3D Periodic Mesh Generation Reference

Mikhail Bogdanov, Aymeric Pellé, Mael Rouxel-Labbé, and Monique Teillaud

This package is devoted to the generation of isotropic simplicial meshes discretizing periodic 3D domains. The domain to be meshed is a region of the three-dimensional flat torus. The periodic mesh generator provides users with the same flexibility that is offered in the PkgMesh3 package.

Introduced in: CGAL 4.13
Depends on: PkgPeriodic3Triangulation3, PkgMesh3, and thirdpartyEigen
BibTeX: cgal:btprl-p3m3-25a
License: GPL

Relation to the 3D Mesh Generation and 3D Periodic Triangulations Packages

A periodic mesh extends, by definition, infinitely in space. To avoid storing and manipulating duplicate points, well-chosen "dummy" points are inserted at the beginning of the meshing process, thus ensuring that the underlying periodic triangulation forms at all times a simplicial complex within a single copy of the periodic space \( \mathbb T_c^3\) (see Sections P3Triangulation3secspace and P3Triangulation3secintro of the package PkgPeriodic3Triangulation3). By identifying a single copy of the flat torus \( \mathbb T_{c_x,c_y,c_y}^3\) (where c_x,c_y, and c_z denote the period along the canonical axes) with a cuboid of side lengths c_x,c_y, andc_z in \( \mathbb R^3\), the meshing process can be exclusively conducted within a cuboid. The mesh within a single copy is created using CGAL's PkgMesh3 package, but because CGAL's PkgMesh3 package aims to mesh traditional (non-periodic) domains, an interface is necessary between CGAL's PkgMesh3 package and CGAL's PkgPeriodic3Triangulation3. This package offers these interfaces.

Classified Reference Pages

Concepts

This package relies entirely on the PkgMesh3Concepts and the PkgMesh3SecondaryConcepts described in the PkgMesh3 package. The following concepts are essentially identical to the concepts MeshDomain_3 and MeshDomainWithFeatures_3 in the package PkgMesh3 :

• Periodic_3MeshDomain_3
• Periodic_3MeshDomainWithFeatures_3

Classes

The following class provides the interface between CGAL's periodic triangulation and CGAL's three-dimensional mesh generator:

• CGAL::Periodic_3_mesh_triangulation_3<MD, K, Vertex_base, Cell_base>

The following class allows to construct a periodic implicit function from an implicit function that is not periodic:

• CGAL::Periodic_3_function_wrapper<Function,BGT>

The following class allows to split the canonical cuboid in two subdomains, separated by the zero-level of an implicit function:

• CGAL::Implicit_to_labeled_subdomains_function_wrapper<Function,BGT>

Function Templates

The following functions handle the generation of a periodic mesh:

• CGAL::make_periodic_3_mesh_3()
• CGAL::refine_periodic_3_mesh_3()
• CGAL::exude_periodic_3_mesh_3()
• CGAL::perturb_periodic_3_mesh_3()
• CGAL::lloyd_optimize_periodic_3_mesh_3()
• CGAL::odt_optimize_periodic_3_mesh_3()

Classes and Functions of Mesh_3

Many classes and functions used by this package are defined within the package PkgMesh3, see PkgMesh3MeshClasses, PkgMesh3Domains, and PkgMesh3Parameters.

Input/Output Functions

• CGAL::IO::output_periodic_mesh_to_medit()

Topics
	Concepts
	The concepts of this package.
	Mesh Classes
	The classes in this group are the main classes involved in the mesh generation process.
	Domain Classes
	The classes in this group are models of domain concepts and their associated classes.
	Mesh Generation Functions
	The two main functions to generate a periodic mesh are make_periodic_3_mesh_3() and refine_periodic_3_mesh_3().
	Input/Output Functions
	The free functions that can be used to read and write periodic meshes.