tensor_product_polynomials_bubbles.cc

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//
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#include <deal.II/base/tensor_product_polynomials_bubbles.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/table.h>

DEAL_II_NAMESPACE_OPEN



/* ------------------- TensorProductPolynomialsBubbles -------------- */


template <int dim>
double
TensorProductPolynomialsBubbles<dim>::compute_value (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>::compute_value(i,p);

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

  //compute \prod_{i=1}^d 4*(1-x_i^2)(p)
  double value=1.;
  for (unsigned int j=0; j<dim; ++j)
    value*=4*p(j)*(1-p(j));
  // and multiply with (2x_i-1)^{r-1}
  for (unsigned int i=0; i<q_degree-1; ++i)
    value*=2*p(comp)-1;
  return value;
}



template <>
double
TensorProductPolynomialsBubbles<0>::compute_value (const unsigned int ,
                                                   const Point<0> &) const
{
  Assert (false, ExcNotImplemented());
  return 0.;
}


template <int dim>
Tensor<1,dim>
TensorProductPolynomialsBubbles<dim>::compute_grad (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>::compute_grad(i,p);

  const unsigned int comp = i - this->n_tensor_pols;
  Tensor<1,dim> grad;

  for (unsigned int d=0; d<dim ; ++d)
    {
      grad[d] = 1.;
      //compute grad(4*\prod_{i=1}^d (x_i(1-x_i)))(p)
      for (unsigned j=0; j<dim; ++j)
        grad[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)
        grad[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;
      grad[comp]+=tmp;
    }

  return grad;
}



template <int dim>
Tensor<2,dim>
TensorProductPolynomialsBubbles<dim>::compute_grad_grad (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>::compute_grad_grad(i,p);

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

  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
  Tensor<2,dim> grad_grad;
  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)
      grad_grad[d1][d2] = grad_grad_1[d1][d2]
                          +grad_grad_2[d1][d2]
                          +grad_grad_3[d1][d2];
  grad_grad[comp][comp]+=psi_value*v[dim][2];

  return grad_grad;
}

template <int dim>
void
TensorProductPolynomialsBubbles<dim>::
compute (const Point<dim>            &p,
         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
{
  const unsigned int q_degree = this->polynomials.size()-1;
  const unsigned int max_q_indices = this->n_tensor_pols;
  (void) max_q_indices;
  const unsigned int n_bubbles = ((q_degree<=1)?1:dim);
  Assert (values.size()==max_q_indices+n_bubbles || values.size()==0,
          ExcDimensionMismatch2(values.size(), max_q_indices+n_bubbles, 0));
  Assert (grads.size()==max_q_indices+n_bubbles     || grads.size()==0,
          ExcDimensionMismatch2(grads.size(), max_q_indices+n_bubbles, 0));
  Assert (grad_grads.size()==max_q_indices+n_bubbles || grad_grads.size()==0,
          ExcDimensionMismatch2(grad_grads.size(), max_q_indices+n_bubbles, 0));
  Assert (third_derivatives.size()==max_q_indices+n_bubbles || third_derivatives.size()==0,
          ExcDimensionMismatch2(third_derivatives.size(), max_q_indices+n_bubbles, 0));
  Assert (fourth_derivatives.size()==max_q_indices+n_bubbles || fourth_derivatives.size()==0,
          ExcDimensionMismatch2(fourth_derivatives.size(), max_q_indices+n_bubbles, 0));

  bool do_values = false, do_grads = false, do_grad_grads = false;
  bool do_3rd_derivatives = false, do_4th_derivatives = false;
  if (values.empty() == false)
    {
      values.resize(this->n_tensor_pols);
      do_values = true;
    }
  if (grads.empty() == false)
    {
      grads.resize(this->n_tensor_pols);
      do_grads = true;
    }
  if (grad_grads.empty() == false)
    {
      grad_grads.resize(this->n_tensor_pols);
      do_grad_grads = true;
    }
  if (third_derivatives.empty() == false)
    {
      third_derivatives.resize(this->n_tensor_pols);
      do_3rd_derivatives = true;
    }
  if (fourth_derivatives.empty() == false)
    {
      fourth_derivatives.resize(this->n_tensor_pols);
      do_4th_derivatives = true;
    }

  this->TensorProductPolynomials<dim>::compute(p, values, grads, grad_grads, third_derivatives, fourth_derivatives);

  for (unsigned int i=this->n_tensor_pols; i<this->n_tensor_pols+n_bubbles; ++i)
    {
      if (do_values)
        values.push_back(compute_value(i,p));
      if (do_grads)
        grads.push_back(compute_grad(i,p));
      if (do_grad_grads)
        grad_grads.push_back(compute_grad_grad(i,p));
      if (do_3rd_derivatives)
        third_derivatives.push_back(compute_derivative<3>(i,p));
      if (do_4th_derivatives)
        fourth_derivatives.push_back(compute_derivative<4>(i,p));
    }
}


/* ------------------- explicit instantiations -------------- */
template class TensorProductPolynomialsBubbles<1>;
template class TensorProductPolynomialsBubbles<2>;
template class TensorProductPolynomialsBubbles<3>;

DEAL_II_NAMESPACE_CLOSE