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Reference documentation for deal.II version 9.3.3
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Reference documentation for deal.II version 9.3.3
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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 | |
| SphericalManifold (const Point< spacedim > center=Point< spacedim >()) | |
| virtual std::unique_ptr< Manifold< dim, spacedim > > | clone () const override |
| virtual Point< spacedim > | get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const override |
| virtual Tensor< 1, spacedim > | get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) const override |
| virtual Tensor< 1, spacedim > | normal_vector (const typename Triangulation< dim, spacedim >::face_iterator &face, const Point< spacedim > &p) const override |
| virtual void | get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, typename Manifold< dim, spacedim >::FaceVertexNormals &face_vertex_normals) 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 > | get_new_point (const ArrayView< const Point< spacedim > > &vertices, const ArrayView< const double > &weights) const override |
| void | get_normals_at_vertices (const Triangulation< 1 >::face_iterator &, Manifold< 1, 1 >::FaceVertexNormals &) const |
| void | get_normals_at_vertices (const Triangulation< 1, 2 >::face_iterator &, Manifold< 1, 2 >::FaceVertexNormals &) const |
| Point< 3 > | get_new_point (const ArrayView< const Tensor< 1, 3 > > &directions, const ArrayView< const double > &distances, const ArrayView< const double > &weights, const Point< 3 > &candidate_point) const |
| Point< 3 > | get_new_point (const ArrayView< const Tensor< 1, 3 > > &directions, const ArrayView< const double > &distances, const ArrayView< const double > &weights, const Point< 3 > &candidate_point) const |
| Point< 3 > | get_new_point (const ArrayView< const Tensor< 1, 3 > > &directions, const ArrayView< const double > &distances, const ArrayView< const double > &weights, const Point< 3 > &candidate_point) 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 void | get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, FaceVertexNormals &face_vertex_normals) const |
Public Attributes | |
| const Point< spacedim > | center |
Private Member Functions | |
| std::pair< double, Tensor< 1, spacedim > > | guess_new_point (const ArrayView< const Tensor< 1, spacedim > > &directions, const ArrayView< const double > &distances, const ArrayView< const double > &weights) const |
| Point< spacedim > | get_new_point (const ArrayView< const Tensor< 1, spacedim > > &directions, const ArrayView< const double > &distances, const ArrayView< const double > &weights, const Point< spacedim > &candidate_point) const |
| virtual void | get_new_points (const ArrayView< const Point< spacedim > > &surrounding_points, const ArrayView< const double > &weights, ArrayView< Point< spacedim > > new_points) const |
Private Attributes | |
| const PolarManifold< spacedim > | polar_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) |
| void | check_no_subscribers () const noexcept |
| 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) |
Manifold description for a spherical space coordinate system.
You can use this Manifold object to describe any sphere, circle, hypersphere or hyperdisc in two or three dimensions. This manifold can be used as a co-dimension one manifold descriptor of a spherical surface embedded in a higher dimensional space, or as a co-dimension zero manifold descriptor for a body with positive volume, provided that the center of the spherical space is excluded from the domain. An example for the use of this function would be in the description of a hyper-shell or hyper-ball geometry, for example after creating a coarse mesh using GridGenerator::hyper_ball(). (However, it is worth mentioning that generating a good mesh for a disk or ball is complicated and requires addition steps. See the "Possibilities for extensions" section of step-6 for an extensive discussion of how one would construct such meshes and what one needs to do for it.)
The two template arguments match the meaning of the two template arguments in Triangulation<dim, spacedim>, however this Manifold can be used to describe both thin and thick objects, and the behavior is identical when dim <= spacedim, i.e., the functionality of SphericalManifold<2,3> is identical to SphericalManifold<3,3>.
While PolarManifold reflects the usual notion of polar coordinates, it may not be suitable for domains that contain either the north or south poles. Consider for instance the pair of points \(x_1=(1,\pi/3,0)\) and \(x_2=(1,\pi/3,\pi)\) in polar coordinates (lying on the surface of a sphere with radius one, on a parallel at height \(\pi/3\)). In this case connecting the points with a straight line in polar coordinates would take the long road around the globe, without passing through the north pole.
These two points would be connected (using a PolarManifold) by the curve
\begin{align*} s: [0,1] & \rightarrow & \mathbb S^3 \\ t & \mapsto & (1,\pi/3,0) + (0,0,t\pi) \end{align*}
This curve is not a geodesic on the sphere, and it is not how we would connect those two points. A better curve, would be the one passing through the North pole:
\[ s(t) = x_1 \cos(\alpha(t)) + \kappa \times x_1 \sin(\alpha(t)) + \kappa ( \kappa \cdot x_1) (1-\cos(\alpha(t))). \]
where \(\kappa = \frac{x_1 \times x_2}{\Vert x_1 \times x_2 \Vert}\) and \(\alpha(t) = t \cdot \arccos(x_1 \cdot x_2)\) for \(t\in[0,1]\). Indeed, this is a geodesic, and it is the natural choice when connecting points on the surface of the sphere. In the examples above, the PolarManifold class implements the first way of connecting two points on the surface of a sphere, while SphericalManifold implements the second way, i.e., this Manifold connects points using geodesics. If more than two points are involved through a SphericalManifold::get_new_points() call, a so-called spherical average is used where the final point minimizes the weighted distance to all other points via geodesics.
In particular, this class implements a Manifold that joins any two points in space by first projecting them onto the surface of a sphere with unit radius, then connecting them with a geodesic, and finally rescaling the final radius so that the resulting one is the weighted average of the starting radii. This Manifold is identical to PolarManifold in dimension two, while for dimension three it returns points that are more uniformly distributed on the sphere, and it is invariant with respect to rotations of the coordinate system, therefore avoiding the problems that PolarManifold has at the poles. Notice, in particular, that computing tangent vectors at the poles with a PolarManifold is not well defined, while it is perfectly fine with this class.
For mathematical reasons, it is impossible to construct a unique map of a sphere using only geodesic curves, and therefore, using this class with MappingManifold is discouraged. If you use this Manifold to describe the geometry of a sphere, you should use MappingQ as the underlying mapping, and not MappingManifold.
This Manifold can be used only on geometries where a ball with finite radius is removed from the center. Indeed, the center is a singular point for this manifold, and if you try to connect two points across the center, they would travel on spherical coordinates, avoiding the center.
The ideal geometry for this Manifold is an HyperShell. If you plan to use this Manifold on a HyperBall, you have to make sure you do not attach this Manifold to the cell containing the center. It is advisable to combine this class with TransfiniteInterpolationManifold to ensure a smooth transition from a curved shape to the straight coordinate system in the center of the ball. (See also the extensive discussion in step-65.)
Definition at line 236 of file manifold_lib.h.
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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.
| SphericalManifold< dim, spacedim >::SphericalManifold | ( | const Point< spacedim > | center = Point<spacedim>() | ) |
The Constructor takes the center of the spherical coordinates.
Definition at line 358 of file manifold_lib.cc.
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Make a clone of this Manifold object.
Implements Manifold< dim, dim >.
Definition at line 368 of file manifold_lib.cc.
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Given any two points in space, first project them on the surface of a sphere with unit radius, then connect them with a geodesic and find the intermediate point, and finally rescale the final radius so that the resulting one is the convex combination of the starting radii.
Reimplemented from Manifold< dim, dim >.
Definition at line 377 of file manifold_lib.cc.
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Compute the derivative of the get_intermediate_point() function with parameter w equal to zero.
Reimplemented from Manifold< dim, dim >.
Definition at line 443 of file manifold_lib.cc.
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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 from Manifold< dim, dim >.
Definition at line 494 of file manifold_lib.cc.
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Compute the normal vectors to the boundary at each vertex.
Definition at line 546 of file manifold_lib.cc.
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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.
This function is optimized to perform on a collection of new points, by collecting operations that are not dependent on the weights outside of the loop over all new points.
The implementation does not allow for surrounding_points and new_points to point to the same array, so make sure to pass different objects into the function.
Reimplemented from Manifold< dim, dim >.
Definition at line 573 of file manifold_lib.cc.
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Return a point on the spherical manifold which is intermediate with respect to the surrounding points.
Reimplemented from Manifold< dim, dim >.
Definition at line 590 of file manifold_lib.cc.
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Return a point on the spherical manifold which is intermediate with respect to the surrounding points. This function uses a linear average of the directions to find an estimated point. It returns a pair of radius and direction from the center point to the candidate point.
Definition at line 817 of file manifold_lib.cc.
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Return a point on the spherical manifold which is intermediate with respect to the surrounding points. This function uses a candidate point as guess, and performs a Newton-style iteration to compute the correct point.
The main part of the implementation uses the ideas in the publication
Buss, Samuel R., and Jay P. Fillmore. "Spherical averages and applications to spherical splines and interpolation." ACM Transactions on Graphics (TOG) 20.2 (2001): 95-126.
and in particular the implementation provided at http://math.ucsd.edu/~sbuss/ResearchWeb/spheremean/
Definition at line 988 of file manifold_lib.cc.
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Compute a new set of points that interpolate between the given points surrounding_points. weights is an array view with as many entries as surrounding_points.size() times new_points.size().
This function is optimized to perform on a collection of new points, by collecting operations that are not dependent on the weights outside of the loop over all new points.
The implementation does not allow for surrounding_points and new_points to point to the same array, so make sure to pass different objects into the function.
Definition at line 608 of file manifold_lib.cc.
| void SphericalManifold< 1, 1 >::get_normals_at_vertices | ( | const Triangulation< 1 >::face_iterator & | , |
| Manifold< 1, 1 >::FaceVertexNormals & | |||
| ) | const |
Definition at line 524 of file manifold_lib.cc.
| void SphericalManifold< 1, 2 >::get_normals_at_vertices | ( | const Triangulation< 1, 2 >::face_iterator & | , |
| Manifold< 1, 2 >::FaceVertexNormals & | |||
| ) | const |
Definition at line 535 of file manifold_lib.cc.
| Point< 3 > SphericalManifold< 1, 3 >::get_new_point | ( | const ArrayView< const Tensor< 1, 3 > > & | directions, |
| const ArrayView< const double > & | distances, | ||
| const ArrayView< const double > & | weights, | ||
| const Point< 3 > & | candidate_point | ||
| ) | const |
Definition at line 1002 of file manifold_lib.cc.
| Point< 3 > SphericalManifold< 2, 3 >::get_new_point | ( | const ArrayView< const Tensor< 1, 3 > > & | directions, |
| const ArrayView< const double > & | distances, | ||
| const ArrayView< const double > & | weights, | ||
| const Point< 3 > & | candidate_point | ||
| ) | const |
Definition at line 1015 of file manifold_lib.cc.
| Point< 3 > SphericalManifold< 3, 3 >::get_new_point | ( | const ArrayView< const Tensor< 1, 3 > > & | directions, |
| const ArrayView< const double > & | distances, | ||
| const ArrayView< const double > & | weights, | ||
| const Point< 3 > & | candidate_point | ||
| ) | const |
Definition at line 1028 of file manifold_lib.cc.
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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 >.
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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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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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Definition at line 419 of file manifold.cc.
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Definition at line 430 of file manifold.cc.
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Definition at line 441 of file manifold.cc.
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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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Definition at line 463 of file manifold.cc.
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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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Definition at line 386 of file manifold.cc.
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Definition at line 397 of file manifold.cc.
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Definition at line 408 of file manifold.cc.
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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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Definition at line 145 of file manifold.cc.
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Definition at line 166 of file manifold.cc.
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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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Definition at line 251 of file manifold.cc.
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Definition at line 273 of file manifold.cc.
| const Point<spacedim> SphericalManifold< dim, spacedim >::center |
The center of the spherical coordinate system.
Definition at line 317 of file manifold_lib.h.
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A manifold description to be used for get_new_point in 2D.
Definition at line 373 of file manifold_lib.h.
1.9.4