GridGenerator Class Reference
[Grid classes]

List of all members.

Classes

class  ExcInvalidRadii
class  ExcInvalidRepetitions
class  ExcInvalidRepetitionsDimension

Static Public Member Functions

template<int dim, int spacedim>
static void hyper_cube (Triangulation< dim, spacedim > &tria, const double left=0., const double right=1.)
template<int dim>
static void subdivided_hyper_cube (Triangulation< dim > &tria, const unsigned int repetitions, const double left=0., const double right=1.)
template<int dim, int spacedim>
static void hyper_rectangle (Triangulation< dim, spacedim > &tria, const Point< spacedim > &p1, const Point< spacedim > &p2, const bool colorize=false)
template<int dim>
static void subdivided_hyper_rectangle (Triangulation< dim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)
template<int dim>
static 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)
template<int dim>
static void subdivided_hyper_rectangle (Triangulation< dim > &tria, const std::vector< std::vector< double > > &spacing, const Point< dim > &p, const Table< dim, unsigned char > &material_id, const bool colorize=false)
template<int dim>
static void parallelogram (Triangulation< dim > &tria, const Tensor< 2, dim > &corners, const bool colorize=false)
template<int dim>
static 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>
static void hyper_ball (Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1.)
template<int dim>
static void half_hyper_ball (Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1.)
template<int dim>
static void cylinder (Triangulation< dim > &tria, const double radius=1., const double half_length=1.)
template<int dim>
static void hyper_L (Triangulation< dim > &tria, const double left=-1., const double right=1.)
template<int dim>
static void hyper_cube_slit (Triangulation< dim > &tria, const double left=0., const double right=1., const bool colorize=false)
template<int dim>
static 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>
static 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)
template<int dim>
static 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>
static 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 repetition=1, const bool colorize=false)
static void moebius (Triangulation< 3, 3 > &tria, const unsigned int n_cells, const unsigned int n_rotations, const double R, const double r)
template<int dim>
static void laplace_transformation (Triangulation< dim > &tria, const std::map< unsigned int, Point< dim > > &new_points)

Static Private Member Functions

template<int dim, int spacedim>
static void colorize_hyper_rectangle (Triangulation< dim, spacedim > &tria)
template<int dim>
static void colorize_subdivided_hyper_rectangle (Triangulation< dim > &tria, const Point< dim > &p1, const Point< dim > &p2, const double epsilon)
template<int dim>
static void colorize_hyper_shell (Triangulation< dim > &tria, const Point< dim > &center, const double inner_radius, const double outer_radius)
static void laplace_solve (const SparseMatrix< double > &S, const std::map< unsigned int, double > &m, Vector< double > &u)


Detailed Description

This class provides a collection of functions for generating basic triangulations. Below, we try to provide some pictures in order to illustrate at least the more complex ones.

Some of these functions receive a flag colorize. If this is set, parts of the boundary receive different boundary numbers, allowing them to be distinguished by application programs. See the documentation of the functions for details.

Additionally this class provides a function (laplace_transformation) that smoothly transforms a grid according to given new boundary points. This can be used to transform (simple-shaped) grids to a more complicated ones, like a shell onto a grid of an airfoil, for example.

No meshes for the codimension one case are provided at the moment.

Author:
Wolfgang Bangerth, Ralf Hartmann, Guido Kanschat, Stefan Nauber, Joerg Weimar, Yaqi Wang, Luca Heltai, 1998, 1999, 2000, 2001, 2002, 2003, 2006, 2007, 2008.

Member Function Documentation

template<int dim, int spacedim>
static void GridGenerator::hyper_cube ( Triangulation< dim, spacedim > &  tria,
const double  left = 0.,
const double  right = 1. 
) [inline, static]

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]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.

hyper_cubes.png

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 needs to be void upon calling this function.

template<int dim>
static void GridGenerator::subdivided_hyper_cube ( Triangulation< dim > &  tria,
const unsigned int  repetitions,
const double  left = 0.,
const double  right = 1. 
) [inline, static]

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 spacedim=dim+1 the same mesh as in the case spacedim=dim is created, but the vertices have an additional coordinate =0. So, if dim=1 one obtains line along the x axis in the xy plane, and if dim=3 one obtains a square in lying in the xy plane in 3d space.

Note:
The triangulation needs to be void upon calling this function.

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

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

If the colorize flag is set, the boundary_indicators of the surfaces 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. 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.

Note:
The triangulation needs to be void upon calling this function.

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

Create a coordinate-parallel parallelepiped from the two diagonally opposite corner points p1 and p2. In dimension i, repetitions[i] cells are generated.

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

If the colorize flag is set, the boundary_indicators of the surfaces 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. 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.

Note:
The triangulation needs to be void upon calling this function.

For an example of the use of this function see the step-28 tutorial program.

template<int dim>
static 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 
) [inline, static]

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 widths 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.

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

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.

template<int dim>
static void GridGenerator::parallelogram ( Triangulation< dim > &  tria,
const Tensor< 2, dim > &  corners,
const bool  colorize = false 
) [inline, static]

A parallelogram. The first corner point is the origin. The dim adjacent points are the one-dimensional subtensors of the tensor provided and additional points will be sums of these two vectors. Colorizing is done according to hyper_rectangle().

Note:
This function is implemented in 2d only.

The triangulation needs to be void upon calling this function.

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

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 needs to be void upon calling this function.

template<int dim>
static void GridGenerator::hyper_ball ( Triangulation< dim > &  tria,
const Point< dim > &  center = Point< dim >(),
const double  radius = 1. 
) [inline, static]

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.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static void GridGenerator::half_hyper_ball ( Triangulation< dim > &  tria,
const Point< dim > &  center = Point< dim >(),
const double  radius = 1. 
) [inline, static]

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 appropriate boundary class is HalfHyperBallBoundary, or HyperBallBoundary.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static void GridGenerator::cylinder ( Triangulation< dim > &  tria,
const double  radius = 1.,
const double  half_length = 1. 
) [inline, static]

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.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static void GridGenerator::hyper_L ( Triangulation< dim > &  tria,
const double  left = -1.,
const double  right = 1. 
) [inline, static]

Initialize the given triangulation with a hyper-L consisting of exactly 2^dim-1 cells. It produces the hypercube with the interval [left,right] without the hypercube made out of the interval [(a+b)/2,b].

hyper_l.png

The triangulation needs to be void upon calling this function.

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

Note:
The triangulation needs to be void upon calling this function.

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

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.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static 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 
) [inline, static]

Produce a hyper-shell, the region between two spheres around center, with given inner_radius and outer_radius.

If the flag colorize is true, then the outer boundary will have the id 1, while the inner boundary has id zero. If the flag is false, both have id zero.

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 two different numbers are meaningful, 6 for a surface based on a hexahedron and 12 for the rhombic dodecahedron.

hypershell3d-6.png
hypershell3d-12.png

This function is declared to exist for triangulations of all space dimensions, but throws an error if called in 1d. It is also currently not implemented in 3d.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static 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 
) [inline, static]

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.

At present, this function only exists in 2d.

Note:
The triangulation needs to be void upon calling this function.

template<int dim>
static 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 
) [inline, static]

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 for this initial triangulation. 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.

The triangulation needs to be void upon calling this function.

template<int dim>
static 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  repetition = 1,
const bool  colorize = false 
) [inline, static]

This class produces a square on the xy-plane with a circular hole in the middle, times the interval [0.L] (only in 3d).

cubes_hole.png

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

  • inner_radius: size of the internal hole
  • outer_radius: size of the biggest enclosed cylinder
  • L: extension on the z-direction
  • repetitions: number of subdivisions along the z-direction
  • colorize: wether 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.

static void GridGenerator::moebius ( Triangulation< 3, 3 > &  tria,
const unsigned int  n_cells,
const unsigned int  n_rotations,
const double  R,
const double  r 
) [static]

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 glueing 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 bend together as loop.

template<int dim>
static void GridGenerator::laplace_transformation ( Triangulation< dim > &  tria,
const std::map< unsigned int, Point< dim > > &  new_points 
) [inline, static]

This function transformes the Triangulation tria smoothly to a domain that is described by the boundary points in the map new_points. This map maps the point indices to the boundary points in the transformed domain.

Note, that the Triangulation is changed in-place, therefore you don't need to keep two triangulations, but the given triangulation is changed (overwritten).

In 1d, this function is not currently implemented.

template<int dim, int spacedim>
static void GridGenerator::colorize_hyper_rectangle ( Triangulation< dim, spacedim > &  tria  )  [inline, static, private]

Perform the action specified by the colorize flag of the hyper_rectangle() function of this class.

template<int dim>
static void GridGenerator::colorize_subdivided_hyper_rectangle ( Triangulation< dim > &  tria,
const Point< dim > &  p1,
const Point< dim > &  p2,
const double  epsilon 
) [inline, static, private]

Perform the action specified by the colorize flag of the subdivided_hyper_rectangle() function of this class. This function is singled out because it is dimension specific.

template<int dim>
static void GridGenerator::colorize_hyper_shell ( Triangulation< dim > &  tria,
const Point< dim > &  center,
const double  inner_radius,
const double  outer_radius 
) [inline, static, private]

Assign boundary number zero to the inner shell boundary and 1 to the outer.

static void GridGenerator::laplace_solve ( const SparseMatrix< double > &  S,
const std::map< unsigned int, double > &  m,
Vector< double > &  u 
) [static, private]

Solve the Laplace equation for laplace_transformation function for one of the dim space dimensions. Externalized into a function of its own in order to allow parallel execution.


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

deal.II documentation generated on Sat Aug 15 16:52:01 2009 by doxygen 1.5.9