gsRemapInterface< T > Class Template Reference

Detailed Description

template<class T>
class gismo::gsRemapInterface< T >

Provides a mapping between the corresponding sides of two patches sharing an interface.

Have two patches \( \Omega_1 = G_1( \widehat \Omega_1 )\) and \( \Omega_2 = G_2( \widehat \Omega_2 )\) with geometry functions \( G_1 \) and \( G_2 \) and parameter domains \( \widehat \Omega_1 \) and \( \widehat \Omega_2 \).

This class represents an interface

\[ \Gamma := G_1( \widehat S_1 ) \cap G_2( \widehat S_2 ), \]

where \( \widehat S_1 \) and \( \widehat S_2 \) are sides of the corresponding parameter domains (i.e., whole edges or whole faces). The indices of the patches, as well as the corresponding sides (as boxSide) are prescribed by the incoming boundaryInterface.

The pre-images of the interface \( \Gamma \) are \( \widehat \Gamma_1 := G_1^{-1}( \Gamma ) \) and \( \widehat \Gamma_2 := G_2^{-1}( \Gamma ) \).

We say the interface is matching if

\[ \Gamma = G_1( \widehat S_1 ) = G_2( \widehat S_2 ). \]

In this case, the pre-images of the interface \( \Gamma \) are the whole sides, i.e., \( \widehat \Gamma_1 = \widehat S_1 \) and \( \widehat \Gamma_2 = \widehat S_2 \). Otherwise, the pre-images are only subsets of the sides.

We say the interface is affine if \( G_1^{-1} \circ G_2 \) or, euqivalently \( G_2^{-1} \circ G_1 \) is an affine-linear function. In this case, this class uses that affine-linear function.

If the mapping is found out not to be affine-linear, this class constructs an approximation of the interface map. This is available only if \( \Omega_1 \) and \( \Omega_2 \) are two dimensional, i.e., the interface itself is one dimensional. We refer to this as the 2D case.

Inheritance diagram for gsRemapInterface< T >:

Collaboration diagram for gsRemapInterface< T >:

Public Types
enum	{   neverAffine,   alwaysAffine }
typedef memory::shared_ptr < gsRemapInterface >	Ptr
	Shared pointer for gsRemapInterface.
typedef memory::unique_ptr < gsRemapInterface >	uPtr
	Unique pointer for gsRemapInterface.

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.
const std::vector< std::vector < T > > &	breakPoints () const
	Returns the break points used in makeDomainIterator.
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...
virtual void	computeMap (gsMapData< T > &InOut) const
	Computes map function data. More...
gsFuncCoordinate< T >	coord (const index_t c) const
	Returns the scalar function giving the i-th coordinate of this function.
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 T	distanceL2 (gsFunction< T > const &) const
	Computes the L2-distance between this function and the field and a function func.
virtual short_t	domainDim () const
	Returns parameter dimension of the domains.
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 interface map. 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.
	gsRemapInterface (const gsMultiPatch< T > &mp, const gsMultiBasis< T > &mb, const boundaryInterface &bi, const gsOptionList &opt=defaultOptions())
	Constructor. More...
const gsFunction< T >::Ptr &	interfaceMap () const
	Returns the interface map iff affine, otherwise the fitting curve. More...
virtual void	invertPoints (const gsMatrix< T > &points, gsMatrix< T > &result, const T accuracy=1e-6, const bool useInitialPoint=false) const
bool	isAffine () const
	Returns true iff the interface is affine.
bool	isMatching () const
	Returns true iff the interface is matching.
gsDomainIterator< T >::uPtr	makeDomainIterator () const
	Returns a domain iterator. More...
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.
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.
virtual std::ostream &	print (std::ostream &os) const
	Prints the state of the object.
void	recoverPoints (gsMatrix< T > &xyz, gsMatrix< T > &uv, index_t k, const T accuracy=1e-6) const
index_t	size () const
	size More...
virtual short_t	targetDim () const
	Dimension of the target space. More...
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	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate derivatives of the function \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\) at points u into result. More...
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 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	hessian_into (const gsMatrix< T > &u, gsMatrix< T > &result, index_t coord=0) 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...

Static Public Member Functions
static gsOptionList	defaultOptions ()

Private Member Functions
void	constructBreaks ()
	Constructs the breakpoints m_breakpoints.
void	constructFittingCurve (index_t numSamplePoints, index_t intervalsOfFittingCurve, index_t degreeOfFittingCurve)
	Constructs the fitting curve m_intfMap in the non-affine case.
void	constructInterfaceBox ()
	Computes the box which represents the intersection of sides of incoming patches.
T	estimateReparamError (index_t steps) const
	Estimates error between geometry1 and geometry2 after repearametrization on physical domain.

Private Attributes
const gsBasis< T > *	m_b1
	Basis on first patch.
const gsBasis< T > *	m_b2
	Basis on second patch.
boundaryInterface	m_bi
	Corresponding boundary interface.
std::vector< std::vector< T > >	m_breakpoints
	Union of breakpoints of both bases.
T	m_equalityTolerance
	Tolerance for considering points to be equal.
const gsGeometry< T > *	m_g1
	Geometry of first patch.
const gsGeometry< T > *	m_g2
	Geometry of second patch.
gsFunction< T >::Ptr	m_intfMap
	Iff affine, interface map, otherwise the fitting curve.
bool	m_isAffine
	True iff the interface is affine.
bool	m_isMatching
	True iff the interface is matching.
T	m_newtonTolerance
	Tolerance for Newton solver.
gsMatrix< T >	m_parameterBounds1
	The bounds of the box that represents \( \widehat \Gamma_1 \). More...
gsMatrix< T >	m_parameterBounds2
	The bounds of the box that represents \( \widehat \Gamma_2 \). More...

Related Functions
(Note that these are not member functions.)
template<class T >
std::ostream &	operator<< (std::ostream &os, const gsRemapInterface< T > &remapIf)
	Prints the state of the object. More...

Member Enumeration Documentation

anonymous enum

Enumerator
neverAffine	Instructs constructor not to use an affine mapping.
alwaysAffine	Instructs constructor always to use an affine mapping.

Constructor & Destructor Documentation

gsRemapInterface	(	const gsMultiPatch< T > &	mp,
		const gsMultiBasis< T > &	mb,
		const boundaryInterface &	bi,
		const gsOptionList &	opt = defaultOptions()
	)

Constructor.

Parameters

mp	The multi-patch object
mb	The multi-basis object
bi	The boundary interface (specifying the patches and their patchSide s)
opt	Options; see below:

Option	Meaning
CheckAffine	The number of interior points (per direction) in point grid used to
	check if the mapping is affine. If set to i, the grid consits of i interior
	points and the two boundary points per direction, so \( (i+2)^d \) points.
	For alwaysAffine (=0) or neverAffine (-1), no checks are performed; default: 1
NumSamplePoints	Number of sample points for computing fitting curve (if not affine); default: 11
IntervalsOfFittingCurve	Number of knot space in the fitting curve (if not affine); default: 5
DegreeOfFittingCurve	Spline degree of fitting curve (if not affine); default: 3
EqualityTolerance	Points are considered equal iff difference is smaller; default 1e-5
NewtonTolerance	Tolerance for Newton solvers (should be significantly smaller than EqualityTolerance);
	default: 1e-10

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 >.

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 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

static gsOptionList defaultOptions ( )

inlinestatic

default options

See constructor for their meaning

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

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (S*m) x N, where S=n*(n+1)/2. Each column in result corresponds to one point (i.e., one column in u) and contains the following values (for n=3, m=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 gsFunctionSet< T >.

Reimplemented in gsGeometry< T >, gsFunctionExpr< T >, gsConstantFunction< T >, gsPreCICEFunction< T >, gsAffineFunction< T >, gsMappedSingleSpline< d, T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

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

virtualinherited

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

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = 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 n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (n * m) 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 gsFunctionSet< T >.

Reimplemented in gsGeometry< T >, gsFunctionExpr< T >, gsConstantFunction< T >, gsPreCICEFunction< T >, gsAffineFunction< T >, gsMappedSingleSpline< d, T >, gsGeometrySlice< T >, gsSquaredDistance< T >, gsBasisFun< T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

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 the interface map.

Interfaces map \( \widehat \Gamma_1 \rightarrow \widehat \Gamma_2 \) that represents \( G_2^{-1} \circ G_1 \)

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

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.

const gsFunction<T>::Ptr& interfaceMap ( ) const

inline

Returns the interface map iff affine, otherwise the fitting curve.

Interfaces map \( \widehat \Gamma_1 \rightarrow \widehat \Gamma_2 \) that represents \( G_2^{-1} \circ G_1 \)

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 >.

gsDomainIterator< T >::uPtr makeDomainIterator

(

)

const

Returns a domain iterator.

The domain iterator lives on \( \widehat \Gamma_1 \). Its break points are the union of the brakpoints of the basis on \( \widehat \Omega_1 \) and the breakpoints of the basis on \( \widehat \Omega_2 \), mapped to \( \widehat \Omega_1 \).

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 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).

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 >.

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 >.

Friends And Related Function Documentation

std::ostream & operator<< ( std::ostream &  os, const gsRemapInterface< T > &  remapIf  )

related

Prints the state of the object.

Member Data Documentation

gsMatrix<T> m_parameterBounds1

private

The bounds of the box that represents \( \widehat \Gamma_1 \).

If \( \widehat \Gamma_1 = (a,b)\times (c,d) \times (e,f) \), the matrix has the structure

\[ \begin{pmatrix} a & c & e \\ b & d & f \end{pmatrix} \]

gsMatrix<T> m_parameterBounds2

private

The bounds of the box that represents \( \widehat \Gamma_2 \).

See m_parameterBounds1