Detailed Description

template<class T>
class gismo::gsPeriodicOverlap< T >

A class for computing periodic parametrizations of closed (cylinder-like) surface meshes. The result will be periodic in the u-direction and the parameter domain will be [0, 1]^2. An alternative implementation is given in gsPeriodicOverlap.

This is the implementation used in the paper L. Groiss, B. Juettler, D. Mokris: 27 variants of Tutte's theorem for plane near-triangulations and an application to periodic spline surface fitting, CAGD, 2021.

Main idea

The user prescribes the u-values of the points on the lower and upper boundaries (i.e., v = 0 and v = 1, respectively). In addition, they are required to provide an overlap mesh: an edge-connected chain of triangles connecting the periodic interface on the bottom and top side of the mesh, which serves as an initial guess for the periodic interface. The triangles corresponding to the overlap are repeated, which allows ``unwrapping" the mesh as.

The equations for the inner vertices (depicted with empty circles in the figure below) are assembled by the methods of the parent class. The vertices from the overlap are set to have their u-coordinate smaller or greater by one than that of their twin, depending on whether they are on the left or the right hand side of the overlap (empty and full squares, respectively).

Inherits gsPeriodicParametrization< T >.

Public Member Functions
void	compute ()
	Computes the periodic parametrization.

gsMesh< T >	createFlatMesh () const

gsMatrix< T >	createUVmatrix ()

gsMatrix< T >	createXYZmatrix ()

	gsPeriodicOverlap (const gsMesh< T > &mesh, const gsMatrix< T > &verticesV0, const gsMatrix< T > &paramsV0, const gsMatrix< T > &verticesV1, const gsMatrix< T > &paramsV1, const gsMesh< T > &overlap, const gsOptionList &list=gsPeriodicParametrization< T >::defaultOptions())

void	restrictMatrices (gsMatrix< T > &uv, const gsMatrix< T > &xyz, real_t uMin=0, real_t uMax=1) const

void	writeTexturedMesh (std::string filename) const

Static Public Member Functions
static gsOptionList	defaultOptions ()
	Returns the list of default options for gsParametrization.

Protected Member Functions
void	calculate (const size_t paraMethod)

void	constructAndSolveEquationSystem (const Neighbourhood &neighbourhood, const size_t n, const size_t N)
	Analogous to the overloaded function from the parent class.

void	constructTwins ()
	Construct the twins.

void	constructTwinsBetween (size_t &currentNrAllVertices, std::list< size_t > vertexIndices, size_t from, size_t to, bool rightHandSide)

void	constructTwinsBetween (size_t &currentNrAllVertices, std::list< size_t > vertexIndices, gsVertexHandle from, gsVertexHandle to, bool rightHandSide)

gsMesh< T >	createExtendedFlatMesh (const std::vector< size_t > &right, const std::vector< size_t > &left) const

size_t	findTwin (size_t vertexId) const
	Finds the twin of the vertex nr. vertexId in the mesh.

const Point2D &	getParameterPoint (size_t vertexIndex) const
	Get parameter point Returns the parameter point with given vertex index. More...

void	initParameterPoints ()

void	updateLambdasWithTwins (std::vector< T > &lambdas, size_t vertexId) const

Protected Attributes
const std::vector< size_t >	m_indicesV0
	indices of the vertices with v=0

const std::vector< size_t >	m_indicesV1
	indicesV1 indices of the vertices with v=1

const gsHalfEdgeMesh< T >	m_mesh
	mesh information

const gsHalfEdgeMesh< T >	m_overlapHEM
	The mesh of the overlap, cf. the introduction to the class.

VectorType	m_parameterPoints
	parameter points

const gsMatrix< T >	m_paramsV0
	u-parameters of the vertices with v=0

const gsMatrix< T >	m_paramsV1
	u-parameters of the vertices with v=1

std::vector< std::pair< size_t, size_t > >	m_twins

Constructor & Destructor Documentation

gsPeriodicOverlap	(	const gsMesh< T > &	mesh,
		const gsMatrix< T > &	verticesV0,
		const gsMatrix< T > &	paramsV0,
		const gsMatrix< T > &	verticesV1,
		const gsMatrix< T > &	paramsV1,
		const gsMesh< T > &	overlap,
		const gsOptionList &	list = `gsPeriodicParametrization<T>::defaultOptions()`
	)

inlineexplicit

Constructor

Parameters

mesh	the surface mesh to be parametrized
verticesV0	vertices on the bottom (i.e., v = 0) boundary
paramsV0	their prescribed parameters
verticessV1	vertices on the upper (i.e., v = 1) boundary
paramsV1	their prescribed parameters
overlap	edge-connected chain of triangles connecting the bottom and top boundary and forming the first guess of the periodic interface
list	list of the method options

Member Function Documentation

void calculate ( const size_t paraMethod )

protected

Analogous to the calculate method of the parent class.

Parameters

paraMethod	which parametrization method to use
indicesV0	indices of the vertices on the bottom (i.e., v = 0) boundary
indicesV1	indices of the vertices on the upper (i.e., v = 1) boundary

void constructTwinsBetween	(	size_t &	currentNrAllVertices,
		std::list< size_t >	vertexIndices,
		size_t	from,
		size_t	to,
		bool	rightHandSide
	)

protected

Helper function to constructTwins. Constructs twins on the left or right side of the overlap.

Parameters

currentNrAllVertices	number of input vertices plus the number of twins constructed so far
vertexIndices	indices of the boundary vertices of the overlap mesh
from	the index (in the overlap mesh) of the starting vertex of the side
to	the index of the end vertex of the side
rightHandSide	flag to decide, whether we are working on the right hand side (true) or left hand side (false) of the overlap.

void constructTwinsBetween	(	size_t &	currentNrAllVertices,
		std::list< size_t >	vertexIndices,
		gsVertexHandle	from,
		gsVertexHandle	to,
		bool	rightHandSide
	)

inlineprotected

Helper function to constructTwins, preprocesses the inputs for the overloaded method by finding the indices of the vertices from and to.

gsMesh< T > createExtendedFlatMesh	(	const std::vector< size_t > &	right,
		const std::vector< size_t > &	left
	)		const

protected

Creates a flat (i.e., 2D) mesh with the overlap triangles on both sides of the parametric domain.

Parameters

left	Indices of the vertices on the left boundary of the overlap mesh (using the same indexing as the twins).
right	Indices of the vertices on the right boundary of the overlap mesh (using the same indexing as the twins).

gsMesh< T > createFlatMesh ( ) const

virtual

Creates a flat (i.e., 2D) mesh out of a periodic parametrization created by the overlap method.

Parameters

restrict If set to true, the mesh is restricted to [0, 1]^2.

Reimplemented from gsParametrization< T >.

gsMatrix< T > createUVmatrix ( )

inherited

Parametric Coordinates u,v from 0..1

Returns

gsMatrix< T > createXYZmatrix ( )

inherited

Corresponding mapped values in R3 to parametric coordinates.

Returns

const gsParametrization< T >::Point2D & getParameterPoint ( size_t vertexIndex ) const

protectedinherited

Get parameter point Returns the parameter point with given vertex index.

Parameters

[in] vertexIndex int - vertex index

Returns: two-dimensional parameter point

void initParameterPoints ( )

protectedinherited

Prepares m_parameterPoints for a computation of a periodic parametrization. Important: Neighbourhood has to be constructed before calling this function.

void restrictMatrices	(	gsMatrix< T > &	uv,
		const gsMatrix< T > &	xyz,
		real_t	uMin = `0`,
		real_t	uMax = `1`
	)		const

inherited

Moves the u-coordinates of parameters outside the interval [uMin, uMax] to inside the interval. Note: it modifies uv!

Parameters

uv	Matrix of the parameters, one column per point
xyz	Matrix of the coordinates, one column per point
uMin	minimal desired u
uMax	maximal desired u

void updateLambdasWithTwins	(	std::vector< T > &	lambdas,
		size_t	vertexId
	)		const

protected

Helper function to constructAndSolveEquationSystem with twins. Takes the coefficients computed by the standard method and shifts them to the twins.

Parameters

lambdas	the coefficients of the linear combination for the current vertex
vertex	the id of the current vertex

void writeTexturedMesh ( std::string filename ) const

inherited

Writes m_mesh into filename.vtk with the vertices coloured according to the parameters.

Parameters

filename The name of the output file (without extension).

Member Data Documentation

std::vector<std::pair<size_t, size_t> > m_twins

protected

Every twin pair is a pair of vertices with (almost) the same coordinates. The first in the pair has the u-coordinate smaller by one than the second.

Detailed Description

template<class T> class gismo::gsPeriodicOverlap< T >

Main idea

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<class T>
class gismo::gsPeriodicOverlap< T >