/usr/include/deal.II/base/tensor_product_polynomials

// ---------------------------------------------------------------------
// $Id$
//
// Copyright (C) 2012 - 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
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#ifndef dealii__tensor_product_polynomials_bubbles_h
#define dealii__tensor_product_polynomials_bubbles_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/tensor_product_polynomials.h>
#include <deal.II/base/utilities.h>

#include <vector>

DEAL_II_NAMESPACE_OPEN


/**
 * @addtogroup Polynomials
 * @{
 */

/**
 * Tensor product of given polynomials and bubble functions of form
 * $(2*x_j-1)^{degree-1}\prod_{i=0}^{dim-1}(x_i(1-x_i))$. This class inherits
 * most of its functionality from TensorProductPolynomials. The bubble
 * enrichments are added for the last indices. index.
 *
 * @author Daniel Arndt, 2015
 */
template <int dim>
class TensorProductPolynomialsBubbles : public TensorProductPolynomials<dim>
{
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 objects that should be derived
   * or otherwise convertible to one-dimensional polynomial objects. It will
   * be copied element by element into a private variable.
   */
  template <class Pol>
  TensorProductPolynomialsBubbles (const std::vector<Pol> &pols);

  /**
   * Computes the value and the first and second derivatives of each tensor
   * product 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.
   *
   * If you need values or derivatives of all tensor product 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 tensor product 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 tensor product polynomial at
   * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
   * numbering.
   *
   * Note, that using this function within a loop over all tensor product
   * polynomials is not efficient, because then each point value of the
   * underlying (one-dimensional) polynomials is (unnecessarily) computed
   * several times.  Instead use the compute() function with
   * <tt>values.size()==</tt>n() to get the point values of all tensor
   * polynomials all at once and in a much more efficient way.
   */
  double compute_value (const unsigned int i,
                        const Point<dim> &p) const;

  /**
   * Computes the order @p order derivative of the <tt>i</tt>th tensor product
   * polynomial at <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor
   * product numbering.
   *
   * Note, that using this function within a loop over all tensor product
   * polynomials is not efficient, because then each derivative value of the
   * underlying (one-dimensional) polynomials is (unnecessarily) computed
   * several times.  Instead use the compute() function, see above, with the
   * size of the appropriate parameter set to n() to get the point value of
   * all tensor polynomials all at once and in a much more efficient way.
   */
  template <int order>
  Tensor<order,dim> compute_derivative (const unsigned int i,
                                        const Point<dim> &p) const;

  /**
   * Computes the grad of the <tt>i</tt>th tensor product polynomial at
   * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
   * numbering.
   *
   * Note, that using this function within a loop over all tensor product
   * polynomials is not efficient, because then each derivative value of the
   * underlying (one-dimensional) polynomials is (unnecessarily) computed
   * several times.  Instead use the compute() function, see above, with
   * <tt>grads.size()==</tt>n() to get the point value of all tensor
   * polynomials all at once and in a much more efficient way.
   */
  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 tensor
   * product polynomial at <tt>unit_point</tt>. Here <tt>i</tt> is given in
   * tensor product numbering.
   *
   * Note, that using this function within a loop over all tensor product
   * polynomials is not efficient, because then each derivative value of the
   * underlying (one-dimensional) polynomials is (unnecessarily) computed
   * several times.  Instead use the compute() function, see above, with
   * <tt>grad_grads.size()==</tt>n() to get the point value of all tensor
   * polynomials all at once and in a much more efficient way.
   */
  Tensor<2,dim> compute_grad_grad (const unsigned int i,
                                   const Point<dim> &p) const;

  /**
   * Returns the number of tensor product polynomials plus the bubble
   * enrichments. For <i>n</i> 1d polynomials this is <i>n<sup>dim</sup>+1</i>
   * if the maximum degree of the polynomials is one and
   * <i>n<sup>dim</sup>+dim</i> otherwise.
   */
  unsigned int n () const;
};

/** @} */


/* ---------------- template and inline functions ---------- */

#ifndef DOXYGEN

template <int dim>
template <class Pol>
inline
TensorProductPolynomialsBubbles<dim>::
TensorProductPolynomialsBubbles(const std::vector<Pol> &pols)
  :
  TensorProductPolynomials<dim>(pols)
{
  const unsigned int q_degree = this->polynomials.size()-1;
  const unsigned int n_bubbles = ((q_degree<=1)?1:dim);
  // append index for renumbering
  for (unsigned int i=0; i<n_bubbles; ++i)
    {
      this->index_map.push_back(i+this->n_tensor_pols);
      this->index_map_inverse.push_back(i+this->n_tensor_pols);
    }
}



template <int dim>
inline
unsigned int
TensorProductPolynomialsBubbles<dim>::n() const
{
  return this->n_tensor_pols+dim;
}



template <>
inline
unsigned int
TensorProductPolynomialsBubbles<0>::n() const
{
  return numbers::invalid_unsigned_int;
}

template <int dim>
template <int order>
Tensor<order,dim>
TensorProductPolynomialsBubbles<dim>::compute_derivative (const unsigned int i,
                                                          const Point<dim> &p) const
{
  const unsigned int q_degree = this->polynomials.size()-1;
  const unsigned int max_q_indices = this->n_tensor_pols;
  const unsigned int n_bubbles = ((q_degree<=1)?1:dim);
  (void)n_bubbles;
  Assert (i<max_q_indices+n_bubbles, ExcInternalError());

  // treat the regular basis functions
  if (i<max_q_indices)
    return this->TensorProductPolynomials<dim>::template compute_derivative<order>(i,p);

  const unsigned int comp = i - this->n_tensor_pols;

  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.;
          //compute grad(4*\prod_{i=1}^d (x_i(1-x_i)))(p)
          for (unsigned j=0; j<dim; ++j)
            derivative_1[d] *= (d==j ? 4*(1-2*p(j)) : 4*p(j)*(1-p(j)));
          // and multiply with (2*x_i-1)^{r-1}
          for (unsigned int i=0; i<q_degree-1; ++i)
            derivative_1[d]*=2*p(comp)-1;
        }

      if (q_degree>=2)
        {
          //add \prod_{i=1}^d 4*(x_i(1-x_i))(p)
          double value=1.;
          for (unsigned int j=0; j < dim; ++j)
            value*=4*p(j)*(1-p(j));
          //and multiply with grad(2*x_i-1)^{r-1}
          double tmp=value*2*(q_degree-1);
          for (unsigned int i=0; i<q_degree-2; ++i)
            tmp*=2*p(comp)-1;
          derivative_1[comp]+=tmp;
        }

      return derivative;
    }
    case 2:
    {
      Tensor<2,dim> &derivative_2 = *reinterpret_cast<Tensor<2,dim>*>(&derivative);

      double v [dim+1][3];
      {
        for (unsigned int c=0; c<dim; ++c)
          {
            v[c][0] = 4*p(c)*(1-p(c));
            v[c][1] = 4*(1-2*p(c));
            v[c][2] = -8;
          }

        double tmp=1.;
        for (unsigned int i=0; i<q_degree-1; ++i)
          tmp *= 2*p(comp)-1;
        v[dim][0] = tmp;

        if (q_degree>=2)
          {
            double tmp = 2*(q_degree-1);
            for (unsigned int i=0; i<q_degree-2; ++i)
              tmp *= 2*p(comp)-1;
            v[dim][1] = tmp;
          }
        else
          v[dim][1] = 0.;

        if (q_degree>=3)
          {
            double tmp=4*(q_degree-2)*(q_degree-1);
            for (unsigned int i=0; i<q_degree-3; ++i)
              tmp *= 2*p(comp)-1;
            v[dim][2] = tmp;
          }
        else
          v[dim][2] = 0.;
      }

      //calculate (\partial_j \partial_k \psi) * monomial
      Tensor<2,dim> grad_grad_1;
      for (unsigned int d1=0; d1<dim; ++d1)
        for (unsigned int d2=0; d2<dim; ++d2)
          {
            grad_grad_1[d1][d2] = v[dim][0];
            for (unsigned int x=0; x<dim; ++x)
              {
                unsigned int derivative=0;
                if (d1==x || d2==x)
                  {
                    if (d1==d2)
                      derivative=2;
                    else
                      derivative=1;
                  }
                grad_grad_1[d1][d2] *= v[x][derivative];
              }
          }

      //calculate (\partial_j  \psi) *(\partial_k monomial)
      // and (\partial_k  \psi) *(\partial_j monomial)
      Tensor<2,dim> grad_grad_2;
      Tensor<2,dim> grad_grad_3;
      for (unsigned int d=0; d<dim; ++d)
        {
          grad_grad_2[d][comp] = v[dim][1];
          grad_grad_3[comp][d] = v[dim][1];
          for (unsigned int x=0; x<dim; ++x)
            {
              grad_grad_2[d][comp] *= v[x][d==x];
              grad_grad_3[comp][d] *= v[x][d==x];
            }
        }

      //calculate \psi *(\partial j \partial_k monomial) and sum
      double psi_value = 1.;
      for (unsigned int x=0; x<dim; ++x)
        psi_value *= v[x][0];

      for (unsigned int d1=0; d1<dim; ++d1)
        for (unsigned int d2=0; d2<dim; ++d2)
          derivative_2[d1][d2] = grad_grad_1[d1][d2]
                                 +grad_grad_2[d1][d2]
                                 +grad_grad_3[d1][d2];
      derivative_2[comp][comp]+=psi_value*v[dim][2];

      return derivative;
    }
    default:
    {
      Assert (false, ExcNotImplemented());
      return derivative;
    }
    }
}


#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE

#endif
libdeal.ii-dev 8.4.2-2+b1 / usr / include / deal.II / base / tensor_product_polynomials_bubbles.h
