const gsMatrix< T > &  points, gsMatrix< T > &  result, const T  accuracy = 1e-6, const bool  useInitialPoint = false  ) const

virtualinherited

Takes the physical points and computes the corresponding parameter values. If the point cannot be inverted (eg. is not part of the geometry) the corresponding parameter values will be undefined

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

virtualinherited

Evaluate the Laplacian at points u.

By default uses central finite differences with h=0.00001

Reimplemented in gsFunctionExpr< T >.

std::vector< boxSide > locateOn ( const gsMatrix< T > &  u, gsVector< bool > &  onG2, gsMatrix< T > &  preIm, bool  lookForBoundary = false, real_t  tol = 1.e-6  ) const

inherited

Get back the side of point u.

Check if points u also lie on the geometry and if required computes the if the points in u lie on one of the boundaries of the geometry

Parameters

u	Matrix of points of the form geoDim() x #points
onG2	gsVector of booleans which indicate if the point is in the domain or outside
preIm	Matrix of preimages of the points u
lookForBoundary	if required the boundaries are computed the points in u lie on

Returns

A std::vector of boxSides containing the numbers of the sides or zero if the points are in the interior If the flag lookForBoundary is not set then a vector containing anything will be returned

int newtonRaphson ( const gsVector< T > &  value, gsVector< T > &  arg, bool  withSupport = true, const T  accuracy = 1e-6, int  max_loop = 100, T  damping_factor = 1  ) const

inherited

Newton-Raphson method to find a solution of the equation f(arg) = value with starting vector arg. If the point cannot be inverted the corresponding parameter values will be undefined

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

inherited

Computes the outer normals at parametric points u.

Assumes that u is a list of points on the boundary of the geometry.

void recoverPoints ( gsMatrix< T > &  xyz, gsMatrix< T > &  uv, index_t  k, const T  accuracy = 1e-6  ) const

inherited

Recovers a point on the (geometry) together with its parameters uv, assuming that the k-th coordinate of the point xyz is not known (and has a random value as input argument).

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

inherited

Refines the basis and adjusts the coefficients to keep the geometry the same.

The syntax of boxes depends on the implementation in the underlying basis. See gsBasis::refineElements_withCoefs() for details.

index_t size ( ) const

inlinevirtualinherited

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 from gsFunctionSet< T >.

Reimplemented in gsPiecewiseFunction< T >.

void slice ( index_t  dir_fixed, T  par, BoundaryGeometryType &  result  ) const

inline

Constucts an isoparametric slice of this HBSpline by fixing par in direction dir_fixed. The resulting HBSpline has one less dimension and is given back in result.

std::vector< gsGeometry< T > * > uniformSplit ( index_t  dir = -1 ) const

virtualinherited

Splits the geometry 2^d parts, where each direction is divided into two parts in in a uniform way, i.e., in the middle of the corresponding side. This method allocated new space for each new geometry, the original one stays unchanged.

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

Member Data Documentation

size_t m_id

protectedinherited

An auxiliary index for this geometry (eg. in case it is part of a multi-patch object)