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//
// Copyright (C) 2002 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__polynomial_space_h
#define dealii__polynomial_space_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/tensor.h>
#include <deal.II/base/point.h>
#include <deal.II/base/polynomial.h>
#include <deal.II/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 vary 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 StreamType>
void output_indices(StreamType &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,
std::vector<Tensor<3,dim> > &third_derivatives,
std::vector<Tensor<4,dim> > &fourth_derivatives) const;
/**
* Computes the value of the <tt>i</tt>th polynomial at unit point
* <tt>p</tt>.
*
* Consider using compute() instead.
*/
double compute_value (const unsigned int i,
const Point<dim> &p) const;
/**
* Computes the <tt>order</tt>th derivative of the <tt>i</tt>th polynomial
* at unit point <tt>p</tt>.
*
* Consider using compute() instead.
*
* @tparam order The order of the derivative.
*/
template <int order>
Tensor<order,dim> compute_derivative (const unsigned int i,
const Point<dim> &p) const;
/**
* Computes the gradient of the <tt>i</tt>th polynomial at unit point
* <tt>p</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 unit point <tt>p</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.
*
* @warning The argument `n` is not the maximal degree, but the number of
* onedimensional polynomials, thus the degree plus one.
*/
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>0?dim:1]) 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 StreamType>
void
PolynomialSpace<dim>::output_indices(StreamType &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;
}
}
template <int dim>
template <int order>
Tensor<order,dim>
PolynomialSpace<dim>::compute_derivative (const unsigned int i,
const Point<dim> &p) const
{
unsigned int indices[dim];
compute_index (i, indices);
double v [dim][order+1];
{
std::vector<double> tmp (order+1);
for (unsigned int d=0; d<dim; ++d)
{
polynomials[indices[d]].value (p(d), tmp);
for (unsigned int j=0; j<order+1; ++j)
v[d][j] = tmp[j];
}
}
Tensor<order,dim> derivative;
switch (order)
{
case 1:
{
Tensor<1,dim> &derivative_1 = *reinterpret_cast<Tensor<1,dim>*>(&derivative);
for (unsigned int d=0; d<dim; ++d)
{
derivative_1[d] = 1.;
for (unsigned int x=0; x<dim; ++x)
{
unsigned int x_order=0;
if (d==x) ++x_order;
derivative_1[d] *= v[x][x_order];
}
}
return derivative;
}
case 2:
{
Tensor<2,dim> &derivative_2 = *reinterpret_cast<Tensor<2,dim>*>(&derivative);
for (unsigned int d1=0; d1<dim; ++d1)
for (unsigned int d2=0; d2<dim; ++d2)
{
derivative_2[d1][d2] = 1.;
for (unsigned int x=0; x<dim; ++x)
{
unsigned int x_order=0;
if (d1==x) ++x_order;
if (d2==x) ++x_order;
derivative_2[d1][d2] *= v[x][x_order];
}
}
return derivative;
}
case 3:
{
Tensor<3,dim> &derivative_3 = *reinterpret_cast<Tensor<3,dim>*>(&derivative);
for (unsigned int d1=0; d1<dim; ++d1)
for (unsigned int d2=0; d2<dim; ++d2)
for (unsigned int d3=0; d3<dim; ++d3)
{
derivative_3[d1][d2][d3] = 1.;
for (unsigned int x=0; x<dim; ++x)
{
unsigned int x_order=0;
if (d1==x) ++x_order;
if (d2==x) ++x_order;
if (d3==x) ++x_order;
derivative_3[d1][d2][d3] *= v[x][x_order];
}
}
return derivative;
}
case 4:
{
Tensor<4,dim> &derivative_4 = *reinterpret_cast<Tensor<4,dim>*>(&derivative);
for (unsigned int d1=0; d1<dim; ++d1)
for (unsigned int d2=0; d2<dim; ++d2)
for (unsigned int d3=0; d3<dim; ++d3)
for (unsigned int d4=0; d4<dim; ++d4)
{
derivative_4[d1][d2][d3][d4] = 1.;
for (unsigned int x=0; x<dim; ++x)
{
unsigned int x_order=0;
if (d1==x) ++x_order;
if (d2==x) ++x_order;
if (d3==x) ++x_order;
if (d4==x) ++x_order;
derivative_4[d1][d2][d3][d4] *= v[x][x_order];
}
}
return derivative;
}
default:
{
Assert (false, ExcNotImplemented());
return derivative;
}
}
}
DEAL_II_NAMESPACE_CLOSE
#endif
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