tensor_product_polynomials_bubbles.h

// ---------------------------------------------------------------------
//
// Copyright (C) 2012 - 2018 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.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------

#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/point.h>
#include <deal.II/base/polynomial.h>
#include <deal.II/base/tensor.h>
#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/base/utilities.h>

#include <vector>

DEAL_II_NAMESPACE_OPEN


template <int dim>
class TensorProductPolynomialsBubbles : public TensorProductPolynomials<dim>
{
public:
  template <class Pol>
  TensorProductPolynomialsBubbles(const std::vector<Pol> &pols);

  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;

  double
  compute_value(const unsigned int i, const Point<dim> &p) const;

  template <int order>
  Tensor<order, dim>
  compute_derivative(const unsigned int i, const Point<dim> &p) const;

  Tensor<1, dim>
  compute_grad(const unsigned int i, const Point<dim> &p) const;

  Tensor<2, dim>
  compute_grad_grad(const unsigned int i, const Point<dim> &p) const;

  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