#include <CMesh3D.hpp>

Inheritance diagram for polyfem::mesh::CMesh3D:

Collaboration diagram for polyfem::mesh::CMesh3D:

Public Member Functions
	CMesh3D ()=default

virtual	~CMesh3D ()=default

	CMesh3D (CMesh3D &&)=default

CMesh3D &	operator= (CMesh3D &&)=default

	CMesh3D (const CMesh3D &)=default

CMesh3D &	operator= (const CMesh3D &)=default

bool	is_conforming () const override
	if the mesh is conforming

void	refine (const int n_refinement, const double t) override
	refine the mesh

int	n_cells () const override
	number of cells

int	n_faces () const override
	number of faces

int	n_edges () const override
	number of edges

int	n_vertices () const override
	number of vertices

int	n_face_vertices (const int f_id) const override
	number of vertices of a face

int	n_cell_vertices (const int c_id) const override
	number of vertices of a cell

int	n_cell_edges (const int c_id) const override

int	n_cell_faces (const int c_id) const override

int	cell_vertex (const int c_id, const int lv_id) const override
	id of the vertex of a cell

int	cell_face (const int c_id, const int lf_id) const override

int	cell_edge (const int c_id, const int le_id) const override

int	face_vertex (const int f_id, const int lv_id) const override
	id of the face vertex

int	edge_vertex (const int e_id, const int lv_id) const override
	id of the edge vertex

void	elements_boxes (std::vector< std::array< Eigen::Vector3d, 2 > > &boxes) const override
	constructs a box around every element (3d cell, 2d face)

void	barycentric_coords (const RowVectorNd &p, const int el_id, Eigen::MatrixXd &coord) const override
	constructs barycentric coodiantes for a point p.

bool	is_boundary_vertex (const int vertex_global_id) const override
	is vertex boundary

bool	is_boundary_edge (const int edge_global_id) const override
	is edge boundary

bool	is_boundary_face (const int face_global_id) const override
	is face boundary

bool	is_boundary_element (const int element_global_id) const override
	is cell boundary

bool	save (const std::string &path) const override

bool	build_from_matrices (const Eigen::MatrixXd &V, const Eigen::MatrixXi &F) override
	build a mesh from matrices

void	attach_higher_order_nodes (const Eigen::MatrixXd &V, const std::vector< std::vector< int > > &nodes) override
	attach high order nodes

void	normalize () override
	normalize the mesh

double	quad_area (const int gid) const override
	area of a quad face of an hex mesh

void	compute_elements_tag () override
	compute element types, see ElementType

RowVectorNd	kernel (const int cell_id) const override

RowVectorNd	point (const int vertex_id) const override
	point coordinates

void	set_point (const int global_index, const RowVectorNd &p) override
	Set the point.

RowVectorNd	edge_barycenter (const int e) const override
	edge barycenter

RowVectorNd	face_barycenter (const int f) const override
	face barycenter

RowVectorNd	cell_barycenter (const int c) const override
	cell barycenter

void	bounding_box (RowVectorNd &min, RowVectorNd &max) const override
	computes the bbox of the mesh

Navigation3D::Index	get_index_from_element (int hi, int lf, int lv) const override

Navigation3D::Index	get_index_from_element (int hi) const override

Navigation3D::Index	get_index_from_element_edge (int hi, int v0, int v1) const override

Navigation3D::Index	get_index_from_element_face (int hi, int v0, int v1, int v2) const override

std::vector< uint32_t >	vertex_neighs (const int v_gid) const override

std::vector< uint32_t >	edge_neighs (const int e_gid) const override

Navigation3D::Index	switch_vertex (Navigation3D::Index idx) const override

Navigation3D::Index	switch_edge (Navigation3D::Index idx) const override

Navigation3D::Index	switch_face (Navigation3D::Index idx) const override

Navigation3D::Index	switch_element (Navigation3D::Index idx) const override

Navigation3D::Index	next_around_edge (Navigation3D::Index idx) const override

Navigation3D::Index	next_around_face (Navigation3D::Index idx) const override

void	get_vertex_elements_neighs (const int v_id, std::vector< int > &ids) const override

void	get_edge_elements_neighs (const int e_id, std::vector< int > &ids) const override

void	compute_boundary_ids (const double eps) override
	computes the selection based on the bbx of the mesh.

void	compute_boundary_ids (const std::function< int(const RowVectorNd &)> &marker) override
	computes boundary selections based on a function

void	compute_boundary_ids (const std::function< int(const RowVectorNd &, bool)> &marker) override
	computes boundary selections based on a function

void	compute_boundary_ids (const std::function< int(const size_t, const RowVectorNd &, bool)> &marker) override
	computes boundary selections based on a function

void	compute_boundary_ids (const std::function< int(const std::vector< int > &, bool)> &marker) override
	computes boundary selections based on a function

void	compute_body_ids (const std::function< int(const size_t, const RowVectorNd &)> &marker) override
	computes boundary selections based on a function

void	compute_boundary_ids (const std::function< int(const size_t, const std::vector< int > &, const RowVectorNd &, bool)> &marker) override
	computes boundary selections based on a function

void	triangulate_faces (Eigen::MatrixXi &tris, Eigen::MatrixXd &pts, std::vector< int > &ranges) const override
	generate a triangular representation of every face

Mesh3DStorage &	mesh_storge ()

void	append (const Mesh &mesh) override
	appends a new mesh to the end of this

std::unique_ptr< Mesh >	copy () const override
	Create a copy of the mesh.

Public Member Functions inherited from polyfem::mesh::Mesh3D
	Mesh3D ()=default

virtual	~Mesh3D ()=default

	Mesh3D (Mesh3D &&)=default

Mesh3D &	operator= (Mesh3D &&)=default

	Mesh3D (const Mesh3D &)=default

Mesh3D &	operator= (const Mesh3D &)=default

bool	is_volume () const override
	checks if mesh is volume

void	elements_boxes (std::vector< std::array< Eigen::Vector3d, 2 > > &boxes) const override
	constructs a box around every element (3d cell, 2d face)

void	barycentric_coords (const RowVectorNd &p, const int el_id, Eigen::MatrixXd &coord) const override
	constructs barycentric coodiantes for a point p.

void	compute_cell_jacobian (const int el_id, const Eigen::MatrixXd &reference_map, Eigen::MatrixXd &jacobian) const

double	tri_area (const int gid) const override
	area of a tri face of a tet mesh

RowVectorNd	edge_node (const Navigation3D::Index &index, const int n_new_nodes, const int i) const

RowVectorNd	face_node (const Navigation3D::Index &index, const int n_new_nodes, const int i, const int j) const

RowVectorNd	cell_node (const Navigation3D::Index &index, const int n_new_nodes, const int i, const int j, const int k) const

void	get_edges (Eigen::MatrixXd &p0, Eigen::MatrixXd &p1) const override
	Get all the edges.

void	get_edges (Eigen::MatrixXd &p0, Eigen::MatrixXd &p1, const std::vector< bool > &valid_elements) const override
	Get all the edges according to valid_elements selection.

void	to_face_functions (std::array< std::function< Navigation3D::Index(Navigation3D::Index)>, 6 > &to_face) const

void	to_vertex_functions (std::array< std::function< Navigation3D::Index(Navigation3D::Index)>, 8 > &to_vertex) const

void	to_edge_functions (std::array< std::function< Navigation3D::Index(Navigation3D::Index)>, 12 > &to_edge) const

std::array< int, 8 >	get_ordered_vertices_from_hex (const int element_index) const

virtual std::array< int, 4 >	get_ordered_vertices_from_tet (const int element_index) const

void	compute_element_barycenters (Eigen::MatrixXd &barycenters) const override
	utility for 2d/3d.

Public Member Functions inherited from polyfem::mesh::Mesh
virtual	~Mesh ()=default
	Destroy the Mesh object.

	Mesh (Mesh &&)=default
	Construct a new Mesh object.

Mesh &	operator= (Mesh &&)=default
	Copy constructor.

	Mesh (const Mesh &)=default
	Construct a new Mesh object.

Mesh &	operator= (const Mesh &)=default

int	dimension () const
	utily for dimension

int	n_elements () const
	utitlity to return the number of elements, cells or faces in 3d and 2d

int	n_boundary_elements () const
	utitlity to return the number of boundary elements, faces or edges in 3d and 2d

int	element_vertex (const int el_id, const int lv_id) const
	id of the vertex of a element

int	boundary_element_vertex (const int primitive_id, const int lv_id) const

const Eigen::MatrixXi &	orders () const
	order of each element

bool	is_rational () const
	check if curved mesh has rational polynomials elements

void	set_is_rational (const bool in_is_rational)
	Set the is rational object.

virtual void	update_elements_tag ()
	Update elements types.

virtual double	edge_length (const int gid) const
	edge length

void	edge_barycenters (Eigen::MatrixXd &barycenters) const
	all edges barycenters

void	face_barycenters (Eigen::MatrixXd &barycenters) const
	all face barycenters

void	cell_barycenters (Eigen::MatrixXd &barycenters) const
	all cells barycenters

bool	is_spline_compatible (const int el_id) const
	checks if element is spline compatible

bool	is_cube (const int el_id) const
	checks if element is cube compatible

bool	is_polytope (const int el_id) const
	checks if element is polygon compatible

bool	is_simplex (const int el_id) const
	checks if element is simples compatible

const std::vector< ElementType > &	elements_tag () const
	Returns the elements types.

void	set_tag (const int el, const ElementType type)
	changes the element type

void	compute_node_ids (const std::function< int(const size_t, const RowVectorNd &, bool)> &marker)
	computes boundary selections based on a function

virtual void	load_boundary_ids (const std::string &path)
	loads the boundary selections for a file

virtual void	set_boundary_ids (const std::vector< int > &boundary_ids)
	Set the boundary selection from a vector.

virtual void	set_body_ids (const std::vector< int > &body_ids)
	Set the volume sections.

virtual int	get_default_boundary_id (const int primitive) const
	Get the default boundary selection of an element (face in 3d, edge in 2d)

virtual int	get_boundary_id (const int primitive) const
	Get the boundary selection of an element (face in 3d, edge in 2d)

virtual int	get_node_id (const int node_id) const
	Get the boundary selection of a node.

void	update_nodes (const Eigen::VectorXi &in_node_to_node)
	Update the node ids to reorder them.

virtual int	get_body_id (const int primitive) const
	Get the volume selection of an element (cell in 3d, face in 2d)

virtual const std::vector< int > &	get_body_ids () const
	Get the volume selection of all elements (cells in 3d, faces in 2d)

bool	has_node_ids () const
	checks if points selections are available

bool	has_boundary_ids () const
	checks if surface selections are available

virtual bool	has_body_ids () const
	checks if volumes selections are available

const std::vector< double > &	cell_weights (const int cell_index) const
	weights for rational polynomial meshes

void	set_cell_weights (const std::vector< std::vector< double > > &in_cell_weights)
	Set the cell weights for rational polynomial meshes.

virtual void	prepare_mesh ()
	method used to finalize the mesh.

bool	has_poly () const
	checks if the mesh has polytopes

bool	is_simplicial () const
	checks if the mesh is simplicial

bool	is_linear () const
	check if the mesh is linear

std::vector< std::pair< int, int > >	edges () const
	list of sorted edges.

std::vector< std::vector< int > >	faces () const
	list of sorted faces.

std::unordered_map< std::pair< int, int >, size_t, polyfem::utils::HashPair >	edges_to_ids () const
	map from edge (pair of v id) to the id of the edge

std::unordered_map< std::vector< int >, size_t, polyfem::utils::HashVector >	faces_to_ids () const
	map from face (tuple of v id) to the id of the face

const Eigen::VectorXi &	in_ordered_vertices () const
	Order of the input vertices.

const Eigen::MatrixXi &	in_ordered_edges () const
	Order of the input edges.

const Eigen::MatrixXi &	in_ordered_faces () const
	Order of the input edges.

void	append (const std::unique_ptr< Mesh > &mesh)
	appends a new mesh to the end of this, utility that takes pointer, calls other one

void	apply_affine_transformation (const MatrixNd &A, const VectorNd &b)
	Apply an affine transformation \(Ax+b\) to the vertex positions \(x\).

Static Public Member Functions
static void	geomesh_2_mesh_storage (const GEO::Mesh &gm, Mesh3DStorage &m)

Static Public Member Functions inherited from polyfem::mesh::Mesh
static std::unique_ptr< Mesh >	create (const std::string &path, const bool non_conforming=false)
	factory to build the proper mesh

static std::unique_ptr< Mesh >	create (GEO::Mesh &M, const bool non_conforming=false)
	factory to build the proper mesh

static std::unique_ptr< Mesh >	create (const Eigen::MatrixXd &vertices, const Eigen::MatrixXi &cells, const bool non_conforming=false)
	factory to build the proper mesh

static std::unique_ptr< Mesh >	create (const int dim, const bool non_conforming=false)
	factory to build the proper empty mesh

Protected Member Functions
bool	load (const std::string &path) override
	loads a mesh from the path

bool	load (const GEO::Mesh &M) override
	loads a mesh from a geo mesh

Protected Member Functions inherited from polyfem::mesh::Mesh
	Mesh ()=default
	Construct a new Mesh object.

Private Attributes
Mesh3DStorage	mesh_

Additional Inherited Members
Protected Attributes inherited from polyfem::mesh::Mesh
std::vector< ElementType >	elements_tag_
	list of element types

std::vector< int >	node_ids_
	list of node labels

std::vector< int >	boundary_ids_
	list of surface labels

std::vector< int >	body_ids_
	list of volume labels

Eigen::MatrixXi	orders_
	list of geometry orders, one per cell

bool	is_rational_ = false
	stores if the mesh is rational

std::vector< EdgeNodes >	edge_nodes_
	high-order nodes associates to edges

std::vector< FaceNodes >	face_nodes_
	high-order nodes associates to faces

std::vector< CellNodes >	cell_nodes_
	high-order nodes associates to cells

std::vector< std::vector< double > >	cell_weights_
	weights associates to cells for rational polynomail meshes

Eigen::VectorXi	in_ordered_vertices_
	Order of the input vertices.

Eigen::MatrixXi	in_ordered_edges_
	Order of the input edges.

Eigen::MatrixXi	in_ordered_faces_
	Order of the input faces, TODO: change to std::vector of Eigen::Vector.

Detailed Description

Definition at line 18 of file CMesh3D.hpp.

Constructor & Destructor Documentation

◆ CMesh3D() [1/3]

polyfem::mesh::CMesh3D::CMesh3D ( )

default

◆ ~CMesh3D()

virtual polyfem::mesh::CMesh3D::~CMesh3D ( )

virtualdefault

◆ CMesh3D() [2/3]

polyfem::mesh::CMesh3D::CMesh3D ( CMesh3D && )

default

◆ CMesh3D() [3/3]

polyfem::mesh::CMesh3D::CMesh3D ( const CMesh3D & )

default

Member Function Documentation

◆ append()

void polyfem::mesh::CMesh3D::append ( const Mesh & mesh )

overridevirtual

appends a new mesh to the end of this

Parameters

[in] mesh to append

Reimplemented from polyfem::mesh::Mesh.

Definition at line 1676 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::append(), polyfem::mesh::Mesh3DStorage::append(), compute_elements_tag(), mesh_, and polyfem::mesh::Navigation3D::prepare_mesh().

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◆ attach_higher_order_nodes()

void polyfem::mesh::CMesh3D::attach_higher_order_nodes	(	const Eigen::MatrixXd &	V,
		const std::vector< std::vector< int > > &	nodes
	)

overridevirtual

attach high order nodes

Parameters

[in]	V	nodes
[in]	nodes	list of nodes per element

Implements polyfem::mesh::Mesh.

Definition at line 512 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::cell_nodes_, cell_vertex(), polyfem::mesh::Navigation3D::Index::edge, polyfem::mesh::Mesh::edge_nodes_, polyfem::mesh::Navigation3D::Index::element, polyfem::mesh::Navigation3D::Index::face, polyfem::mesh::Mesh::face_nodes_, face_vertex(), polyfem::mesh::Mesh::faces(), get_index_from_element(), n_cells(), n_edges(), n_faces(), next_around_face(), polyfem::mesh::Mesh::orders_, switch_edge(), switch_face(), switch_vertex(), V, and polyfem::mesh::Navigation3D::Index::vertex.

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◆ barycentric_coords()

void polyfem::mesh::CMesh3D::barycentric_coords	(	const RowVectorNd &	p,
		const int	el_id,
		Eigen::MatrixXd &	coord
	)		const

overridevirtual

constructs barycentric coodiantes for a point p.

WARNING works only for simplices

Parameters

[in]	p	query point
[in]	el_id	element id
[out]	coord	matrix containing the barycentric coodinates

Implements polyfem::mesh::Mesh.

Definition at line 1662 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3D::get_ordered_vertices_from_tet(), polyfem::mesh::Mesh::is_simplex(), and point().

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◆ bounding_box()

void polyfem::mesh::CMesh3D::bounding_box	(	RowVectorNd &	min,
		RowVectorNd &	max
	)		const

overridevirtual

computes the bbox of the mesh

Parameters

[out]	min	min coodiante
[out]	max	max coodiante

Implements polyfem::mesh::Mesh.

Definition at line 1004 of file CMesh3D.cpp.

References mesh_, polyfem::mesh::Mesh3DStorage::points, and V.

Referenced by compute_boundary_ids(), and normalize().

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◆ build_from_matrices()

bool polyfem::mesh::CMesh3D::build_from_matrices	(	const Eigen::MatrixXd &	V,
		const Eigen::MatrixXi &	F
	)

overridevirtual

build a mesh from matrices

Parameters

[in]	V	vertices
[in]	F	connectivity

Returns: if success

Implements polyfem::mesh::Mesh.

Definition at line 469 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::cell_nodes_, polyfem::mesh::Mesh::edge_nodes_, polyfem::F, polyfem::mesh::Mesh::face_nodes_, load(), and V.

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◆ cell_barycenter()

RowVectorNd polyfem::mesh::CMesh3D::cell_barycenter ( const int c ) const

overridevirtual

cell barycenter

Parameters

[in] c global cell index

Returns: cell barycenter

Implements polyfem::mesh::Mesh.

Definition at line 1585 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::elements, mesh_, n_cell_vertices(), n_vertices(), and point().

Referenced by compute_body_ids(), and load().

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◆ cell_edge()

int polyfem::mesh::CMesh3D::cell_edge	(	const int	c_id,
		const int	le_id
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 43 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

◆ cell_face()

int polyfem::mesh::CMesh3D::cell_face	(	const int	c_id,
		const int	lf_id
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 42 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

◆ cell_vertex()

int polyfem::mesh::CMesh3D::cell_vertex	(	const int	f_id,
		const int	lv_id
	)		const

inlineoverridevirtual

id of the vertex of a cell

Parameters

[in]	f_id	global cell id
[in]	lv_id	local vertex id

Returns: vertex id

Implements polyfem::mesh::Mesh.

Definition at line 41 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

Referenced by attach_higher_order_nodes(), and elements_boxes().

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◆ compute_body_ids()

void polyfem::mesh::CMesh3D::compute_body_ids ( const std::function< int(const size_t, const RowVectorNd &)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the id and barycenter and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1305 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::body_ids_, cell_barycenter(), and polyfem::mesh::Mesh::n_elements().

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◆ compute_boundary_ids() [1/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const double eps )

overridevirtual

computes the selection based on the bbx of the mesh.

Left gets 1, bottom 2, right 3, top 4, front 5, back 6

Parameters

[in] eps tolerance for proximity

Implements polyfem::mesh::Mesh.

Definition at line 1272 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, bounding_box(), face_barycenter(), is_boundary_face(), and n_faces().

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◆ compute_boundary_ids() [2/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const std::function< int(const RowVectorNd &)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the barycenter and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1199 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, face_barycenter(), is_boundary_face(), and n_faces().

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◆ compute_boundary_ids() [3/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const std::function< int(const RowVectorNd &, bool)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the barycenter and true/false if the element is on the boundary and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1214 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, face_barycenter(), is_boundary_face(), and n_faces().

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◆ compute_boundary_ids() [4/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const std::function< int(const size_t, const RowVectorNd &, bool)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the id, barycenter, and true/false if the element is on the boundary and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1226 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, face_barycenter(), is_boundary_face(), and n_faces().

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◆ compute_boundary_ids() [5/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const std::function< int(const size_t, const std::vector< int > &, const RowVectorNd &, bool)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the id, the list of vertices, the barycenter, and true/false if the element is on the boundary and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1254 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, face_barycenter(), face_vertex(), is_boundary_face(), n_face_vertices(), and n_faces().

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◆ compute_boundary_ids() [6/6]

void polyfem::mesh::CMesh3D::compute_boundary_ids ( const std::function< int(const std::vector< int > &, bool)> & marker )

overridevirtual

computes boundary selections based on a function

Parameters

[in] marker lambda function that takes the list of vertices and true/false if the element is on the boundary and returns an integer

Implements polyfem::mesh::Mesh.

Definition at line 1238 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::boundary_ids_, face_vertex(), is_boundary_face(), n_face_vertices(), and n_faces().

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◆ compute_elements_tag()

void polyfem::mesh::CMesh3D::compute_elements_tag ( )

overridevirtual

compute element types, see ElementType

Implements polyfem::mesh::Mesh.

Definition at line 1341 of file CMesh3D.cpp.

References polyfem::mesh::BOUNDARY_POLYTOPE, polyfem::mesh::Mesh3DStorage::edges, polyfem::mesh::Mesh3DStorage::elements, polyfem::mesh::Mesh::elements_tag_, polyfem::mesh::Mesh3DStorage::faces, polyfem::mesh::INTERFACE_CUBE, polyfem::mesh::INTERIOR_POLYTOPE, mesh_, polyfem::mesh::MULTI_SINGULAR_BOUNDARY_CUBE, polyfem::mesh::MULTI_SINGULAR_INTERIOR_CUBE, polyfem::mesh::REGULAR_BOUNDARY_CUBE, polyfem::mesh::REGULAR_INTERIOR_CUBE, polyfem::mesh::SIMPLE_SINGULAR_BOUNDARY_CUBE, polyfem::mesh::SIMPLE_SINGULAR_INTERIOR_CUBE, polyfem::mesh::SIMPLEX, and polyfem::mesh::Mesh3DStorage::vertices.

Referenced by append(), load(), load(), and refine().

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◆ copy()

std::unique_ptr< Mesh > polyfem::mesh::CMesh3D::copy ( ) const

overridevirtual

Create a copy of the mesh.

Returns: pointer to the new copy mesh

Implements polyfem::mesh::Mesh.

Definition at line 1688 of file CMesh3D.cpp.

◆ edge_barycenter()

RowVectorNd polyfem::mesh::CMesh3D::edge_barycenter ( const int e ) const

overridevirtual

edge barycenter

Parameters

[in] e global edge index

Returns: edge barycenter

Implements polyfem::mesh::Mesh.

Definition at line 1564 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::edges, mesh_, and point().

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◆ edge_neighs()

std::vector< uint32_t > polyfem::mesh::CMesh3D::edge_neighs ( const int e_gid ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 84 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::edges, and mesh_.

◆ edge_vertex()

int polyfem::mesh::CMesh3D::edge_vertex	(	const int	e_id,
		const int	lv_id
	)		const

inlineoverridevirtual

id of the edge vertex

Parameters

[in]	e_id	global edge id
[in]	lv_id	local vertex index

Returns: int id of the face vertex

Implements polyfem::mesh::Mesh.

Definition at line 45 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::edges, and mesh_.

◆ elements_boxes()

void polyfem::mesh::CMesh3D::elements_boxes ( std::vector< std::array< Eigen::Vector3d, 2 > > & boxes ) const

overridevirtual

constructs a box around every element (3d cell, 2d face)

Parameters

[out] boxes axis aligned bounding boxes

Implements polyfem::mesh::Mesh.

Definition at line 1640 of file CMesh3D.cpp.

References cell_vertex(), n_cell_vertices(), polyfem::mesh::Mesh::n_elements(), and point().

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◆ face_barycenter()

RowVectorNd polyfem::mesh::CMesh3D::face_barycenter ( const int f ) const

overridevirtual

face barycenter

Parameters

[in] f global face index

Returns: face barycenter

Implements polyfem::mesh::Mesh.

Definition at line 1571 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::faces, mesh_, n_face_vertices(), n_vertices(), and point().

Referenced by compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), and compute_boundary_ids().

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◆ face_vertex()

int polyfem::mesh::CMesh3D::face_vertex	(	const int	f_id,
		const int	lv_id
	)		const

inlineoverridevirtual

id of the face vertex

Parameters

[in]	f_id	global face id
[in]	lv_id	local vertex index

Returns: int id of the face vertex

Implements polyfem::mesh::Mesh.

Definition at line 44 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::faces, and mesh_.

Referenced by attach_higher_order_nodes(), compute_boundary_ids(), and compute_boundary_ids().

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◆ geomesh_2_mesh_storage()

void polyfem::mesh::CMesh3D::geomesh_2_mesh_storage	(	const GEO::Mesh &	gm,
		Mesh3DStorage &	m
	)

static

Definition at line 1599 of file CMesh3D.cpp.

References polyfem::mesh::MeshProcessing3D::build_connectivity(), polyfem::mesh::Mesh3DStorage::edges, polyfem::mesh::Mesh3DStorage::faces, polyfem::mesh::H_SUR, polyfem::mesh::Vertex::id, polyfem::mesh::Mesh3DStorage::points, polyfem::mesh::QUA, polyfem::mesh::TRI, polyfem::mesh::Mesh3DStorage::type, polyfem::mesh::Vertex::v, and polyfem::mesh::Mesh3DStorage::vertices.

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◆ get_edge_elements_neighs()

void polyfem::mesh::CMesh3D::get_edge_elements_neighs	(	const int	e_id,
		std::vector< int > &	ids
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 101 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::edges, and mesh_.

◆ get_index_from_element() [1/2]

Navigation3D::Index polyfem::mesh::CMesh3D::get_index_from_element ( int hi ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 78 of file CMesh3D.hpp.

References polyfem::mesh::Navigation3D::get_index_from_element_face(), and mesh_.

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◆ get_index_from_element() [2/2]

Navigation3D::Index polyfem::mesh::CMesh3D::get_index_from_element	(	int	hi,
		int	lf,
		int	lv
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 77 of file CMesh3D.hpp.

References polyfem::mesh::Navigation3D::get_index_from_element_face(), and mesh_.

Referenced by attach_higher_order_nodes().

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◆ get_index_from_element_edge()

Navigation3D::Index polyfem::mesh::CMesh3D::get_index_from_element_edge	(	int	hi,
		int	v0,
		int	v1
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 80 of file CMesh3D.hpp.

References polyfem::mesh::Navigation3D::get_index_from_element_edge(), and mesh_.

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◆ get_index_from_element_face()

Navigation3D::Index polyfem::mesh::CMesh3D::get_index_from_element_face	(	int	hi,
		int	v0,
		int	v1,
		int	v2
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 81 of file CMesh3D.hpp.

References polyfem::mesh::Navigation3D::get_index_from_element_tri(), and mesh_.

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◆ get_vertex_elements_neighs()

void polyfem::mesh::CMesh3D::get_vertex_elements_neighs	(	const int	v_id,
		std::vector< int > &	ids
	)		const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 96 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Mesh3DStorage::vertices.

◆ is_boundary_edge()

bool polyfem::mesh::CMesh3D::is_boundary_edge ( const int edge_global_id ) const

inlineoverridevirtual

is edge boundary

Parameters

[in] edge_global_id global edge id

Returns: is edge boundary

Implements polyfem::mesh::Mesh.

Definition at line 51 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::edges, and mesh_.

◆ is_boundary_element()

bool polyfem::mesh::CMesh3D::is_boundary_element ( const int element_global_id ) const

overridevirtual

is cell boundary

Parameters

[in] element_global_id global cell id

Returns: is cell boundary

Implements polyfem::mesh::Mesh.

Definition at line 1178 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::elements, is_boundary_face(), is_boundary_vertex(), and mesh_.

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◆ is_boundary_face()

bool polyfem::mesh::CMesh3D::is_boundary_face ( const int face_global_id ) const

inlineoverridevirtual

is face boundary

Parameters

[in] face_global_id global face id

Returns: is face boundary

Implements polyfem::mesh::Mesh.

Definition at line 52 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::faces, and mesh_.

Referenced by compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), and is_boundary_element().

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◆ is_boundary_vertex()

bool polyfem::mesh::CMesh3D::is_boundary_vertex ( const int vertex_global_id ) const

inlineoverridevirtual

is vertex boundary

Parameters

[in] vertex_global_id global vertex id

Returns: is vertex boundary

Implements polyfem::mesh::Mesh.

Definition at line 50 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Mesh3DStorage::vertices.

Referenced by is_boundary_element().

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◆ is_conforming()

bool polyfem::mesh::CMesh3D::is_conforming ( ) const

inlineoverridevirtual

if the mesh is conforming

Returns: if the mesh is conforming

Implements polyfem::mesh::Mesh.

Definition at line 28 of file CMesh3D.hpp.

◆ kernel()

RowVectorNd polyfem::mesh::CMesh3D::kernel ( const int cell_id ) const

overridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 1334 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

◆ load() [1/2]

bool polyfem::mesh::CMesh3D::load ( const GEO::Mesh & M )

overrideprotectedvirtual

loads a mesh from a geo mesh

Parameters

[in] M geo mesh

Returns: if success

Implements polyfem::mesh::Mesh.

Definition at line 204 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::cell_nodes_, compute_elements_tag(), polyfem::mesh::Mesh::edge_nodes_, polyfem::mesh::Mesh3DStorage::elements, polyfem::mesh::Mesh::face_nodes_, polyfem::mesh::Mesh3DStorage::faces, polyfem::mesh::Element::fs, polyfem::mesh::Element::fs_flag, polyfem::mesh::Element::hex, polyfem::mesh::HEX, polyfem::mesh::HYB, polyfem::mesh::Vertex::id, polyfem::mesh::Face::id, polyfem::mesh::Element::id, mesh_, polyfem::mesh::Mesh3DStorage::points, polyfem::mesh::Navigation3D::prepare_mesh(), polyfem::mesh::TET, polyfem::mesh::Mesh3DStorage::type, polyfem::mesh::Element::v_in_Kernel, polyfem::mesh::Mesh3DStorage::vertices, polyfem::mesh::Face::vs, and polyfem::mesh::Element::vs.

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◆ load() [2/2]

bool polyfem::mesh::CMesh3D::load ( const std::string & path )

overrideprotectedvirtual

loads a mesh from the path

Parameters

[in] path file location

Returns: if success

Implements polyfem::mesh::Mesh.

Definition at line 80 of file CMesh3D.cpp.

References cell_barycenter(), polyfem::mesh::Mesh::cell_nodes_, compute_elements_tag(), polyfem::mesh::Mesh::edge_nodes_, polyfem::mesh::Mesh3DStorage::elements, polyfem::utils::StringUtils::endswith(), polyfem::mesh::Mesh::face_nodes_, polyfem::mesh::Mesh3DStorage::faces, polyfem::mesh::find(), polyfem::mesh::Element::fs, polyfem::mesh::Element::fs_flag, polyfem::mesh::Vertex::id, polyfem::mesh::Face::id, polyfem::mesh::Element::id, load(), mesh_, n_cells(), polyfem::mesh::Mesh3DStorage::points, polyfem::mesh::Navigation3D::prepare_mesh(), polyfem::mesh::Mesh3DStorage::vertices, polyfem::mesh::Face::vs, polyfem::mesh::Element::vs, x, y, and z.

Referenced by build_from_matrices(), and load().

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◆ mesh_storge()

Mesh3DStorage & polyfem::mesh::CMesh3D::mesh_storge ( )

inline

Definition at line 118 of file CMesh3D.hpp.

References mesh_.

◆ n_cell_edges()

int polyfem::mesh::CMesh3D::n_cell_edges ( const int c_id ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 39 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

◆ n_cell_faces()

int polyfem::mesh::CMesh3D::n_cell_faces ( const int c_id ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 40 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

◆ n_cell_vertices()

int polyfem::mesh::CMesh3D::n_cell_vertices ( const int c_id ) const

inlineoverridevirtual

number of vertices of a cell

Parameters

[in] c_id global cell id (face for 2d meshes)

Returns: number of vertices

Implements polyfem::mesh::Mesh.

Definition at line 38 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

Referenced by cell_barycenter(), and elements_boxes().

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◆ n_cells()

int polyfem::mesh::CMesh3D::n_cells ( ) const

inlineoverridevirtual

number of cells

Returns: number of cells

Implements polyfem::mesh::Mesh.

Definition at line 32 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::elements, and mesh_.

Referenced by attach_higher_order_nodes(), load(), and normalize().

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◆ n_edges()

int polyfem::mesh::CMesh3D::n_edges ( ) const

inlineoverridevirtual

number of edges

Returns: number of edges

Implements polyfem::mesh::Mesh.

Definition at line 34 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::edges, and mesh_.

Referenced by attach_higher_order_nodes().

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◆ n_face_vertices()

int polyfem::mesh::CMesh3D::n_face_vertices ( const int f_id ) const

inlineoverridevirtual

number of vertices of a face

Parameters

[in] f_id global face id

Returns: number of vertices

Implements polyfem::mesh::Mesh.

Definition at line 37 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::faces, and mesh_.

Referenced by compute_boundary_ids(), compute_boundary_ids(), face_barycenter(), quad_area(), and triangulate_faces().

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◆ n_faces()

int polyfem::mesh::CMesh3D::n_faces ( ) const

inlineoverridevirtual

number of faces

Returns: number of faces

Implements polyfem::mesh::Mesh.

Definition at line 33 of file CMesh3D.hpp.

References polyfem::mesh::Mesh3DStorage::faces, and mesh_.

Referenced by attach_higher_order_nodes(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), compute_boundary_ids(), and triangulate_faces().

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◆ n_vertices()

int polyfem::mesh::CMesh3D::n_vertices ( ) const

inlineoverridevirtual

number of vertices

Returns: number of vertices

Implements polyfem::mesh::Mesh.

Definition at line 35 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Mesh3DStorage::points.

Referenced by cell_barycenter(), face_barycenter(), quad_area(), refine(), and triangulate_faces().

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◆ next_around_edge()

Navigation3D::Index polyfem::mesh::CMesh3D::next_around_edge ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 93 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::next_around_3Dedge().

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◆ next_around_face()

Navigation3D::Index polyfem::mesh::CMesh3D::next_around_face ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 94 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::next_around_2Dface().

Referenced by attach_higher_order_nodes().

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◆ normalize()

void polyfem::mesh::CMesh3D::normalize ( )

overridevirtual

normalize the mesh

Implements polyfem::mesh::Mesh.

Definition at line 1011 of file CMesh3D.cpp.

References bounding_box(), polyfem::mesh::Mesh::cell_nodes_, polyfem::mesh::Mesh::edge_nodes_, polyfem::mesh::Mesh3DStorage::elements, polyfem::mesh::Mesh::face_nodes_, polyfem::mesh::Mesh3D::get_edges(), polyfem::logger(), mesh_, n_cells(), polyfem::mesh::Mesh3DStorage::points, V, and val.

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◆ operator=() [1/2]

CMesh3D & polyfem::mesh::CMesh3D::operator= ( CMesh3D && )

default

◆ operator=() [2/2]

CMesh3D & polyfem::mesh::CMesh3D::operator= ( const CMesh3D & )

default

◆ point()

RowVectorNd polyfem::mesh::CMesh3D::point ( const int global_index ) const

overridevirtual

point coordinates

Parameters

[in] global_index global vertex index

Returns: RowVectorNd

Implements polyfem::mesh::Mesh.

Definition at line 1328 of file CMesh3D.cpp.

References mesh_, and polyfem::mesh::Mesh3DStorage::points.

Referenced by barycentric_coords(), cell_barycenter(), edge_barycenter(), elements_boxes(), face_barycenter(), quad_area(), and triangulate_faces().

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◆ quad_area()

double polyfem::mesh::CMesh3D::quad_area ( const int gid ) const

overridevirtual

area of a quad face of an hex mesh

Parameters

[in] gid global face id

Returns: face area

Reimplemented from polyfem::mesh::Mesh.

Definition at line 1543 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::faces, mesh_, n_face_vertices(), n_vertices(), and point().

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◆ refine()

void polyfem::mesh::CMesh3D::refine	(	const int	n_refinement,
		const double	t
	)

overridevirtual

refine the mesh

Parameters

[in]	n_refinement	number of refinements
[in]	t	position of the refinement location (0.5 for standard refinement)

Implements polyfem::mesh::Mesh.

Definition at line 19 of file CMesh3D.cpp.

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◆ save()

bool polyfem::mesh::CMesh3D::save ( const std::string & path ) const

overridevirtual

Implements polyfem::mesh::Mesh.

Definition at line 416 of file CMesh3D.cpp.

References polyfem::mesh::Mesh3DStorage::elements, polyfem::utils::StringUtils::endswith(), polyfem::mesh::Mesh3DStorage::faces, mesh_, polyfem::mesh::Mesh3DStorage::points, and polyfem::mesh::to_geogram_mesh().

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◆ set_point()

void polyfem::mesh::CMesh3D::set_point	(	const int	global_index,
		const RowVectorNd &	p
	)

overridevirtual

Set the point.

Parameters

[in]	global_index	global vertex index
[in]	p	value

Implements polyfem::mesh::Mesh.

Definition at line 1317 of file CMesh3D.cpp.

References mesh_, polyfem::mesh::Mesh3DStorage::points, and polyfem::mesh::Mesh3DStorage::vertices.

◆ switch_edge()

Navigation3D::Index polyfem::mesh::CMesh3D::switch_edge ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 88 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::switch_edge().

Referenced by attach_higher_order_nodes().

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◆ switch_element()

Navigation3D::Index polyfem::mesh::CMesh3D::switch_element ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 90 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::switch_element().

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◆ switch_face()

Navigation3D::Index polyfem::mesh::CMesh3D::switch_face ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 89 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::switch_face().

Referenced by attach_higher_order_nodes().

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◆ switch_vertex()

Navigation3D::Index polyfem::mesh::CMesh3D::switch_vertex ( Navigation3D::Index idx ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 87 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Navigation3D::switch_vertex().

Referenced by attach_higher_order_nodes().

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◆ triangulate_faces()

void polyfem::mesh::CMesh3D::triangulate_faces	(	Eigen::MatrixXi &	tris,
		Eigen::MatrixXd &	pts,
		std::vector< int > &	ranges
	)		const

overridevirtual

generate a triangular representation of every face

Parameters

[out]	tris	triangles connectivity
[out]	pts	triangles vertices
[out]	ranges	connection to original faces

Implements polyfem::mesh::Mesh.

Definition at line 1071 of file CMesh3D.cpp.

References polyfem::mesh::Mesh::cell_barycenters(), polyfem::mesh::Mesh3DStorage::elements, polyfem::mesh::Mesh::face_barycenters(), polyfem::mesh::Mesh3DStorage::faces, polyfem::mesh::Element::fs, mesh_, n_face_vertices(), n_faces(), n_vertices(), point(), polyfem::mesh::Mesh3DStorage::points, and polyfem::mesh::Element::vs.

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◆ vertex_neighs()

std::vector< uint32_t > polyfem::mesh::CMesh3D::vertex_neighs ( const int v_gid ) const

inlineoverridevirtual

Implements polyfem::mesh::Mesh3D.

Definition at line 83 of file CMesh3D.hpp.

References mesh_, and polyfem::mesh::Mesh3DStorage::vertices.

Member Data Documentation

◆ mesh_

Mesh3DStorage polyfem::mesh::CMesh3D::mesh_

private

Definition at line 134 of file CMesh3D.hpp.

Referenced by append(), bounding_box(), cell_barycenter(), cell_edge(), cell_face(), cell_vertex(), compute_elements_tag(), edge_barycenter(), edge_neighs(), edge_vertex(), face_barycenter(), face_vertex(), get_edge_elements_neighs(), get_index_from_element(), get_index_from_element(), get_index_from_element_edge(), get_index_from_element_face(), get_vertex_elements_neighs(), is_boundary_edge(), is_boundary_element(), is_boundary_face(), is_boundary_vertex(), kernel(), load(), load(), mesh_storge(), n_cell_edges(), n_cell_faces(), n_cell_vertices(), n_cells(), n_edges(), n_face_vertices(), n_faces(), n_vertices(), next_around_edge(), next_around_face(), normalize(), point(), quad_area(), refine(), save(), set_point(), switch_edge(), switch_element(), switch_face(), switch_vertex(), triangulate_faces(), and vertex_neighs().

The documentation for this class was generated from the following files:

src/polyfem/mesh/mesh3D/CMesh3D.hpp
src/polyfem/mesh/mesh3D/CMesh3D.cpp

Public Member Functions

Static Public Member Functions

Protected Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ CMesh3D() [1/3]

◆ ~CMesh3D()

◆ CMesh3D() [2/3]

◆ CMesh3D() [3/3]

Member Function Documentation

◆ append()

◆ attach_higher_order_nodes()

◆ barycentric_coords()

◆ bounding_box()

◆ build_from_matrices()

◆ cell_barycenter()

◆ cell_edge()

◆ cell_face()

◆ cell_vertex()

◆ compute_body_ids()

◆ compute_boundary_ids() [1/6]

◆ compute_boundary_ids() [2/6]

◆ compute_boundary_ids() [3/6]

◆ compute_boundary_ids() [4/6]

◆ compute_boundary_ids() [5/6]

◆ compute_boundary_ids() [6/6]

◆ compute_elements_tag()

◆ copy()

◆ edge_barycenter()

◆ edge_neighs()

◆ edge_vertex()

◆ elements_boxes()

◆ face_barycenter()

◆ face_vertex()

◆ geomesh_2_mesh_storage()

◆ get_edge_elements_neighs()

◆ get_index_from_element() [1/2]

◆ get_index_from_element() [2/2]

◆ get_index_from_element_edge()

◆ get_index_from_element_face()

◆ get_vertex_elements_neighs()

◆ is_boundary_edge()

◆ is_boundary_element()

◆ is_boundary_face()

◆ is_boundary_vertex()

◆ is_conforming()

◆ kernel()

◆ load() [1/2]

◆ load() [2/2]

◆ mesh_storge()

◆ n_cell_edges()

◆ n_cell_faces()

◆ n_cell_vertices()

◆ n_cells()

◆ n_edges()

◆ n_face_vertices()

◆ n_faces()

◆ n_vertices()

◆ next_around_edge()

◆ next_around_face()

◆ normalize()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ point()

◆ quad_area()

◆ refine()

◆ save()

◆ set_point()

◆ switch_edge()

◆ switch_element()

◆ switch_face()

◆ switch_vertex()

◆ triangulate_faces()

◆ vertex_neighs()

Member Data Documentation

◆ mesh_