Reference documentation for deal.II version 9.1.1

#include <deal.II/grid/manifold.h>
Public Types  
using  FaceVertexNormals = std::array< Tensor< 1, spacedim >, GeometryInfo< dim >::vertices_per_face > 
Public Member Functions  
virtual  ~Manifold () override=default 
virtual std::unique_ptr< Manifold< dim, spacedim > >  clone () const =0 
Computing the location of points.  
virtual Point< spacedim >  get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const 
virtual Point< spacedim >  get_new_point (const ArrayView< const Point< spacedim >> &surrounding_points, const ArrayView< const double > &weights) const 
virtual void  get_new_points (const ArrayView< const Point< spacedim >> &surrounding_points, const Table< 2, double > &weights, ArrayView< Point< spacedim >> new_points) const 
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 tangent vectors  
virtual Tensor< 1, spacedim >  get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) 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 
Public Member Functions inherited from Subscriptor  
Subscriptor ()  
Subscriptor (const Subscriptor &)  
Subscriptor (Subscriptor &&) noexcept  
virtual  ~Subscriptor () 
Subscriptor &  operator= (const Subscriptor &) 
Subscriptor &  operator= (Subscriptor &&) noexcept 
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) 
Additional Inherited Members  
Static Public Member Functions inherited from Subscriptor  
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
Manifolds are used to describe the geometry of boundaries of domains as well as the geometry of the interior. Manifold objects are therefore associated with cells, faces, and/or edges, either by direct user action or, if a user program does not do this explicitly, a default manifold object is used.
Manifolds are best understood by using the language of differential geometry, but their common uses can be easily described simply through examples.
In the most essential use of manifolds, manifold descriptions are used to create a "point between other points". For example, when a triangulation creates a new vertex on a cell, face, or edge, it determines the new vertex' coordinates through the following function call:
Here, points
is a collection of points in spacedim
dimension, and a
collection of corresponding weights. The points in this context will then be the vertices of the cell, face, or edge, and the weights are typically one over the number of points when a new midpoint of the cell, face, or edge is needed. Derived classes then will implement the Manifold::get_new_point() function in a way that computes the location of this new point. In the simplest case, for example in the FlatManifold class, the function simply computes the arithmetic average (with given weights) of the given points. However, other classes do something differently; for example, the SphericalManifold class, which is used to describe domains that form (part of) the sphere, will ensure that, given the two vertices of an edge at the boundary, the new returned point will lie on the grand circle that connects the two points, rather than choosing a point that is halfway between the two points in \({\mathbb R}^d\).
Manifold::get_new_point() has a default implementation that can simplify this process somewhat: Internally, the function calls the Manifold::get_intermediate_point() to compute pairwise intermediate points. Internally the Manifold::get_intermediate_point() calls the Manifold::project_to_manifold() function after computing the convex combination of the given points. This allows derived classes to only overload Manifold::project_to_manifold() for simple situations. This is often useful when describing manifolds that are embedded in higher dimensional space, e.g., the surface of a sphere. In those cases, the desired new point may be computed simply by the (weighted) average of the provided points, projected back out onto the sphere.
The second use of this class is in computing directions on domains and boundaries. For example, we may need to compute the normal vector to a face in order to impose the noflow boundary condition \(\mathbf u \cdot \mathbf n = 0\) (see the VectorTools::compute_no_normal_flux_constraints() as an example). Similarly, we may need normal vectors in the computation of the normal component of the gradient of the numerical solution in order to compute the jump in the gradient of the solution in error estimators (see, for example, the KellyErrorEstimator class).
To make this possible, the Manifold class provides a member function (to be implemented by derived classes) that computes a "vector tangent to the manifold at one point, in direction of another point" via the Manifold::get_tangent_vector() function. For example, in 2d, one would use this function with the two vertices of an edge at the boundary to compute a "tangential" vector along the edge, and then get the normal vector by rotation by 90 degrees. In 3d, one would compute the two vectors "tangential" to the two edges of a boundary face adjacent to a boundary vertex, and then take the cross product of these two to obtain a vector normal to the boundary.
For reasons that are more difficult to understand, these direction vectors are normalized in a very specific way, rather than to have unit norm. See the documentation of Manifold::get_tangent_vector(), as well as below, for more information.
In the simplest case (namely, the FlatManifold class), these tangent vectors are just the difference vector between the two given points. However, in more complicated (and more interesting) cases, the direction may be different. For example, for the SphericalManifold case, if the two given points lie on a common grand circle around the origin, then the tangent vector will be tangential to the grand circle, rather than pointing straight from one point to the other.
The "real" way to understand what this class does is to see it in the framework of differential geometry. More specifically, differential geometry is fundamentally based on the assumption that two sufficiently close points are connected via a line of "shortest distance". This line is called a "geodesic", and it is selected from all other lines that connect the two points by the property that it is shortest if distances are measured in terms of the "metric" that describes a manifold. To give examples, recall that the geodesics of a flat manifold (implemented in the FlatManifold class) are simply the straight lines connecting two points, whereas for spherical manifolds (see the SphericalManifold class) geodesics between two points of same distance are the grand circles, and are in general curved lines when connecting two lines of different distance from the origin.
In the following discussion, and for the purposes of implementing the current class, the concept of "metrics" that is so fundamental to differential geometry is no longer of great importance to us. Rather, everything can simply be described by postulating the existence of geodesics connecting points on a manifold.
Given geodesics, the operations discussed in the previous two sections can be described in a more formal way. In essence, they rely on the fact that we can assume that a geodesic is parameterized by a "time" like variable \(t\) so that \(\mathbf s(t)\) describes the curve and so that \(\mathbf s(0)\) is the location of the first and \(\mathbf s(1)\) the location of the second point. Furthermore, \(\mathbf s(t)\) traces out the geodesic at constant speed, covering equal distance in equal time (as measured by the metric). Note that this parameterization uses time, not arc length to denote progress along the geodesic.
In this picture, computing a midpoint between points \(\mathbf x_1\) and \(\mathbf x_2\), with weights \(w_1\) and \(w_2=1w_1\), simply requires computing the point \(\mathbf s(w_1)\). Computing a new point as a weighted average of more than two points can be done by considering pairwise geodesics, finding suitable points on the geodetic between the first two points, then on the geodetic between this new point and the third given point, etc.
Likewise, the "tangential" vector described above is simply the velocity vector, \(\mathbf s'(t)\), evaluated at one of the end points of a geodesic (i.e., at \(t=0\) or \(t=1\)). In the case of a flat manifold, the geodesic is simply the straight line connecting two points, and the velocity vector is just the connecting vector in that case. On the other hand, for two points on a spherical manifold, the geodesic is a grand circle, and the velocity vector is tangent to the spherical surface.
Note that if we wanted to, we could use this to compute the length of the geodesic that connects two points \(\mathbf x_1\) and \(\mathbf x_2\) by computing \(\int_0^1 \\mathbf s'(t)\ dt\) along the geodesic that connects them, but this operation will not be of use to us in practice. One could also conceive computing the direction vector using the "new point" operation above, using the formula \(\mathbf s'(0)=\lim_{w\rightarrow 0} \frac{\mathbf s(w)\mathbf s(0)}{w}\) where all we need to do is compute the new point \(\mathbf s(w)\) with weights \(w\) and \(1w\) along the geodesic connecting \(\mathbf x_1\) and \(\mathbf x_2\). The default implementation of the function does this, by evaluating the quotient for a small but finite weight \(w\). In practice, however, it is almost always possible to explicitly compute the direction vector, i.e., without the need to numerically approximate the limit process, and derived classes should do so.
Definition at line 329 of file manifold.h.
using Manifold< dim, spacedim >::FaceVertexNormals = std::array<Tensor<1, spacedim>, GeometryInfo<dim>::vertices_per_face> 
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 350 of file manifold.h.

overridevirtualdefault 
Destructor. Does nothing here, but needs to be declared virtual to make class hierarchies derived from this class possible.

pure virtual 
Return a copy of this manifold.
Every derived class should implement this operation in a sensible manner.
Implemented in TransfiniteInterpolationManifold< dim, spacedim >, TorusManifold< dim >, FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, FlatManifold< dim >, FunctionManifold< dim, spacedim, chartdim >, EllipticalManifold< dim, spacedim >, CylindricalManifold< dim, spacedim >, OpenCASCADE::NURBSPatchManifold< dim, spacedim >, OpenCASCADE::ArclengthProjectionLineManifold< dim, spacedim >, OpenCASCADE::NormalToMeshProjectionManifold< dim, spacedim >, SphericalManifold< dim, spacedim >, OpenCASCADE::NormalToMeshProjectionBoundary< dim, spacedim >, OpenCASCADE::DirectionalProjectionManifold< dim, spacedim >, OpenCASCADE::DirectionalProjectionBoundary< dim, spacedim >, TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >, CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >, OpenCASCADE::NormalProjectionManifold< dim, spacedim >, PolarManifold< dim, spacedim >, PolarManifold< spacedim >, and OpenCASCADE::NormalProjectionBoundary< dim, spacedim >.

virtual 
Return an intermediate point between two given points. Overloading this function allows the default pairwise reduction implementation of the method get_new_point() that takes a Quadrature object as input to work properly.
An implementation of this function should returns a parametric curve on the manifold, joining the points p1
and p2
, with parameter w
in the interval [0,1]. In particular get_intermediate_point(p1, p2, 0.0)
should return p1
and get_intermediate_point(p1, p2, 1.0)
should return p2
.
In its default implementation, this function calls the project_to_manifold() method with the convex combination of p1
and p2
. User classes can get away by simply implementing the project_to_manifold() method.
Reimplemented in ChartManifold< dim, spacedim, chartdim >, ChartManifold< dim, spacedim, 2 >, ChartManifold< dim, spacedim, 3 >, ChartManifold< dim_B, spacedim_B, chartdim_B >, ChartManifold< dim, spacedim, 1 >, ChartManifold< dim_A, spacedim_A, chartdim_A >, ChartManifold< dim1, intermediate_dim, chartdim >, ChartManifold< dim, spacedim, spacedim >, ChartManifold< dim2, spacedim, intermediate_dim >, ChartManifold< dim, dim, dim >, ChartManifold< dim, 3, 3 >, ChartManifold< dim, spacedim_A+spacedim_B, chartdim_A+chartdim_B >, and SphericalManifold< dim, spacedim >.
Definition at line 50 of file manifold.cc.

virtual 
Return the point which shall become the new vertex surrounded by the given points surrounding_points
. weights
contains appropriate weights for the surrounding points according to which the manifold determines the new point's position.
In its default implementation it uses a pairwise reduction of the points by calling the function get_intermediate_point() on the first two points, then on the resulting point and the next, until all points in the vector have been taken into account. User classes can get away by simply implementing the get_intermediate_point() function. Notice that by default the get_intermediate_point() function calls the project_to_manifold() function with the convex combination of its arguments. For simple situations you may get away by implementing only the project_to_manifold() function.
Reimplemented in TransfiniteInterpolationManifold< dim, spacedim >, ChartManifold< dim, spacedim, chartdim >, ChartManifold< dim, spacedim, 2 >, ChartManifold< dim, spacedim, 3 >, ChartManifold< dim_B, spacedim_B, chartdim_B >, ChartManifold< dim, spacedim, 1 >, ChartManifold< dim_A, spacedim_A, chartdim_A >, ChartManifold< dim1, intermediate_dim, chartdim >, ChartManifold< dim, spacedim, spacedim >, ChartManifold< dim2, spacedim, intermediate_dim >, ChartManifold< dim, dim, dim >, ChartManifold< dim, 3, 3 >, ChartManifold< dim, spacedim_A+spacedim_B, chartdim_A+chartdim_B >, FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, CylindricalManifold< dim, spacedim >, and SphericalManifold< dim, spacedim >.
Definition at line 63 of file manifold.cc.

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
must match the length of new_points
.
In its default implementation, this function simply calls get_new_point() on each row of weights
and writes those points into the output array new_points
. However, this function is more efficient if multiple new points need to be generated like in MappingQGeneric and the manifold does expensive transformations between a chart space and the physical space, such as ChartManifold. For this function, the surrounding points need to be transformed back to the chart sparse only once, rather than for every call to get_new_point(). If efficiency is not important, you may get away by implementing only the get_new_point() function.
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 in ChartManifold< dim, spacedim, chartdim >, ChartManifold< dim, spacedim, 2 >, ChartManifold< dim, spacedim, 3 >, ChartManifold< dim_B, spacedim_B, chartdim_B >, ChartManifold< dim, spacedim, 1 >, ChartManifold< dim_A, spacedim_A, chartdim_A >, ChartManifold< dim1, intermediate_dim, chartdim >, ChartManifold< dim, spacedim, spacedim >, ChartManifold< dim2, spacedim, intermediate_dim >, ChartManifold< dim, dim, dim >, ChartManifold< dim, 3, 3 >, ChartManifold< dim, spacedim_A+spacedim_B, chartdim_A+chartdim_B >, TransfiniteInterpolationManifold< dim, spacedim >, FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, and SphericalManifold< dim, spacedim >.
Definition at line 117 of file manifold.cc.

virtual 
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 >, OpenCASCADE::NormalToMeshProjectionManifold< dim, spacedim >, OpenCASCADE::DirectionalProjectionManifold< dim, spacedim >, and OpenCASCADE::NormalProjectionManifold< dim, spacedim >.
Definition at line 38 of file manifold.cc.

virtual 
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_quadrature() 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 310 of file manifold.cc.

virtual 
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_quadrature() 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 324 of file manifold.cc.

virtual 
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_quadrature() 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 444 of file manifold.cc.
Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_face  (  const typename Triangulation< dim, spacedim >::face_iterator &  face  )  const 
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 338 of file manifold.cc.
Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_cell  (  const typename Triangulation< dim, spacedim >::cell_iterator &  cell  )  const 
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 358 of file manifold.cc.

virtual 
Return a vector that, at \(\mathbf x_1\), is tangential to the geodesic that connects two points \(\mathbf x_1,\mathbf x_2\). The geodesic is the shortest line between these two points, where "shortest" is defined via a metric specific to a particular implementation of this class in a derived class. For example, in the case of a FlatManifold, the shortest line between two points is just the straight line, and in this case the tangent vector is just the difference \(\mathbf d=\mathbf x_2\mathbf x_1\). On the other hand, for a manifold that describes a surface embedded in a higher dimensional space (e.g., the surface of a sphere), then the tangent vector is tangential to the surface, and consequently may point in a different direction than the straight line that connects the two points.
While tangent vectors are often normalized to unit length, the vectors returned by this function are normalized as described in the introduction of this class. Specifically, if \(\mathbf s(t)\) traces out the geodesic between the two points where \(\mathbf x_1 = \mathbf s(0)\) and \(\mathbf x_2 = \mathbf s(1)\), then the returned vector must equal \(\mathbf s'(0)\). In other words, the norm of the returned vector also encodes, in some sense, the length of the geodesic because a curve \(\mathbf s(t)\) must move "faster" if the two points it connects between arguments \(t=0\) and \(t=1\) are farther apart.
The default implementation of this function approximates \(\mathbf s'(0) \approx \frac{\mathbf s(\epsilon)\mathbf x_1}{\epsilon}\) for a small value of \(\epsilon\), and the evaluation of \(\mathbf s(\epsilon)\) is done by calling get_new_point(). If possible, derived classes should override this function by an implementation of the exact derivative.
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 in ChartManifold< dim, spacedim, chartdim >, ChartManifold< dim, spacedim, 2 >, ChartManifold< dim, spacedim, 3 >, ChartManifold< dim_B, spacedim_B, chartdim_B >, ChartManifold< dim, spacedim, 1 >, ChartManifold< dim_A, spacedim_A, chartdim_A >, ChartManifold< dim1, intermediate_dim, chartdim >, ChartManifold< dim, spacedim, spacedim >, ChartManifold< dim2, spacedim, intermediate_dim >, ChartManifold< dim, dim, dim >, ChartManifold< dim, 3, 3 >, ChartManifold< dim, spacedim_A+spacedim_B, chartdim_A+chartdim_B >, FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, and SphericalManifold< dim, spacedim >.
Definition at line 469 of file manifold.cc.

virtual 
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 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 in FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, and SphericalManifold< dim, spacedim >.
Definition at line 231 of file manifold.cc.

virtual 
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 295 of file manifold.cc.