Reference documentation for deal.II version Git 6454e3f 2017-06-26 22:46:01 +0200
FlatManifold< dim, spacedim > Class Template Reference

#include <deal.II/grid/manifold.h>

Inheritance diagram for FlatManifold< dim, spacedim >:
[legend]

## Public Member Functions

FlatManifold (const Tensor< 1, spacedim > &periodicity=Tensor< 1, spacedim >(), const double tolerance=1e-10)

virtual Point< spacedim > get_new_point (const std::vector< Point< spacedim > > &surrounding_points, const std::vector< double > &weights) const

virtual void add_new_points (const std::vector< Point< spacedim > > &surrounding_points, const Table< 2, double > &weights, std::vector< Point< spacedim > > &new_points) const

virtual Point< spacedim > project_to_manifold (const std::vector< Point< spacedim > > &points, const Point< spacedim > &candidate) const

virtual Tensor< 1, spacedim > get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) const

const Tensor< 1, spacedim > & get_periodicity () const

Public Member Functions inherited from Manifold< dim, spacedim >
virtual ~Manifold ()

virtual Point< spacedim > get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const

virtual Point< spacedim > get_new_point_on_line (const typename Triangulation< dim, spacedim >::line_iterator &line) const

virtual Point< spacedim > get_new_point_on_hex (const typename Triangulation< dim, spacedim >::hex_iterator &hex) const

Point< spacedim > get_new_point_on_face (const typename Triangulation< dim, spacedim >::face_iterator &face) const

Point< spacedim > get_new_point_on_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const

virtual Tensor< 1, spacedim > normal_vector (const typename Triangulation< dim, spacedim >::face_iterator &face, const Point< spacedim > &p) const

virtual void get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, FaceVertexNormals &face_vertex_normals) const

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&)

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&)

void subscribe (const char *identifier=nullptr) const

void unsubscribe (const char *identifier=nullptr) const

unsigned int n_subscriptions () const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

## Static Private Member Functions

static::ExceptionBase & ExcPeriodicBox (int arg1, Point< spacedim > arg2, double arg3)

## Private Attributes

const Tensor< 1, spacedim > periodicity

const double tolerance

Public Types inherited from Manifold< dim, spacedim >
typedef Tensor< 1, spacedim > FaceVertexNormals[GeometryInfo< dim >::vertices_per_face]

Static Public Member Functions inherited from Subscriptor
static::ExceptionBase & ExcInUse (int arg1, char *arg2, std::string &arg3)

static::ExceptionBase & ExcNoSubscriber (char *arg1, char *arg2)

## Detailed Description

### template<int dim, int spacedim = dim> class FlatManifold< dim, spacedim >

Specialization of Manifold<dim,spacedim>, which represent a possibly periodic Euclidean space of dimension dim embedded in the Euclidean space of spacedim dimensions. The main characteristic of this Manifold is the fact that the function FlatManifold<dim,spacedim>::project_to_manifold() is the identity function.

Definition at line 651 of file manifold.h.

## Constructor & Destructor Documentation

template<int dim, int spacedim>
 FlatManifold< dim, spacedim >::FlatManifold ( const Tensor< 1, spacedim > & periodicity = Tensor<1,spacedim>(), const double tolerance = 1e-10 )

Default constructor. The optional argument can be used to specify the periodicity of the spacedim-dimensional manifold (one period per direction). A periodicity value of zero means that along that direction there is no periodicity. By default no periodicity is assumed.

Periodicity affects the way a middle point is computed. It is assumed that if two points are more than half period distant, then the distance should be computed by crossing the periodicity boundary, i.e., the average is computed by adding a full period to the sum of the two. For example, if along direction 0 we have 2*pi periodicity, then the average of (2*pi-eps) and (eps) is not pi, but 2*pi (or zero), since, on a periodic manifold, these two points are at distance 2*eps and not (2*pi- eps). Special cases are taken into account, to ensure that the behavior is always as expected. The third argument is used as a relative tolerance when computing distances.

Periodicity will be intended in the following way: the domain is considered to be the box contained in [Point<spacedim>(), periodicity) where the right extreme is excluded. If any of the components of this box has zero length, then no periodicity is assumed in that direction. Whenever a function that tries to compute averages is called, an exception will be thrown if one of the points which you are using for the average lies outside the periodicity box. The return points are guaranteed to lie in the periodicity box plus or minus tolerance*periodicity.norm().

Definition at line 535 of file manifold.cc.

## Member Function Documentation

template<int dim, int spacedim>
 Point< spacedim > FlatManifold< dim, spacedim >::get_new_point ( const std::vector< Point< spacedim > > & surrounding_points, const std::vector< double > & weights ) const
virtual

Let the new point be the average sum of surrounding vertices.

This particular implementation constructs the weighted average of the surrounding points, and then calls internally the function project_to_manifold(). The reason why we do it this way, is to allow lazy programmers to implement only the project_to_manifold() function for their own Manifold classes which are small (or trivial) perturbations of a flat manifold. This is the case whenever the coarse mesh is a decent approximation of the manifold geometry. In this case, the middle point of a cell is close to true middle point of the manifold, and a projection may suffice.

For most simple geometries, it is possible to get reasonable results by deriving your own Manifold class from FlatManifold, and write a new interface only for the project_to_manifold function. You will have good approximations also with large deformations, as long as in the coarsest mesh size you are trying to refine, the middle point is not too far from the manifold mid point, i.e., as long as the coarse mesh size is small enough.

Reimplemented from Manifold< dim, spacedim >.

Definition at line 547 of file manifold.cc.

template<int dim, int spacedim>
 void FlatManifold< dim, spacedim >::add_new_points ( const std::vector< Point< spacedim > > & surrounding_points, const Table< 2, double > & weights, std::vector< Point< spacedim > > & new_points ) const
virtual

Compute a new set of points that interpolate between the given points surrounding_points. weights is a table with as many columns as surrounding_points.size(). The number of rows in weights determines how many new points will be computed and appended to the last input argument new_points. After exit of this function, the size of new_points equals the size at entry plus the number of rows in weights.

For this particular implementation, the interpolation of the surrounding_points according to the weights is simply performed in Cartesian space.

Reimplemented from Manifold< dim, spacedim >.

Definition at line 602 of file manifold.cc.

template<int dim, int spacedim>
 Point< spacedim > FlatManifold< dim, spacedim >::project_to_manifold ( const std::vector< Point< spacedim > > & points, const Point< spacedim > & candidate ) const
virtual

Project to FlatManifold. This is the identity function for flat, Euclidean spaces. Note however that this function can be overloaded by derived classes, which will then benefit from the logic behind the get_new_point() function which are often very similar (if not identical) to the one implemented in this class.

Reimplemented from Manifold< dim, spacedim >.

Definition at line 670 of file manifold.cc.

template<int dim, int spacedim>
 Tensor< 1, spacedim > FlatManifold< dim, spacedim >::get_tangent_vector ( const Point< spacedim > & x1, const Point< spacedim > & x2 ) const
virtual

Return a vector that, at $$\mathbf x_1$$, is tangential to the geodesic that connects two points $$\mathbf x_1,\mathbf x_2$$. For the current class, we assume that the manifold is flat, so the geodesic is the straight line between the two points, and we return $$\mathbf x_2-\mathbf x_1$$. The normalization of the vector is chosen so that it fits the convention described in Manifold::get_tangent_vector().

Note
If you use this class as a stepping stone to build a manifold that only "slightly" deviates from a flat manifold, by overloading the project_to_manifold() function.
Parameters
 x1 The first point that describes the geodesic, and the one at which the "direction" is to be evaluated. x2 The second point that describes the geodesic.
Returns
A "direction" vector tangential to the geodesic. Here, this is $$\mathbf x_2-\mathbf x_1$$, possibly modified by the periodicity of the domain as set in the constructor, to use the "shortest" connection between the points through the periodic boundary as necessary.

Reimplemented from Manifold< dim, spacedim >.

Definition at line 689 of file manifold.cc.

template<int dim, int spacedim>
 const Tensor< 1, spacedim > & FlatManifold< dim, spacedim >::get_periodicity ( ) const

Return the periodicity of this Manifold.

Definition at line 680 of file manifold.cc.

## Member Data Documentation

template<int dim, int spacedim = dim>
 const Tensor<1,spacedim> FlatManifold< dim, spacedim >::periodicity
private

The periodicity of this Manifold. Periodicity affects the way a middle point is computed. It is assumed that if two points are more than half period distant, then the distance should be computed by crossing the periodicity boundary, i.e., the average is computed by adding a full period to the sum of the two. For example, if along direction 0 we have 2*pi periodicity, then the average of (2*pi-eps) and (eps) is not pi, but 2*pi (or zero), since, on a periodic manifold, these two points are at distance 2*eps and not (2*pi-eps).

A periodicity 0 along one direction means no periodicity. This is the default value for all directions.

Definition at line 786 of file manifold.h.

template<int dim, int spacedim = dim>
 const double FlatManifold< dim, spacedim >::tolerance
private

Relative tolerance. This tolerance is used to compute distances in double precision.

Definition at line 796 of file manifold.h.

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