Reference documentation for deal.II version 8.5.0

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 dsimplex 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_cxx11::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 > ¢er=Point< dim >(), const double radius=1.) 
template<int dim, int spacedim>  
void  hyper_sphere (Triangulation< dim, spacedim > &tria, const Point< spacedim > ¢er=Point< spacedim >(), const double radius=1.) 
template<int dim>  
void  quarter_hyper_ball (Triangulation< dim > &tria, const Point< dim > ¢er=Point< dim >(), const double radius=1.) 
template<int dim>  
void  half_hyper_ball (Triangulation< dim > &tria, const Point< dim > ¢er=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 > ¢er, 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 > ¢er, 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 > ¢er, 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) 
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) 
template<int dim, int spacedim1, int spacedim2>  
void  flatten_triangulation (const Triangulation< dim, spacedim1 > &in_tria, Triangulation< dim, spacedim2 > &out_tria) 
Creating lowerdimensional meshes from parts of higherdimensional  
meshes  
template<template< int, int > class MeshType, int dim, int spacedim>  
std::map< typename MeshType< dim1, spacedim >::cell_iterator, typename MeshType< dim, spacedim >::face_iterator >  extract_boundary_mesh (const MeshType< dim, spacedim > &volume_mesh, MeshType< dim1, spacedim > &surface_mesh, const std::set< types::boundary_id > &boundary_ids=std::set< types::boundary_id >()) 
Exceptions  
static::ExceptionBase &  ExcInvalidRadii () 
static::ExceptionBase &  ExcInvalidRepetitions (int arg1) 
static::ExceptionBase &  ExcInvalidRepetitionsDimension (int arg1) 
static::ExceptionBase &  ExcInvalidInputOrientation () 
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 attaching geometry objects and evaluating different boundary conditions.
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.
Definition at line 404 of file grid_generator.cc.
void GridGenerator::simplex  (  Triangulation< dim, dim > &  tria, 
const std::vector< Point< dim > > &  vertices  
) 
Triangulation of a dsimplex 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 righthanded system. And I am not happy about the discrimination involved here.
The meshes generated in two and three dimensions are
tria  The Triangulation to create. It needs to be empty upon calling this function. 
vertices  The dim+1 corners of the simplex. 
Triangulation<2,2>
, Triangulation<3,3>
.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 repetitions^{dim}
.
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.
Definition at line 1108 of file grid_generator.cc.
void GridGenerator::hyper_rectangle  (  Triangulation< dim, spacedim > &  tria, 
const Point< dim > &  p1,  
const Point< dim > &  p2,  
const bool  colorize = false 

) 
Create a coordinateparallel 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 xdirection
is 0, the upper one is 1. The indicators for the surfaces in ydirection
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. This may later on lead to problems if one wants to assign boundary or manifold objects to parts of the boundary with certain boundary indicators since then the boundary object may not apply to the edges bounding the face it is meant to describe.
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 x_{i} adds 2^{i}. 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.
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
.
Definition at line 341 of file grid_generator.cc.
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 coordinateparallel 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 xdirection
is 0, the upper one is 1 (the left and the right vertical face). The indicators for the surfaces in ydirection
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 x_{i} adds 2^{i}. 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 step15 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
.
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 1132 of file grid_generator.cc.
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 j
th 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 1281 of file grid_generator.cc.
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 dimdimensional 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.
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.
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. 
Definition at line 1769 of file grid_generator.cc.
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 756 of file grid_generator.cc.
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().
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().
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.Definition at line 816 of file grid_generator.cc.
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().
Definition at line 832 of file grid_generator.cc.
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().
Definition at line 851 of file grid_generator.cc.
void GridGenerator::subdivided_parallelepiped  (  Triangulation< dim, spacedim > &  tria, 
const Point< spacedim > &  origin,  
const std_cxx11::array< Tensor< 1, spacedim >, dim > &  edges,  
const std::vector< unsigned int > &  subdivisions = std::vector<unsigned int>() , 

const bool  colorize = false 

) 
A subdivided parallelepiped.
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. 
dim
and spacedim
.Definition at line 882 of file grid_generator.cc.
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 (+/) xdirection: 1/2. In ydirection 4/8, in zdirection 16/32. The cells at corners and edges (3d) get these values bitwise or'd.
Presently only available in 2d and 3d.
void GridGenerator::hyper_ball  (  Triangulation< dim > &  tria, 
const Point< dim > &  center = Point< dim >() , 

const double  radius = 1. 

) 
Initialize the given triangulation with 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. 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.
You should attach a SphericalManifold to the cells and faces for correct placement of vertices upon refinement and to be able to use higher order mappings.
void GridGenerator::hyper_sphere  (  Triangulation< dim, spacedim > &  tria, 
const Point< spacedim > &  center = Point<spacedim>() , 

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.
You should attach a SphericalManifold to the cells and faces for correct placement of vertices upon refinement and to be able to use higher order mappings.
The following pictures are generated with:
See the documentation module on manifolds for more details.
Definition at line 3065 of file grid_generator.cc.
void GridGenerator::quarter_hyper_ball  (  Triangulation< dim > &  tria, 
const Point< dim > &  center = Point< dim >() , 

const double  radius = 1. 

) 
This class produces a hyperball 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 appropriate boundary class is HyperBallBoundary.
void GridGenerator::half_hyper_ball  (  Triangulation< dim > &  tria, 
const Point< dim > &  center = Point< dim >() , 

const double  radius = 1. 

) 
This class produces a half hyperball around center
, which contains four elements in 2d and 6 in 3d. The cut plane is perpendicular to the xaxis.
The boundary indicators for the final triangulation are 0 for the curved boundary and 1 for the cut plane.
The appropriate boundary class is HalfHyperBallBoundary, or HyperBallBoundary.
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 yzplane
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.
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 xaxis. The cone extends from x=half_length
to x=half_length
and its projection into the yzplane
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.
An example of use can be found in the documentation of the ConeBoundary class, with which you probably want to associate boundary indicator 0 (the hull of the cone).
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.
tria  A Triangulation object which has to be empty. 
sizes  A vector of integers of dimension GeometryInfo<dim>::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 Tshaped 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
Definition at line 1885 of file grid_generator.cc.
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 hyperL (in 2d or 3d) consisting of exactly 2^dim1
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., nonconvex) 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 set with growing number accordingly to the counterclockwise. Colorize option works only with 2dimensional problem. This function will create the classical Lshape in 2d and it will look like the following in 3d:
This function exists for triangulations of all space dimensions, but throws an error if called in 1d.
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 zdirection, 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.
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 hypershell, 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.
You should attach a SphericalManifold to the cells and faces for correct placement of vertices upon refinement and to be able to use higher order mappings. Alternatively, it is also possible to attach a HyperShellBoundary to the inner and outer boundary. This will create inferior meshes as described below.
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 (see below).
While the SphericalManifold, that is demonstrated in the documentation of the documentation module on manifolds, creates reasonable meshes for any number of n_cells
if attached to all cells and boundaries, the situation is less than ideal when only attaching a HyperShellBoundary. Then, only vertices on the boundaries are placed at the correct distance from the center. As an example, the 3d meshes give rise to the following meshes upon one refinement:
Neither of these meshes is particularly good since one ends up with poorly shaped cells at the inner edge upon refinement. For example, this is the middle plane of the mesh for the n_cells=6
:
The mesh generated with n_cells=12
is better but still not good. As a consequence, you may also specify n_cells=96
as a third option. The mesh generated in this way is based on a once refined version of the one with n_cells=12
, where all internal nodes are replaced along a shell somewhere between the inner and outer boundary of the domain. The following two images compare half of the hyper shell for n_cells=12
and n_cells=96
(note that the doubled radial lines on the cross section are artifacts of the visualization):
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 hypershell, 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 nonnegative 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.
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 hypershell 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.
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
.
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. By default, the boundary faces will have manifold id 0 and you should attach a TorusManifold to it. The cells will have manifold id 1 and you should attach a SphericalManifold to it.
If dim
is 2, the mesh will describe the surface of the torus. All cells and faces will have manifold id 0 and you should attach a TorusManifold to it.
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. 
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 xyplane with a circular hole in the middle. Square and circle are centered at the origin. In 3d, this geometry is extruded in \(z\) direction to the interval \([0,L]\).
It is implemented in 2d and 3d, and takes the following arguments:
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 zdirection (only used in 3d). 
repetitions  Number of subdivisions along the zdirection . 
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 moebiusloop.
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. 
void GridGenerator::merge_triangulations  (  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 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 step35 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).
Definition at line 3872 of file grid_generator.cc.
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.
Definition at line 3930 of file grid_generator.cc.
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.
[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 . 
Definition at line 3986 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.
input
must be a coarse mesh that has no refined cells. Definition at line 4030 of file grid_generator.cc.
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.
Definition at line 4376 of file grid_generator.cc.
std::map< typename MeshType< dim1, spacedim >::cell_iterator, typename MeshType< dim, spacedim >::face_iterator > GridGenerator::extract_boundary_mesh  (  const MeshType< dim, spacedim > &  volume_mesh, 
MeshType< dim1, spacedim > &  surface_mesh,  
const std::set< types::boundary_id > &  boundary_ids = std::set<types::boundary_id>() 

) 
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 step38.
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.
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 rebuild 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 higherdimensional space. The dimension of the cells of the surface mesh is consequently dim1. 
spacedim  The dimension of the space in which both the volume and the surface mesh live. 
[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. 
Definition at line 4467 of file grid_generator.cc.