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// $Id: polynomial_space.h 20161 2009-11-24 22:54:51Z kanschat $
// Version: $Name$
//
// Copyright (C) 2002, 2003, 2004, 2005, 2006, 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__polynomial_space_h
#define __deal2__polynomial_space_h
#include <base/config.h>
#include <base/exceptions.h>
#include <base/tensor.h>
#include <base/point.h>
#include <base/polynomial.h>
#include <base/smartpointer.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/**
* Representation of the space of polynomials of degree at most n in
* higher dimensions.
*
* Given a vector of <i>n</i> one-dimensional polynomials
* <i>P<sub>0</sub></i> to <i>P<sub>n</sub></i>, where
* <i>P<sub>i</sub></i> has degree <i>i</i>, this class generates all
* dim-dimensional polynomials of the form <i>
* P<sub>ijk</sub>(x,y,z) =
* P<sub>i</sub>(x)P<sub>j</sub>(y)P<sub>k</sub>(z)</i>, where the sum
* of <i>i</i>, <i>j</i> and <i>k</i> is less than or equal <i>n</i>.
*
* The output_indices() function prints the ordering of the
* polynomials, i.e. for each dim-dimensional polynomial in the
* polynomial space it gives the indices i,j,k of the one-dimensional
* polynomials in x,y and z direction. The ordering of the
* dim-dimensional polynomials can be changed by using the
* set_numbering() function.
*
* The standard ordering of polynomials is that indices for the first
* space dimension vasry fastest and the last space dimension is
* slowest. In particular, if we take for simplicity the vector of
* monomials <i>x<sup>0</sup>, x<sup>1</sup>, x<sup>2</sup>,...,
* x<sup>n</sup></i>, we get
*
* <dl>
* <dt> 1D <dd> <i> x<sup>0</sup>, x<sup>1</sup>,...,x<sup>n</sup></i>
* <dt> 2D: <dd> <i> x<sup>0</sup>y<sup>0</sup>,
* x<sup>1</sup>y<sup>0</sup>,...,
* x<sup>n</sup>y<sup>0</sup>,<br>
* x<sup>0</sup>y<sup>1</sup>,
* x<sup>1</sup>y<sup>1</sup>,...,
* x<sup>n-1</sup>y<sup>1</sup>,<br>
* x<sup>0</sup>y<sup>2</sup>,...
* x<sup>n-2</sup>y<sup>2</sup>,<br>...<br>
* x<sup>0</sup>y<sup>n-1</sup>,
* x<sup>1</sup>y<sup>n-1</sup>,<br>
* x<sup>0</sup>y<sup>n</sup>
* </i>
* <dt> 3D: <dd> <i> x<sup>0</sup>y<sup>0</sup>z<sup>0</sup>,...,
* x<sup>n</sup>y<sup>0</sup>z<sup>0</sup>,<br>
* x<sup>0</sup>y<sup>1</sup>z<sup>0</sup>,...,
* x<sup>n-1</sup>y<sup>1</sup>z<sup>0</sup>,<br>...<br>
* x<sup>0</sup>y<sup>n</sup>z<sup>0</sup>,<br>
* x<sup>0</sup>y<sup>0</sup>z<sup>1</sup>,...
* x<sup>n-1</sup>y<sup>0</sup>z<sup>1</sup>,<br>...<br>
* x<sup>0</sup>y<sup>n-1</sup>z<sup>1</sup>,<br>
* x<sup>0</sup>y<sup>0</sup>z<sup>2</sup>,...
* x<sup>n-2</sup>y<sup>0</sup>z<sup>2</sup>,<br>...<br>
* x<sup>0</sup>y<sup>0</sup>z<sup>n</sup>
* </i>
* </dl>
*
* @ingroup Polynomials
* @author Guido Kanschat, Wolfgang Bangerth, Ralf Hartmann 2002, 2003, 2004, 2005
*/
template <int dim>
class PolynomialSpace
{
public:
/**
* Access to the dimension of
* this object, for checking and
* automatic setting of dimension
* in other classes.
*/
static const unsigned int dimension = dim;
/**
* Constructor. <tt>pols</tt> is a
* vector of pointers to
* one-dimensional polynomials
* and will be copied into a
* private member variable. The static
* type of the template argument
* <tt>pols</tt> needs to be
* convertible to
* Polynomials::Polynomial@<double@>,
* i.e. should usually be a
* derived class of
* Polynomials::Polynomial@<double@>.
*/
template <class Pol>
PolynomialSpace (const std::vector<Pol> &pols);
/**
* Prints the list of the indices
* to <tt>out</tt>.
*/
template <class STREAM>
void output_indices(STREAM &out) const;
/**
* Sets the ordering of the
* polynomials. Requires
* <tt>renumber.size()==n()</tt>.
* Stores a copy of
* <tt>renumber</tt>.
*/
void set_numbering(const std::vector<unsigned int> &renumber);
/**
* Computes the value and the
* first and second derivatives
* of each polynomial at
* <tt>unit_point</tt>.
*
* The size of the vectors must
* either be equal 0 or equal
* n(). In the first case,
* the function will not compute
* these values, i.e. you
* indicate what you want to have
* computed by resizing those
* vectors which you want filled.
*
* If you need values or
* derivatives of all polynomials
* then use this function, rather
* than using any of the
* compute_value(),
* compute_grad() or
* compute_grad_grad()
* functions, see below, in a
* loop over all polynomials.
*/
void compute (const Point<dim> &unit_point,
std::vector<double> &values,
std::vector<Tensor<1,dim> > &grads,
std::vector<Tensor<2,dim> > &grad_grads) const;
/**
* Computes the value of the
* <tt>i</tt>th polynomial at
* <tt>unit_point</tt>.
*
* Consider using compute() instead.
*/
double compute_value (const unsigned int i,
const Point<dim> &p) const;
/**
* Computes the gradient of the
* <tt>i</tt>th polynomial at
* <tt>unit_point</tt>.
*
* Consider using compute() instead.
*/
Tensor<1,dim> compute_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* Computes the second derivative
* (grad_grad) of the <tt>i</tt>th
* polynomial at
* <tt>unit_point</tt>.
*
* Consider using compute() instead.
*/
Tensor<2,dim> compute_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the number of
* polynomials spanning the space
* represented by this
* class. Here, if <tt>N</tt> is the
* number of one-dimensional
* polynomials given, then the
* result of this function is
* <i>N</i> in 1d, <i>N(N+1)/2</i> in
* 2d, and <i>N(N+1)(N+2)/6</i> in
* 3d.
*/
unsigned int n () const;
/**
* Degree of the space. This is
* by definition the number of
* polynomials given to the
* constructor, NOT the maximal
* degree of a polynomial in this
* vector. The latter value is
* never checked and therefore
* left to the application.
*/
unsigned int degree () const;
/**
* Static function used in the
* constructor to compute the
* number of polynomials.
*/
static unsigned int compute_n_pols (const unsigned int n);
protected:
/**
* Compute numbers in x, y and z
* direction. Given an index
* <tt>n</tt> in the d-dimensional
* polynomial space, compute the
* indices i,j,k such that
* <i>p<sub>n</sub>(x,y,z) =
* p<sub>i</sub>(x)p<sub>j</sub>(y)p<sub>k</sub>(z)</i>.
*/
void compute_index (const unsigned int n,
unsigned int (&index)[dim]) const;
private:
/**
* Copy of the vector <tt>pols</tt> of
* polynomials given to the
* constructor.
*/
const std::vector<Polynomials::Polynomial<double> > polynomials;
/**
* Store the precomputed value
* which the <tt>n()</tt> function
* returns.
*/
const unsigned int n_pols;
/**
* Index map for reordering the
* polynomials.
*/
std::vector<unsigned int> index_map;
/**
* Index map for reordering the
* polynomials.
*/
std::vector<unsigned int> index_map_inverse;
};
/* -------------- declaration of explicit specializations --- */
template <>
void PolynomialSpace<1>::compute_index(const unsigned int n,
unsigned int (&index)[1]) const;
template <>
void PolynomialSpace<2>::compute_index(const unsigned int n,
unsigned int (&index)[2]) const;
template <>
void PolynomialSpace<3>::compute_index(const unsigned int n,
unsigned int (&index)[3]) const;
/* -------------- inline and template functions ------------- */
template <int dim>
template <class Pol>
PolynomialSpace<dim>::PolynomialSpace (const std::vector<Pol> &pols)
:
polynomials (pols.begin(), pols.end()),
n_pols (compute_n_pols(polynomials.size())),
index_map(n_pols),
index_map_inverse(n_pols)
{
// per default set this index map
// to identity. This map can be
// changed by the user through the
// set_numbering function
for (unsigned int i=0; i<n_pols; ++i)
{
index_map[i]=i;
index_map_inverse[i]=i;
}
}
template<int dim>
inline
unsigned int
PolynomialSpace<dim>::n() const
{
return n_pols;
}
template<int dim>
inline
unsigned int
PolynomialSpace<dim>::degree() const
{
return polynomials.size();
}
template <int dim>
template <class STREAM>
void
PolynomialSpace<dim>::output_indices(STREAM &out) const
{
unsigned int ix[dim];
for (unsigned int i=0; i<n_pols; ++i)
{
compute_index(i,ix);
out << i << "\t";
for (unsigned int d=0; d<dim; ++d)
out << ix[d] << " ";
out << std::endl;
}
}
DEAL_II_NAMESPACE_CLOSE
#endif
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