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// $Id: quadrature.h 20107 2009-11-13 15:40:05Z kronbichler $
// Version: $Name$
//
// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2009 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__quadrature_h
#define __deal2__quadrature_h
#include <base/config.h>
#include <base/point.h>
#include <base/subscriptor.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup Quadrature */
/*@{*/
/**
* Base class for quadrature formulæ in arbitrary dimensions. This class
* stores quadrature points and weights on the unit line [0,1], unit
* square [0,1]x[0,1], etc.
*
* There are a number of derived classes, denoting concrete
* integration formulæ. Their names names prefixed by
* <tt>Q</tt>. Refer to the list of derived classes for more details.
*
* The schemes for higher dimensions are typically tensor products of the
* one-dimensional formulæ, but refer to the section on implementation
* detail below.
*
* In order to allow for dimension independent programming, a
* quadrature formula of dimension zero exists. Since an integral over
* zero dimensions is the evaluation at a single point, any
* constructor of such a formula initializes to a single quadrature
* point with weight one. Access to the weight is possible, while
* access to the quadrature point is not permitted, since a Point of
* dimension zero contains no information. The main purpose of these
* formulæ is their use in QProjector, which will create a useful
* formula of dimension one out of them.
*
* <h3>Mathematical background</h3>
*
* For each quadrature formula we denote by <tt>m</tt>, the maximal
* degree of polynomials integrated exactly. This number is given in
* the documentation of each formula. The order of the integration
* error is <tt>m+1</tt>, that is, the error is the size of the cell
* two the <tt>m+1</tt> by the Bramble-Hilbert Lemma. The number
* <tt>m</tt> is to be found in the documentation of each concrete
* formula. For the optimal formulæ QGauss we have $m = 2N-1$, where
* N is the constructor parameter to QGauss. The tensor product
* formulæ are exact on tensor product polynomials of degree
* <tt>m</tt> in each space direction, but they are still only of
* <tt>m+1</tt>st order.
*
* <h3>Implementation details</h3>
*
* Most integration formulæ in more than one space dimension are
* tensor products of quadrature formulæ in one space dimension, or
* more generally the tensor product of a formula in <tt>(dim-1)</tt>
* dimensions and one in one dimension. There is a special constructor
* to generate a quadrature formula from two others. For example, the
* QGauss@<dim@> formulæ include <i>N<sup>dim</sup></i> quadrature
* points in <tt>dim</tt> dimensions, where N is the constructor
* parameter of QGauss.
*
* @note Instantiations for this template are provided for dimensions
* 0, 1, 2, and 3 (see the section on @ref Instantiations).
*
* @author Wolfgang Bangerth, Guido Kanschat, 1998, 1999, 2000, 2005, 2009
*/
template <int dim>
class Quadrature : public Subscriptor
{
public:
/**
* Define a typedef for a
* quadrature that acts on an
* object of one dimension
* less. For cells, this would
* then be a face quadrature.
*/
typedef Quadrature<dim-1> SubQuadrature;
/**
* @deprecated Use size()
* instead.
*
* Number of quadrature points.
*
* @warning After introduction of
* the assignment operator, this
* number is not constant anymore
* and erroneous assignment can
* compromise integrity of the
* Quadrature object. Since
* direct data access should be
* considered a design flaw
* anyway, it is strongly
* suggested to use size()
* instead.
*/
unsigned int n_quadrature_points;
/**
* Constructor.
*
* This constructor is marked as
* explicit to avoid involuntary
* accidents like in
* <code>hp::QCollection@<dim@>
* q_collection(3)</code> where
* <code>hp::QCollection@<dim@>
* q_collection(QGauss@<dim@>(3))</code>
* was meant.
*/
explicit Quadrature (const unsigned int n_quadrature_points = 0);
/**
* Build this quadrature formula
* as the tensor product of a
* formula in a dimension one
* less than the present and a
* formula in one dimension.
*
* <tt>SubQuadrature<dim>::type</tt>
* expands to
* <tt>Quadrature<dim-1></tt>.
*/
Quadrature (const SubQuadrature &,
const Quadrature<1> &);
/**
* Build this quadrature formula
* as the <tt>dim</tt>-fold
* tensor product of a formula in
* one dimension.
*
* Assuming that the points in
* the one-dimensional rule are in
* ascending order, the points of
* the resulting rule are ordered
* lexicographically with
* <i>x</i> running fastest.
*
* In order to avoid a conflict
* with the copy constructor in
* 1d, we let the argument be a
* 0d quadrature formula for
* dim==1, and a 1d quadrature
* formula for all other space
* dimensions.
*/
explicit Quadrature (const Quadrature<dim != 1 ? 1 : 0> &quadrature_1d);
/**
* Copy constructor.
*/
Quadrature (const Quadrature<dim> &q);
/**
* Construct a quadrature formula
* from given vectors of
* quadrature points (which
* should really be in the unit
* cell) and the corresponding
* weights. You will want to have
* the weights sum up to one, but
* this is not checked.
*/
Quadrature (const std::vector<Point<dim> > &points,
const std::vector<double> &weights);
/**
* Construct a dummy quadrature
* formula from a list of points,
* with weights set to
* infinity. The resulting object
* is therefore not meant to
* actually perform integrations,
* but rather to be used with
* FEValues objects in
* order to find the position of
* some points (the quadrature
* points in this object) on the
* transformed cell in real
* space.
*/
Quadrature (const std::vector<Point<dim> > &points);
/**
* Constructor for a one-point
* quadrature. Sets the weight of
* this point to one.
*/
Quadrature (const Point<dim> &point);
/**
* Assignment operator. Copies
* contents of #weights and
* #quadrature_points as well as
* size.
*/
Quadrature& operator = (const Quadrature<dim>&);
/**
* Virtual destructor.
*/
virtual ~Quadrature ();
/**
* Number of quadrature points.
*/
unsigned int size () const;
/**
* Return the <tt>i</tt>th quadrature
* point.
*/
const Point<dim> & point (const unsigned int i) const;
/**
* Return a reference to the
* whole array of quadrature
* points.
*/
const std::vector<Point<dim> > & get_points () const;
/**
* Return the weight of the <tt>i</tt>th
* quadrature point.
*/
double weight (const unsigned int i) const;
/**
* Return a reference to the whole array
* of weights.
*/
const std::vector<double> & get_weights () const;
/**
* Determine an estimate for
* the memory consumption (in
* bytes) of this
* object.
*/
unsigned int memory_consumption () const;
protected:
/**
* List of quadrature points. To
* be filled by the constructors
* of derived classes.
*/
std::vector<Point<dim> > quadrature_points;
/**
* List of weights of the
* quadrature points. To be
* filled by the constructors of
* derived classes.
*/
std::vector<double> weights;
};
/**
* Quadrature formula implementing anisotropic distributions of
* quadrature points on the reference cell. To this end, the tensor
* product of <tt>dim</tt> one-dimensional quadrature formulas is
* generated.
*
* @note Each constructor can only be used in the dimension matching
* the number of arguments.
*
* @author Guido Kanschat, 2005
*/
template <int dim>
class QAnisotropic : public Quadrature<dim>
{
public:
/**
* Constructor for a
* one-dimensional formula. This
* one just copies the given
* quadrature rule.
*/
QAnisotropic(const Quadrature<1>& qx);
/**
* Constructor for a
* two-dimensional formula.
*/
QAnisotropic(const Quadrature<1>& qx,
const Quadrature<1>& qy);
/**
* Constructor for a
* three-dimensional formula.
*/
QAnisotropic(const Quadrature<1>& qx,
const Quadrature<1>& qy,
const Quadrature<1>& qz);
};
/**
* Quadrature formula constructed by iteration of another quadrature formula in
* each direction. In more than one space dimension, the resulting quadrature
* formula is constructed in the usual way by building the tensor product of
* the respective iterated quadrature formula in one space dimension.
*
* In one space dimension, the given base formula is copied and scaled onto
* a given number of subintervals of length <tt>1/n_copies</tt>. If the quadrature
* formula uses both end points of the unit interval, then in the interior
* of the iterated quadrature formula there would be quadrature points which
* are used twice; we merge them into one with a weight which is the sum
* of the weights of the left- and the rightmost quadrature point.
*
* Since all dimensions higher than one are built up by tensor products of
* one dimensional and <tt>dim-1</tt> dimensional quadrature formulæ, the
* argument given to the constructor needs to be a quadrature formula in
* one space dimension, rather than in <tt>dim</tt> dimensions.
*
* The aim of this class is to provide a
* low order formula, where the error constant can be tuned by
* increasing the number of quadrature points. This is useful in
* integrating non-differentiable functions on cells.
*
* @author Wolfgang Bangerth 1999
*/
template <int dim>
class QIterated : public Quadrature<dim>
{
public:
/**
* Constructor. Iterate the given
* quadrature formula <tt>n_copies</tt> times in
* each direction.
*/
QIterated (const Quadrature<1> &base_quadrature,
const unsigned int n_copies);
/**
* Exception
*/
DeclException0 (ExcInvalidQuadratureFormula);
private:
/**
* Check whether the given
* quadrature formula has quadrature
* points at the left and right end points
* of the interval.
*/
static bool
uses_both_endpoints (const Quadrature<1> &base_quadrature);
};
/*@}*/
#ifndef DOXYGEN
// ------------------- inline and template functions ----------------
template<int dim>
inline
unsigned int
Quadrature<dim>::size () const
{
return weights.size();
}
template <int dim>
inline
const Point<dim> &
Quadrature<dim>::point (const unsigned int i) const
{
Assert (dim != 0, ExcNotImplemented());
AssertIndexRange(i, size());
return quadrature_points[i];
}
template <int dim>
double
Quadrature<dim>::weight (const unsigned int i) const
{
Assert (dim != 0, ExcNotImplemented());
AssertIndexRange(i, size());
return weights[i];
}
template <int dim>
inline
const std::vector<Point<dim> > &
Quadrature<dim>::get_points () const
{
Assert (dim > 0, ExcInternalError());
return quadrature_points;
}
template <int dim>
inline
const std::vector<double> &
Quadrature<dim>::get_weights () const
{
return weights;
}
/* -------------- declaration of explicit specializations ------------- */
template <>
Quadrature<0>::Quadrature (const unsigned int);
template <>
Quadrature<0>::Quadrature (const Quadrature<-1> &,
const Quadrature<1> &);
template <>
Quadrature<0>::Quadrature (const Quadrature<1> &);
template <>
Quadrature<0>::~Quadrature ();
template <>
Quadrature<1>::Quadrature (const Quadrature<0> &,
const Quadrature<1> &);
template <>
Quadrature<1>::Quadrature (const Quadrature<0> &);
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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