Reference documentation for deal.II version GIT 5b2d3c498b 20220516 09:15:01+00:00

#include <deal.II/grid/manifold.h>
Public Types  
using  FaceVertexNormals = std::array< Tensor< 1, spacedim >, GeometryInfo< dim >::vertices_per_face > 
Public Member Functions  
ChartManifold (const Tensor< 1, chartdim > &periodicity=Tensor< 1, chartdim >())  
virtual  ~ChartManifold () override=default 
virtual Point< spacedim >  get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const override 
virtual Point< spacedim >  get_new_point (const ArrayView< const Point< spacedim >> &surrounding_points, const ArrayView< const double > &weights) 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< chartdim >  pull_back (const Point< spacedim > &space_point) const =0 
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 
virtual std::unique_ptr< Manifold< dim, spacedim > >  clone () const=0 
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 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 
This class describes mappings that can be expressed in terms of charts. Specifically, this class with its template arguments describes a chart of dimension chartdim, which is part of a Manifold<dim,spacedim> and is used in an object of type Triangulation<dim,spacedim>: It specializes a Manifold of dimension chartdim embedded in a manifold of dimension spacedim, for which you have explicit pull_back() and push_forward() transformations. Its use is explained in great detail in step53.
This is a helper class which is useful when you have an explicit map from an Euclidean space of dimension chartdim to an Euclidean space of dimension spacedim which represents your manifold, i.e., when your manifold \(\mathcal{M}\) can be represented by a map
\[ F: \mathcal{B} \subset R^{\text{chartdim}} \mapsto \mathcal{M} \subset R^{\text{spacedim}} \]
(the push_forward() function) and that admits the inverse transformation
\[ F^{1}: \mathcal{M} \subset R^{\text{spacedim}} \mapsto \mathcal{B} \subset R^{\text{chartdim}} \]
(the pull_back() function).
The get_new_point() function of the ChartManifold class is implemented by calling the pull_back() method for all surrounding_points
, computing their weighted average in the chartdim Euclidean space, and calling the push_forward() method with the resulting point, i.e.,
\[ \mathbf x^{\text{new}} = F(\sum_i w_i F^{1}(\mathbf x_i)). \]
Derived classes are required to implement the push_forward() and the pull_back() methods. All other functions (with the exception of the push_forward_gradient() function, see below) that are required by mappings will then be provided by this class.
In order to compute vectors that are tangent to the manifold (for example, tangent to a surface embedded in higher dimensional space, or simply the three unit vectors of \({\mathbb R}^3\)), one needs to also have access to the gradient of the pushforward function \(F\). The gradient is the matrix \((\nabla F)_{ij}=\partial_j F_i\), where we take the derivative with regard to the chartdim reference coordinates on the flat Euclidean space in which \(\mathcal B\) is located. In other words, at a point \(\mathbf x\), \(\nabla F(\mathbf x)\) is a matrix of size spacedim
times chartdim
.
Only the ChartManifold::get_tangent_vector() function uses the gradient of the pushforward, but only a subset of all finite element codes actually require the computation of tangent vectors. Consequently, while derived classes need to implement the abstract virtual push_forward() and pull_back() functions of this class, they do not need to implement the virtual push_forward_gradient() function. Rather, that function has a default implementation (and consequently is not abstract, therefore not forcing derived classes to overload it), but the default implementation clearly can not compute anything useful and therefore simply triggers and exception.
The dimension arguments chartdim
, dim
and spacedim
must satisfy the following relationships:
However, there is no a priori relationship between dim
and chartdim
. For example, if you want to describe a mapping for an edge (a 1d object) in a 2d triangulation embedded in 3d space, you could do so by parameterizing it via a line
\[ F: [0,1] \rightarrow {\mathbb R}^3 \]
in which case chartdim
is 1. On the other hand, there is no reason why one can't describe this as a mapping
\[ F: {\mathbb R}^3 \rightarrow {\mathbb R}^3 \]
in such a way that the line \([0,1]\times \{0\}\times \{0\}\) happens to be mapped onto the edge in question. Here, chartdim
is 3. This may seem cumbersome but satisfies the requirements of an invertible function \(F\) just fine as long as it is possible to get from the edge to the pullback space and then back again. Finally, given that we are dealing with a 2d triangulation in 3d, one will often have a mapping from, say, the 2d unit square or unit disk to the domain in 3d space, and the edge in question may simply be the mapped edge of the unit domain in 2d space. In this case, chartdim
is 2.
Definition at line 899 of file manifold.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.
ChartManifold< dim, spacedim, chartdim >::ChartManifold  (  const Tensor< 1, chartdim > &  periodicity = Tensor<1, chartdim>()  ) 
Constructor. The optional argument can be used to specify the periodicity of the chartdimdimensional manifold (one period per direction). A periodicity value of zero means that along that direction there is no periodicity. By default no periodicity is assumed.
Periodicity affects the way a middle point is computed. It is assumed that if two points are more than half period distant, then the distance should be computed by crossing the periodicity boundary, i.e., then the average is computed by adding a full period to the sum of the two. For example, if along direction 0 we have 2*pi periodicity, then the average of (2*pieps) and (eps) is not pi, but 2*pi (or zero), since, on the manifold, these two points are at distance 2*eps and not (2*pieps)
Definition at line 975 of file manifold.cc.

overridevirtualdefault 
Destructor. Does nothing here, but needs to be declared to make it virtual.

overridevirtual 
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, dim >.
Definition at line 984 of file manifold.cc.

overridevirtual 
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, dim >.
Reimplemented in CylindricalManifold< dim, spacedim >.
Definition at line 999 of file manifold.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
.
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 nontrivial manifolds.
Reimplemented from Manifold< dim, dim >.
Definition at line 1020 of file manifold.cc.

pure virtual 
Pull back the given point in spacedim to the Euclidean chartdim dimensional space.
Refer to the general documentation of this class for more information.
Implemented in OpenCASCADE::NURBSPatchManifold< dim, spacedim >, OpenCASCADE::ArclengthProjectionLineManifold< dim, spacedim >, TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >, FunctionManifold< dim, spacedim, chartdim >, EllipticalManifold< dim, spacedim >, CylindricalManifold< dim, spacedim >, PolarManifold< dim, spacedim >, and CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >.

pure virtual 
Given a point in the chartdim dimensional Euclidean space, this method returns a point on the manifold embedded in the spacedim Euclidean space.
Refer to the general documentation of this class for more information.
Implemented in TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >, FunctionManifold< dim, spacedim, chartdim >, and CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >.

virtual 
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function \(F\) that maps from the chartdimdimensional to the spacedimdimensional 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 TensorProductManifold< dim, dim_A, spacedim_A, chartdim_A, dim_B, spacedim_B, chartdim_B >, FunctionManifold< dim, spacedim, chartdim >, and CompositionManifold< dim, spacedim, chartdim, intermediate_dim, dim1, dim2 >.
Definition at line 1049 of file manifold.cc.

overridevirtual 
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 preimages of x1
and x2
(where preimages 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 chartdimdimensional 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 pushforward mapping. Here, \(F^{1}(\mathbf x_2)F^{1}(\mathbf x_1)\) is a chartdimdimensional vector, and \(\nabla_\xi F\left(F^{1}(\mathbf x_1)\right) = \nabla_\xi F\left(\xi_1\right)\) is a spacedimtimeschartdimdimensional matrix. Consequently, and as desired, the operation results in a spacedimdimensional 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, dim >.
Definition at line 1062 of file manifold.cc.
const Tensor< 1, chartdim > & ChartManifold< dim, spacedim, chartdim >::get_periodicity 
Return the periodicity associated with the submanifold.
Definition at line 1094 of file manifold.cc.

pure virtualinherited 
Return a copy of this manifold.
Every derived class should implement this operation in a sensible manner.
Implemented in TransfiniteInterpolationManifold< dim, spacedim >, SphericalManifold< dim, spacedim >, and FlatManifold< dim, spacedim >.

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 418 of file manifold.cc.

inherited 
Definition at line 429 of file manifold.cc.

inherited 
Definition at line 440 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 462 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 385 of file manifold.cc.

inherited 
Definition at line 396 of file manifold.cc.

inherited 
Definition at line 407 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.

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 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 SphericalManifold< dim, spacedim >, and FlatManifold< dim, spacedim >.
Definition at line 595 of file manifold.cc.

inherited 
Definition at line 144 of file manifold.cc.

inherited 
Definition at line 165 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 614 of file manifold.cc.

inherited 
Definition at line 250 of file manifold.cc.

inherited 
Definition at line 272 of file manifold.cc.

private 
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.