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Reference documentation for deal.II version 8.4.2
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Reference documentation for deal.II version 8.4.2
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#include <deal.II/grid/manifold.h>
Public Member Functions | |
| ChartManifold (const Point< chartdim > periodicity=Point< chartdim >()) | |
| virtual | ~ChartManifold () |
| virtual Point< spacedim > | get_new_point (const Quadrature< spacedim > &quad) const |
| virtual Point< chartdim > | pull_back (const Point< spacedim > &space_point) const =0 |
| virtual Point< spacedim > | push_forward (const Point< chartdim > &chart_point) const =0 |
 Public Member Functions inherited from Manifold< dim, spacedim > | |
| virtual | ~Manifold () |
| virtual Point< spacedim > | project_to_manifold (const std::vector< Point< spacedim > > &surrounding_points, const Point< spacedim > &candidate) 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_quad (const typename Triangulation< dim, spacedim >::quad_iterator &quad) 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 |
| Public Member Functions inherited from Subscriptor | |
| Subscriptor () | |
| Subscriptor (const Subscriptor &) | |
| virtual | ~Subscriptor () |
| Subscriptor & | operator= (const Subscriptor &) |
| void | subscribe (const char *identifier=0) const |
| void | unsubscribe (const char *identifier=0) const |
| unsigned int | n_subscriptions () const |
| void | list_subscribers () const |
| DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects."<< "\"<< "(Additional information: "<< arg3<< ")\"<< "See the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "a lot more information on what this error means and "<< "how to fix programs in which it happens.") | |
| DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier <"<< arg2<< "> subscribes to this object of class "<< arg1<< ". Consequently, it cannot be unsubscribed.") | |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
Private Attributes | |
| const FlatManifold< dim, chartdim > | sub_manifold |
This class describes mappings that can be expressed in terms of charts. Specifically, this class with its template arguments describes a chart of dimension chartdim, which is part of a Manifold<dim,spacedim> and is used in an object of type Triangulation<dim,spacedim>: It specializes a Manifold of dimension chartdim embedded in a manifold of dimension spacedim, for which you have explicit pull_back() and push_forward() transformations. Its use is explained in great detail in step-53.
This is a helper class which is useful when you have an explicit map from an Euclidean space of dimension chartdim to an Euclidean space of dimension spacedim which represents your manifold, i.e., when your manifold 
can be represented by a map
![\[ F: \mathcal{B} \subset R^{\text{chartdim}} \mapsto \mathcal{M} \subset R^{\text{spacedim}} \]](form_695.png)
(the push_forward() function) and that admits the inverse transformation
![\[ F^{-1}: \mathcal{M} \subset R^{\text{spacedim}} \mapsto \mathcal{B} \subset R^{\text{chartdim}} \]](form_696.png)
(the pull_back() function).
The get_new_point() function of the ChartManifold class is implemented by calling the pull_back() method for all surrounding_points, computing their weighted average in the chartdim Euclidean space, and calling the push_forward() method with the resulting point, i.e.,
![\[ p^{\text{new}} = F(\sum_i w_i F^{-1}(p_i)). \]](form_697.png)
Derived classes are required to implement the push_forward() and the pull_back() methods. All other functions required by mappings will then be provided by this class.
The dimension arguments chartdim, dim and spacedim must satisfy the following relationships:
However, there is no a priori relationship between dim and chartdim. For example, if you want to describe a mapping for an edge (a 1d object) in a 2d triangulation embedded in 3d space, you could do so by parameterizing it via a line
![\[ F: [0,1] \rightarrow {\mathbb R}^3 \]](form_698.png)
in which case chartdim is 1. On the other hand, there is no reason why one can't describe this as a mapping
![\[ F: {\mathbb R}^3 \rightarrow {\mathbb R}^3 \]](form_699.png)
in such a way that the line ![$[0,1]\times \{0\}\times \{0\}$](form_700.png)
happens to be mapped onto the edge in question. Here, chartdim is 3. This may seem cumbersome but satisfies the requirements of an invertible function 
just fine as long as it is possible to get from the edge to the pull-back space and then back again. Finally, given that we are dealing with a 2d triangulation in 3d, one will often have a mapping from, say, the 2d unit square or unit disk to the domain in 3d space, and the edge in question may simply be the mapped edge of the unit domain in 2d space. In this case, chartdim is 2.
Definition at line 377 of file manifold.h.
| ChartManifold< dim, spacedim, chartdim >::ChartManifold | ( | const Point< chartdim > | periodicity = Point<chartdim>() | ) |
Constructor. The optional argument can be used to specify the periodicity of the chartdim-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., then 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 the manifold, these two points are at distance 2*eps and not (2*pi-eps)
Definition at line 286 of file manifold.cc.
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virtual |
Destructor. Does nothing here, but needs to be declared to make it virtual.
Definition at line 282 of file manifold.cc.
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virtual |
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, spacedim >.
Reimplemented in SphericalManifold< dim, spacedim >.
Definition at line 294 of file manifold.cc.
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pure virtual |
Pull back the given point in spacedim to the Euclidean chartdim dimensional space.
Refer to the general documentation of this class for more information.
Implemented in OpenCASCADE::ArclengthProjectionLineManifold< dim, spacedim >, FunctionManifold< dim, spacedim, chartdim >, and SphericalManifold< dim, spacedim >.
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pure virtual |
Given a point in the chartdim dimensional Euclidean space, this method returns a point on the manifold embedded in the spacedim Euclidean space.
Refer to the general documentation of this class for more information.
Implemented in FunctionManifold< dim, spacedim, chartdim >.
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private |
The sub_manifold object is used to compute the average of the points in the chart coordinates system.
Definition at line 433 of file manifold.h.
1.8.12