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 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
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...

ctype	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...

Protected Member Functions
const Implementation &	impl () const
	return reference to the implementation

Interface for grid implementers
Implementation	realGeometry

	Geometry (const Implementation &impl)
	copy constructor from implementation

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

mydim	Dimension of the domain
cdim	Dimension of the range
GridImp	Type that is a model of Dune::Grid
GeometryImp	Class 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 also: 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

◆ 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

global( referenceElement.position( i, mydimension ) )

Dune::Geometry::global

GlobalCoordinate global(const LocalCoordinate &local) const

Evaluate the map .

Definition: geometry.hh:157

Dune::Geometry::mydimension

@ mydimension

Definition: geometry.hh:85

Parameters

[in] i number of the corner (with respect to the reference element)

Returns: position of the i-th corner

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

◆ 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] local Position 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().

◆ 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] local Position \(x\in D\)

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] local position \(x\in D\)

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] local position \(x\in D\)

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] global Position 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().

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

dune/grid/common/geometry.hh

DUNE Distributed and Unified Numerics Environment

Dune Core Modules (2.5.0)

Public Types

Public Member Functions

Protected Member Functions

Interface for grid implementers

Detailed Description

Maps

Engine Concept

Terminology

Grid refinement

Member Typedef Documentation

◆ JacobianInverseTransposed

◆ JacobianTransposed

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Member Function Documentation

◆ center()

◆ corner()

◆ corners()

◆ global()

◆ integrationElement()

◆ jacobianInverseTransposed()

◆ jacobianTransposed()

◆ LocalCoordinate()