Reference documentation for deal.II version Git 65001fc 2018-06-18 17:06:20 -0400
GridGenerator Namespace Reference

## Functions

Creating meshes for basic geometries
template<int dim, int spacedim>
void hyper_cube (Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)

template<int dim>
void simplex (Triangulation< dim, dim > &tria, const std::vector< Point< dim >> &vertices)
Triangulation of a d-simplex with (d+1) vertices and mesh cells. More...

template<int dim, int spacedim>
void subdivided_hyper_cube (Triangulation< dim, spacedim > &tria, const unsigned int repetitions, const double left=0., const double right=1.)

template<int dim, int spacedim>
void hyper_rectangle (Triangulation< dim, spacedim > &tria, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)

template<int dim, int spacedim>
void subdivided_hyper_rectangle (Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)

template<int dim>
void subdivided_hyper_rectangle (Triangulation< dim > &tria, const std::vector< std::vector< double >> &step_sizes, const Point< dim > &p_1, const Point< dim > &p_2, const bool colorize=false)

template<int dim>
void subdivided_hyper_rectangle (Triangulation< dim > &tria, const std::vector< std::vector< double >> &spacing, const Point< dim > &p, const Table< dim, types::material_id > &material_id, const bool colorize=false)

template<int dim, int spacedim>
void cheese (Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &holes)
Rectangular domain with rectangular pattern of holes. More...

template<int dim>
void general_cell (Triangulation< dim > &tria, const std::vector< Point< dim >> &vertices, const bool colorize=false)

template<int dim>
void parallelogram (Triangulation< dim > &tria, const Point< dim >(&corners)[dim], const bool colorize=false)

template<int dim>
void parallelepiped (Triangulation< dim > &tria, const Point< dim >(&corners)[dim], const bool colorize=false)

template<int dim>
void subdivided_parallelepiped (Triangulation< dim > &tria, const unsigned int n_subdivisions, const Point< dim >(&corners)[dim], const bool colorize=false)

template<int dim>
void subdivided_parallelepiped (Triangulation< dim > &tria, const unsigned int(&n_subdivisions)[dim], const Point< dim >(&corners)[dim], const bool colorize=false)

template<int dim, int spacedim>
void subdivided_parallelepiped (Triangulation< dim, spacedim > &tria, const Point< spacedim > &origin, const std::array< Tensor< 1, spacedim >, dim > &edges, const std::vector< unsigned int > &subdivisions=std::vector< unsigned int >(), const bool colorize=false)

template<int dim>
void enclosed_hyper_cube (Triangulation< dim > &tria, const double left=0., const double right=1., const double thickness=1., const bool colorize=false)

template<int dim>
void hyper_ball (Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1., const bool attach_spherical_manifold_on_boundary_cells=false)

template<int spacedim>
void hyper_sphere (Triangulation< spacedim-1, spacedim > &tria, const Point< spacedim > &center=Point< spacedim >(), const double radius=1.)

template<int dim>
void quarter_hyper_ball (Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1.)

template<int dim>
void half_hyper_ball (Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1.)

template<int dim>
void cylinder (Triangulation< dim > &tria, const double radius=1., const double half_length=1.)

template<int dim>
void truncated_cone (Triangulation< dim > &tria, const double radius_0=1.0, const double radius_1=0.5, const double half_length=1.0)

template<int dim, int spacedim>
void hyper_cross (Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &sizes, const bool colorize_cells=false)
A center cell with stacks of cell protruding from each surface. More...

template<int dim>
void hyper_L (Triangulation< dim > &tria, const double left=-1., const double right=1., const bool colorize=false)

template<int dim>
void hyper_cube_slit (Triangulation< dim > &tria, const double left=0., const double right=1., const bool colorize=false)

template<int dim>
void hyper_shell (Triangulation< dim > &tria, const Point< dim > &center, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, bool colorize=false)

template<int dim>
void half_hyper_shell (Triangulation< dim > &tria, const Point< dim > &center, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, const bool colorize=false)

template<int dim>
void quarter_hyper_shell (Triangulation< dim > &tria, const Point< dim > &center, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, const bool colorize=false)

template<int dim>
void cylinder_shell (Triangulation< dim > &tria, const double length, const double inner_radius, const double outer_radius, const unsigned int n_radial_cells=0, const unsigned int n_axial_cells=0)

template<int dim, int spacedim>
void torus (Triangulation< dim, spacedim > &tria, const double R, const double r)

template<int dim>
void hyper_cube_with_cylindrical_hole (Triangulation< dim > &triangulation, const double inner_radius=.25, const double outer_radius=.5, const double L=.5, const unsigned int repetitions=1, const bool colorize=false)

void moebius (Triangulation< 3, 3 > &tria, const unsigned int n_cells, const unsigned int n_rotations, const double R, const double r)

Creating meshes from other meshes
template<int dim, int spacedim>
void merge_triangulations (const Triangulation< dim, spacedim > &triangulation_1, const Triangulation< dim, spacedim > &triangulation_2, Triangulation< dim, spacedim > &result, const double duplicated_vertex_tolerance=1.0e-12)

template<int dim, int spacedim>
void create_union_triangulation (const Triangulation< dim, spacedim > &triangulation_1, const Triangulation< dim, spacedim > &triangulation_2, Triangulation< dim, spacedim > &result)

template<int dim, int spacedim>
void create_triangulation_with_removed_cells (const Triangulation< dim, spacedim > &input_triangulation, const std::set< typename Triangulation< dim, spacedim >::active_cell_iterator > &cells_to_remove, Triangulation< dim, spacedim > &result)

void extrude_triangulation (const Triangulation< 2, 2 > &input, const unsigned int n_slices, const double height, Triangulation< 3, 3 > &result)

void extrude_triangulation (const Triangulation< 2, 2 > &input, const std::vector< double > &slice_coordinates, Triangulation< 3, 3 > &result)

template<int dim, int spacedim1, int spacedim2>
void flatten_triangulation (const Triangulation< dim, spacedim1 > &in_tria, Triangulation< dim, spacedim2 > &out_tria)

Creating lower-dimensional meshes from parts of higher-dimensional

meshes

template<template< int, int > class MeshType, int dim, int spacedim>
std::map< typename MeshType< dim-1, spacedim >::cell_iterator, typename MeshType< dim, spacedim >::face_iterator > extract_boundary_mesh (const MeshType< dim, spacedim > &volume_mesh, MeshType< dim-1, spacedim > &surface_mesh, const std::set< types::boundary_id > &boundary_ids=std::set< types::boundary_id >())

Exceptions

static::ExceptionBase & ExcInvalidRepetitions (int arg1)

static::ExceptionBase & ExcInvalidRepetitionsDimension (int arg1)

static::ExceptionBase & ExcInvalidInputOrientation ()

## Detailed Description

This namespace provides a collection of functions for generating triangulations for some basic geometries.

Some of these functions receive a flag colorize. If this is set, parts of the boundary receive different boundary indicators allowing them to be distinguished for the purpose of evaluating different boundary conditions.

If the domain is curved, each of the domain parts that should be refined by following an appropriate Manifold description will receive a different manifold indicator, and the correct Manifold descriptor will be attached to the Triangulation. Notice that if you later transform the triangulation, you have to make sure you attach the correct new Manifold to the triangulation.

## Function Documentation

template<int dim, int spacedim>
 void GridGenerator::hyper_cube ( Triangulation< dim, spacedim > & tria, const double left = 0., const double right = 1., const bool colorize = false )

Initialize the given triangulation with a hypercube (line in 1D, square in 2D, etc) consisting of exactly one cell. The hypercube volume is the tensor product interval $$[left,right]^{\text{dim}}$$ in the present number of dimensions, where the limits are given as arguments. They default to zero and unity, then producing the unit hypercube.

If the argument colorize is false, all boundary indicators are set to zero ("not colorized") for 2d and 3d. If it is true, the boundary is colorized as in hyper_rectangle(). In 1d the indicators are always colorized, see hyper_rectangle().

If dim < spacedim, this will create a dim dimensional object in the first dim coordinate directions embedded into the spacedim dimensional space with the remaining entries set to zero. For example, a Triangulation<2,3> will be a square in the xy plane with z=0.

See also subdivided_hyper_cube() for a coarse mesh consisting of several cells. See hyper_rectangle(), if different lengths in different ordinate directions are required.

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 420 of file grid_generator.cc.

template<int dim>
 void GridGenerator::simplex ( Triangulation< dim, dim > & tria, const std::vector< Point< dim >> & vertices )

Triangulation of a d-simplex with (d+1) vertices and mesh cells.

The vertices argument contains a vector with all d+1 vertices of the simplex. They must be given in an order such that the vectors from the first vertex to each of the others form a right-handed system. And I am not happy about the discrimination involved here.

The meshes generated in two and three dimensions are

Parameters
 tria The Triangulation to create. It needs to be empty upon calling this function. vertices The dim+1 corners of the simplex.
Note
Implemented for Triangulation<2,2>, Triangulation<3,3>.
Date
2015
template<int dim, int spacedim>
 void GridGenerator::subdivided_hyper_cube ( Triangulation< dim, spacedim > & tria, const unsigned int repetitions, const double left = 0., const double right = 1. )

Same as hyper_cube(), but with the difference that not only one cell is created but each coordinate direction is subdivided into repetitions cells. Thus, the number of cells filling the given volume is repetitionsdim.

If dim < spacedim, this will create a dim dimensional object in the first dim coordinate directions embedded into the spacedim dimensional space with the remaining entries set to zero. For example, a Triangulation<2,3> will be a square in the xy plane with z=0.

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 1139 of file grid_generator.cc.

template<int dim, int spacedim>
 void GridGenerator::hyper_rectangle ( Triangulation< dim, spacedim > & tria, const Point< dim > & p1, const Point< dim > & p2, const bool colorize = false )

Create a coordinate-parallel brick from the two diagonally opposite corner points p1 and p2.

If the colorize flag is true, the boundary_ids of the boundary faces are assigned, such that the lower one in x-direction is 0, the upper one is 1. The indicators for the surfaces in y-direction are 2 and 3, the ones for z are 4 and 5. This corresponds to the numbers of faces of the unit square of cube as laid out in the documentation of the GeometryInfo class. Importantly, however, in 3d colorization does not set boundary_ids of edges, but only of faces, because each boundary edge is shared between two faces and it is not clear how the boundary id of an edge should be set in that case.

Additionally, if colorize is true, material ids are assigned to the cells according to the octant their center is in: being in the right half space for any coordinate direction xi adds 2i. For instance, a cell with center point (1,-1,1) yields a material id 5, assuming that the center of the hyper rectangle lies at the origin. No manifold id is set for the cells.

If dim < spacedim, this will create a dim dimensional object in the first dim coordinate directions embedded into the spacedim dimensional space with the remaining entries set to zero. For example, a Triangulation<2,3> will be a rectangle in the xy plane with z=0, defined by the two opposing corners p1 and p2.

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 356 of file grid_generator.cc.

template<int dim, int spacedim>
 void GridGenerator::subdivided_hyper_rectangle ( Triangulation< dim, spacedim > & tria, const std::vector< unsigned int > & repetitions, const Point< dim > & p1, const Point< dim > & p2, const bool colorize = false )

Create a coordinate-parallel brick from the two diagonally opposite corner points p1 and p2. The number of cells in coordinate direction i is given by the integer repetitions[i].

To get cells with an aspect ratio different from that of the domain, use different numbers of subdivisions, given by repetitions, in different coordinate directions. The minimum number of subdivisions in each direction is 1.

If the colorize flag is set, the boundary_ids of the surfaces are assigned, such that the lower one in x-direction is 0, the upper one is 1 (the left and the right vertical face). The indicators for the surfaces in y-direction are 2 and 3, the ones for z are 4 and 5. Additionally, material ids are assigned to the cells according to the octant their center is in: being in the right half plane for any coordinate direction xi adds 2i. For instance, the center point (1,-1,1) yields a material id 5 (this means that in 2d only material ids 0,1,2,3 are assigned independent from the number of repetitions).

Note that the colorize flag is ignored in 1d and is assumed to always be true. That means the boundary indicator is 0 on the left and 1 on the right. See step-15 for details.

If dim < spacedim, this will create a dim dimensional object in the first dim coordinate directions embedded into the spacedim dimensional space with the remaining entries set to zero. For example, a Triangulation<2,3> will be a rectangle in the xy plane with z=0, defined by the two opposing corners p1 and p2.

Note
For an example of the use of this function see the step-28 tutorial program.
Parameters
 tria The Triangulation to create. It needs to be empty upon calling this function. repetitions A vector of dim positive values denoting the number of cells to generate in that direction. p1 First corner point. p2 Second corner opposite to p1. colorize Assign different boundary ids if set to true. The same comments apply as for the hyper_rectangle() function.

Definition at line 1163 of file grid_generator.cc.

template<int dim>
 void GridGenerator::subdivided_hyper_rectangle ( Triangulation< dim > & tria, const std::vector< std::vector< double >> & step_sizes, const Point< dim > & p_1, const Point< dim > & p_2, const bool colorize = false )

Like the previous function. However, here the second argument does not denote the number of subdivisions in each coordinate direction, but a sequence of step sizes for each coordinate direction. The domain will therefore be subdivided into step_sizes[i].size() cells in coordinate direction i, with width step_sizes[i][j] for the jth cell.

This function is therefore the right one to generate graded meshes where cells are concentrated in certain areas, rather than a uniformly subdivided mesh as the previous function generates.

The step sizes have to add up to the dimensions of the hyper rectangle specified by the points p1 and p2.

Definition at line 1314 of file grid_generator.cc.

template<int dim>
 void GridGenerator::subdivided_hyper_rectangle ( Triangulation< dim > & tria, const std::vector< std::vector< double >> & spacing, const Point< dim > & p, const Table< dim, types::material_id > & material_id, const bool colorize = false )

Like the previous function, but with the following twist: the material_id argument is a dim-dimensional array that, for each cell, indicates which material_id should be set. In addition, and this is the major new functionality, if the material_id of a cell is (unsigned char)(-1), then that cell is deleted from the triangulation, i.e. the domain will have a void there.

Note
If you need a lot of holes, you may consider cheese().
template<int dim, int spacedim>
 void GridGenerator::cheese ( Triangulation< dim, spacedim > & tria, const std::vector< unsigned int > & holes )

Rectangular domain with rectangular pattern of holes.

The domain itself is rectangular, very much as if it had been generated by subdivided_hyper_rectangle(). The argument holes specifies how many square holes the domain should have in each coordinate direction. The total number of mesh cells in that direction is then twice this number plus one.

The number of holes in one direction must be at least one.

An example with two by three holes is

If dim < spacedim, this will create a dim dimensional object in the first dim coordinate directions embedded into the spacedim dimensional space with the remaining entries set to zero.

Parameters
 tria The Triangulation to create. It needs to be empty upon calling this function. holes Positive number of holes in each of the dim directions.
Date
2015

Definition at line 1796 of file grid_generator.cc.

template<int dim>
 void GridGenerator::general_cell ( Triangulation< dim > & tria, const std::vector< Point< dim >> & vertices, const bool colorize = false )

A general quadrilateral in 2d or a general hexahedron in 3d. It is the responsibility of the user to provide the vertices in the right order (see the documentation of the GeometryInfo class) because the vertices are stored in the same order as they are given. It is also important to make sure that the volume of the cell is positive.

If the argument colorize is false, all boundary indicators are set to zero ("not colorized") for 2d and 3d. If it is true, the boundary is colorized as in hyper_rectangle(). In 1d the indicators are always colorized, see hyper_rectangle().

Definition at line 775 of file grid_generator.cc.

template<int dim>
 void GridGenerator::parallelogram ( Triangulation< dim > & tria, const Point< dim >(&) corners[dim], const bool colorize = false )

A parallelogram. The first corner point is the origin. The dim adjacent points are the ones given in the second argument and the fourth point will be the sum of these two vectors. Colorizing is done in the same way as in hyper_rectangle().

Note
This function is implemented in 2d only.
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::parallelepiped ( Triangulation< dim > & tria, const Point< dim >(&) corners[dim], const bool colorize = false )

A parallelepiped. The first corner point is the origin. The dim adjacent points are vectors describing the edges of the parallelepiped with respect to the origin. Additional points are sums of these dim vectors. Colorizing is done according to hyper_rectangle().

Note
This function silently reorders the vertices on the cells to lexicographic ordering (see GridReordering::reorder_grid). In other words, if reordering of the vertices does occur, the ordering of vertices in the array of corners will no longer refer to the same triangulation.
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 835 of file grid_generator.cc.

template<int dim>
 void GridGenerator::subdivided_parallelepiped ( Triangulation< dim > & tria, const unsigned int n_subdivisions, const Point< dim >(&) corners[dim], const bool colorize = false )

A subdivided parallelepiped. The first corner point is the origin. The dim adjacent points are vectors describing the edges of the parallelepiped with respect to the origin. Additional points are sums of these dim vectors. The variable n_subdivisions designates the number of subdivisions in each of the dim directions. Colorizing is done according to hyper_rectangle().

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 849 of file grid_generator.cc.

template<int dim>
 void GridGenerator::subdivided_parallelepiped ( Triangulation< dim > & tria, const unsigned int(&) n_subdivisions[dim], const Point< dim >(&) corners[dim], const bool colorize = false )

A subdivided parallelepiped, i.e., the same as above, but where the number of subdivisions in each of the dim directions may vary. Colorizing is done according to hyper_rectangle().

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 866 of file grid_generator.cc.

template<int dim, int spacedim>
 void GridGenerator::subdivided_parallelepiped ( Triangulation< dim, spacedim > & tria, const Point< spacedim > & origin, const std::array< Tensor< 1, spacedim >, dim > & edges, const std::vector< unsigned int > & subdivisions = std::vector(), const bool colorize = false )

A subdivided parallelepiped.

Parameters
 tria The Triangulation to create. It needs to be empty upon calling this function. origin First corner of the parallelepiped. edges An array of dim tensors describing the length and direction of the edges from origin. subdivisions Number of subdivisions in each of the dim directions. Each entry must be positive. An empty vector is equivalent to one subdivision in each direction. colorize Assign different boundary ids if set to true.
Note
Implemented for all combinations of dim and spacedim.
You likely need to help the compiler by explicitly specifying the two template parameters when calling this function.

Definition at line 898 of file grid_generator.cc.

template<int dim>
 void GridGenerator::enclosed_hyper_cube ( Triangulation< dim > & tria, const double left = 0., const double right = 1., const double thickness = 1., const bool colorize = false )

Hypercube with a layer of hypercubes around it. The first two parameters give the lower and upper bound of the inner hypercube in all coordinate directions. thickness marks the size of the layer cells.

If the flag colorize is set, the outer cells get material id's according to the following scheme: extending over the inner cube in (+/-) x-direction: 1/2. In y-direction 4/8, in z-direction 16/32. The cells at corners and edges (3d) get these values bitwise or'd.

Presently only available in 2d and 3d.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::hyper_ball ( Triangulation< dim > & tria, const Point< dim > & center = Point< dim >(), const double radius = 1., const bool attach_spherical_manifold_on_boundary_cells = false )

Initialize the given triangulation with several coarse mesh cells that cover a hyperball, i.e. a circle or a ball around center with given radius.

In order to avoid degenerate cells at the boundaries, the circle is triangulated by five cells, the ball by seven cells. Specifically, these cells are one cell in the center plus one "cap" cell on each of the faces of this center cell. This ensures that under repeated refinement, none of the cells at the outer boundary will degenerate to have an interior angle approaching 180 degrees. The diameter of the center cell is chosen so that the aspect ratio of the boundary cells after one refinement is optimized.

This function is declared to exist for triangulations of all space dimensions, but throws an error if called in 1d.

By default, the manifold_id is set to 0 on the boundary faces, 1 on the the boundary cells, and types::flat_manifold_id on the central cell and on internal faces.

A SphericalManifold is attached by default to the boundary faces for correct placement of boundary vertices upon refinement and to be able to use higher order mappings. However, it turns out that this strategy may not be the optimal one to create a good a mesh for a hyperball. The "Possibilities for extensions" section of step-6 has an extensive discussion of how one would construct better meshes and what one needs to do for it. Selecting the argument attach_spherical_manifold_on_boundary_cells to true attaches a SphericalManifold manifold also to the boundary cells, and not only to the boundary faces.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int spacedim>
 void GridGenerator::hyper_sphere ( Triangulation< spacedim-1, spacedim > & tria, const Point< spacedim > & center = Point(), const double radius = 1. )

Creates a hyper sphere, i.e., a surface of a ball in spacedim dimensions. This function only exists for dim+1=spacedim in 2 and 3 space dimensions. (To create a mesh of a ball, use GridGenerator::hyper_ball().)

By default, all manifold ids of the triangulation are set to zero, and a SphericalManifold is attached to the grid.

The following pictures are generated with:

Triangulation<2,3> triangulation;
triangulation.refine_global(3);

See the documentation module on manifolds for more details.

Note
The triangulation passed as argument needs to be empty when calling this function.

Definition at line 3068 of file grid_generator.cc.

template<int dim>
 void GridGenerator::quarter_hyper_ball ( Triangulation< dim > & tria, const Point< dim > & center = Point< dim >(), const double radius = 1. )

This class produces a hyper-ball intersected with the positive orthant relative to center, which contains three elements in 2d and four in 3d.

The boundary indicators for the final triangulation are 0 for the curved boundary and 1 for the cut plane. The manifold id for the curved boundary is set to zero, and a SphericalManifold is attached to it.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::half_hyper_ball ( Triangulation< dim > & tria, const Point< dim > & center = Point< dim >(), const double radius = 1. )

This class produces a half hyper-ball around center, which contains four elements in 2d and 6 in 3d. The cut plane is perpendicular to the x-axis.

The boundary indicators for the final triangulation are 0 for the curved boundary and 1 for the cut plane. The manifold id for the curved boundary is set to zero, and a SphericalManifold is attached to it.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::cylinder ( Triangulation< dim > & tria, const double radius = 1., const double half_length = 1. )

Create a cylinder around the $$x$$-axis. The cylinder extends from x=-half_length to x=+half_length and its projection into the yz-plane is a circle of radius radius.

In two dimensions, the cylinder is a rectangle from x=-half_length to x=+half_length and from y=-radius to y=radius.

The boundaries are colored according to the following scheme: 0 for the hull of the cylinder, 1 for the left hand face and 2 for the right hand face.

If you want the cylinder to revolve around a different axis than the $$x$$-axis, then simply rotate the mesh generated by this function using the GridTools::transform() function using a rotation operator as argument.

The manifold id for the hull of the cylinder is set to zero, and a CylindricalManifold is attached to it.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::truncated_cone ( Triangulation< dim > & tria, const double radius_0 = 1.0, const double radius_1 = 0.5, const double half_length = 1.0 )

Create a cut cone around the x-axis. The cone extends from x=-half_length to x=half_length and its projection into the yz-plane is a circle of radius radius_0 at x=-half_length and a circle of radius radius_1 at x=+half_length. In between the radius is linearly decreasing.

In two dimensions, the cone is a trapezoid from x=-half_length to x=+half_length and from y=-radius_0 to y=radius_0 at x=-half_length and from y=-radius_1 to y=radius_1 at x=+half_length. In between the range of y is linearly decreasing.

The boundaries are colored according to the following scheme: 0 for the hull of the cone, 1 for the left hand face and 2 for the right hand face. Both the boundary indicators and the manifold indicators are set.

In three dimensions, the manifold id of the hull is set to zero, and a CylindricalManifold is attached to it.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim, int spacedim>
 void GridGenerator::hyper_cross ( Triangulation< dim, spacedim > & tria, const std::vector< unsigned int > & sizes, const bool colorize_cells = false )

A center cell with stacks of cell protruding from each surface.

Each of the square mesh cells is Cartesian and has size one in each coordinate direction. The center of cell number zero is the origin.

Parameters
 tria A Triangulation object which has to be empty. sizes A vector of integers of dimension GeometryInfo::faces_per_cell with the following meaning: the legs of the cross are stacked on the faces of the center cell, in the usual order of deal.II cells, namely first $$-x$$, then $$x$$, then $$-y$$ and so on. The corresponding entries in sizes name the number of cells stacked on this face. All numbers may be zero, thus L- and T-shaped domains are specializations of this domain. colorize_cells If colorization is chosen, then the material id of a cells corresponds to the leg it is in. The id of the center cell is zero, and then the legs are numbered starting at one.

Examples in two and three dimensions are

Date
2015

Definition at line 1915 of file grid_generator.cc.

template<int dim>
 void GridGenerator::hyper_L ( Triangulation< dim > & tria, const double left = -1., const double right = 1., const bool colorize = false )

Initialize the given triangulation with a hyper-L (in 2d or 3d) consisting of exactly 2^dim-1 cells. It produces the hypercube with the interval [left,right] without the hypercube made out of the interval [(left+right)/2,right] for each coordinate. Because the domain is about the simplest one with a reentrant (i.e., non-convex) corner, solutions of many partial differential equation have singularities at this corner. That is, at the corner, the gradient or a higher derivative (depending on the boundary conditions chosen) does not remain bounded. As a consequence, this domain is often used to test convergence of schemes when the solution lacks regularity.

If the colorize flag is set, the boundary_ids of the surfaces are assigned such that the left boundary is 0 and the others are assigned counterclockwise in ascending order. The colorize option only works in two dimensions.

This function will create the classical L-shape in 2d and it will look like the following in 3d:

Note
The 3d domain is also often referred to as the "Fichera corner", named after Gaetano Fichera (1922-1996) who first computed an approximation of the corner singularity exponent of the lowest eigenfunction of the domain.

This function exists for triangulations of all space dimensions, but throws an error if called in 1d.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::hyper_cube_slit ( Triangulation< dim > & tria, const double left = 0., const double right = 1., const bool colorize = false )

Initialize the given Triangulation with a hypercube with a slit. In each coordinate direction, the hypercube extends from left to right.

In 2d, the split goes in vertical direction from x=(left+right)/2, y=left to the center of the square at x=y=(left+right)/2.

In 3d, the 2d domain is just extended in the z-direction, such that a plane cuts the lower half of a rectangle in two. This function is declared to exist for triangulations of all space dimensions, but throws an error if called in 1d.

If colorize is set to true, the faces forming the slit are marked with boundary id 1 and 2, respectively.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::hyper_shell ( Triangulation< dim > & tria, const Point< dim > & center, const double inner_radius, const double outer_radius, const unsigned int n_cells = 0, bool colorize = false )

Produce a hyper-shell, the region between two spheres around center, with given inner_radius and outer_radius. The number n_cells indicates the number of cells of the resulting triangulation, i.e., how many cells form the ring (in 2d) or the shell (in 3d).

If the flag colorize is true, then the outer boundary will have the indicator 1, while the inner boundary has id zero. In 3d, this applies to both the faces and the edges of these boundaries. If the flag is false, both have indicator zero.

All manifold ids are set to zero, and a SphericalManifold is attached to every cell and face of the triangulation.

In 2d, the number n_cells of elements for this initial triangulation can be chosen arbitrarily. If the number of initial cells is zero (as is the default), then it is computed adaptively such that the resulting elements have the least aspect ratio.

In 3d, only certain numbers are allowed, 6 (or the default 0) for a surface based on a hexahedron (i.e. 6 panels on the inner sphere extruded in radial direction to form 6 cells), 12 for the rhombic dodecahedron, and 96. This choice dates from an older version of deal.II before the Manifold classes were implemented: today all three choices are roughly equivalent (after performing global refinement, of course).

The grids with 12 and 96 cells are plotted below:

Note
This function is declared to exist for triangulations of all space dimensions, but throws an error if called in 1d.
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::half_hyper_shell ( Triangulation< dim > & tria, const Point< dim > & center, const double inner_radius, const double outer_radius, const unsigned int n_cells = 0, const bool colorize = false )

Produce a half hyper-shell, i.e. the space between two circles in two space dimensions and the region between two spheres in 3d, with given inner and outer radius and a given number of elements for this initial triangulation. However, opposed to the previous function, it does not produce a whole shell, but only one half of it, namely that part for which the first component is restricted to non-negative values. The purpose of this class is to enable computations for solutions which have rotational symmetry, in which case the half shell in 2d represents a shell in 3d.

If the number of initial cells is zero (as is the default), then it is computed adaptively such that the resulting elements have the least aspect ratio.

If colorize is set to true, the inner, outer, and the part of the boundary where $$x=0$$, get indicator 0, 1, and 2, respectively. Otherwise all indicators are set to 0.

All manifold ids are set to zero, and a SphericalManifold is attached to the triangulation.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::quarter_hyper_shell ( Triangulation< dim > & tria, const Point< dim > & center, const double inner_radius, const double outer_radius, const unsigned int n_cells = 0, const bool colorize = false )

Produce a domain that is the intersection between a hyper-shell with given inner and outer radius, i.e. the space between two circles in two space dimensions and the region between two spheres in 3d, and the positive quadrant (in 2d) or octant (in 3d). In 2d, this is indeed a quarter of the full annulus, while the function is a misnomer in 3d because there the domain is not a quarter but one eighth of the full shell.

If the number of initial cells is zero (as is the default), then it is computed adaptively such that the resulting elements have the least aspect ratio in 2d.

If colorize is set to true, the inner, outer, left, and right boundary get indicator 0, 1, 2, and 3 in 2d, respectively. Otherwise all indicators are set to 0. In 3d indicator 2 is at the face x=0, 3 at y=0, 4 at z=0.

All manifold ids are set to zero, and a SphericalManifold is attached to the triangulation.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim>
 void GridGenerator::cylinder_shell ( Triangulation< dim > & tria, const double length, const double inner_radius, const double outer_radius, const unsigned int n_radial_cells = 0, const unsigned int n_axial_cells = 0 )

Produce a domain that is the space between two cylinders in 3d, with given length, inner and outer radius and a given number of elements. The cylinder shell is built around the $$z$$-axis with the two faces located at $$z = 0$$ and $$z =$$ length.

If n_radial_cells is zero (as is the default), then it is computed adaptively such that the resulting elements have the least aspect ratio. The same holds for n_axial_cells.

Note
Although this function is declared as a template, it does not make sense in 1D and 2D. Also keep in mind that this object is rotated and positioned differently than the one created by cylinder().

All manifold ids are set to zero, and a CylindricalManifold is attached to the triangulation.

Note
The triangulation passed as argument needs to be empty when calling this function.
template<int dim, int spacedim>
 void GridGenerator::torus ( Triangulation< dim, spacedim > & tria, const double R, const double r )

Produce the volume or surface mesh of a torus. The axis of the torus is the $$y$$-axis while the plane of the torus is the $$x$$- $$z$$ plane.

If dim is 3, the mesh will be the volume of the torus and this function attaches a TorusManifold to all boundary cells and faces (which are marked with a manifold id of 0).

If dim is 2, the mesh will describe the surface of the torus and this function attaches a TorusManifold to all cells and faces (which are marked with a manifold id of 0).

Parameters
 tria The triangulation to be filled. R The radius of the circle, which forms the middle line of the torus containing the loop of cells. Must be greater than r. r The inner radius of the torus.
Note
Implemented for Triangulation<2,3> and Triangulation<3,3>.
template<int dim>
 void GridGenerator::hyper_cube_with_cylindrical_hole ( Triangulation< dim > & triangulation, const double inner_radius = .25, const double outer_radius = .5, const double L = .5, const unsigned int repetitions = 1, const bool colorize = false )

This class produces a square in the xy-plane with a cylindrical hole in the middle. The square and the circle are centered at the origin. In 3d, this geometry is extruded in $$z$$ direction to the interval $$[0,L]$$.

The inner boundary has a manifold id of $$0$$ and a boundary id of $$6$$. This function attaches a PolarManifold or CylindricalManifold to the interior boundary in 2d and 3d respectively. The other faces have boundary ids of $$0, 1, 2, 3, 4$$, or $$5$$ given in the standard order of faces in 2d or 3d.

It is implemented in 2d and 3d, and takes the following arguments:

Parameters
 triangulation The triangulation to be filled. inner_radius Radius of the internal hole. outer_radius Half of the edge length of the square. L Extension in z-direction (only used in 3d). repetitions Number of subdivisions along the z-direction. colorize Whether to assign different boundary indicators to different faces. The colors are given in lexicographic ordering for the flat faces (0 to 3 in 2d, 0 to 5 in 3d) plus the curved hole (4 in 2d, and 6 in 3d). If colorize is set to false, then flat faces get the number 0 and the hole gets number 1.
 void GridGenerator::moebius ( Triangulation< 3, 3 > & tria, const unsigned int n_cells, const unsigned int n_rotations, const double R, const double r )

Produce a ring of cells in 3d that is cut open, twisted and glued together again. This results in a kind of moebius-loop.

Parameters
 tria The triangulation to be worked on. n_cells The number of cells in the loop. Must be greater than 4. n_rotations The number of rotations (Pi/2 each) to be performed before gluing the loop together. R The radius of the circle, which forms the middle line of the torus containing the loop of cells. Must be greater than r. r The radius of the cylinder bent together as a loop.
template<int dim, int spacedim>
 void GridGenerator::merge_triangulations ( const Triangulation< dim, spacedim > & triangulation_1, const Triangulation< dim, spacedim > & triangulation_2, Triangulation< dim, spacedim > & result, const double duplicated_vertex_tolerance = 1.0e-12 )

Given the two triangulations specified as the first two arguments, create the triangulation that contains the cells of both triangulation and store it in the third parameter. Previous content of result will be deleted.

This function is most often used to compose meshes for more complicated geometries if the geometry can be composed of simpler parts for which functions exist to generate coarse meshes. For example, the channel mesh used in step-35 could in principle be created using a mesh created by the GridGenerator::hyper_cube_with_cylindrical_hole function and several rectangles, and merging them using the current function. The rectangles will have to be translated to the right for this, a task that can be done using the GridTools::shift function (other tools to transform individual mesh building blocks are GridTools::transform, GridTools::rotate, and GridTools::scale).

Vertices that are less than duplicated_vertex_tolerance apart will be merged together. It is usually necessary to set this value to something that depends on the input triangulations in some way. One reasonable choice is to use the minimum distance between all adjacent vertices of the input mesh divided by some constant:

auto min_line_length = [](const Triangulation<dim> &tria) -> double
{
double length = std::numeric_limits<double>::max();
for (const auto cell : tria.active_cell_iterators())
for (unsigned int n = 0; n < GeometryInfo<dim>::lines_per_cell; ++n)
length = std::min(length, (cell->line(n)->vertex(0) -
cell->line(n)->vertex(1)).norm());
return length;
};
const double tolerance = std::min(min_line_length(triangulation_1),
min_line_length(triangulation_2)) / 2.0;

This will merge any vertices that are closer than any pair of vertices on the input meshes.

Note
The two input triangulations must be coarse meshes that have no refined cells.
The function copies the material ids of the cells of the two input triangulations into the output triangulation but it currently makes no attempt to do the same for boundary ids. In other words, if the two coarse meshes have anything but the default boundary indicators, then you will currently have to set boundary indicators again by hand in the output triangulation.
Unlike most GridGenerator functions, this function does not attach any manifolds to result, nor does it set any manifold ids.
For a related operation on refined meshes when both meshes are derived from the same coarse mesh, see GridGenerator::create_union_triangulation().

Definition at line 3793 of file grid_generator.cc.

template<int dim, int spacedim>
 void GridGenerator::create_union_triangulation ( const Triangulation< dim, spacedim > & triangulation_1, const Triangulation< dim, spacedim > & triangulation_2, Triangulation< dim, spacedim > & result )

Given the two triangulations specified as the first two arguments, create the triangulation that contains the finest cells of both triangulation and store it in the third parameter. Previous content of result will be deleted.

Note
This function is intended to create an adaptively refined triangulation that contains the most refined cells from two input triangulations that were derived from the same coarse grid by adaptive refinement. This is an operation sometimes needed when one solves for two variables of a coupled problem on separately refined meshes on the same domain (for example because these variables have boundary layers in different places) but then needs to compute something that involves both variables or wants to output the result into a single file. In both cases, in order not to lose information, the two solutions can not be interpolated onto the respectively other mesh because that may be coarser than the ones on which the variable was computed. Rather, one needs to have a mesh for the domain that is at least as fine as each of the two initial meshes. This function computes such a mesh.
If you want to create a mesh that is the merger of two other coarse meshes, for example in order to compose a mesh for a complicated geometry from meshes for simpler geometries, then this is not the function for you. Instead, consider GridGenerator::merge_triangulations().
This function assumes that both triangulation_1 and triangulation_2 have the same manifold descriptions. The output Triangulation has the same manifold ids as these two triangulations.
Precondition
Both of the source conditions need to be available entirely locally. In other words, they can not be objects of type parallel::distributed::Triangulation.

Definition at line 3859 of file grid_generator.cc.

template<int dim, int spacedim>
 void GridGenerator::create_triangulation_with_removed_cells ( const Triangulation< dim, spacedim > & input_triangulation, const std::set< typename Triangulation< dim, spacedim >::active_cell_iterator > & cells_to_remove, Triangulation< dim, spacedim > & result )

This function creates a triangulation that consists of the same cells as are present in the first argument, except those cells that are listed in the second argument. The purpose of the function is to generate geometries subtractively from the geometry described by an existing triangulation. A prototypical case is a 2d domain with rectangular holes. This can be achieved by first meshing the entire domain and then using this function to get rid of the cells that are located at the holes. Likewise, you could create the mesh that GridGenerator::hyper_L() produces by starting with a GridGenerator::hyper_cube(), refining it once, and then calling the current function with a single cell in the second argument.

Parameters
 [in] input_triangulation The original triangulation that serves as the template from which the new one is to be created. [in] cells_to_remove A list of cells of the triangulation provided as first argument that should be removed (i.e., that should not show up in the result. [out] result The resulting triangulation that consists of the same cells as are in input_triangulation, with the exception of the cells listed in cells_to_remove.
Note
Unlike most GridGenerator functions, this function does not attach any manifolds to result, nor does it set any manifold ids.
Precondition
Because we cannot create triangulations de novo that contain adaptively refined cells, the input triangulation needs to have all of its cells on the same level. Oftentimes, this will in fact be the coarsest level, but it is allowed to pass in a triangulation that has been refined globally a number of times. The output triangulation will in that case simply be a mesh with only one level that consists of the active cells of the input minus the ones listed in the second argument. However, the input triangulation must not have been adaptively refined.

Definition at line 3920 of file grid_generator.cc.

 void GridGenerator::extrude_triangulation ( const Triangulation< 2, 2 > & input, const unsigned int n_slices, const double height, Triangulation< 3, 3 > & result )

Take a 2d Triangulation that is being extruded in z direction by the total height of height using n_slices slices (minimum is 2). The boundary indicators of the faces of input are going to be assigned to the corresponding side walls in z direction. The bottom and top get the next two free boundary indicators.

Note
The 2d input triangulation input must be a coarse mesh that has no refined cells.
Since input and output have different spatial dimensions no manifold objects are copied (nor are any manifold ids set) by this function.

Definition at line 3972 of file grid_generator.cc.

 void GridGenerator::extrude_triangulation ( const Triangulation< 2, 2 > & input, const std::vector< double > & slice_coordinates, Triangulation< 3, 3 > & result )

Overload of the previous function. Take a 2d Triangulation that is being extruded. Differing from the previous function taking height and number of slices for uniform extrusion, this function takes z-axis values slice_coordinates where the slicing will happen. The boundary indicators of the faces of input are going to be assigned to the corresponding side walls in z direction. The bottom and top get the next two free boundary indicators.

Note
The 2d input triangulation input must be a coarse mesh that has no refined cells.
Since input and output have different spatial dimensions no manifold objects are copied (nor are any manifold ids set) by this function.

Definition at line 3997 of file grid_generator.cc.

template<int dim, int spacedim1, int spacedim2>
 void GridGenerator::flatten_triangulation ( const Triangulation< dim, spacedim1 > & in_tria, Triangulation< dim, spacedim2 > & out_tria )

Given an input triangulation in_tria, this function makes a new flat triangulation out_tria which contains a single level with all active cells of the input triangulation. If spacedim1 and spacedim2 are different, only the smallest spacedim components of the vertices are copied over. This is useful to create a Triangulation<2,3> out of a Triangulation<2,2>, or to project a Triangulation<2,3> into a Triangulation<2,2>, by neglecting the z components of the vertices.

No internal checks are performed on the vertices, which are assumed to make sense topologically in the target spacedim2 dimensional space. If this is not the case, you will encounter problems when using the triangulation later on.

All information about cell manifold_ids and material ids are copied from one triangulation to the other, and only the boundary manifold_ids and boundary_ids are copied over from the faces of in_tria to the faces of out_tria. If you need to specify manifold ids on interior faces, they have to be specified manually after the triangulation is created.

This function will fail if the input Triangulation is of type parallel::distributed::Triangulation, as well as when the input Triangulation contains hanging nodes.

Note
Since input and output have different spatial dimensions no manifold objects are copied by this function: you must attach new manifold objects to out_tria.

Definition at line 4363 of file grid_generator.cc.

template<template< int, int > class MeshType, int dim, int spacedim>
 std::map< typename MeshType< dim-1, spacedim >::cell_iterator, typename MeshType< dim, spacedim >::face_iterator > GridGenerator::extract_boundary_mesh ( const MeshType< dim, spacedim > & volume_mesh, MeshType< dim-1, spacedim > & surface_mesh, const std::set< types::boundary_id > & boundary_ids = std::set() )

This function implements a boundary subgrid extraction. Given a <dim,spacedim>-Triangulation (the "volume mesh") the function extracts a subset of its boundary (the "surface mesh"). The boundary to be extracted is specified by a list of boundary_ids. If none is specified the whole boundary will be extracted. The function is used in step-38.

The function also builds a mapping linking the cells on the surface mesh to the corresponding faces on the volume one. This mapping is the return value of the function.

Note
The function builds the surface mesh by creating a coarse mesh from the selected faces of the coarse cells of the volume mesh. It copies the boundary indicators of these faces to the cells of the coarse surface mesh. The surface mesh is then refined in the same way as the faces of the volume mesh are. In order to ensure that the surface mesh has the same vertices as the volume mesh, it is therefore important that you assign appropriate boundary descriptions through Triangulation::set_manifold() to the surface mesh object before calling this function. If you don't, the refinement will happen under the assumption that all faces are straight (i.e using the FlatManifold class) rather than utilizing the Manifold object you may want to use to determine the location of new vertices.
Template Parameters
 MeshType A type that satisfies the requirements of the MeshType concept. The map that is returned will be between cell iterators pointing into the container describing the surface mesh and face iterators of the volume mesh container. If MeshType is DoFHandler or hp::DoFHandler, then the function will re-build the triangulation underlying the second argument and return a map between appropriate iterators into the MeshType arguments. However, the function will not actually distribute degrees of freedom on this newly created surface mesh. dim The dimension of the cells of the volume mesh. For example, if dim==2, then the cells are quadrilaterals that either live in the plane, or form a surface in a higher-dimensional space. The dimension of the cells of the surface mesh is consequently dim-1. spacedim The dimension of the space in which both the volume and the surface mesh live.
Parameters
 [in] volume_mesh A container of cells that define the volume mesh. [out] surface_mesh A container whose associated triangulation will be built to consist of the cells that correspond to the (selected portion of) the boundary of the volume mesh. [in] boundary_ids A list of boundary indicators denoting that subset of faces of volume cells for which this function should extract the surface mesh. If left at its default, i.e., if the set is empty, then the function operates on all boundary faces.
Returns
A map that for each cell of the surface mesh (key) returns an iterator to the corresponding face of a cell of the volume mesh (value). The keys include both active and non-active cells of the surface mesh. For dim=2 (i.e., where volume cells are quadrilaterals and surface cells are lines), the order of vertices of surface cells and the corresponding volume faces match. For dim=3 (i.e., where volume cells are hexahedra and surface cells are quadrilaterals), the order of vertices may not match in order to ensure that each surface cell has a right-handed coordinate system when viewed from one of the two sides of the surface connecting the cells of the surface mesh.
Note
The algorithm outlined above assumes that all faces on higher refinement levels always have exactly the same boundary indicator as their parent face. Consequently, we can start with coarse level faces and build the surface mesh based on that. It would not be very difficult to extend the function to also copy boundary indicators from finer level faces to their corresponding surface mesh cells, for example to accommodate different geometry descriptions in the case of curved boundaries (but this is not currently implemented).
Since volume_mesh and surface_mesh have different spatial dimensions no manifold objects are copied by this function: you must attach new manifold objects to surface_mesh.

Definition at line 4467 of file grid_generator.cc.