Dune Core Modules (unstable)

Dune::Geometry< mydim, cdim, GridImp, GeometryImp > Class Template Reference

Wrapper class for geometries. More...

#include <dune/grid/common/geometry.hh>

Public Types

enum  { mydimension =mydim }
 export geometry dimension More...
 
enum  { coorddimension =cdim }
 export coordinate dimension More...
 
typedef GeometryImp< mydim, cdim, GridImp > Implementation
 type of underlying implementation More...
 
typedef GridImp::ctype ctype
 define type used for coordinates in grid module
 
typedef FieldVector< ctype, mydim > LocalCoordinate
 type of local coordinates
 
typedef FieldVector< ctype, cdim > GlobalCoordinate
 type of the global coordinates
 
typedef Implementation::JacobianInverseTransposed JacobianInverseTransposed
 type of jacobian inverse transposed More...
 
typedef Implementation::JacobianTransposed JacobianTransposed
 type of jacobian transposed More...
 

Public Member Functions

Implementationimpl ()
 access to the underlying implementation More...
 
const Implementationimpl () const
 access to the underlying implementation More...
 
GeometryType type () const
 Return the type of the reference element. The type can be used to access the Dune::ReferenceElement.
 
bool affine () const
 Return true if the geometry mapping is affine and false otherwise.
 
int corners () const
 Return the number of corners of the reference element. More...
 
GlobalCoordinate corner (int i) const
 Obtain a corner of the geometry. More...
 
GlobalCoordinate global (const LocalCoordinate &local) const
 Evaluate the map \( g\). More...
 
 LocalCoordinate (const GlobalCoordinate &global) const
 Evaluate the inverse map \( g^{-1}\). More...
 
ctype integrationElement (const LocalCoordinate &local) const
 Return the factor appearing in the integral transformation formula. More...
 
Volume volume () const
 return volume of geometry
 
GlobalCoordinate center () const
 return center of geometry More...
 
JacobianTransposed jacobianTransposed (const LocalCoordinate &local) const
 Return the transposed of the Jacobian. More...
 
JacobianInverseTransposed jacobianInverseTransposed (const LocalCoordinate &local) const
 Return inverse of transposed of Jacobian. More...
 

Public Attributes

decltype(std::declval< Implementation >().volume()) typedef Volume
 Number type used for the geometry volume.
 

Related Functions

(Note that these are not member functions.)

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp, typename Impl >
auto referenceElement (const Geometry< mydim, cdim, GridImp, GeometryImp > &geo, const Impl &impl) -> decltype(referenceElement< typename GridImp::ctype, mydim >(geo.type()))
 Second-level dispatch to select the correct reference element for a grid geometry. More...
 

Interface for grid implementers

Implementation realGeometry
 
 Geometry (const Implementation &impl)
 copy constructor from implementation
 
const Geometryoperator= (const Geometry &rhs)=delete
 

Detailed Description

template<int mydim, int cdim, class GridImp, template< int, int, class > class GeometryImp>
class Dune::Geometry< mydim, cdim, GridImp, GeometryImp >

Wrapper class for geometries.

Template Parameters
mydimDimension of the domain
cdimDimension of the range
GridImpType that is a model of Dune::Grid
GeometryImpClass template that is a model of Dune::Geometry

Maps

A Geometry defines a map

\[ g : D \to W\]

where \(D\subseteq\mathbf{R}^\textrm{mydim}\) and \(W\subseteq\mathbf{R}^\textrm{cdim}\). The domain \(D\) is one of a set of predefined convex polytopes, the so-called reference elements (see Dune::ReferenceElement). The dimensionality of \(D\) is mydim. In general \(\textrm{mydim}\leq\textrm{cdim}\), i.e. the convex polytope may be mapped to a manifold. Moreover, we require that \( g\in \left( C^1(D) \right)^\textrm{cdim}\) and one-to-one.

Engine Concept

The Geometry class template wraps an object of type GeometryImp and forwards all member function calls to corresponding members of this class. In that sense Geometry defines the interface and GeometryImp supplies the implementation.

@addtogroup Grid Grid

The Dune Grid module defines a general interface to a parallel, in general

nonconforming, locally refined and hierarchical finite element mesh. The interface is independent of dimension and element type.

Terminology

@subsection subs1 Entity

An entity is a geometric object that is part of a grid. It is

generalized polytope that has the same dimensionality as the grid or a lower dimension.

@subsection subs20 Dimension

A grid has a fixed dimension \f$d\f$ which is the number of coordinates

required to specify any point in the grid. The dimension is a template parameter of a grid.

@subsection subs21 Codimension of an entity

Each entity has a codimension \f$c\f$ where \f$0 \leq c \leq d\f$ (the dimension of the grid).
An entity with codimension \f$ c\f$ in a grid of dimension \f$ d\f$ is a \f$d-c\f$-dimensional
object.


@subsection subs5 Subentity

Entities are hierarchically constructed in the sense that entities of
codimension 0 are made up of entities of codimension 1 which are themselves
made up of entities of codimension 2 etc. until entities of codimension \f$d-1\f$
which consist of entities of codimension \f$ d\f$.


@subsection subs3 Element

An element is an entity of codimension 0.


@subsection subs4 Vertex

A vertex is an entity of codimension \f$ d\f$ (the same as the grid's dimension).


@subsection subs22 World dimension

Each grid has a world dimension \f$ w\f$ with \f$ w\geq d\f$. This is the number

of coordinates of the positions of the grid's vertices.

@subsection subs33 Hierarchical grid

The %Dune grid interface describes not only a single grid but a sequence of
grids with different resolution. This is achieved by beginning with an
intentionally coarse grid, the so-called macro grid. Then each

element may be individually subdivided to yield new (smaller) elements. This construction is recursive such that each macro element and all the elements that resulted from subdividing it form a tree structure.

Grid refinement

The grid can only be modified in special phases, the so-called refinement phase. In between refinement phases the entities of the grid can not be modified in any way. During refinement currently only the hierachic subdivision can be modified.

@subsection subs3333 Grid level

All elements of the macro grid form level 0 of the grid structure. All
elements that are obtained from an \f$ l\f$-fold subdivision of a macro
element form level \f$ l\f$ of the grid structure.

@subsection subs333 Leaf grid

All elements of a grid that are not subdivided any further make up
the leaf grid. The leaf grid is the mesh with the finest resolution.

@subsection subs6 Assignable

A type is said to be assignable if it has a (public) copy constructor and
assignment operator. Note that this definition requires always both methods.


@subsection subs7 Default-constructible

A type is said to be default-constructible if it has a constructor without arguments.


@subsection subs8 Copy-constructible from type X

A type is said to be copy constructible from some other type X if it has
a copy constructor that takes a reference to an object of type X.


@subsection subs9 Equality-comparable

A type is said to be equality-comparable if it has an operator==.


@subsection subs10 LessThan-comparable

A type is lessthan-comparable if it has an operator<.


@subsection subs11 Dereferenceable

A type is dereferenceable if it has an operator* that delivers
a reference to a value type.

@subsection subs11 Iterator

An iterator is a type that can be dereferenced to yield an object of
its value type, i.e. it behaves like a pointer, and that can be incremented to
point to the next element in a linear sequence. In that respect it is comparable to
ForwardIterator in the Standard Template Library.


@subsection subs12 Mutable iterator

An iterator is called mutable if the value it refers to can be changed, i.e. it is
assignable.


@subsection subs13 Immutable iterator

An iterator is called immutable if the value referenced by the iterator can not
be changed, i. e. the value is not assignable and only methods marked const on the value
can be called.


@subsection subs14 Model

A type M is called a model of another type X if it implements all the methods
of X with the intended semantics. Typically X is a type that describes an interface.


@section Grid3 Types common to all grid implementations

- Dune::ReferenceElement describes the topology and geometry of standard entities.
Any given entity of the grid can be completely specified by a reference element
and a map from this reference element to world coordinate space.

- Dune::GeometryType defines names for the reference elements.

- Dune::CollectiveCommunication defines an interface to global communication
operations in a portable and transparent way. In particular also for sequential grids.



@section Grid2 Types making up a grid implementation

Each implementation of the Dune grid interface consist of a number of related types which together form a model of the grid interface. These types are the following:

  • Grid which is a model of Dune::Grid where the template parameters are at least the dimension and the world dimension. It is a container of entities that allows to access these entities and that knows the number of entities.
  • Entity which is a model of Dune::Entity. This class is parametrized by dimension and codimension. The entity encapsulates the topological part of an entity, i.e. its hierarchical construction from subentities and the relation to other entities. Entities cannot be created, copied or modified by the user. They can only be read-accessed through immutable iterators.
  • Geometry which is a model of Dune::Geometry. This class encapsulates the geometric part of an entity by mapping local coordinates in a reference element to world coordinates.
  • LevelIterator which is a model of Dune::LevelIterator is an immutable iterator that provides access to all entities of a given codimension and level of the grid.
  • LeafIterator which is a model of Dune::LeafIterator is an immutable iterator that provides access to all entities of a given codimension of the leaf grid.
  • HierarchicIterator which is a model of Dune::HierarchicIterator is an immutable iterator that provides access to all entities of codimension 0 that resulted from subdivision of a given entity of codimension 0.
  • Intersection which is a model of Dune::Intersection provides access an intersection of codimension 1 of two entity of codimension 0 or one entity and the boundary. In a conforming mesh this is a face of an element. For two entities with a common intersection the Intersection also provides information about the geometric location of the intersection. Furthermore it also provides information about intersections of an entity with the internal or external boundaries.
  • IntersectionIterator which is a model of Dune::IntersectionIterator provides access to all intersections of a given entity of codimension 0.
  • LevelIndexSet and LeafIndexSet which are both models of Dune::IndexSet are used to attach any kind of user-defined data to (subsets of) entities of the grid. This data is supposed to be stored in one-dimensional arrays for reasons of efficiency.
  • LocalIdSet and GlobalIdSet which are both models of Dune::IdSet are used to save user data during a grid refinement phase and during dynamic load balancing in the parallel case.
@section Grid22 Overview of basic capabilities of the types

<TABLE>
<TR>
<TD>Class</TD>
<TD>Assignable</TD>
<TD>DefaultConstructible</TD>
<TD>EqualityComparable</TD>
<TD>LessThanComparable</TD>
</TR>
<TR>
<TD>Grid</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
</TR>
<TR>
<TD>Entity</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
</TR>
<TR>
<TD>GeometryType</TD>
<TD>yes</TD>
<TD>yes</TD>
<TD>yes</TD>
<TD>yes</TD>
</TR>
<TR>
<TD>Geometry</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
</TR>
<TR>
<TD>LevelIterator</TD>
<TD>yes</TD>
<TD>no</TD>
<TD>yes</TD>
<TD>no</TD>
</TR>
<TR>
<TD>LeafIterator</TD>
<TD>yes</TD>
<TD>no</TD>
<TD>yes</TD>
<TD>no</TD>
</TR>
<TR>
<TD>HierarchicIterator</TD>
<TD>yes</TD>
<TD>no</TD>
<TD>yes</TD>
<TD>no</TD>
</TR>
<TR>
<TD>Intersection</TD>
<TD>yes</TD>
<TD>no</TD>
<TD>yes</TD>
<TD>no</TD>
</TR>
<TR>
<TD>IntersectionIterator</TD>
<TD>yes</TD>
<TD>no</TD>
<TD>yes</TD>
<TD>no</TD>
</TR>
<TR>
<TD>IndexSet</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
</TR>
<TR>
<TD>IdSet</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
<TD>no</TD>
</TR>
</TABLE>

Member Typedef Documentation

◆ Implementation

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
typedef GeometryImp< mydim, cdim, GridImp > Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::Implementation

type of underlying implementation

Warning
Implementation details may change without prior notification.

◆ JacobianInverseTransposed

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
typedef Implementation::JacobianInverseTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::JacobianInverseTransposed

type of jacobian inverse transposed

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

  • It satisfies the ConstMatrix interface.
  • It is copy constructible and copy assignable.

◆ JacobianTransposed

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
typedef Implementation::JacobianTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::JacobianTransposed

type of jacobian transposed

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

  • It satisfies the ConstMatrix interface.
  • It is copy constructible and copy assignable.

Member Enumeration Documentation

◆ anonymous enum

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
anonymous enum

export geometry dimension

Enumerator
mydimension 

geometry dimension

◆ anonymous enum

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
anonymous enum

export coordinate dimension

Enumerator
coorddimension 

dimension of embedding coordinate system

Member Function Documentation

◆ center()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::center ( ) const
inline

return center of geometry

Note that this method is still subject to a change of name and semantics. At the moment, the center is not required to be the centroid of the geometry, or even the centroid of its corners. This makes the current default implementation acceptable, which maps the centroid of the reference element to the geometry. We may change the name (and semantic) of the method to centroid() if we find reasonably efficient ways to implement it properly.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ corner()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::corner ( int  i) const
inline

Obtain a corner of the geometry.

This method is for convenient access to the corners of the geometry. The same result could be achieved by calling

Parameters
[in]inumber of the corner (with respect to the reference element)
Returns
position of the i-th corner

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::EdgeS0_5Basis< Geometry, RF >::EdgeS0_5Basis(), and Dune::EdgeS0_5Interpolation< Geometry, typename Basis::Traits >::EdgeS0_5Interpolation().

◆ corners()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
int Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::corners ( ) const
inline

Return the number of corners of the reference element.

Since a geometry is a convex polytope the number of corners is a well-defined concept. The method is redundant because this information is also available via the reference element. It is here for efficiency and ease of use.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ global()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::global ( const LocalCoordinate local) const
inline

Evaluate the map \( g\).

Parameters
[in]localPosition in the reference element \(D\)
Returns
Position in \(W\)

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::HierarchicSearch< Grid, IS >::findEntity(), and Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::LocalCoordinate().

◆ impl() [1/2]

◆ impl() [2/2]

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
const Implementation& Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl ( ) const
inline

access to the underlying implementation

Warning
Implementation details may change without prior notification.

◆ integrationElement()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
ctype Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::integrationElement ( const LocalCoordinate local) const
inline

Return the factor appearing in the integral transformation formula.

Let \( g : D \to W\) denote the transformation described by the Geometry. Then the jacobian of the transformation is defined as the \(\textrm{cdim}\times\textrm{mydim}\) matrix

\[ J_g(x) = \left( \begin{array}{ccc} \frac{\partial g_0}{\partial x_0} & \cdots & \frac{\partial g_0}{\partial x_{n-1}} \\ \vdots & \ddots & \vdots \\ \frac{\partial g_{m-1}}{\partial x_0} & \cdots & \frac{\partial g_{m-1}}{\partial x_{n-1}} \end{array} \right).\]

Here we abbreviated \(m=\textrm{cdim}\) and \(n=\textrm{mydim}\) for ease of readability.

The integration element \(\mu(x)\) for any \(x\in D\) is then defined as

\[ \mu(x) = \sqrt{|\det J_g^T(x)J_g(x)|}.\]

Parameters
[in]localPosition \(x\in D\)
Returns
integration element \(\mu(x)\)
Note
Each implementation computes the integration element with optimal efficiency. For example in an equidistant structured mesh it may be as simple as \(h^\textrm{mydim}\).

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ jacobianInverseTransposed()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
JacobianInverseTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobianInverseTransposed ( const LocalCoordinate local) const
inline

Return inverse of transposed of Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters
[in]localposition \(x\in D\)
Returns
\(J_g^{-T}(x)\)

The use of this function is to compute the gradient of some function \(f : W \to \textbf{R}\) at some position \(y=g(x)\), where \(x\in D\) and \(g\) the transformation of the Geometry. When we set \(\hat{f}(x) = f(g(x))\) and apply the chain rule we obtain

\[\nabla f(g(x)) = J_g^{-T}(x) \nabla \hat{f}(x).\]

Note
In the non-quadratic case \(\textrm{cdim} \neq \textrm{mydim}\), the pseudoinverse of \(J_g^T(x)\) is returned. This means that it is inverse for all tangential vectors in \(g(x)\) while mapping all normal vectors to zero.
The exact return type is implementation defined.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ jacobianTransposed()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
JacobianTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobianTransposed ( const LocalCoordinate local) const
inline

Return the transposed of the Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters
[in]localposition \(x\in D\)
Returns
\(J_g^T(x)\)
Note
The exact return type is implementation defined.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ LocalCoordinate()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::LocalCoordinate ( const GlobalCoordinate global) const
inline

Evaluate the inverse map \( g^{-1}\).

Parameters
[in]globalPosition in \(W\)
Returns
Position in \(D\) that maps to global

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::global(), and Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ operator=()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>
const Geometry& Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::operator= ( const Geometry< mydim, cdim, GridImp, GeometryImp > &  rhs)
delete

delete assignment operator

Friends And Related Function Documentation

◆ referenceElement()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp, typename Impl >
auto referenceElement ( const Geometry< mydim, cdim, GridImp, GeometryImp > &  geo,
const Impl &  impl 
) -> decltype(referenceElement<typename GridImp::ctype,mydim>(geo.type()))
related

Second-level dispatch to select the correct reference element for a grid geometry.

This function is the default implementation of the second-level reference element dispatch performed by Geometry.

Note
This function is only important for grid implementors with geometries that do not have a standard reference element.

When referenceElement() is called with a Geometry, it will forward the call to referenceElement(const Geometry&,const GeometryImplementation&). This default implementation will do the right thing as long as your geometry is based on a standard Dune ReferenceElement. If it is not and you want to supply your own reference element implementation, provide an override of this function for your specific geometry implementation.


The documentation for this class was generated from the following file:
GlobalCoordinate global(const LocalCoordinate &local) const
Evaluate the map .
Definition: geometry.hh:165
@ mydimension
Definition: geometry.hh:90
auto referenceElement(const Geometry< mydim, cdim, GridImp, GeometryImp > &geo, const Impl &impl) -> decltype(referenceElement< typename GridImp::ctype, mydim >(geo.type()))
Second-level dispatch to select the correct reference element for a grid geometry.
Definition: geometry.hh:440
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