tensor_product_polynomials_bubbles.h

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//
// Copyright (C) 2012 - 2022 by the deal.II authors
//
// This file is part of the deal.II library.
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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/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 ScalarPolynomialsBase<dim>
{
public:
  static constexpr unsigned int dimension = dim;
 
  template <class Pol>
  TensorProductPolynomialsBubbles(const std::vector<Pol> &pols);
 
  void
  output_indices(std::ostream &out) const;
 
  void
  set_numbering(const std::vector<unsigned int> &renumber);
 
  const std::vector<unsigned int> &
  get_numbering() const;
 
  const std::vector<unsigned int> &
  get_numbering_inverse() const;
 
  void
  evaluate(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 override;
 
  double
  compute_value(const unsigned int i, const Point<dim> &p) const override;
 
  template <int order>
  Tensor<order, dim>
  compute_derivative(const unsigned int i, const Point<dim> &p) const;
 
  virtual Tensor<1, dim>
  compute_1st_derivative(const unsigned int i,
                         const Point<dim> & p) const override;
 
  virtual Tensor<2, dim>
  compute_2nd_derivative(const unsigned int i,
                         const Point<dim> & p) const override;
 
  virtual Tensor<3, dim>
  compute_3rd_derivative(const unsigned int i,
                         const Point<dim> & p) const override;
 
  virtual Tensor<4, dim>
  compute_4th_derivative(const unsigned int i,
                         const Point<dim> & p) const override;
 
  Tensor<1, dim>
  compute_grad(const unsigned int i, const Point<dim> &p) const override;
 
  Tensor<2, dim>
  compute_grad_grad(const unsigned int i, const Point<dim> &p) const override;
 
  unsigned int
  n() const;
 
  std::string
  name() const override;
 
  virtual std::unique_ptr<ScalarPolynomialsBase<dim>>
  clone() const override;
 
private:
  TensorProductPolynomials<dim> tensor_polys;
 
  std::vector<unsigned int> index_map;
 
  std::vector<unsigned int> index_map_inverse;
};
 
/* ---------------- template and inline functions ---------- */
 
#ifndef DOXYGEN
 
template <int dim>
template <class Pol>
inline TensorProductPolynomialsBubbles<dim>::TensorProductPolynomialsBubbles(
  const std::vector<Pol> &pols)
  : ScalarPolynomialsBase<dim>(1,
                               Utilities::fixed_power<dim>(pols.size()) + dim)
  , tensor_polys(pols)
  , index_map(tensor_polys.n() +
              ((tensor_polys.polynomials.size() <= 2) ? 1 : dim))
  , index_map_inverse(tensor_polys.n() +
                      ((tensor_polys.polynomials.size() <= 2) ? 1 : dim))
{
  const unsigned int q_degree  = tensor_polys.polynomials.size() - 1;
  const unsigned int n_bubbles = ((q_degree <= 1) ? 1 : dim);
  // append index for renumbering
  for (unsigned int i = 0; i < tensor_polys.n() + n_bubbles; ++i)
    {
      index_map[i]         = i;
      index_map_inverse[i] = i;
    }
}
 
 
template <int dim>
inline unsigned int
TensorProductPolynomialsBubbles<dim>::n() const
{
  return tensor_polys.n() + dim;
}
 
 
template <>
inline unsigned int
TensorProductPolynomialsBubbles<0>::n() const
{
  return numbers::invalid_unsigned_int;
}
 
 
template <int dim>
inline const std::vector<unsigned int> &
TensorProductPolynomialsBubbles<dim>::get_numbering() const
{
  return index_map;
}
 
 
template <int dim>
inline const std::vector<unsigned int> &
TensorProductPolynomialsBubbles<dim>::get_numbering_inverse() const
{
  return index_map_inverse;
}
 
 
template <int dim>
inline std::string
TensorProductPolynomialsBubbles<dim>::name() const
{
  return "TensorProductPolynomialsBubbles";
}
 
 
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      = tensor_polys.polynomials.size() - 1;
  const unsigned int max_q_indices = tensor_polys.n();
  Assert(i < max_q_indices + /* n_bubbles= */ ((q_degree <= 1) ? 1 : dim),
         ExcInternalError());
 
  // treat the regular basis functions
  if (i < max_q_indices)
    return tensor_polys.template compute_derivative<order>(i, p);
 
  const unsigned int comp = i - tensor_polys.n();
 
  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;
        }
    }
}
 
 
 
template <int dim>
inline Tensor<1, dim>
TensorProductPolynomialsBubbles<dim>::compute_1st_derivative(
  const unsigned int i,
  const Point<dim> & p) const
{
  return compute_derivative<1>(i, p);
}
 
 
 
template <int dim>
inline Tensor<2, dim>
TensorProductPolynomialsBubbles<dim>::compute_2nd_derivative(
  const unsigned int i,
  const Point<dim> & p) const
{
  return compute_derivative<2>(i, p);
}
 
 
 
template <int dim>
inline Tensor<3, dim>
TensorProductPolynomialsBubbles<dim>::compute_3rd_derivative(
  const unsigned int i,
  const Point<dim> & p) const
{
  return compute_derivative<3>(i, p);
}
 
 
 
template <int dim>
inline Tensor<4, dim>
TensorProductPolynomialsBubbles<dim>::compute_4th_derivative(
  const unsigned int i,
  const Point<dim> & p) const
{
  return compute_derivative<4>(i, p);
}
 
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
 
#endif