Reference documentation for deal.II version GIT b8135fa6eb 20230325 15:55:02+00:00

#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 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
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 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
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 
Static Public Member Functions  
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
Public Attributes  
const Point< spacedim >  center 
Private Types  
using  map_value_type = decltype(counter_map)::value_type 
using  map_iterator = decltype(counter_map)::iterator 
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 
void  check_no_subscribers () const noexcept 
Private Attributes  
const PolarManifold< spacedim >  polar_manifold 
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 
Static Private Attributes  
static std::mutex  mutex 
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 codimension one manifold descriptor of a spherical surface embedded in a higher dimensional space, or as a codimension 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 hypershell or hyperball 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 step6 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 socalled 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 step65.)
Definition at line 236 of file manifold_lib.h.

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.

privateinherited 
The data type used in counter_map.
Definition at line 230 of file subscriptor.h.

privateinherited 
The iterator type used in counter_map.
Definition at line 235 of file subscriptor.h.
SphericalManifold< dim, spacedim >::SphericalManifold  (  const Point< spacedim >  center = Point<spacedim>()  ) 
The Constructor takes the center of the spherical coordinates.
Definition at line 359 of file manifold_lib.cc.

overridevirtual 
Make a clone of this Manifold object.
Implements Manifold< dim, dim >.
Definition at line 369 of file manifold_lib.cc.

overridevirtual 
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 378 of file manifold_lib.cc.

overridevirtual 
Compute the derivative of the get_intermediate_point() function with parameter w equal to zero.
Reimplemented from Manifold< dim, dim >.
Definition at line 444 of file manifold_lib.cc.

overridevirtual 
Return the normal vector to a face embedded in this manifold, at the point p. It is not required that the normals actually point outward from the domain even if the face iterator given points to a face on the boundary of the domain. If p is not in fact on the surface, but only closeby, 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 (spacedimdim+1)dimensional tangent space of vectors that are all both normal to the face and normal to the dimdimensional surface that lives in spacedimdimensional space. For example, think of a twodimensional mesh that covers a twodimensional surface in threedimensional space. In that case, each face (edge) is onedimensional, 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 495 of file manifold_lib.cc.

overridevirtual 
Compute the normal vectors to the boundary at each vertex.
Definition at line 547 of file manifold_lib.cc.

overridevirtual 
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 574 of file manifold_lib.cc.

overridevirtual 
Return a point on the spherical manifold which is intermediate with respect to the surrounding points.
Reimplemented from Manifold< dim, dim >.
Definition at line 591 of file manifold_lib.cc.

private 
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 818 of file manifold_lib.cc.

private 
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 Newtonstyle 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): 95126.
and in particular the implementation provided at http://math.ucsd.edu/~sbuss/ResearchWeb/spheremean/
Definition at line 989 of file manifold_lib.cc.

privatevirtual 
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 609 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 525 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 536 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 1003 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 1016 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 1029 of file manifold_lib.cc.

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 >.
Definition at line 409 of file manifold.cc.

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().
Definition at line 427 of file manifold.cc.

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().
Definition at line 448 of file manifold.cc.

inherited 
Definition at line 453 of file manifold.cc.

inherited 
Definition at line 464 of file manifold.cc.

inherited 
Definition at line 475 of file manifold.cc.

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().
Definition at line 470 of file manifold.cc.

inherited 
Definition at line 497 of file manifold.cc.

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.
Definition at line 481 of file manifold.cc.

inherited 
Definition at line 420 of file manifold.cc.

inherited 
Definition at line 431 of file manifold.cc.

inherited 
Definition at line 442 of file manifold.cc.

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.
Definition at line 492 of file manifold.cc.

inherited 
Definition at line 143 of file manifold.cc.

inherited 
Definition at line 164 of file manifold.cc.

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 onesided limits, i.e. limit with respect to points inside the given face.
Definition at line 617 of file manifold.cc.

inherited 
Definition at line 285 of file manifold.cc.

inherited 
Definition at line 307 of file manifold.cc.

inherited 
Subscribes a user of the object by storing the pointer validity
. The subscriber may be identified by text supplied as identifier
.
Definition at line 136 of file subscriptor.cc.

inherited 
Unsubscribes a user from the object.
identifier
and the validity
pointer must be the same as the one supplied to subscribe(). Definition at line 156 of file subscriptor.cc.

inlineinherited 
Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.
Definition at line 300 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 317 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 204 of file subscriptor.cc.

inlineinherited 
Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.
This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.
Definition at line 309 of file subscriptor.h.

privatenoexceptinherited 
Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.
Definition at line 53 of file subscriptor.cc.
const Point<spacedim> SphericalManifold< dim, spacedim >::center 
The center of the spherical coordinate system.
Definition at line 317 of file manifold_lib.h.

private 
A manifold description to be used for get_new_point in 2d.
Definition at line 373 of file manifold_lib.h.

mutableprivateinherited 
Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).
The creator (and owner) of an object is counted in the map below if HE manages to supply identification.
We use the mutable
keyword in order to allow subscription to constant objects also.
This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic
class template.
Definition at line 219 of file subscriptor.h.

mutableprivateinherited 
In this map, we count subscriptions for each different identification string supplied to subscribe().
Definition at line 225 of file subscriptor.h.

mutableprivateinherited 
In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.
Definition at line 241 of file subscriptor.h.

mutableprivateinherited 
Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.
Definition at line 249 of file subscriptor.h.

staticprivateinherited 
A mutex used to ensure data consistency when printing out the list of subscribers.
Definition at line 271 of file subscriptor.h.