CGAL 6.2 - Tetrahedral Remeshing

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Modules | Functions
Reference Manual

##

Jane Tournois, Noura Faraj, Jean-Marc Thiery, and Tamy Boubekeur
The package provides a function for remeshing tetrahedral meshes, targeting high quality meshes with respect to dihedral angles. This practical iterative remeshing algorithm is designed to remesh multi-material tetrahedral meshes, by iteratively performing a sequence of elementary operations such as edge splits, edge collapses, edge flips, and vertex relocations following a Laplacian smoothing. The algorithm results in high-quality uniform isotropic meshes, with the desired mesh density, while preserving the input geometric curve and surface features.
Introduced in: CGAL 5.1
Depends on: 3D Triangulations
BibTeX: cgal:tftb-tr-26a
License: GPL
Windows Demo: CGAL Lab

## Classified Reference Pages

## Concepts

- `RemeshingTriangulationTraits_3`
- `RemeshingVertexBase_3`
- `RemeshingCellBase_3`
- `RemeshingSizingField_3`

## Classes

- `CGAL::Tetrahedral_remeshing::Remeshing_vertex_base_3`
- `CGAL::Tetrahedral_remeshing::Remeshing_cell_base_3`
- `CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3`
- `CGAL::Uniform_sizing_field`
- `CGAL::Adaptive_remeshing_sizing_field`

## Function Templates

- `CGAL::tetrahedral_isotropic_remeshing()`
- `CGAL::convert_to_triangulation_3()`
- `CGAL::create_adaptive_remeshing_sizing_field()`

## Modules
| Concepts
| Classes
| Sizing Fields

## Functions
template<typename Traits , typename TDS , typename SLDS , typename SizingFunction , typename NamedParameters = parameters::Default_named_parameters>
void | CGAL::tetrahedral_isotropic_remeshing (CGAL::Triangulation_3< Traits, TDS, SLDS > &tr, const SizingFunction &sizing, const NamedParameters &np=parameters::default_values())
| remeshes a tetrahedral mesh.
template<typename Tr , typename CornerIndex , typename CurveIndex , typename NamedParameters = parameters::Default_named_parameters>
CGAL::Triangulation_3< typename Tr::Geom_traits, typename Tr::Triangulation_data_structure > | CGAL::convert_to_triangulation_3 (CGAL::Mesh_complex_3_in_triangulation_3< Tr, CornerIndex, CurveIndex > c3t3, const NamedParameters &np=parameters::default_values())
| converts the triangulation contained in the input to a `Triangulation_3`.

## Function Documentation

## ◆ convert_to_triangulation_3()

template<typename Tr , typename CornerIndex , typename CurveIndex , typename NamedParameters = parameters::Default_named_parameters>

CGAL::Triangulation_3< typename Tr::Geom_traits, typename Tr::Triangulation_data_structure > CGAL::convert_to_triangulation_3 | ( | CGAL::Mesh_complex_3_in_triangulation_3< Tr, CornerIndex, CurveIndex > | c3t3,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/tetrahedral_remeshing.h>`

converts the triangulation contained in the input to a `Triangulation_3`.

This function should be used to generate a valid triangulation for tetrahedral remeshing, when the input triangulation is generated with the tetrahedral mesh generation package.

Template Parameters

  Tr | is the underlying triangulation for `Mesh_complex_3_in_triangulation_3`. It can be instantiated with any 3D regular triangulation of CGAL provided that its vertex and cell base classes are models of the concepts `SimplicialMeshCellBase_3` (refined by `RemeshingCellBase_3`) and `SimplicialMeshVertexBase_3` (refined by `RemeshingVertexBase_3`), respectively.
  CornerIndex | is the type of the indices for feature corners. If `c3t3` has been generated using `CGAL::make_mesh_3()`, it must match `MeshDomainWithFeatures_3::Corner_index`.
  CurveIndex | is the type of the indices for feature curves. If `c3t3` has been generated using `CGAL::make_mesh_3()`, it must match `MeshDomainWithFeatures_3::Curve_index`.
  NamedParameters | a sequence of Named Parameters

Parameters

  c3t3 | the complex containing the triangulation to be remeshed.
  np | optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map containing the constrained-or-not status of each edge of `c3t3.triangulation()`. For each edge `e` for which `c3t3.is_in_complex(e)` returns `true`, the constrained status of `e` is set to `true`. Type: a class model of `ReadWritePropertyMap` with `std::pair<Triangulation_3::Vertex_handle, Triangulation_3::Vertex_handle>` as key type and `bool` as value type. It must be default constructible. Default: a default property map where no edge is constrained |
  a property map containing the constrained-or-not status of each vertex of `c3t3.triangulation()`. For each vertex `v` for which `c3t3.is_in_complex(v)` returns `true`, the constrained status of `v` is set to `true`. Type: a class model of `ReadWritePropertyMap` with `Triangulation_3::Vertex_handle` as key type and `bool` as value type. It must be default constructible. Default: a default property map where no vertex is constrained |

Examples
  Tetrahedral_remeshing/mesh_and_remesh_c3t3.cpp, Tetrahedral_remeshing/mesh_and_remesh_polyhedral_domain_with_features.cpp, Tetrahedral_remeshing/mesh_and_remesh_with_adaptive_sizing.cpp, and Tetrahedral_remeshing/mesh_and_remesh_with_sizing.cpp.

## ◆ tetrahedral_isotropic_remeshing()

template<typename Traits , typename TDS , typename SLDS , typename SizingFunction , typename NamedParameters = parameters::Default_named_parameters>

void CGAL::tetrahedral_isotropic_remeshing | ( | CGAL::Triangulation_3< Traits, TDS, SLDS > & | tr,
|  | const SizingFunction & | sizing,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/tetrahedral_remeshing.h>`

remeshes a tetrahedral mesh.

It is recommended to use `CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3` for the first parameter, as it inherits from `Triangulation_3` with a `TDS` suitable for this function.

This function takes as input a 3-dimensional triangulation and performs a sequence of atomic operations in order to generate as output a high quality mesh with a prescribed uniform density. These atomic operations are performed as follows:

- edge splits, until all edges satisfy a prescribed length criterion,
- edge collapses, until all edges satisfy a prescribed length criterion,
- edge flips, to locally improve dihedral angles, until they can no longer be improved by flipping,
- global smoothing by vertex relocations,
- re-projection of boundary vertices to the initial surface.

This remeshing function can deal with multi-domains, boundaries, and features. It preserves the geometry of subdomains throughout the remeshing process.

Subdomains are defined by indices that are stored in the cells of the input triangulation, following the `SimplicialMeshCellBase_3` concept (refined by `RemeshingCellBase_3`). The surfacic interfaces between subdomains are formed by facets whose two incident cells have different subdomain indices. The edges where three or more subdomains meet form feature polylines, and are considered as constrained edges.

Template Parameters

  Traits | is the geometric traits, model of `RemeshingTriangulationTraits_3`
  TDS | is the triangulation data structure for `Triangulation_3`, model of `TriangulationDataStructure_3`, with cell base model of `RemeshingCellBase_3` and vertex base model of `RemeshingVertexBase_3`.
  SLDS | is an optional parameter for `Triangulation_3`, that specifies the type of the spatial lock data structure.
  SizingFunction | a sizing field functional, model of `RemeshingSizingField_3`, or type convertible to `double`.
  NamedParameters | a sequence of Named Parameters

Parameters

  tr | the triangulation to be remeshed, of type `Triangulation_3<Traits, TDS, SLDS>`. `CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3` is a helper class that satisfies all the requirements of its template parameters.
  sizing | the target edge length. This parameter provides a mesh density target for the remeshing algorithm. It can be a number convertible to `double`, or an instance of a model of `RemeshingSizingField_3`.
  np | optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  the number of iterations for the full sequence of atomic operations performed (listed in the above description) Type: unsigned int Default: `1` |
  If `false`, none of the input volume boundaries can be modified. Otherwise, the topology is preserved, but atomic operations can be performed on the surfaces, and along feature polylines, such that boundaries are remeshed. Type: `bool` Default: `true` Extra: Boundaries are between the exterior and the interior, between two subdomains, between the areas selected or not for remeshing (cf `Remeshing_cell_is_selected_map`), or defined by `Remeshing_edge_is_constrained_map` and `Remeshing_facet_is_constrained_map`. |
  a property map containing the constrained-or-not status of each facet of `tr`. Type: a class model of `ReadablePropertyMap` with `Triangulation_3::Facet` as key type and `bool` as value type. It must be default constructible. Default: a default property map where no facet is constrained Extra: A constrained facet can be split or collapsed, but not flipped. Extra: This map, contrary to the others, is not updated throughout the remeshing process. |
  a property map containing the constrained-or-not status of each edge of `tr`. Type: a class model of `ReadWritePropertyMap` with `std::pair<Triangulation_3::Vertex_handle, Triangulation_3::Vertex_handle>` as key type and `bool` as value type. It must be default constructible. Default: a default property map where no edge is constrained Extra: A constrained edge can be split or collapsed, but not flipped. Extra: The pairs must be ordered to ensure consistency. Extra: During the meshing process, the set of constrained edges evolves consistently with edge splits and collapses, so the property map must be writable. |
  a property map containing the constrained-or-not status of each vertex of `tr`. Type: a class model of `ReadWritePropertyMap` with `Triangulation_3::Vertex_handle` as key type and `bool` as value type. It must be default constructible. Default: a default property map where no vertex is constrained Extra: A constrained vertex cannot be removed by collapse, nor moved by smoothing. |
  a property map containing the selected - or - not status for each cell of `tr` for remeshing. Only selected cells are modified (and possibly their neighbors if surfaces are modified) by remeshing. Type: a class model of `ReadWritePropertyMap` with `Triangulation_3::Cell_handle` as key type and `bool` as value type. It must be default constructible. Default: a default property map where all cells of the domain (i.e. with a non-zero `Subdomain_index`) are selected. Extra: During the meshing process, the set of selected cells evolves consistently with the atomic operations that are performed, so the property map must be writable. |
  If `true`, the end vertices of the edges set as constrained in `edge_is_constrained_map` move along the constrained polylines they belong to. Type: `bool` Default: `false` Extra: The endvertices of constraints listed by `edge_is_constrained_map`, and edges incident to at least three subdomains are made eligible to one dimensional smoothing, along the constrained polylines they belong to. Corners (i.e. vertices listed by `vertex_is_constrained_map` or incident to more than 2 constrained edges) are not allowed to move at all. Note that activating the smoothing step on polyline constraints tends to reduce the quality of the minimal dihedral angle in the mesh. If `remesh_boundaries` is set to `false`, this parameter is ignored. |

See also
  `CGAL::Tetrahedral_remeshing::Remeshing_triangulation_3`

Examples
  Tetrahedral_remeshing/mesh_and_remesh_c3t3.cpp, Tetrahedral_remeshing/mesh_and_remesh_polyhedral_domain_with_features.cpp, Tetrahedral_remeshing/mesh_and_remesh_with_adaptive_sizing.cpp, Tetrahedral_remeshing/mesh_and_remesh_with_sizing.cpp, Tetrahedral_remeshing/tetrahedral_remeshing_example.cpp, Tetrahedral_remeshing/tetrahedral_remeshing_from_mesh.cpp, Tetrahedral_remeshing/tetrahedral_remeshing_of_one_subdomain.cpp, and Tetrahedral_remeshing/tetrahedral_remeshing_with_features.cpp.