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Reference documentation for deal.II version 9.4.1
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Reference documentation for deal.II version 9.4.1
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#include <deal.II/grid/manifold_lib.h>
Public Types | |
| using | FaceVertexNormals = std::array< Tensor< 1, spacedim >, GeometryInfo< dim >::vertices_per_face > |
Public Member Functions | |
| CylindricalManifold (const unsigned int axis=0, const double tolerance=1e-10) | |
| CylindricalManifold (const Tensor< 1, spacedim > &direction, const Point< spacedim > &point_on_axis, const double tolerance=1e-10) | |
| virtual std::unique_ptr< Manifold< dim, spacedim > > | clone () const override |
| virtual Point< 3 > | pull_back (const Point< spacedim > &space_point) const override |
| virtual Point< spacedim > | push_forward (const Point< 3 > &chart_point) const override |
| virtual DerivativeForm< 1, 3, spacedim > | push_forward_gradient (const Point< 3 > &chart_point) const override |
| virtual Point< spacedim > | get_new_point (const ArrayView< const Point< spacedim > > &surrounding_points, const ArrayView< const double > &weights) const override |
| virtual Point< spacedim > | get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const override |
| virtual void | get_new_points (const ArrayView< const Point< spacedim > > &surrounding_points, const Table< 2, double > &weights, ArrayView< Point< spacedim > > new_points) const override |
| virtual Point< spacedim > | push_forward (const Point< chartdim > &chart_point) const =0 |
| virtual DerivativeForm< 1, chartdim, spacedim > | push_forward_gradient (const Point< chartdim > &chart_point) const |
| virtual Tensor< 1, spacedim > | get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) const override |
| const Tensor< 1, chartdim > & | get_periodicity () const |
| Point< 1 > | get_new_point_on_quad (const Triangulation< 1, 1 >::quad_iterator &) const |
| Point< 2 > | get_new_point_on_quad (const Triangulation< 1, 2 >::quad_iterator &) const |
| Point< 3 > | get_new_point_on_quad (const Triangulation< 1, 3 >::quad_iterator &) const |
| Point< 3 > | get_new_point_on_hex (const Triangulation< 3, 3 >::hex_iterator &hex) const |
| Point< 1 > | get_new_point_on_face (const Triangulation< 1, 1 >::face_iterator &) const |
| Point< 2 > | get_new_point_on_face (const Triangulation< 1, 2 >::face_iterator &) const |
| Point< 3 > | get_new_point_on_face (const Triangulation< 1, 3 >::face_iterator &) const |
| Tensor< 1, 2 > | normal_vector (const Triangulation< 2, 2 >::face_iterator &face, const Point< 2 > &p) const |
| Tensor< 1, 3 > | normal_vector (const Triangulation< 3, 3 >::face_iterator &face, const Point< 3 > &p) const |
| void | get_normals_at_vertices (const Triangulation< 2, 2 >::face_iterator &face, FaceVertexNormals &n) const |
| void | get_normals_at_vertices (const Triangulation< 3, 3 >::face_iterator &face, FaceVertexNormals &n) const |
Computing the location of points. | |
| virtual Point< spacedim > | project_to_manifold (const ArrayView< const 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 |
Computing normal vectors | |
| 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 |
Private Attributes | |
| const Tensor< 1, spacedim > | normal_direction |
| const Tensor< 1, spacedim > | direction |
| const Point< spacedim > | point_on_axis |
| const double | tolerance |
| const Tensor< 1, spacedim > | dxn |
| const FlatManifold< chartdim, chartdim > | sub_manifold |
Subscriptor functionality | |
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class. | |
| std::atomic< unsigned int > | counter |
| std::map< std::string, unsigned int > | counter_map |
| std::vector< std::atomic< bool > * > | validity_pointers |
| const std::type_info * | object_info |
| void | subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
| void | unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
| unsigned int | n_subscriptions () const |
| template<typename StreamType > | |
| void | list_subscribers (StreamType &stream) const |
| void | list_subscribers () const |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
| using | map_value_type = decltype(counter_map)::value_type |
| using | map_iterator = decltype(counter_map)::iterator |
| static std::mutex | mutex |
| static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
| static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
| void | check_no_subscribers () const noexcept |
Cylindrical Manifold description. In three dimensions, points are transformed using a cylindrical coordinate system along the x-, y- or z-axis (when using the first constructor of this class), or an arbitrarily oriented cylinder described by the direction of its axis and a point located on the axis.
This class was developed to be used in conjunction with the cylinder or cylinder_shell functions of GridGenerator. This function will throw a run time exception whenever spacedim is not equal to three.
Definition at line 390 of file manifold_lib.h.
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inherited |
Type keeping information about the normals at the vertices of a face of a cell. Thus, there are GeometryInfo<dim>::vertices_per_face normal vectors, that define the tangent spaces of the boundary at the vertices. Note that the vectors stored in this object are not required to be normalized, nor to actually point outward, as one often will only want to check for orthogonality to define the tangent plane; if a function requires the normals to be normalized, then it must do so itself.
For obvious reasons, this type is not useful in 1d.
Definition at line 306 of file manifold.h.
| CylindricalManifold< dim, spacedim >::CylindricalManifold | ( | const unsigned int | axis = 0, |
| const double | tolerance = 1e-10 |
||
| ) |
Constructor. Using default values for the constructor arguments yields a cylinder along the x-axis (axis=0). Choose axis=1 or axis=2 for a tube along the y- or z-axis, respectively. The tolerance value is used to determine if a point is on the axis.
Definition at line 1045 of file manifold_lib.cc.
| CylindricalManifold< dim, spacedim >::CylindricalManifold | ( | const Tensor< 1, spacedim > & | direction, |
| const Point< spacedim > & | point_on_axis, | ||
| const double | tolerance = 1e-10 |
||
| ) |
Constructor. If constructed with this constructor, the manifold described is a cylinder with an axis that points in direction direction and goes through the given point_on_axis. The direction may be arbitrarily scaled, and the given point may be any point on the axis. The tolerance value is used to determine if a point is on the axis.
Definition at line 1060 of file manifold_lib.cc.
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overridevirtual |
Make a clone of this Manifold object.
Implements Manifold< dim, spacedim >.
Definition at line 1081 of file manifold_lib.cc.
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overridevirtual |
Compute the cylindrical coordinates \((r, \phi, \lambda)\) for the given space point where \(r\) denotes the distance from the axis, \(\phi\) the angle between the given point and the computed normal direction, and \(\lambda\) the axial position.
Implements ChartManifold< dim, spacedim, chartdim >.
Definition at line 1119 of file manifold_lib.cc.
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overridevirtual |
Compute the Cartesian coordinates for a chart point given in cylindrical coordinates \((r, \phi, \lambda)\), where \(r\) denotes the distance from the axis, \(\phi\) the angle between the given point and the computed normal direction, and \(\lambda\) the axial position.
Definition at line 1145 of file manifold_lib.cc.
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overridevirtual |
Compute the derivatives of the mapping from cylindrical coordinates \((r, \phi, \lambda)\) to cartesian coordinates where \(r\) denotes the distance from the axis, \(\phi\) the angle between the given point and the computed normal direction, and \(\lambda\) the axial position.
Definition at line 1165 of file manifold_lib.cc.
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overridevirtual |
Compute new points on the CylindricalManifold. See the documentation of the base class for a detailed description of what this function does.
Reimplemented from ChartManifold< dim, spacedim, chartdim >.
Definition at line 1090 of file manifold_lib.cc.
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overridevirtualinherited |
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, spacedim >.
Definition at line 984 of file manifold.cc.
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overridevirtualinherited |
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 must match the length of new_points.
The implementation of this function first transforms the surrounding_points to the chart space by calling pull_back(). Then, new points are computed on the chart by usual interpolation according to the given weights, which are finally transformed to the image space by push_forward().
This implementation can be much more efficient for computing multiple new points from the same surrounding points than separate calls to get_new_point() in case the pull_back() operation is expensive. This is because pull_back() is only called once for the surrounding points and the interpolation is done for all given weights using this set of points. Often, pull_back() is also more expensive than push_forward() because the former might involve some kind of Newton iteration in non-trivial manifolds.
Reimplemented from Manifold< dim, spacedim >.
Definition at line 1020 of file manifold.cc.
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pure virtualinherited |
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 CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >, FunctionManifold< dim, spacedim, chartdim >, and TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >.
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virtualinherited |
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function \(F\) that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size \(\text{spacedim}\times\text{chartdim}\).
This function is used in the computations required by the get_tangent_vector() function. Since not all users of the Manifold class interface will require calling that function, the current function is implemented but will trigger an exception whenever called. This allows derived classes to avoid implementing the push_forward_gradient function if this functionality is not needed in the user program.
Refer to the general documentation of this class for more information.
Reimplemented in CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >, FunctionManifold< dim, spacedim, chartdim >, and TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >.
Definition at line 1049 of file manifold.cc.
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overridevirtualinherited |
Return a vector that, at \(\mathbf x_1\), is tangential to the geodesic that connects two points \(\mathbf x_1,\mathbf x_2\). See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are \(\xi_1=F^{-1}(\mathbf x_1), \xi_2=F^{-1}(\mathbf x_2)\), then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
\begin{align*} \zeta(t) &= \xi_1 + t (\xi_2-\xi_1) \\ &= F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right] \end{align*}
In image space, i.e., in the space in which we operate, this leads to the curve
\begin{align*} \mathbf s(t) &= F(\zeta(t)) \\ &= F(\xi_1 + t (\xi_2-\xi_1)) \\ &= F\left(F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]\right). \end{align*}
What the current function is supposed to return is \(\mathbf s'(0)\). By the chain rule, this is equal to
\begin{align*} \mathbf s'(0) &= \frac{d}{dt}\left. F\left(F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]\right) \right|_{t=0} \\ &= \nabla_\xi F\left(F^{-1}(\mathbf x_1)\right) \left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]. \end{align*}
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
Thus, the computation of tangent vectors also requires the implementation of derivatives \(\nabla_\xi F(\xi)\) of the push-forward mapping. Here, \(F^{-1}(\mathbf x_2)-F^{-1}(\mathbf x_1)\) is a chartdim-dimensional vector, and \(\nabla_\xi F\left(F^{-1}(\mathbf x_1)\right) = \nabla_\xi F\left(\xi_1\right)\) is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
| 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. |
Reimplemented from Manifold< dim, spacedim >.
Definition at line 1062 of file manifold.cc.
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inherited |
Return the periodicity associated with the submanifold.
Definition at line 1094 of file manifold.cc.
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virtualinherited |
Given a point which lies close to the given manifold, it modifies it and projects it to manifold itself.
This class is used by the default implementation of the function get_new_point() and should be implemented by derived classes. The default implementation simply throws an exception if called.
If your manifold is simple, you could implement this function only, and the default behavior should work out of the box.
Reimplemented in FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, FlatManifold< dim, spacedim >, FlatManifold< dim, dim >, OpenCASCADE::NormalProjectionManifold< dim, spacedim >, OpenCASCADE::DirectionalProjectionManifold< dim, spacedim >, and OpenCASCADE::NormalToMeshProjectionManifold< dim, spacedim >.
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virtualinherited |
Backward compatibility interface. Return the point which shall become the new middle vertex of the two children of a regular line. In 2D, this line is a line at the boundary, while in 3d, it is bounding a face at the boundary (the lines therefore is also on the boundary).
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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virtualinherited |
Backward compatibility interface. Return the point which shall become the common point of the four children of a quad at the boundary in three or more spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.
This function is called after the four lines bounding the given quad are refined, so you may want to use the information provided by quad->line(i)->child(j), i=0...3, j=0,1.
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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inherited |
Definition at line 418 of file manifold.cc.
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inherited |
Definition at line 429 of file manifold.cc.
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inherited |
Definition at line 440 of file manifold.cc.
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virtualinherited |
Backward compatibility interface. Return the point which shall become the common point of the eight children of a hex in three or spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.
This function is called after the all the bounding objects of the given hex are refined, so you may want to use the information provided by hex->quad(i)->line(j)->child(k), i=0...5, j=0...3, k=0,1.
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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inherited |
Definition at line 462 of file manifold.cc.
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inherited |
Backward compatibility interface. Depending on dim=2 or dim=3 this function calls the get_new_point_on_line or the get_new_point_on_quad function. It throws an exception for dim=1. This wrapper allows dimension independent programming.
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inherited |
Definition at line 385 of file manifold.cc.
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inherited |
Definition at line 396 of file manifold.cc.
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inherited |
Definition at line 407 of file manifold.cc.
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inherited |
Backward compatibility interface. Depending on dim=1, dim=2 or dim=3 this function calls the get_new_point_on_line, get_new_point_on_quad or the get_new_point_on_hex function. This wrapper allows dimension independent programming.
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virtualinherited |
Return the normal vector to a face embedded in this manifold, at the point p. If p is not in fact on the surface, but only close-by, try to return something reasonable, for example the normal vector at the surface point closest to p. (The point p will in fact not normally lie on the actual surface, but rather be a quadrature point mapped by some polynomial mapping; the mapped surface, however, will not usually coincide with the actual surface.)
This function only makes sense if dim==spacedim because otherwise there is no unique normal vector but in fact a (spacedim-dim+1)-dimensional tangent space of vectors that are all both normal to the face and normal to the dim-dimensional surface that lives in spacedim-dimensional space. For example, think of a two-dimensional mesh that covers a two-dimensional surface in three-dimensional space. In that case, each face (edge) is one-dimensional, and there are two linearly independent vectors that are both normal to the edge: one is normal to the edge and tangent to the surface (intuitively, that would be the one that points from the current cell to the neighboring one, if the surface was locally flat), and the other one is rooted in the edge but points perpendicular to the surface (which is also perpendicular to the edge that lives within the surface). Thus, because there are no obviously correct semantics for this function if spacedim is greater than dim, the function will simply throw an error in that situation.
The face iterator gives an indication which face this function is supposed to compute the normal vector for. This is useful if the boundary of the domain is composed of different nondifferential pieces (for example when using the FlatManifold class to approximate a geometry that is completely described by the coarse mesh, with piecewise (bi-)linear components between the vertices, but where the boundary may have a kink at the vertices itself).
Reimplemented in FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, FlatManifold< dim, spacedim >, FlatManifold< dim, dim >, PolarManifold< dim, spacedim >, and SphericalManifold< dim, spacedim >.
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inherited |
Definition at line 144 of file manifold.cc.
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inherited |
Definition at line 165 of file manifold.cc.
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virtualinherited |
Compute the normal vectors to the boundary at each vertex of the given face embedded in the Manifold. It is not required that the normal vectors be normed somehow. Neither is it required that the normals actually point outward.
This function is needed to compute data for C1 mappings. The default implementation calls normal_vector() on each vertex.
Note that when computing normal vectors at a vertex where the boundary is not differentiable, you have to make sure that you compute the one-sided limits, i.e. limit with respect to points inside the given face.
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inherited |
Definition at line 250 of file manifold.cc.
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inherited |
Definition at line 272 of file manifold.cc.
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private |
A vector orthogonal to the normal direction.
Definition at line 458 of file manifold_lib.h.
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private |
The direction vector of the axis.
Definition at line 463 of file manifold_lib.h.
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private |
An arbitrary point on the axis.
Definition at line 468 of file manifold_lib.h.
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private |
Relative tolerance to measure zero distances.
Definition at line 473 of file manifold_lib.h.
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private |
The direction vector perpendicular to both direction and normal_direction.
Definition at line 478 of file manifold_lib.h.
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privateinherited |
The sub_manifold object is used to compute the average of the points in the chart coordinates system.
In an ideal world, it would have type FlatManifold<dim,chartdim>. However, this would instantiate cases where dim>spacedim, which leads to invalid situations. We instead use <chartdim,chartdim>, which is (i) always valid, and (ii) does not matter at all since the first (dim) argument of manifolds is, in fact, ignored as far as manifold functionality is concerned.
Definition at line 1085 of file manifold.h.
1.9.5