Reference documentation for deal.II version 9.4.1
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FunctionManifold< dim, spacedim, chartdim > Class Template Reference

#include <deal.II/grid/manifold_lib.h>

Inheritance diagram for FunctionManifold< dim, spacedim, chartdim >:
[legend]

## 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< 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_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 intcounter

std::map< std::string, unsigned intcounter_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 ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

void check_no_subscribers () const noexcept

## Detailed Description

template<int dim, int spacedim = dim, int chartdim = dim>
class FunctionManifold< dim, spacedim, chartdim >

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.

## ◆ FaceVertexNormals

template<int dim, int spacedim = dim>
 using Manifold< dim, spacedim >::FaceVertexNormals = std::array, GeometryInfo::vertices_per_face>
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() [1/3]

template<int dim, int spacedim, int chartdim>
 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>(), const double tolerance = 1e-10 )

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() [2/3]

template<int dim, int spacedim, int chartdim>
 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>(), const double tolerance = 1e-10 )

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() [3/3]

template<int dim, int spacedim, int chartdim>
 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>(), const typename FunctionParser< spacedim >::ConstMap const_map = typename FunctionParser::ConstMap(), const std::string chart_vars = FunctionParser::default_variable_names(), const std::string space_vars = FunctionParser::default_variable_names(), const double tolerance = 1e-10, const double h = 1e-8 )

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.

## ◆ ~FunctionManifold()

template<int dim, int spacedim, int chartdim>
 FunctionManifold< dim, spacedim, chartdim >::~FunctionManifold
overridevirtual

If needed, we delete the pointers we own.

Definition at line 1433 of file manifold_lib.cc.

## ◆ clone()

template<int dim, int spacedim, int chartdim>
 std::unique_ptr< Manifold< dim, spacedim > > FunctionManifold< dim, spacedim, chartdim >::clone
overridevirtual

Make a clone of this Manifold object.

Implements Manifold< dim, spacedim >.

Definition at line 1451 of file manifold_lib.cc.

## ◆ push_forward()

template<int dim, int spacedim, int chartdim>
 Point< spacedim > FunctionManifold< dim, spacedim, chartdim >::push_forward ( const Point< chartdim > & chart_point ) const
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.

template<int dim, int spacedim, int chartdim>
 DerivativeForm< 1, chartdim, spacedim > FunctionManifold< dim, spacedim, chartdim >::push_forward_gradient ( const Point< chartdim > & chart_point ) const
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.

Reimplemented from ChartManifold< dim, spacedim, chartdim >.

Definition at line 1519 of file manifold_lib.cc.

## ◆ pull_back()

template<int dim, int spacedim, int chartdim>
 Point< chartdim > FunctionManifold< dim, spacedim, chartdim >::pull_back ( const Point< spacedim > & space_point ) const
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.

## ◆ get_intermediate_point()

template<int dim, int spacedim, int chartdim>
 Point< spacedim > ChartManifold< dim, spacedim, chartdim >::get_intermediate_point ( const Point< spacedim > & p1, const Point< spacedim > & p2, const double w ) const
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.

## ◆ get_new_point()

template<int dim, int spacedim, int chartdim>
 Point< spacedim > ChartManifold< dim, spacedim, chartdim >::get_new_point ( const ArrayView< const Point< spacedim > > & surrounding_points, const ArrayView< const double > & weights ) const
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.

## ◆ get_new_points()

template<int dim, int spacedim, int chartdim>
 void ChartManifold< dim, spacedim, chartdim >::get_new_points ( const ArrayView< const Point< spacedim > > & surrounding_points, const Table< 2, double > & weights, ArrayView< Point< spacedim > > new_points ) const
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.

## ◆ get_tangent_vector()

template<int dim, int spacedim, int chartdim>
 Tensor< 1, spacedim > ChartManifold< dim, spacedim, chartdim >::get_tangent_vector ( const Point< spacedim > & x1, const Point< spacedim > & x2 ) const
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.

Parameters
 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.
Returns
A "direction" vector tangential to the geodesic.

Reimplemented from Manifold< dim, spacedim >.

Definition at line 1062 of file manifold.cc.

## ◆ get_periodicity()

template<int dim, int spacedim, int chartdim>
 const Tensor< 1, chartdim > & ChartManifold< dim, spacedim, chartdim >::get_periodicity
inherited

Return the periodicity associated with the submanifold.

Definition at line 1094 of file manifold.cc.

## ◆ project_to_manifold()

template<int dim, int spacedim = dim>
 virtual Point< spacedim > Manifold< dim, spacedim >::project_to_manifold ( const ArrayView< const Point< spacedim > > & surrounding_points, const Point< spacedim > & candidate ) const
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.

## ◆ get_new_point_on_line()

template<int dim, int spacedim = dim>
 virtual Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_line ( const typename Triangulation< dim, spacedim >::line_iterator & line ) const
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().

template<int dim, int spacedim = dim>
 virtual Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_quad ( const typename Triangulation< dim, spacedim >::quad_iterator & quad ) const
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().

 Point< 1 > Manifold< 1, 1 >::get_new_point_on_quad ( const Triangulation< 1, 1 >::quad_iterator & ) const
inherited

Definition at line 418 of file manifold.cc.

 Point< 2 > Manifold< 1, 2 >::get_new_point_on_quad ( const Triangulation< 1, 2 >::quad_iterator & ) const
inherited

Definition at line 429 of file manifold.cc.

 Point< 3 > Manifold< 1, 3 >::get_new_point_on_quad ( const Triangulation< 1, 3 >::quad_iterator & ) const
inherited

Definition at line 440 of file manifold.cc.

## ◆ get_new_point_on_hex() [1/2]

template<int dim, int spacedim = dim>
 virtual Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_hex ( const typename Triangulation< dim, spacedim >::hex_iterator & hex ) const
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().

## ◆ get_new_point_on_hex() [2/2]

 Point< 3 > Manifold< 3, 3 >::get_new_point_on_hex ( const Triangulation< 3, 3 >::hex_iterator & hex ) const
inherited

Definition at line 462 of file manifold.cc.

## ◆ get_new_point_on_face() [1/4]

template<int dim, int spacedim = dim>
 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_face ( const typename Triangulation< dim, spacedim >::face_iterator & face ) const
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.

## ◆ get_new_point_on_face() [2/4]

 Point< 1 > Manifold< 1, 1 >::get_new_point_on_face ( const Triangulation< 1, 1 >::face_iterator & ) const
inherited

Definition at line 385 of file manifold.cc.

## ◆ get_new_point_on_face() [3/4]

 Point< 2 > Manifold< 1, 2 >::get_new_point_on_face ( const Triangulation< 1, 2 >::face_iterator & ) const
inherited

Definition at line 396 of file manifold.cc.

## ◆ get_new_point_on_face() [4/4]

 Point< 3 > Manifold< 1, 3 >::get_new_point_on_face ( const Triangulation< 1, 3 >::face_iterator & ) const
inherited

Definition at line 407 of file manifold.cc.

## ◆ get_new_point_on_cell()

template<int dim, int spacedim = dim>
 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
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.

## ◆ normal_vector() [1/3]

template<int dim, int spacedim = dim>
 virtual Tensor< 1, spacedim > Manifold< dim, spacedim >::normal_vector ( const typename Triangulation< dim, spacedim >::face_iterator & face, const Point< spacedim > & p ) const
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).

Note
In 2d, the default implementation of this function computes the normal vector by taking the tangent direction from p to the further one of the two vertices that make up an edge, and then rotates it outward (with respect to the coordinate system of the edge) by 90 degrees. In 3d, the default implementation is more complicated, aiming at avoiding problems with numerical round-off for points close to one of the vertices, and avoiding tangent directions that are linearly dependent.

## ◆ normal_vector() [2/3]

 Tensor< 1, 2 > Manifold< 2, 2 >::normal_vector ( const Triangulation< 2, 2 >::face_iterator & face, const Point< 2 > & p ) const
inherited

Definition at line 144 of file manifold.cc.

## ◆ normal_vector() [3/3]

 Tensor< 1, 3 > Manifold< 3, 3 >::normal_vector ( const Triangulation< 3, 3 >::face_iterator & face, const Point< 3 > & p ) const
inherited

Definition at line 165 of file manifold.cc.

## ◆ get_normals_at_vertices() [1/3]

template<int dim, int spacedim = dim>
 virtual void Manifold< dim, spacedim >::get_normals_at_vertices ( const typename Triangulation< dim, spacedim >::face_iterator & face, FaceVertexNormals & face_vertex_normals ) const
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.

## ◆ get_normals_at_vertices() [2/3]

 void Manifold< 2, 2 >::get_normals_at_vertices ( const Triangulation< 2, 2 >::face_iterator & face, FaceVertexNormals & n ) const
inherited

Definition at line 250 of file manifold.cc.

## ◆ get_normals_at_vertices() [3/3]

 void Manifold< 3, 3 >::get_normals_at_vertices ( const Triangulation< 3, 3 >::face_iterator & face, FaceVertexNormals & n ) const
inherited

Definition at line 272 of file manifold.cc.

## ◆ const_map

template<int dim, int spacedim = dim, int chartdim = dim>
 const FunctionParser::ConstMap FunctionManifold< dim, spacedim, chartdim >::const_map
private

Constants for the FunctionParser classes.

Definition at line 714 of file manifold_lib.h.

## ◆ push_forward_function

template<int dim, int spacedim = dim, int chartdim = dim>
 SmartPointer, FunctionManifold > FunctionManifold< dim, spacedim, chartdim >::push_forward_function
private

Pointer to the push_forward function.

Definition at line 721 of file manifold_lib.h.

## ◆ pull_back_function

template<int dim, int spacedim = dim, int chartdim = dim>
 SmartPointer, FunctionManifold > FunctionManifold< dim, spacedim, chartdim >::pull_back_function
private

Pointer to the pull_back function.

Definition at line 728 of file manifold_lib.h.

## ◆ tolerance

template<int dim, int spacedim = dim, int chartdim = dim>
 const double FunctionManifold< dim, spacedim, chartdim >::tolerance
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.

## ◆ owns_pointers

template<int dim, int spacedim = dim, int chartdim = dim>
 bool FunctionManifold< dim, spacedim, chartdim >::owns_pointers
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.

## ◆ push_forward_expression

template<int dim, int spacedim = dim, int chartdim = dim>
 const std::string FunctionManifold< dim, spacedim, chartdim >::push_forward_expression
private

The expression used to construct the push_forward function.

Definition at line 749 of file manifold_lib.h.

## ◆ pull_back_expression

template<int dim, int spacedim = dim, int chartdim = dim>
 const std::string FunctionManifold< dim, spacedim, chartdim >::pull_back_expression
private

The expression used to construct the pull_back function.

Definition at line 754 of file manifold_lib.h.

## ◆ chart_vars

template<int dim, int spacedim = dim, int chartdim = dim>
 const std::string FunctionManifold< dim, spacedim, chartdim >::chart_vars
private

Variable names in the chart domain.

Definition at line 759 of file manifold_lib.h.

## ◆ space_vars

template<int dim, int spacedim = dim, int chartdim = dim>
 const std::string FunctionManifold< dim, spacedim, chartdim >::space_vars
private

Variable names in the space domain.

Definition at line 764 of file manifold_lib.h.

## ◆ finite_difference_step

template<int dim, int spacedim = dim, int chartdim = dim>
 const double FunctionManifold< dim, spacedim, chartdim >::finite_difference_step
private

The finite difference step to use internally.

Definition at line 769 of file manifold_lib.h.

## ◆ sub_manifold

template<int dim, int spacedim = dim, int chartdim = dim>
 const FlatManifold ChartManifold< dim, spacedim, chartdim >::sub_manifold
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.

The documentation for this class was generated from the following files: