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Reference documentation for deal.II version 9.4.1
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Reference documentation for deal.II version 9.4.1
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#include <deal.II/grid/manifold_lib.h>
Public Types | |
| using | FaceVertexNormals = std::array< Tensor< 1, spacedim >, GeometryInfo< dim >::vertices_per_face > |
Public Member Functions | |
| FunctionManifold (const Function< chartdim > &push_forward_function, const Function< spacedim > &pull_back_function, const Tensor< 1, chartdim > &periodicity=Tensor< 1, chartdim >(), const double tolerance=1e-10) | |
| FunctionManifold (std::unique_ptr< Function< chartdim > > push_forward, std::unique_ptr< Function< spacedim > > pull_back, const Tensor< 1, chartdim > &periodicity=Tensor< 1, chartdim >(), const double tolerance=1e-10) | |
| FunctionManifold (const std::string push_forward_expression, const std::string pull_back_expression, const Tensor< 1, chartdim > &periodicity=Tensor< 1, chartdim >(), const typename FunctionParser< spacedim >::ConstMap=typename FunctionParser< spacedim >::ConstMap(), const std::string chart_vars=FunctionParser< chartdim >::default_variable_names(), const std::string space_vars=FunctionParser< spacedim >::default_variable_names(), const double tolerance=1e-10, const double h=1e-8) | |
| virtual | ~FunctionManifold () override |
| virtual std::unique_ptr< Manifold< dim, spacedim > > | clone () const override |
| virtual Point< spacedim > | push_forward (const Point< chartdim > &chart_point) const override |
| virtual DerivativeForm< 1, chartdim, spacedim > | push_forward_gradient (const Point< chartdim > &chart_point) const override |
| virtual Point< chartdim > | pull_back (const Point< spacedim > &space_point) const override |
| 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 Tensor< 1, spacedim > | get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) const override |
| const Tensor< 1, chartdim > & | get_periodicity () const |
| Point< 1 > | get_new_point_on_quad (const Triangulation< 1, 1 >::quad_iterator &) const |
| Point< 2 > | get_new_point_on_quad (const Triangulation< 1, 2 >::quad_iterator &) const |
| Point< 3 > | get_new_point_on_quad (const Triangulation< 1, 3 >::quad_iterator &) const |
| Point< 3 > | get_new_point_on_hex (const Triangulation< 3, 3 >::hex_iterator &hex) const |
| Point< 1 > | get_new_point_on_face (const Triangulation< 1, 1 >::face_iterator &) const |
| Point< 2 > | get_new_point_on_face (const Triangulation< 1, 2 >::face_iterator &) const |
| Point< 3 > | get_new_point_on_face (const Triangulation< 1, 3 >::face_iterator &) const |
| Tensor< 1, 2 > | normal_vector (const Triangulation< 2, 2 >::face_iterator &face, const Point< 2 > &p) const |
| Tensor< 1, 3 > | normal_vector (const Triangulation< 3, 3 >::face_iterator &face, const Point< 3 > &p) const |
| void | get_normals_at_vertices (const Triangulation< 2, 2 >::face_iterator &face, FaceVertexNormals &n) const |
| void | get_normals_at_vertices (const Triangulation< 3, 3 >::face_iterator &face, FaceVertexNormals &n) const |
Computing the location of points. | |
| virtual Point< spacedim > | project_to_manifold (const ArrayView< const Point< spacedim > > &surrounding_points, const Point< spacedim > &candidate) const |
| virtual Point< spacedim > | get_new_point_on_line (const typename Triangulation< dim, spacedim >::line_iterator &line) const |
| virtual Point< spacedim > | get_new_point_on_quad (const typename Triangulation< dim, spacedim >::quad_iterator &quad) const |
| virtual Point< spacedim > | get_new_point_on_hex (const typename Triangulation< dim, spacedim >::hex_iterator &hex) const |
| Point< spacedim > | get_new_point_on_face (const typename Triangulation< dim, spacedim >::face_iterator &face) const |
| Point< spacedim > | get_new_point_on_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const |
Computing normal vectors | |
| virtual Tensor< 1, spacedim > | normal_vector (const typename Triangulation< dim, spacedim >::face_iterator &face, const Point< spacedim > &p) const |
| virtual void | get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, FaceVertexNormals &face_vertex_normals) const |
Private Attributes | |
| const FunctionParser< spacedim >::ConstMap | const_map |
| SmartPointer< const Function< chartdim >, FunctionManifold< dim, spacedim, chartdim > > | push_forward_function |
| SmartPointer< const Function< spacedim >, FunctionManifold< dim, spacedim, chartdim > > | pull_back_function |
| const double | tolerance |
| bool | owns_pointers |
| const std::string | push_forward_expression |
| const std::string | pull_back_expression |
| const std::string | chart_vars |
| const std::string | space_vars |
| const double | finite_difference_step |
| 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 |
Manifold description derived from ChartManifold, based on explicit Function<spacedim> and Function<chartdim> objects describing the push_forward() and pull_back() functions.
You can use this Manifold object to describe any arbitrary shape domain, as long as you can express it in terms of an invertible map, for which you provide both the forward expression, and the inverse expression.
In debug mode, a check is performed to verify that the transformations are actually one the inverse of the other.
Definition at line 594 of file manifold_lib.h.
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inherited |
Type keeping information about the normals at the vertices of a face of a cell. Thus, there are GeometryInfo<dim>::vertices_per_face normal vectors, that define the tangent spaces of the boundary at the vertices. Note that the vectors stored in this object are not required to be normalized, nor to actually point outward, as one often will only want to check for orthogonality to define the tangent plane; if a function requires the normals to be normalized, then it must do so itself.
For obvious reasons, this type is not useful in 1d.
Definition at line 306 of file manifold.h.
| FunctionManifold< dim, spacedim, chartdim >::FunctionManifold | ( | const Function< chartdim > & | push_forward_function, |
| const Function< spacedim > & | pull_back_function, | ||
| const Tensor< 1, chartdim > & | periodicity = Tensor<1, chartdim>(), |
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| const double | tolerance = 1e-10 |
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| ) |
Explicit functions constructor. Takes a push_forward function of spacedim components, and a pull_back function of chartdim components. See the documentation of the base class ChartManifold for the meaning of the optional periodicity argument.
The tolerance argument is used in debug mode to actually check that the two functions are one the inverse of the other.
Note: the object constructed in this way stores pointers to the push_forward and pull_back functions. Therefore, one must guarantee that the function objects are destroyed only after the constructed manifold.
Definition at line 1363 of file manifold_lib.cc.
| FunctionManifold< dim, spacedim, chartdim >::FunctionManifold | ( | std::unique_ptr< Function< chartdim > > | push_forward, |
| std::unique_ptr< Function< spacedim > > | pull_back, | ||
| const Tensor< 1, chartdim > & | periodicity = Tensor<1, chartdim>(), |
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| const double | tolerance = 1e-10 |
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| ) |
Same as previous, except this constructor takes ownership of the Function objects passed as first and second argument, and is ultimately in charge of deleting the pointers when the FunctionManifold object is destroyed.
This constructor is useful because it allows creating function objects at the place of calling the constructor without having to name and later delete these objects. This allows the following idiom: FunctionManifold<dim> manifold(std::make_unique<MyPushForward>(...), std::make_unique<MyPullBack>(...));
Definition at line 1383 of file manifold_lib.cc.
| FunctionManifold< dim, spacedim, chartdim >::FunctionManifold | ( | const std::string | push_forward_expression, |
| const std::string | pull_back_expression, | ||
| const Tensor< 1, chartdim > & | periodicity = Tensor<1, chartdim>(), |
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| const typename FunctionParser< spacedim >::ConstMap | const_map = typename FunctionParser<spacedim>::ConstMap(), |
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| const std::string | chart_vars = FunctionParser<chartdim>::default_variable_names(), |
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| const std::string | space_vars = FunctionParser<spacedim>::default_variable_names(), |
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| const double | tolerance = 1e-10, |
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| const double | h = 1e-8 |
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| ) |
Expressions constructor. Takes the expressions of the push_forward function of spacedim components, and of the pull_back function of chartdim components. See the documentation of the base class ChartManifold for the meaning of the optional periodicity argument.
The strings should be the readable by the default constructor of the FunctionParser classes. You can specify custom variable expressions with the last two optional arguments. If you don't, the default names are used, i.e., "x,y,z".
The tolerance argument is used in debug mode to actually check that the two functions are one the inverse of the other.
Definition at line 1403 of file manifold_lib.cc.
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overridevirtual |
If needed, we delete the pointers we own.
Definition at line 1433 of file manifold_lib.cc.
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overridevirtual |
Make a clone of this Manifold object.
Implements Manifold< dim, spacedim >.
Definition at line 1451 of file manifold_lib.cc.
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overridevirtual |
Given a point in the chartdim coordinate system, uses the push_forward_function to compute the push_forward of points in chartdim space dimensions to spacedim space dimensions.
Implements ChartManifold< dim, spacedim, chartdim >.
Definition at line 1491 of file manifold_lib.cc.
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overridevirtual |
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function \(F\) that maps from the sub_manifold coordinate system to the Euclidean coordinate system. 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. The default implementation calls the get_gradient() method of the FunctionManifold::push_forward_function() member class. If you construct this object using the constructor that takes two string expression, then the default implementation of this method uses a finite difference scheme to compute the gradients(see the
AutoDerivativeFunction() class for details), and you can specify the size of the spatial step size at construction time with the h parameter.
Refer to the general documentation of this class for more information.
Reimplemented from ChartManifold< dim, spacedim, chartdim >.
Definition at line 1519 of file manifold_lib.cc.
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overridevirtual |
Given a point in the spacedim coordinate system, uses the pull_back_function to compute the pull_back of points in spacedim space dimensions to chartdim space dimensions.
Implements ChartManifold< dim, spacedim, chartdim >.
Definition at line 1536 of file manifold_lib.cc.
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overridevirtualinherited |
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, spacedim >.
Definition at line 984 of file manifold.cc.
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overridevirtualinherited |
Refer to the general documentation of this class and the documentation of the base class for more information.
Reimplemented from Manifold< dim, spacedim >.
Reimplemented in CylindricalManifold< dim, spacedim >.
Definition at line 999 of file manifold.cc.
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overridevirtualinherited |
Compute a new set of points that interpolate between the given points surrounding_points. weights is a table with as many columns as surrounding_points.size(). The number of rows in weights must match the length of new_points.
The implementation of this function first transforms the surrounding_points to the chart space by calling pull_back(). Then, new points are computed on the chart by usual interpolation according to the given weights, which are finally transformed to the image space by push_forward().
This implementation can be much more efficient for computing multiple new points from the same surrounding points than separate calls to get_new_point() in case the pull_back() operation is expensive. This is because pull_back() is only called once for the surrounding points and the interpolation is done for all given weights using this set of points. Often, pull_back() is also more expensive than push_forward() because the former might involve some kind of Newton iteration in non-trivial manifolds.
Reimplemented from Manifold< dim, spacedim >.
Definition at line 1020 of file manifold.cc.
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overridevirtualinherited |
Return a vector that, at \(\mathbf x_1\), is tangential to the geodesic that connects two points \(\mathbf x_1,\mathbf x_2\). See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are \(\xi_1=F^{-1}(\mathbf x_1), \xi_2=F^{-1}(\mathbf x_2)\), then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
\begin{align*} \zeta(t) &= \xi_1 + t (\xi_2-\xi_1) \\ &= F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right] \end{align*}
In image space, i.e., in the space in which we operate, this leads to the curve
\begin{align*} \mathbf s(t) &= F(\zeta(t)) \\ &= F(\xi_1 + t (\xi_2-\xi_1)) \\ &= F\left(F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]\right). \end{align*}
What the current function is supposed to return is \(\mathbf s'(0)\). By the chain rule, this is equal to
\begin{align*} \mathbf s'(0) &= \frac{d}{dt}\left. F\left(F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]\right) \right|_{t=0} \\ &= \nabla_\xi F\left(F^{-1}(\mathbf x_1)\right) \left[F^{-1}(\mathbf x_2) -F^{-1}(\mathbf x_1)\right]. \end{align*}
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
Thus, the computation of tangent vectors also requires the implementation of derivatives \(\nabla_\xi F(\xi)\) of the push-forward mapping. Here, \(F^{-1}(\mathbf x_2)-F^{-1}(\mathbf x_1)\) is a chartdim-dimensional vector, and \(\nabla_\xi F\left(F^{-1}(\mathbf x_1)\right) = \nabla_\xi F\left(\xi_1\right)\) is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
| x1 | The first point that describes the geodesic, and the one at which the "direction" is to be evaluated. |
| x2 | The second point that describes the geodesic. |
Reimplemented from Manifold< dim, spacedim >.
Definition at line 1062 of file manifold.cc.
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inherited |
Return the periodicity associated with the submanifold.
Definition at line 1094 of file manifold.cc.
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virtualinherited |
Given a point which lies close to the given manifold, it modifies it and projects it to manifold itself.
This class is used by the default implementation of the function get_new_point() and should be implemented by derived classes. The default implementation simply throws an exception if called.
If your manifold is simple, you could implement this function only, and the default behavior should work out of the box.
Reimplemented in FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, FlatManifold< dim, spacedim >, FlatManifold< dim, dim >, OpenCASCADE::NormalProjectionManifold< dim, spacedim >, OpenCASCADE::DirectionalProjectionManifold< dim, spacedim >, and OpenCASCADE::NormalToMeshProjectionManifold< dim, spacedim >.
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virtualinherited |
Backward compatibility interface. Return the point which shall become the new middle vertex of the two children of a regular line. In 2D, this line is a line at the boundary, while in 3d, it is bounding a face at the boundary (the lines therefore is also on the boundary).
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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virtualinherited |
Backward compatibility interface. Return the point which shall become the common point of the four children of a quad at the boundary in three or more spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.
This function is called after the four lines bounding the given quad are refined, so you may want to use the information provided by quad->line(i)->child(j), i=0...3, j=0,1.
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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inherited |
Definition at line 418 of file manifold.cc.
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inherited |
Definition at line 429 of file manifold.cc.
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inherited |
Definition at line 440 of file manifold.cc.
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virtualinherited |
Backward compatibility interface. Return the point which shall become the common point of the eight children of a hex in three or spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.
This function is called after the all the bounding objects of the given hex are refined, so you may want to use the information provided by hex->quad(i)->line(j)->child(k), i=0...5, j=0...3, k=0,1.
The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().
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inherited |
Definition at line 462 of file manifold.cc.
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inherited |
Backward compatibility interface. Depending on dim=2 or dim=3 this function calls the get_new_point_on_line or the get_new_point_on_quad function. It throws an exception for dim=1. This wrapper allows dimension independent programming.
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inherited |
Definition at line 385 of file manifold.cc.
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inherited |
Definition at line 396 of file manifold.cc.
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inherited |
Definition at line 407 of file manifold.cc.
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inherited |
Backward compatibility interface. Depending on dim=1, dim=2 or dim=3 this function calls the get_new_point_on_line, get_new_point_on_quad or the get_new_point_on_hex function. This wrapper allows dimension independent programming.
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virtualinherited |
Return the normal vector to a face embedded in this manifold, at the point p. If p is not in fact on the surface, but only close-by, try to return something reasonable, for example the normal vector at the surface point closest to p. (The point p will in fact not normally lie on the actual surface, but rather be a quadrature point mapped by some polynomial mapping; the mapped surface, however, will not usually coincide with the actual surface.)
This function only makes sense if dim==spacedim because otherwise there is no unique normal vector but in fact a (spacedim-dim+1)-dimensional tangent space of vectors that are all both normal to the face and normal to the dim-dimensional surface that lives in spacedim-dimensional space. For example, think of a two-dimensional mesh that covers a two-dimensional surface in three-dimensional space. In that case, each face (edge) is one-dimensional, and there are two linearly independent vectors that are both normal to the edge: one is normal to the edge and tangent to the surface (intuitively, that would be the one that points from the current cell to the neighboring one, if the surface was locally flat), and the other one is rooted in the edge but points perpendicular to the surface (which is also perpendicular to the edge that lives within the surface). Thus, because there are no obviously correct semantics for this function if spacedim is greater than dim, the function will simply throw an error in that situation.
The face iterator gives an indication which face this function is supposed to compute the normal vector for. This is useful if the boundary of the domain is composed of different nondifferential pieces (for example when using the FlatManifold class to approximate a geometry that is completely described by the coarse mesh, with piecewise (bi-)linear components between the vertices, but where the boundary may have a kink at the vertices itself).
Reimplemented in FlatManifold< dim, spacedim >, FlatManifold< chartdim, chartdim >, FlatManifold< dim, spacedim >, FlatManifold< dim, dim >, PolarManifold< dim, spacedim >, and SphericalManifold< dim, spacedim >.
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inherited |
Definition at line 144 of file manifold.cc.
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inherited |
Definition at line 165 of file manifold.cc.
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virtualinherited |
Compute the normal vectors to the boundary at each vertex of the given face embedded in the Manifold. It is not required that the normal vectors be normed somehow. Neither is it required that the normals actually point outward.
This function is needed to compute data for C1 mappings. The default implementation calls normal_vector() on each vertex.
Note that when computing normal vectors at a vertex where the boundary is not differentiable, you have to make sure that you compute the one-sided limits, i.e. limit with respect to points inside the given face.
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inherited |
Definition at line 250 of file manifold.cc.
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inherited |
Definition at line 272 of file manifold.cc.
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private |
Constants for the FunctionParser classes.
Definition at line 714 of file manifold_lib.h.
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private |
Pointer to the push_forward function.
Definition at line 721 of file manifold_lib.h.
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private |
Pointer to the pull_back function.
Definition at line 728 of file manifold_lib.h.
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private |
Relative tolerance. In debug mode, we check that the two functions provided at construction time are actually one the inverse of the other. This value is used as relative tolerance in this check.
Definition at line 735 of file manifold_lib.h.
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private |
Check ownership of the smart pointers. Indicates whether this class is the owner of the objects pointed to by the previous two member variables. This value is set in the constructor of the class. If true, then the destructor will delete the function objects pointed to be the two pointers.
Definition at line 744 of file manifold_lib.h.
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private |
The expression used to construct the push_forward function.
Definition at line 749 of file manifold_lib.h.
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private |
The expression used to construct the pull_back function.
Definition at line 754 of file manifold_lib.h.
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private |
Variable names in the chart domain.
Definition at line 759 of file manifold_lib.h.
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private |
Variable names in the space domain.
Definition at line 764 of file manifold_lib.h.
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private |
The finite difference step to use internally.
Definition at line 769 of file manifold_lib.h.
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privateinherited |
The sub_manifold object is used to compute the average of the points in the chart coordinates system.
In an ideal world, it would have type FlatManifold<dim,chartdim>. However, this would instantiate cases where dim>spacedim, which leads to invalid situations. We instead use <chartdim,chartdim>, which is (i) always valid, and (ii) does not matter at all since the first (dim) argument of manifolds is, in fact, ignored as far as manifold functionality is concerned.
Definition at line 1085 of file manifold.h.
1.9.5