mapping_q_internal.h

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// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 2020 - 2024 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Part of the source code is dual licensed under Apache-2.0 WITH
// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
// governing the source code and code contributions can be found in
// LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
//
// ------------------------------------------------------------------------
 
#ifndef dealii_mapping_q_internal_h
#define dealii_mapping_q_internal_h
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/array_view.h>
#include <deal.II/base/derivative_form.h>
#include <deal.II/base/geometry_info.h>
#include <deal.II/base/point.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/table.h>
#include <deal.II/base/tensor.h>
 
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_update_flags.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q.h>
 
#include <deal.II/grid/grid_tools_geometry.h>
 
#include <deal.II/matrix_free/evaluation_flags.h>
#include <deal.II/matrix_free/evaluation_template_factory.h>
#include <deal.II/matrix_free/fe_evaluation_data.h>
#include <deal.II/matrix_free/mapping_info_storage.h>
#include <deal.II/matrix_free/shape_info.h>
#include <deal.II/matrix_free/tensor_product_point_kernels.h>
 
#include <array>
#include <limits>
 
 
DEAL_II_NAMESPACE_OPEN
 
namespace internal
{
  namespace MappingQ1
  {
    // These are left as templates on the spatial dimension (even though dim
    // == spacedim must be true for them to make sense) because templates are
    // expanded before the compiler eliminates code due to the 'if (dim ==
    // spacedim)' statement (see the body of the general
    // transform_real_to_unit_cell).
    template <int spacedim>
    inline Point<1>
    transform_real_to_unit_cell(
      const std::array<Point<spacedim>, GeometryInfo<1>::vertices_per_cell>
                            &vertices,
      const Point<spacedim> &p)
    {
      Assert(spacedim == 1, ExcInternalError());
      return Point<1>((p[0] - vertices[0][0]) /
                      (vertices[1][0] - vertices[0][0]));
    }
 
 
 
    template <int spacedim>
    inline Point<2>
    transform_real_to_unit_cell(
      const std::array<Point<spacedim>, GeometryInfo<2>::vertices_per_cell>
                            &vertices,
      const Point<spacedim> &p)
    {
      Assert(spacedim == 2, ExcInternalError());
 
      // For accuracy reasons, we do all arithmetic in extended precision
      // (long double). This has a noticeable effect on the hit rate for
      // borderline cases and thus makes the algorithm more robust.
      const long double x = p[0];
      const long double y = p[1];
 
      const long double x0 = vertices[0][0];
      const long double x1 = vertices[1][0];
      const long double x2 = vertices[2][0];
      const long double x3 = vertices[3][0];
 
      const long double y0 = vertices[0][1];
      const long double y1 = vertices[1][1];
      const long double y2 = vertices[2][1];
      const long double y3 = vertices[3][1];
 
      const long double a = (x1 - x3) * (y0 - y2) - (x0 - x2) * (y1 - y3);
      const long double b = -(x0 - x1 - x2 + x3) * y + (x - 2 * x1 + x3) * y0 -
                            (x - 2 * x0 + x2) * y1 - (x - x1) * y2 +
                            (x - x0) * y3;
      const long double c = (x0 - x1) * y - (x - x1) * y0 + (x - x0) * y1;
 
      const long double discriminant = b * b - 4 * a * c;
      // exit if the point is not in the cell (this is the only case where the
      // discriminant is negative)
      AssertThrow(
        discriminant > 0.0,
        (typename Mapping<spacedim, spacedim>::ExcTransformationFailed()));
 
      long double       eta1;
      long double       eta2;
      const long double sqrt_discriminant = std::sqrt(discriminant);
      // special case #1: if a is near-zero to make the discriminant exactly
      // equal b, then use the linear formula
      if (b != 0.0 && std::abs(b) == sqrt_discriminant)
        {
          eta1 = -c / b;
          eta2 = -c / b;
        }
      // special case #2: a is zero for parallelograms and very small for
      // near-parallelograms:
      else if (std::abs(a) < 1e-8 * std::abs(b))
        {
          // if both a and c are very small then the root should be near
          // zero: this first case will capture that
          eta1 = 2 * c / (-b - sqrt_discriminant);
          eta2 = 2 * c / (-b + sqrt_discriminant);
        }
      // finally, use the plain version:
      else
        {
          eta1 = (-b - sqrt_discriminant) / (2 * a);
          eta2 = (-b + sqrt_discriminant) / (2 * a);
        }
      // pick the one closer to the center of the cell.
      const long double eta =
        (std::abs(eta1 - 0.5) < std::abs(eta2 - 0.5)) ? eta1 : eta2;
 
      /*
       * There are two ways to compute xi from eta, but either one may have a
       * zero denominator.
       */
      const long double subexpr0        = -eta * x2 + x0 * (eta - 1);
      const long double xi_denominator0 = eta * x3 - x1 * (eta - 1) + subexpr0;
      const long double max_x = std::max(std::max(std::abs(x0), std::abs(x1)),
                                         std::max(std::abs(x2), std::abs(x3)));
 
      if (std::abs(xi_denominator0) > 1e-10 * max_x)
        {
          const double xi = (x + subexpr0) / xi_denominator0;
          return {xi, static_cast<double>(eta)};
        }
      else
        {
          const long double max_y =
            std::max(std::max(std::abs(y0), std::abs(y1)),
                     std::max(std::abs(y2), std::abs(y3)));
          const long double subexpr1 = -eta * y2 + y0 * (eta - 1);
          const long double xi_denominator1 =
            eta * y3 - y1 * (eta - 1) + subexpr1;
          if (std::abs(xi_denominator1) > 1e-10 * max_y)
            {
              const double xi = (subexpr1 + y) / xi_denominator1;
              return {xi, static_cast<double>(eta)};
            }
          else // give up and try Newton iteration
            {
              AssertThrow(
                false,
                (typename Mapping<spacedim,
                                  spacedim>::ExcTransformationFailed()));
            }
        }
      // bogus return to placate compiler. It should not be possible to get
      // here.
      DEAL_II_ASSERT_UNREACHABLE();
      return {std::numeric_limits<double>::quiet_NaN(),
              std::numeric_limits<double>::quiet_NaN()};
    }
 
 
 
    template <int spacedim>
    inline Point<3>
    transform_real_to_unit_cell(
      const std::array<Point<spacedim>, GeometryInfo<3>::vertices_per_cell>
        & /*vertices*/,
      const Point<spacedim> & /*p*/)
    {
      // It should not be possible to get here
      DEAL_II_ASSERT_UNREACHABLE();
      return {std::numeric_limits<double>::quiet_NaN(),
              std::numeric_limits<double>::quiet_NaN(),
              std::numeric_limits<double>::quiet_NaN()};
    }
  } // namespace MappingQ1
 
 
 
  namespace MappingQImplementation
  {
    template <int dim>
    std::vector<Point<dim>>
    unit_support_points(const std::vector<Point<1>>     &line_support_points,
                        const std::vector<unsigned int> &renumbering)
    {
      AssertDimension(Utilities::pow(line_support_points.size(), dim),
                      renumbering.size());
      std::vector<Point<dim>> points(renumbering.size());
      const unsigned int      n1 = line_support_points.size();
      for (unsigned int q2 = 0, q = 0; q2 < (dim > 2 ? n1 : 1); ++q2)
        for (unsigned int q1 = 0; q1 < (dim > 1 ? n1 : 1); ++q1)
          for (unsigned int q0 = 0; q0 < n1; ++q0, ++q)
            {
              points[renumbering[q]][0] = line_support_points[q0][0];
              if (dim > 1)
                points[renumbering[q]][1] = line_support_points[q1][0];
              if (dim > 2)
                points[renumbering[q]][2] = line_support_points[q2][0];
            }
      return points;
    }
 
 
 
    inline ::Table<2, double>
    compute_support_point_weights_on_quad(const unsigned int polynomial_degree)
    {
      ::Table<2, double> loqvs;
 
      // we are asked to compute weights for interior support points, but
      // there are no interior points if degree==1
      if (polynomial_degree == 1)
        return loqvs;
 
      const unsigned int M          = polynomial_degree - 1;
      const unsigned int n_inner_2d = M * M;
      const unsigned int n_outer_2d = 4 + 4 * M;
 
      // set the weights of transfinite interpolation
      loqvs.reinit(n_inner_2d, n_outer_2d);
      QGaussLobatto<2> gl(polynomial_degree + 1);
      for (unsigned int i = 0; i < M; ++i)
        for (unsigned int j = 0; j < M; ++j)
          {
            const Point<2> &p =
              gl.point((i + 1) * (polynomial_degree + 1) + (j + 1));
            const unsigned int index_table = i * M + j;
            for (unsigned int v = 0; v < 4; ++v)
              loqvs(index_table, v) =
                -GeometryInfo<2>::d_linear_shape_function(p, v);
            loqvs(index_table, 4 + i)         = 1. - p[0];
            loqvs(index_table, 4 + i + M)     = p[0];
            loqvs(index_table, 4 + j + 2 * M) = 1. - p[1];
            loqvs(index_table, 4 + j + 3 * M) = p[1];
          }
 
      // the sum of weights of the points at the outer rim should be one.
      // check this
      for (unsigned int unit_point = 0; unit_point < n_inner_2d; ++unit_point)
        Assert(std::fabs(std::accumulate(loqvs[unit_point].begin(),
                                         loqvs[unit_point].end(),
                                         0.) -
                         1) < 1e-13 * polynomial_degree,
               ExcInternalError());
 
      return loqvs;
    }
 
 
 
    inline ::Table<2, double>
    compute_support_point_weights_on_hex(const unsigned int polynomial_degree)
    {
      ::Table<2, double> lohvs;
 
      // we are asked to compute weights for interior support points, but
      // there are no interior points if degree==1
      if (polynomial_degree == 1)
        return lohvs;
 
      const unsigned int M = polynomial_degree - 1;
 
      const unsigned int n_inner = Utilities::fixed_power<3>(M);
      const unsigned int n_outer = 8 + 12 * M + 6 * M * M;
 
      // set the weights of transfinite interpolation
      lohvs.reinit(n_inner, n_outer);
      QGaussLobatto<3> gl(polynomial_degree + 1);
      for (unsigned int i = 0; i < M; ++i)
        for (unsigned int j = 0; j < M; ++j)
          for (unsigned int k = 0; k < M; ++k)
            {
              const Point<3>    &p = gl.point((i + 1) * (M + 2) * (M + 2) +
                                           (j + 1) * (M + 2) + (k + 1));
              const unsigned int index_table = i * M * M + j * M + k;
 
              // vertices
              for (unsigned int v = 0; v < 8; ++v)
                lohvs(index_table, v) =
                  GeometryInfo<3>::d_linear_shape_function(p, v);
 
              // lines
              {
                constexpr std::array<unsigned int, 4> line_coordinates_y(
                  {{0, 1, 4, 5}});
                const Point<2> py(p[0], p[2]);
                for (unsigned int l = 0; l < 4; ++l)
                  lohvs(index_table, 8 + line_coordinates_y[l] * M + j) =
                    -GeometryInfo<2>::d_linear_shape_function(py, l);
              }
 
              {
                constexpr std::array<unsigned int, 4> line_coordinates_x(
                  {{2, 3, 6, 7}});
                const Point<2> px(p[1], p[2]);
                for (unsigned int l = 0; l < 4; ++l)
                  lohvs(index_table, 8 + line_coordinates_x[l] * M + k) =
                    -GeometryInfo<2>::d_linear_shape_function(px, l);
              }
 
              {
                constexpr std::array<unsigned int, 4> line_coordinates_z(
                  {{8, 9, 10, 11}});
                const Point<2> pz(p[0], p[1]);
                for (unsigned int l = 0; l < 4; ++l)
                  lohvs(index_table, 8 + line_coordinates_z[l] * M + i) =
                    -GeometryInfo<2>::d_linear_shape_function(pz, l);
              }
 
              // quads
              lohvs(index_table, 8 + 12 * M + 0 * M * M + i * M + j) =
                1. - p[0];
              lohvs(index_table, 8 + 12 * M + 1 * M * M + i * M + j) = p[0];
              lohvs(index_table, 8 + 12 * M + 2 * M * M + k * M + i) =
                1. - p[1];
              lohvs(index_table, 8 + 12 * M + 3 * M * M + k * M + i) = p[1];
              lohvs(index_table, 8 + 12 * M + 4 * M * M + j * M + k) =
                1. - p[2];
              lohvs(index_table, 8 + 12 * M + 5 * M * M + j * M + k) = p[2];
            }
 
      // the sum of weights of the points at the outer rim should be one.
      // check this
      for (unsigned int unit_point = 0; unit_point < n_inner; ++unit_point)
        Assert(std::fabs(std::accumulate(lohvs[unit_point].begin(),
                                         lohvs[unit_point].end(),
                                         0.) -
                         1) < 1e-13 * polynomial_degree,
               ExcInternalError());
 
      return lohvs;
    }
 
 
 
    inline std::vector<::Table<2, double>>
    compute_support_point_weights_perimeter_to_interior(
      const unsigned int polynomial_degree,
      const unsigned int dim)
    {
      Assert(dim > 0 && dim <= 3, ExcImpossibleInDim(dim));
      std::vector<::Table<2, double>> output(dim);
      if (polynomial_degree <= 1)
        return output;
 
      // fill the 1d interior weights
      QGaussLobatto<1> quadrature(polynomial_degree + 1);
      output[0].reinit(polynomial_degree - 1,
                       GeometryInfo<1>::vertices_per_cell);
      for (unsigned int q = 0; q < polynomial_degree - 1; ++q)
        for (const unsigned int i : GeometryInfo<1>::vertex_indices())
          output[0](q, i) =
            GeometryInfo<1>::d_linear_shape_function(quadrature.point(q + 1),
                                                     i);
 
      if (dim > 1)
        output[1] = compute_support_point_weights_on_quad(polynomial_degree);
 
      if (dim > 2)
        output[2] = compute_support_point_weights_on_hex(polynomial_degree);
 
      return output;
    }
 
 
 
    template <int dim>
    inline ::Table<2, double>
    compute_support_point_weights_cell(const unsigned int polynomial_degree)
    {
      Assert(dim > 0 && dim <= 3, ExcImpossibleInDim(dim));
      if (polynomial_degree <= 1)
        return ::Table<2, double>();
 
      QGaussLobatto<dim>              quadrature(polynomial_degree + 1);
      const std::vector<unsigned int> h2l =
        FETools::hierarchic_to_lexicographic_numbering<dim>(polynomial_degree);
 
      ::Table<2, double> output(quadrature.size() -
                                        GeometryInfo<dim>::vertices_per_cell,
                                      GeometryInfo<dim>::vertices_per_cell);
      for (unsigned int q = 0; q < output.size(0); ++q)
        for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
          output(q, i) = GeometryInfo<dim>::d_linear_shape_function(
            quadrature.point(h2l[q + GeometryInfo<dim>::vertices_per_cell]), i);
 
      return output;
    }
 
 
 
    template <int dim, int spacedim>
    inline Point<spacedim>
    compute_mapped_location_of_point(
      const typename ::MappingQ<dim, spacedim>::InternalData &data)
    {
      AssertDimension(data.shape_values.size(),
                      data.mapping_support_points.size());
 
      // use now the InternalData to compute the point in real space.
      Point<spacedim> p_real;
      for (unsigned int i = 0; i < data.mapping_support_points.size(); ++i)
        p_real += data.mapping_support_points[i] * data.shape(0, i);
 
      return p_real;
    }
 
 
 
    template <int dim, int spacedim, typename Number>
    inline Point<dim, Number>
    do_transform_real_to_unit_cell_internal(
      const Point<spacedim, Number>                      &p,
      const Point<dim, Number>                           &initial_p_unit,
      const ArrayView<const Point<spacedim>>             &points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int>                    &renumber,
      const bool print_iterations_to_deallog = false)
    {
      if (print_iterations_to_deallog)
        deallog << "Start MappingQ::do_transform_real_to_unit_cell for real "
                << "point [ " << p << " ] " << std::endl;
 
      AssertDimension(points.size(),
                      Utilities::pow(polynomials_1d.size(), dim));
 
      // Newton iteration to solve
      //    f(x)=p(x)-p=0
      // where we are looking for 'x' and p(x) is the forward transformation
      // from unit to real cell. We solve this using a Newton iteration
      //    x_{n+1}=x_n-[f'(x)]^{-1}f(x)
      // The start value is set to be the linear approximation to the cell
 
      // The shape values and derivatives of the mapping at this point are
      // previously computed.
 
      Point<dim, Number> p_unit = initial_p_unit;
      auto p_real = internal::evaluate_tensor_product_value_and_gradient(
        polynomials_1d, points, p_unit, polynomials_1d.size() == 2, renumber);
 
      Tensor<1, spacedim, Number> f = p_real.first - p;
 
      // early out if we already have our point in all SIMD lanes, i.e.,
      // f.norm_square() < 1e-24 * p_real.second[0].norm_square(). To enable
      // this step also for VectorizedArray where we do not have operator <,
      // compare the result to zero which is defined for SIMD types
      if (std::max(Number(0.),
                   f.norm_square() - 1e-24 * p_real.second[0].norm_square()) ==
          Number(0.))
        return p_unit;
 
      // we need to compare the position of the computed p(x) against the
      // given point 'p'. We will terminate the iteration and return 'x' if
      // they are less than eps apart. The question is how to choose eps --
      // or, put maybe more generally: in which norm we want these 'p' and
      // 'p(x)' to be eps apart.
      //
      // the question is difficult since we may have to deal with very
      // elongated cells where we may achieve 1e-12*h for the distance of
      // these two points in the 'long' direction, but achieving this
      // tolerance in the 'short' direction of the cell may not be possible
      //
      // what we do instead is then to terminate iterations if
      //    \| p(x) - p \|_A < eps
      // where the A-norm is somehow induced by the transformation of the
      // cell. in particular, we want to measure distances relative to the
      // sizes of the cell in its principal directions.
      //
      // to define what exactly A should be, note that to first order we have
      // the following (assuming that x* is the solution of the problem, i.e.,
      // p(x*)=p):
      //    p(x) - p = p(x) - p(x*)
      //             = -grad p(x) * (x*-x) + higher order terms
      // This suggest to measure with a norm that corresponds to
      //    A = {[grad p(x)]^T [grad p(x)]}^{-1}
      // because then
      //    \| p(x) - p \|_A  \approx  \| x - x* \|
      // Consequently, we will try to enforce that
      //    \| p(x) - p \|_A  =  \| f \|  <=  eps
      //
      // Note that using this norm is a bit dangerous since the norm changes
      // in every iteration (A isn't fixed by depending on xk). However, if
      // the cell is not too deformed (it may be stretched, but not twisted)
      // then the mapping is almost linear and A is indeed constant or
      // nearly so.
      const double       eps                    = 1.e-11;
      const unsigned int newton_iteration_limit = 20;
 
      Point<dim, Number> invalid_point;
      invalid_point[0]                = std::numeric_limits<double>::infinity();
      bool tried_project_to_unit_cell = false;
 
      unsigned int newton_iteration            = 0;
      Number       f_weighted_norm_square      = 1.;
      Number       last_f_weighted_norm_square = 1.;
 
      do
        {
          if (print_iterations_to_deallog)
            deallog << "Newton iteration " << newton_iteration
                    << " for unit point guess " << p_unit << std::endl;
 
          // f'(x)
          Tensor<2, spacedim, Number> df;
          for (unsigned int d = 0; d < spacedim; ++d)
            for (unsigned int e = 0; e < dim; ++e)
              df[d][e] = p_real.second[e][d];
 
          // check determinand(df) > 0 on all SIMD lanes
          if (!(std::min(determinant(df),
                         Number(std::numeric_limits<double>::min())) ==
                Number(std::numeric_limits<double>::min())))
            {
              // We allow to enter this function with an initial guess
              // outside the unit cell. We might have run into invalid
              // Jacobians due to that, so we should at least try once to go
              // back to the unit cell and go on with the Newton iteration
              // from there. Since the outside case is unlikely, we can
              // afford spending the extra effort at this place.
              if (tried_project_to_unit_cell == false)
                {
                  p_unit = GeometryInfo<dim>::project_to_unit_cell(p_unit);
                  p_real = internal::evaluate_tensor_product_value_and_gradient(
                    polynomials_1d,
                    points,
                    p_unit,
                    polynomials_1d.size() == 2,
                    renumber);
                  f                           = p_real.first - p;
                  f_weighted_norm_square      = 1.;
                  last_f_weighted_norm_square = 1;
                  tried_project_to_unit_cell  = true;
                  continue;
                }
              else
                return invalid_point;
            }
 
          // Solve  [f'(x)]d=f(x)
          const Tensor<2, spacedim, Number> df_inverse = invert(df);
          const Tensor<1, spacedim, Number> delta      = df_inverse * f;
          last_f_weighted_norm_square                  = delta.norm_square();
 
          if (print_iterations_to_deallog)
            deallog << "   delta=" << delta << std::endl;
 
          // do a line search
          double step_length = 1.0;
          do
            {
              // update of p_unit. The spacedim-th component of transformed
              // point is simply ignored in codimension one case. When this
              // component is not zero, then we are projecting the point to
              // the surface or curve identified by the cell.
              Point<dim, Number> p_unit_trial = p_unit;
              for (unsigned int i = 0; i < dim; ++i)
                p_unit_trial[i] -= step_length * delta[i];
 
              // shape values and derivatives at new p_unit point
              const auto p_real_trial =
                internal::evaluate_tensor_product_value_and_gradient(
                  polynomials_1d,
                  points,
                  p_unit_trial,
                  polynomials_1d.size() == 2,
                  renumber);
              const Tensor<1, spacedim, Number> f_trial =
                p_real_trial.first - p;
              f_weighted_norm_square = (df_inverse * f_trial).norm_square();
 
              if (print_iterations_to_deallog)
                {
                  deallog << "     step_length=" << step_length << std::endl;
                  if (step_length == 1.0)
                    deallog << "       ||f ||   =" << f.norm() << std::endl;
                  deallog << "       ||f*||   =" << f_trial.norm() << std::endl
                          << "       ||f*||_A ="
                          << std::sqrt(f_weighted_norm_square) << std::endl;
                }
 
              // See if we are making progress with the current step length
              // and if not, reduce it by a factor of two and try again.
              //
              // Strictly speaking, we should probably use the same norm as we
              // use for the outer algorithm. In practice, line search is just
              // a crutch to find a "reasonable" step length, and so using the
              // l2 norm is probably just fine.
              //
              // check f_trial.norm() < f.norm() in SIMD form. This is a bit
              // more complicated because some SIMD lanes might not be doing
              // any progress any more as they have already reached roundoff
              // accuracy. We define that as the case of updates less than
              // 1e-6. The tolerance might seem coarse but since we are
              // dealing with a Newton iteration of a polynomial function we
              // either converge quadratically or not any more. Thus, our
              // selection is to terminate if either the norm of f is
              // decreasing or that threshold of 1e-6 is reached.
              if (std::max(f_weighted_norm_square - 1e-6 * 1e-6, Number(0.)) *
                    std::max(f_trial.norm_square() - f.norm_square(),
                             Number(0.)) ==
                  Number(0.))
                {
                  p_real = p_real_trial;
                  p_unit = p_unit_trial;
                  f      = f_trial;
                  break;
                }
              else if (step_length > 0.05)
                step_length *= 0.5;
              else
                break;
            }
          while (true);
 
          // In case we terminated the line search due to the step becoming
          // too small, we give the iteration another try with the
          // projection of the initial guess to the unit cell before we give
          // up, just like for the negative determinant case.
          if (step_length <= 0.05 && tried_project_to_unit_cell == false)
            {
              p_unit = GeometryInfo<dim>::project_to_unit_cell(p_unit);
              p_real = internal::evaluate_tensor_product_value_and_gradient(
                polynomials_1d,
                points,
                p_unit,
                polynomials_1d.size() == 2,
                renumber);
              f                           = p_real.first - p;
              f_weighted_norm_square      = 1.;
              last_f_weighted_norm_square = 1;
              tried_project_to_unit_cell  = true;
              continue;
            }
          else if (step_length <= 0.05)
            return invalid_point;
 
          ++newton_iteration;
          if (newton_iteration > newton_iteration_limit)
            return invalid_point;
        }
      // Stop if f_weighted_norm_square <= eps^2 on all SIMD lanes or if the
      // weighted norm is less than 1e-6 and the improvement against the
      // previous step was less than a factor of 10 (in that regime, we
      // either have quadratic convergence or roundoff errors due to a bad
      // mapping)
      while (
        !(std::max(f_weighted_norm_square - eps * eps, Number(0.)) *
            std::max(last_f_weighted_norm_square -
                       std::min(f_weighted_norm_square, Number(1e-6 * 1e-6)) *
                         100.,
                     Number(0.)) ==
          Number(0.)));
 
      if (print_iterations_to_deallog)
        deallog << "Iteration converged for p_unit = [ " << p_unit
                << " ] and iteration error "
                << std::sqrt(f_weighted_norm_square) << std::endl;
 
      return p_unit;
    }
 
 
 
    template <int dim>
    inline Point<dim>
    do_transform_real_to_unit_cell_internal_codim1(
      const Point<dim + 1>                               &p,
      const Point<dim>                                   &initial_p_unit,
      const ArrayView<const Point<dim + 1>>              &points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int>                    &renumber)
    {
      const unsigned int spacedim = dim + 1;
 
      AssertDimension(points.size(),
                      Utilities::pow(polynomials_1d.size(), dim));
 
      Point<dim> p_unit = initial_p_unit;
 
      const double       eps        = 1.e-12;
      const unsigned int loop_limit = 10;
 
      unsigned int loop                   = 0;
      double       f_weighted_norm_square = 1.;
 
      while (f_weighted_norm_square > eps * eps && loop++ < loop_limit)
        {
          const auto p_real =
            internal::evaluate_tensor_product_value_and_gradient(
              polynomials_1d,
              points,
              p_unit,
              polynomials_1d.size() == 2,
              renumber);
          Tensor<1, spacedim>                    p_minus_F = p - p_real.first;
          const DerivativeForm<1, spacedim, dim> DF        = p_real.second;
 
          const auto hessian = internal::evaluate_tensor_product_hessian(
            polynomials_1d, points, p_unit, renumber);
          Point<dim>     f;
          Tensor<2, dim> df;
          for (unsigned int j = 0; j < dim; ++j)
            {
              f[j] = DF[j] * p_minus_F;
              for (unsigned int l = 0; l < dim; ++l)
                df[j][l] = -DF[j] * DF[l] + hessian[j][l] * p_minus_F;
            }
 
          // Solve  [df(x)]d=f(x)
          const Tensor<1, dim> d =
            invert(df) * static_cast<const Tensor<1, dim> &>(f);
          f_weighted_norm_square = d.norm_square();
          p_unit -= d;
        }
 
 
      // Here we check that in the last execution of while the first
      // condition was already wrong, meaning the residual was below
      // eps. Only if the first condition failed, loop will have been
      // increased and tested, and thus have reached the limit.
      AssertThrow(loop < loop_limit,
                  (typename Mapping<dim, spacedim>::ExcTransformationFailed()));
 
      return p_unit;
    }
 
 
 
    template <int dim, int spacedim>
    class InverseQuadraticApproximation
    {
    public:
      static constexpr unsigned int n_functions =
        (spacedim == 1 ? 3 : (spacedim == 2 ? 6 : 10));
 
      InverseQuadraticApproximation(
        const ArrayView<const Point<spacedim>> &real_support_points,
        const std::vector<Point<dim>>          &unit_support_points)
        : normalization_shift(real_support_points[0])
        , normalization_length(
            1. / real_support_points[0].distance(real_support_points[1]))
        , is_affine(true)
      {
        AssertDimension(real_support_points.size(), unit_support_points.size());
 
        // For the bi-/trilinear approximation, we cannot build a quadratic
        // polynomial due to a lack of points (interpolation matrix would
        // get singular). Similarly, it is not entirely clear how to gather
        // enough information for the case dim < spacedim.
        //
        // In both cases we require the vector real_support_points to
        // contain the vertex positions and fall back to an affine
        // approximation:
        Assert(dim == spacedim || real_support_points.size() ==
                                    GeometryInfo<dim>::vertices_per_cell,
               ExcInternalError());
        if (real_support_points.size() == GeometryInfo<dim>::vertices_per_cell)
          {
            const auto affine = GridTools::affine_cell_approximation<dim>(
              make_array_view(real_support_points));
            DerivativeForm<1, spacedim, dim> A_inv =
              affine.first.covariant_form().transpose();
 
            // The code for evaluation assumes an additional transformation of
            // the form (x - normalization_shift) * normalization_length --
            // account for this in the definition of the coefficients.
            coefficients[0] =
              apply_transformation(A_inv, normalization_shift - affine.second);
            for (unsigned int d = 0; d < spacedim; ++d)
              for (unsigned int e = 0; e < dim; ++e)
                coefficients[1 + d][e] =
                  A_inv[e][d] * (1.0 / normalization_length);
            is_affine = true;
            return;
          }
 
        Tensor<2, n_functions>          matrix;
        std::array<double, n_functions> shape_values;
        for (unsigned int q = 0; q < unit_support_points.size(); ++q)
          {
            // Evaluate quadratic shape functions in point, with the
            // normalization applied in order to avoid roundoff issues with
            // scaling far away from 1.
            shape_values[0] = 1.;
            const Tensor<1, spacedim> p_scaled =
              normalization_length *
              (real_support_points[q] - normalization_shift);
            for (unsigned int d = 0; d < spacedim; ++d)
              shape_values[1 + d] = p_scaled[d];
            for (unsigned int d = 0, c = 0; d < spacedim; ++d)
              for (unsigned int e = 0; e <= d; ++e, ++c)
                shape_values[1 + spacedim + c] = p_scaled[d] * p_scaled[e];
 
            // Build lower diagonal of least squares matrix and rhs, the
            // essential part being that we construct the matrix with the
            // real points and the right hand side by comparing to the
            // reference point positions which sets up an inverse
            // interpolation.
            for (unsigned int i = 0; i < n_functions; ++i)
              for (unsigned int j = 0; j < n_functions; ++j)
                matrix[i][j] += shape_values[i] * shape_values[j];
            for (unsigned int i = 0; i < n_functions; ++i)
              coefficients[i] += shape_values[i] * unit_support_points[q];
          }
 
        // Factorize the matrix A = L * L^T in-place with the
        // Cholesky-Banachiewicz algorithm. The implementation is similar to
        // FullMatrix::cholesky() but re-implemented to avoid memory
        // allocations and some unnecessary divisions which we can do here as
        // we only need to solve with dim right hand sides.
        for (unsigned int i = 0; i < n_functions; ++i)
          {
            double Lij_sum = 0;
            for (unsigned int j = 0; j < i; ++j)
              {
                double Lik_Ljk_sum = 0;
                for (unsigned int k = 0; k < j; ++k)
                  Lik_Ljk_sum += matrix[i][k] * matrix[j][k];
                matrix[i][j] = matrix[j][j] * (matrix[i][j] - Lik_Ljk_sum);
                Lij_sum += matrix[i][j] * matrix[i][j];
              }
            AssertThrow(matrix[i][i] - Lij_sum >= 0,
                        ExcMessage("Matrix of normal equations not positive "
                                   "definite"));
 
            // Store the inverse in the diagonal since that is the quantity
            // needed later in the factorization as well as the forward and
            // backward substitution, minimizing the number of divisions.
            matrix[i][i] = 1. / std::sqrt(matrix[i][i] - Lij_sum);
          }
 
        // Solve lower triangular part, L * y = rhs.
        for (unsigned int i = 0; i < n_functions; ++i)
          {
            Point<dim> sum = coefficients[i];
            for (unsigned int j = 0; j < i; ++j)
              sum -= matrix[i][j] * coefficients[j];
            coefficients[i] = sum * matrix[i][i];
          }
 
        // Solve upper triangular part, L^T * x = y (i.e., x = A^{-1} * rhs)
        for (unsigned int i = n_functions; i > 0;)
          {
            --i;
            Point<dim> sum = coefficients[i];
            for (unsigned int j = i + 1; j < n_functions; ++j)
              sum -= matrix[j][i] * coefficients[j];
            coefficients[i] = sum * matrix[i][i];
          }
 
        // Check whether the approximation is indeed affine, allowing to
        // skip the quadratic terms.
        is_affine = true;
        for (unsigned int i = dim + 1; i < n_functions; ++i)
          if (coefficients[i].norm_square() > 1e-20)
            {
              is_affine = false;
              break;
            }
      }
 
      InverseQuadraticApproximation(const InverseQuadraticApproximation &) =
        default;
 
      template <typename Number>
      Point<dim, Number>
      compute(const Point<spacedim, Number> &p) const
      {
        Point<dim, Number> result;
        for (unsigned int d = 0; d < dim; ++d)
          result[d] = coefficients[0][d];
 
        // Apply the normalization to ensure a good conditioning. Since Number
        // might be a vectorized array whereas the normalization is a point of
        // doubles, we cannot use the overload of operator- and must instead
        // loop over the components of the point.
        Point<spacedim, Number> p_scaled;
        for (unsigned int d = 0; d < spacedim; ++d)
          p_scaled[d] = (p[d] - normalization_shift[d]) * normalization_length;
 
        for (unsigned int d = 0; d < spacedim; ++d)
          result += coefficients[1 + d] * p_scaled[d];
 
        if (!is_affine)
          {
            Point<dim, Number> result_affine = result;
            for (unsigned int d = 0, c = 0; d < spacedim; ++d)
              for (unsigned int e = 0; e <= d; ++e, ++c)
                result +=
                  coefficients[1 + spacedim + c] * (p_scaled[d] * p_scaled[e]);
 
            // Check if the quadratic approximation ends up considerably
            // farther outside the unit cell on some or all SIMD lanes than
            // the affine approximation - in that case, we switch those
            // components back to the affine approximation. Note that the
            // quadratic approximation will grow more quickly away from the
            // unit cell. We make the selection for each SIMD lane with a
            // ternary operation.
            const Number distance_to_unit_cell = result.distance_square(
              GeometryInfo<dim>::project_to_unit_cell(result));
            const Number affine_distance_to_unit_cell =
              result_affine.distance_square(
                GeometryInfo<dim>::project_to_unit_cell(result_affine));
            for (unsigned int d = 0; d < dim; ++d)
              result[d] = compare_and_apply_mask<SIMDComparison::greater_than>(
                distance_to_unit_cell,
                affine_distance_to_unit_cell + 0.5,
                result_affine[d],
                result[d]);
          }
        return result;
      }
 
    private:
      const Point<spacedim> normalization_shift;
 
      const double normalization_length;
 
      std::array<Point<dim>, n_functions> coefficients;
 
      bool is_affine;
    };
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_q_points_Jacobians_and_grads_tensor(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      std::vector<Point<spacedim>>                  &quadrature_points,
      std::vector<DerivativeForm<1, dim, spacedim>> &jacobians,
      std::vector<DerivativeForm<1, spacedim, dim>> &inverse_jacobians,
      std::vector<DerivativeForm<2, dim, spacedim>> &jacobian_grads)
    {
      const UpdateFlags update_flags = data.update_each;
 
      using VectorizedArrayType =
        typename ::MappingQ<dim,
                                  spacedim>::InternalData::VectorizedArrayType;
      const unsigned int     n_shape_values = data.n_shape_functions;
      const unsigned int     n_q_points     = data.shape_info.n_q_points;
      constexpr unsigned int n_lanes        = VectorizedArrayType::size();
      constexpr unsigned int n_comp         = 1 + (spacedim - 1) / n_lanes;
      constexpr unsigned int n_hessians     = (dim * (dim + 1)) / 2;
 
      // Since MappingQ::InternalData does not have separate arrays for the
      // covariant and contravariant transformations, but uses the arrays in
      // the `MappingRelatedData`, it can happen that vectors do not have the
      // right size
      if (update_flags & update_contravariant_transformation)
        jacobians.resize(n_q_points);
      if (update_flags & update_covariant_transformation)
        inverse_jacobians.resize(n_q_points);
 
      EvaluationFlags::EvaluationFlags evaluation_flag =
        (update_flags & update_quadrature_points ? EvaluationFlags::values :
                                                   EvaluationFlags::nothing) |
        ((cell_similarity != CellSimilarity::translation) &&
             (update_flags & update_contravariant_transformation) ?
           EvaluationFlags::gradients :
           EvaluationFlags::nothing) |
        ((cell_similarity != CellSimilarity::translation) &&
             (update_flags & update_jacobian_grads) ?
           EvaluationFlags::hessians :
           EvaluationFlags::nothing);
 
      Assert(!(evaluation_flag & EvaluationFlags::values) || n_q_points > 0,
             ExcInternalError());
      Assert(!(evaluation_flag & EvaluationFlags::values) ||
               n_q_points == quadrature_points.size(),
             ExcDimensionMismatch(n_q_points, quadrature_points.size()));
      Assert(!(evaluation_flag & EvaluationFlags::gradients) ||
               data.n_shape_functions > 0,
             ExcInternalError());
      Assert(!(evaluation_flag & EvaluationFlags::hessians) ||
               n_q_points == jacobian_grads.size(),
             ExcDimensionMismatch(n_q_points, jacobian_grads.size()));
 
      // shortcut in case we have an identity interpolation and only request
      // the quadrature points
      if (evaluation_flag == EvaluationFlags::values &&
          data.shape_info.element_type ==
            internal::MatrixFreeFunctions::tensor_symmetric_collocation)
        {
          for (unsigned int q = 0; q < n_q_points; ++q)
            quadrature_points[q] =
              data.mapping_support_points[data.shape_info
                                            .lexicographic_numbering[q]];
          return;
        }
 
      FEEvaluationData<dim, VectorizedArrayType, false> eval(data.shape_info);
 
      // prepare arrays
      if (evaluation_flag != EvaluationFlags::nothing)
        {
          eval.set_data_pointers(&data.scratch, n_comp);
 
          // make sure to initialize on all lanes also when some are unused in
          // the code below
          for (unsigned int i = 0; i < n_shape_values * n_comp; ++i)
            eval.begin_dof_values()[i] = VectorizedArrayType();
 
          const std::vector<unsigned int> &renumber_to_lexicographic =
            data.shape_info.lexicographic_numbering;
          for (unsigned int i = 0; i < n_shape_values; ++i)
            for (unsigned int d = 0; d < spacedim; ++d)
              {
                const unsigned int in_comp  = d % n_lanes;
                const unsigned int out_comp = d / n_lanes;
                eval
                  .begin_dof_values()[out_comp * n_shape_values + i][in_comp] =
                  data.mapping_support_points[renumber_to_lexicographic[i]][d];
              }
 
          // do the actual tensorized evaluation
          internal::FEEvaluationFactory<dim, VectorizedArrayType>::evaluate(
            n_comp, evaluation_flag, eval.begin_dof_values(), eval);
        }
 
      // do the postprocessing
      if (evaluation_flag & EvaluationFlags::values)
        {
          for (unsigned int out_comp = 0; out_comp < n_comp; ++out_comp)
            for (unsigned int i = 0; i < n_q_points; ++i)
              for (unsigned int in_comp = 0;
                   in_comp < n_lanes && in_comp < spacedim - out_comp * n_lanes;
                   ++in_comp)
                quadrature_points[i][out_comp * n_lanes + in_comp] =
                  eval.begin_values()[out_comp * n_q_points + i][in_comp];
        }
 
      if (evaluation_flag & EvaluationFlags::gradients)
        {
          // We need to reinterpret the data after evaluate has been applied.
          for (unsigned int out_comp = 0; out_comp < n_comp; ++out_comp)
            for (unsigned int point = 0; point < n_q_points; ++point)
              for (unsigned int in_comp = 0;
                   in_comp < n_lanes && in_comp < spacedim - out_comp * n_lanes;
                   ++in_comp)
                for (unsigned int j = 0; j < dim; ++j)
                  {
                    jacobians[point][out_comp * n_lanes + in_comp][j] =
                      eval.begin_gradients()[(out_comp * n_q_points + point) *
                                               dim +
                                             j][in_comp];
                  }
        }
 
      if (update_flags & update_volume_elements)
        if (cell_similarity != CellSimilarity::translation)
          for (unsigned int point = 0; point < n_q_points; ++point)
            data.volume_elements[point] = jacobians[point].determinant();
 
      // copy values from InternalData to vector given by reference
      if (update_flags & update_covariant_transformation)
        {
          AssertDimension(jacobians.size(), n_q_points);
          AssertDimension(inverse_jacobians.size(), n_q_points);
          if (cell_similarity != CellSimilarity::translation)
            for (unsigned int point = 0; point < n_q_points; ++point)
              inverse_jacobians[point] =
                jacobians[point].covariant_form().transpose();
        }
 
      if (evaluation_flag & EvaluationFlags::hessians)
        {
          constexpr int desymmetrize_3d[6][2] = {
            {0, 0}, {1, 1}, {2, 2}, {0, 1}, {0, 2}, {1, 2}};
          constexpr int desymmetrize_2d[3][2] = {{0, 0}, {1, 1}, {0, 1}};
 
          // We need to reinterpret the data after evaluate has been applied.
          for (unsigned int out_comp = 0; out_comp < n_comp; ++out_comp)
            for (unsigned int point = 0; point < n_q_points; ++point)
              for (unsigned int j = 0; j < n_hessians; ++j)
                for (unsigned int in_comp = 0;
                     in_comp < n_lanes &&
                     in_comp < spacedim - out_comp * n_lanes;
                     ++in_comp)
                  {
                    const unsigned int total_number = point * n_hessians + j;
                    const unsigned int new_point    = total_number % n_q_points;
                    const unsigned int new_hessian_comp =
                      total_number / n_q_points;
                    const unsigned int new_hessian_comp_i =
                      dim == 2 ? desymmetrize_2d[new_hessian_comp][0] :
                                 desymmetrize_3d[new_hessian_comp][0];
                    const unsigned int new_hessian_comp_j =
                      dim == 2 ? desymmetrize_2d[new_hessian_comp][1] :
                                 desymmetrize_3d[new_hessian_comp][1];
                    const double value =
                      eval.begin_hessians()[(out_comp * n_q_points + point) *
                                              n_hessians +
                                            j][in_comp];
                    jacobian_grads[new_point][out_comp * n_lanes + in_comp]
                                  [new_hessian_comp_i][new_hessian_comp_j] =
                                    value;
                    jacobian_grads[new_point][out_comp * n_lanes + in_comp]
                                  [new_hessian_comp_j][new_hessian_comp_i] =
                                    value;
                  }
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_q_points_Jacobians_generic(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int> &renumber_lexicographic_to_hierarchic,
      std::vector<Point<spacedim>>    &quadrature_points,
      std::vector<DerivativeForm<1, dim, spacedim>> &jacobians,
      std::vector<DerivativeForm<1, spacedim, dim>> &inverse_jacobians)
    {
      const UpdateFlags                      update_flags = data.update_each;
      const ArrayView<const Point<spacedim>> support_points(
        data.mapping_support_points);
 
      const unsigned int n_points = unit_points.size();
      const unsigned int n_lanes  = VectorizedArray<double>::size();
 
      // Since MappingQ::InternalData does not have separate arrays for the
      // covariant and contravariant transformations, but uses the arrays in
      // the `MappingRelatedData`, it can happen that vectors do not have the
      // right size
      if (update_flags & update_contravariant_transformation)
        jacobians.resize(n_points);
      if (update_flags & update_covariant_transformation)
        inverse_jacobians.resize(n_points);
 
      const bool is_translation =
        cell_similarity == CellSimilarity::translation;
 
      const bool needs_gradient =
        !is_translation &&
        (update_flags &
         (update_covariant_transformation |
          update_contravariant_transformation | update_volume_elements));
 
      // Use the more heavy VectorizedArray code path if there is more than
      // one point left to compute
      for (unsigned int i = 0; i < n_points; i += n_lanes)
        if (n_points - i > 1)
          {
            Point<dim, VectorizedArray<double>> p_vec;
            for (unsigned int j = 0; j < n_lanes; ++j)
              if (i + j < n_points)
                for (unsigned int d = 0; d < dim; ++d)
                  p_vec[d][j] = unit_points[i + j][d];
              else
                for (unsigned int d = 0; d < dim; ++d)
                  p_vec[d][j] = unit_points[i][d];
 
            Tensor<1, spacedim, VectorizedArray<double>> value;
            DerivativeForm<1, dim, spacedim, VectorizedArray<double>>
              derivative;
            if (needs_gradient)
              {
                const auto result =
                  internal::evaluate_tensor_product_value_and_gradient(
                    polynomials_1d,
                    support_points,
                    p_vec,
                    polynomials_1d.size() == 2,
                    renumber_lexicographic_to_hierarchic);
 
                value = result.first;
 
                for (unsigned int d = 0; d < spacedim; ++d)
                  for (unsigned int e = 0; e < dim; ++e)
                    derivative[d][e] = result.second[e][d];
              }
            else
              value = internal::evaluate_tensor_product_value(
                polynomials_1d,
                support_points,
                p_vec,
                polynomials_1d.size() == 2,
                renumber_lexicographic_to_hierarchic);
 
            if (update_flags & update_quadrature_points)
              for (unsigned int j = 0; j < n_lanes && i + j < n_points; ++j)
                for (unsigned int d = 0; d < spacedim; ++d)
                  quadrature_points[i + j][d] = value[d][j];
 
            if (is_translation)
              continue;
 
            if (update_flags & update_contravariant_transformation)
              for (unsigned int j = 0; j < n_lanes && i + j < n_points; ++j)
                for (unsigned int d = 0; d < spacedim; ++d)
                  for (unsigned int e = 0; e < dim; ++e)
                    jacobians[i + j][d][e] = derivative[d][e][j];
 
            if (update_flags & update_volume_elements)
              {
                const auto determinant = derivative.determinant();
                for (unsigned int j = 0; j < n_lanes && i + j < n_points; ++j)
                  data.volume_elements[i + j] = determinant[j];
              }
 
            if (update_flags & update_covariant_transformation)
              {
                const auto covariant = derivative.covariant_form();
                for (unsigned int j = 0; j < n_lanes && i + j < n_points; ++j)
                  for (unsigned int d = 0; d < dim; ++d)
                    for (unsigned int e = 0; e < spacedim; ++e)
                      inverse_jacobians[i + j][d][e] = covariant[e][d][j];
              }
          }
        else
          {
            Tensor<1, spacedim, double>              value;
            DerivativeForm<1, dim, spacedim, double> derivative;
            if (needs_gradient)
              {
                const auto result =
                  internal::evaluate_tensor_product_value_and_gradient(
                    polynomials_1d,
                    support_points,
                    unit_points[i],
                    polynomials_1d.size() == 2,
                    renumber_lexicographic_to_hierarchic);
 
                value = result.first;
 
                for (unsigned int d = 0; d < spacedim; ++d)
                  for (unsigned int e = 0; e < dim; ++e)
                    derivative[d][e] = result.second[e][d];
              }
            else
              value = internal::evaluate_tensor_product_value(
                polynomials_1d,
                support_points,
                unit_points[i],
                polynomials_1d.size() == 2,
                renumber_lexicographic_to_hierarchic);
 
            if (update_flags & update_quadrature_points)
              quadrature_points[i] = value;
 
            if (is_translation)
              continue;
 
            if (dim == spacedim && update_flags & update_volume_elements)
              data.volume_elements[i] = derivative.determinant();
 
            if (update_flags & update_contravariant_transformation)
              jacobians[i] = derivative;
 
            if (update_flags & update_covariant_transformation)
              inverse_jacobians[i] = jacobians[i].covariant_form().transpose();
          }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_grads(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int> &renumber_lexicographic_to_hierarchic,
      std::vector<DerivativeForm<2, dim, spacedim>> &jacobian_grads)
    {
      if (data.update_each & update_jacobian_grads)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points = jacobian_grads.size();
 
          if (cell_similarity != CellSimilarity::translation)
            for (unsigned int point = 0; point < n_q_points; ++point)
              {
                const SymmetricTensor<2, dim, Tensor<1, spacedim>> second =
                  internal::evaluate_tensor_product_hessian(
                    polynomials_1d,
                    support_points,
                    unit_points[point],
                    renumber_lexicographic_to_hierarchic);
 
                for (unsigned int i = 0; i < spacedim; ++i)
                  for (unsigned int j = 0; j < dim; ++j)
                    for (unsigned int l = 0; l < dim; ++l)
                      jacobian_grads[point][i][j][l] = second[j][l][i];
              }
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_pushed_forward_grads(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int>  &renumber_lexicographic_to_hierarchic,
      std::vector<Tensor<3, spacedim>> &jacobian_pushed_forward_grads)
    {
      if (data.update_each & update_jacobian_pushed_forward_grads)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points = jacobian_pushed_forward_grads.size();
 
          if (cell_similarity != CellSimilarity::translation)
            {
              double tmp[spacedim][spacedim][spacedim];
              for (unsigned int point = 0; point < n_q_points; ++point)
                {
                  const SymmetricTensor<2, dim, Tensor<1, spacedim>> second =
                    internal::evaluate_tensor_product_hessian(
                      polynomials_1d,
                      support_points,
                      unit_points[point],
                      renumber_lexicographic_to_hierarchic);
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[point].transpose();
 
                  // first push forward the j-components
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < dim; ++l)
                        {
                          tmp[i][j][l] = second[0][l][i] * covariant[j][0];
                          for (unsigned int jr = 1; jr < dim; ++jr)
                            {
                              tmp[i][j][l] +=
                                second[jr][l][i] * covariant[j][jr];
                            }
                        }
 
                  // now, pushing forward the l-components
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        {
                          jacobian_pushed_forward_grads[point][i][j][l] =
                            tmp[i][j][0] * covariant[l][0];
                          for (unsigned int lr = 1; lr < dim; ++lr)
                            {
                              jacobian_pushed_forward_grads[point][i][j][l] +=
                                tmp[i][j][lr] * covariant[l][lr];
                            }
                        }
                }
            }
        }
    }
 
 
 
    template <int dim, int spacedim, int length_tensor>
    inline DerivativeForm<3, dim, spacedim>
    expand_3rd_derivatives(
      const Tensor<1, length_tensor, Tensor<1, spacedim>> &compressed)
    {
      Assert(dim >= 1 && dim <= 3, ExcNotImplemented());
      DerivativeForm<3, dim, spacedim> result;
      for (unsigned int i = 0; i < spacedim; ++i)
        {
          if (dim == 1)
            result[i][0][0][0] = compressed[0][i];
          else if (dim >= 2)
            {
              for (unsigned int d = 0; d < 2; ++d)
                for (unsigned int e = 0; e < 2; ++e)
                  for (unsigned int f = 0; f < 2; ++f)
                    result[i][d][e][f] = compressed[d + e + f][i];
              if (dim == 3)
                {
                  AssertDimension(length_tensor, 10);
 
                  // the derivatives are ordered in rows by increasing z
                  // derivative, and in each row we have x^{(n-j)} y^{(j)} as
                  // j walks through the columns
                  for (unsigned int d = 0; d < 2; ++d)
                    for (unsigned int e = 0; e < 2; ++e)
                      {
                        result[i][d][e][2] = compressed[4 + d + e][i];
                        result[i][d][2][e] = compressed[4 + d + e][i];
                        result[i][2][d][e] = compressed[4 + d + e][i];
                      }
                  for (unsigned int d = 0; d < 2; ++d)
                    {
                      result[i][d][2][2] = compressed[7 + d][i];
                      result[i][2][d][2] = compressed[7 + d][i];
                      result[i][2][2][d] = compressed[7 + d][i];
                    }
                  result[i][2][2][2] = compressed[9][i];
                }
            }
        }
      return result;
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_2nd_derivatives(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int> &renumber_lexicographic_to_hierarchic,
      std::vector<DerivativeForm<3, dim, spacedim>> &jacobian_2nd_derivatives)
    {
      if (data.update_each & update_jacobian_2nd_derivatives)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points = jacobian_2nd_derivatives.size();
 
          if (cell_similarity != CellSimilarity::translation)
            {
              for (unsigned int point = 0; point < n_q_points; ++point)
                {
                  jacobian_2nd_derivatives[point] = expand_3rd_derivatives<dim>(
                    internal::evaluate_tensor_product_higher_derivatives<3>(
                      polynomials_1d,
                      support_points,
                      unit_points[point],
                      renumber_lexicographic_to_hierarchic));
                }
            }
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_pushed_forward_2nd_derivatives(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int>  &renumber_lexicographic_to_hierarchic,
      std::vector<Tensor<4, spacedim>> &jacobian_pushed_forward_2nd_derivatives)
    {
      if (data.update_each & update_jacobian_pushed_forward_2nd_derivatives)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points =
            jacobian_pushed_forward_2nd_derivatives.size();
 
          if (cell_similarity != CellSimilarity::translation)
            {
              ::ndarray<double, spacedim, spacedim, dim, dim>      tmp;
              ::ndarray<double, spacedim, spacedim, spacedim, dim> tmp2;
              for (unsigned int point = 0; point < n_q_points; ++point)
                {
                  const DerivativeForm<3, dim, spacedim> third =
                    expand_3rd_derivatives<dim>(
                      internal::evaluate_tensor_product_higher_derivatives<3>(
                        polynomials_1d,
                        support_points,
                        unit_points[point],
                        renumber_lexicographic_to_hierarchic));
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[point].transpose();
 
                  // push forward the j-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < dim; ++l)
                        for (unsigned int m = 0; m < dim; ++m)
                          {
                            tmp[i][j][l][m] =
                              third[i][0][l][m] * covariant[j][0];
                            for (unsigned int jr = 1; jr < dim; ++jr)
                              tmp[i][j][l][m] +=
                                third[i][jr][l][m] * covariant[j][jr];
                          }
 
                  // push forward the l-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        for (unsigned int m = 0; m < dim; ++m)
                          {
                            tmp2[i][j][l][m] =
                              tmp[i][j][0][m] * covariant[l][0];
                            for (unsigned int lr = 1; lr < dim; ++lr)
                              tmp2[i][j][l][m] +=
                                tmp[i][j][lr][m] * covariant[l][lr];
                          }
 
                  // push forward the m-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        for (unsigned int m = 0; m < spacedim; ++m)
                          {
                            jacobian_pushed_forward_2nd_derivatives
                              [point][i][j][l][m] =
                                tmp2[i][j][l][0] * covariant[m][0];
                            for (unsigned int mr = 1; mr < dim; ++mr)
                              jacobian_pushed_forward_2nd_derivatives[point][i]
                                                                     [j][l]
                                                                     [m] +=
                                tmp2[i][j][l][mr] * covariant[m][mr];
                          }
                }
            }
        }
    }
 
 
 
    template <int dim, int spacedim, int length_tensor>
    inline DerivativeForm<4, dim, spacedim>
    expand_4th_derivatives(
      const Tensor<1, length_tensor, Tensor<1, spacedim>> &compressed)
    {
      Assert(dim >= 1 && dim <= 3, ExcNotImplemented());
      DerivativeForm<4, dim, spacedim> result;
      for (unsigned int i = 0; i < spacedim; ++i)
        {
          if (dim == 1)
            result[i][0][0][0][0] = compressed[0][i];
          else if (dim >= 2)
            {
              for (unsigned int d = 0; d < 2; ++d)
                for (unsigned int e = 0; e < 2; ++e)
                  for (unsigned int f = 0; f < 2; ++f)
                    for (unsigned int g = 0; g < 2; ++g)
                      result[i][d][e][f][g] = compressed[d + e + f + g][i];
              if (dim == 3)
                {
                  AssertDimension(length_tensor, 15);
 
                  // the derivatives are ordered in rows by increasing z
                  // derivative, and in each row we have x^{(n-j)} y^{(j)} as
                  // j walks through the columns
                  for (unsigned int d = 0; d < 2; ++d)
                    for (unsigned int e = 0; e < 2; ++e)
                      for (unsigned int f = 0; f < 2; ++f)
                        {
                          result[i][d][e][f][2] = compressed[5 + d + e + f][i];
                          result[i][d][e][2][f] = compressed[5 + d + e + f][i];
                          result[i][d][2][e][f] = compressed[5 + d + e + f][i];
                          result[i][2][d][e][f] = compressed[5 + d + e + f][i];
                        }
                  for (unsigned int d = 0; d < 2; ++d)
                    for (unsigned int e = 0; e < 2; ++e)
                      {
                        result[i][d][e][2][2] = compressed[9 + d + e][i];
                        result[i][d][2][e][2] = compressed[9 + d + e][i];
                        result[i][d][2][2][e] = compressed[9 + d + e][i];
                        result[i][2][d][e][2] = compressed[9 + d + e][i];
                        result[i][2][d][2][e] = compressed[9 + d + e][i];
                        result[i][2][2][d][e] = compressed[9 + d + e][i];
                      }
                  for (unsigned int d = 0; d < 2; ++d)
                    {
                      result[i][d][2][2][2] = compressed[12 + d][i];
                      result[i][2][d][2][2] = compressed[12 + d][i];
                      result[i][2][2][d][2] = compressed[12 + d][i];
                      result[i][2][2][2][d] = compressed[12 + d][i];
                    }
                  result[i][2][2][2][2] = compressed[14][i];
                }
            }
        }
      return result;
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_3rd_derivatives(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int> &renumber_lexicographic_to_hierarchic,
      std::vector<DerivativeForm<4, dim, spacedim>> &jacobian_3rd_derivatives)
    {
      if (data.update_each & update_jacobian_3rd_derivatives)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points = jacobian_3rd_derivatives.size();
 
          if (cell_similarity != CellSimilarity::translation)
            {
              for (unsigned int point = 0; point < n_q_points; ++point)
                {
                  jacobian_3rd_derivatives[point] = expand_4th_derivatives<dim>(
                    internal::evaluate_tensor_product_higher_derivatives<4>(
                      polynomials_1d,
                      support_points,
                      unit_points[point],
                      renumber_lexicographic_to_hierarchic));
                }
            }
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_update_jacobian_pushed_forward_3rd_derivatives(
      const CellSimilarity::Similarity cell_similarity,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const ArrayView<const Point<dim>>                            &unit_points,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int>  &renumber_lexicographic_to_hierarchic,
      std::vector<Tensor<5, spacedim>> &jacobian_pushed_forward_3rd_derivatives)
    {
      if (data.update_each & update_jacobian_pushed_forward_3rd_derivatives)
        {
          const ArrayView<const Point<spacedim>> support_points(
            data.mapping_support_points);
          const unsigned int n_q_points =
            jacobian_pushed_forward_3rd_derivatives.size();
 
          if (cell_similarity != CellSimilarity::translation)
            {
              ::
                ndarray<double, spacedim, spacedim, spacedim, spacedim, dim>
                  tmp;
              ::ndarray<double, spacedim, spacedim, spacedim, dim, dim>
                tmp2;
              for (unsigned int point = 0; point < n_q_points; ++point)
                {
                  const DerivativeForm<4, dim, spacedim> fourth =
                    expand_4th_derivatives<dim>(
                      internal::evaluate_tensor_product_higher_derivatives<4>(
                        polynomials_1d,
                        support_points,
                        unit_points[point],
                        renumber_lexicographic_to_hierarchic));
 
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[point].transpose();
 
                  // push-forward the j-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < dim; ++l)
                        for (unsigned int m = 0; m < dim; ++m)
                          for (unsigned int n = 0; n < dim; ++n)
                            {
                              tmp[i][j][l][m][n] =
                                fourth[i][0][l][m][n] * covariant[j][0];
                              for (unsigned int jr = 1; jr < dim; ++jr)
                                tmp[i][j][l][m][n] +=
                                  fourth[i][jr][l][m][n] * covariant[j][jr];
                            }
 
                  // push-forward the l-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        for (unsigned int m = 0; m < dim; ++m)
                          for (unsigned int n = 0; n < dim; ++n)
                            {
                              tmp2[i][j][l][m][n] =
                                tmp[i][j][0][m][n] * covariant[l][0];
                              for (unsigned int lr = 1; lr < dim; ++lr)
                                tmp2[i][j][l][m][n] +=
                                  tmp[i][j][lr][m][n] * covariant[l][lr];
                            }
 
                  // push-forward the m-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        for (unsigned int m = 0; m < spacedim; ++m)
                          for (unsigned int n = 0; n < dim; ++n)
                            {
                              tmp[i][j][l][m][n] =
                                tmp2[i][j][l][0][n] * covariant[m][0];
                              for (unsigned int mr = 1; mr < dim; ++mr)
                                tmp[i][j][l][m][n] +=
                                  tmp2[i][j][l][mr][n] * covariant[m][mr];
                            }
 
                  // push-forward the n-coordinate
                  for (unsigned int i = 0; i < spacedim; ++i)
                    for (unsigned int j = 0; j < spacedim; ++j)
                      for (unsigned int l = 0; l < spacedim; ++l)
                        for (unsigned int m = 0; m < spacedim; ++m)
                          for (unsigned int n = 0; n < spacedim; ++n)
                            {
                              jacobian_pushed_forward_3rd_derivatives
                                [point][i][j][l][m][n] =
                                  tmp[i][j][l][m][0] * covariant[n][0];
                              for (unsigned int nr = 1; nr < dim; ++nr)
                                jacobian_pushed_forward_3rd_derivatives[point]
                                                                       [i][j][l]
                                                                       [m][n] +=
                                  tmp[i][j][l][m][nr] * covariant[n][nr];
                            }
                }
            }
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    maybe_compute_face_data(
      const ::MappingQ<dim, spacedim> &mapping,
      const typename ::Triangulation<dim, spacedim>::cell_iterator &cell,
      const unsigned int                                            face_no,
      const unsigned int                                            subface_no,
      const unsigned int                                            n_q_points,
      const std::vector<double>                                    &weights,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
        &output_data)
    {
      const UpdateFlags update_flags = data.update_each;
 
      if (update_flags &
          (update_boundary_forms | update_normal_vectors | update_JxW_values))
        {
          if (update_flags & update_boundary_forms)
            AssertDimension(output_data.boundary_forms.size(), n_q_points);
          if (update_flags & update_normal_vectors)
            AssertDimension(output_data.normal_vectors.size(), n_q_points);
          if (update_flags & update_JxW_values)
            AssertDimension(output_data.JxW_values.size(), n_q_points);
 
          Assert(data.aux.size() + 1 >= dim, ExcInternalError());
 
          // first compute some common data that is used for evaluating
          // all of the flags below
 
          // map the unit tangentials to the real cell. checking for d!=dim-1
          // eliminates compiler warnings regarding unsigned int expressions <
          // 0.
          for (unsigned int d = 0; d != dim - 1; ++d)
            {
              const unsigned int vector_index =
                face_no + GeometryInfo<dim>::faces_per_cell * d;
              Assert(vector_index < data.unit_tangentials.size(),
                     ExcInternalError());
              Assert(data.aux[d].size() <=
                       data.unit_tangentials[vector_index].size(),
                     ExcInternalError());
              mapping.transform(make_array_view(
                                  data.unit_tangentials[vector_index]),
                                mapping_contravariant,
                                data,
                                make_array_view(data.aux[d]));
            }
 
          if (update_flags & update_boundary_forms)
            {
              // if dim==spacedim, we can use the unit tangentials to compute
              // the boundary form by simply taking the cross product
              if (dim == spacedim)
                {
                  for (unsigned int i = 0; i < n_q_points; ++i)
                    switch (dim)
                      {
                        case 1:
                          // in 1d, we don't have access to any of the
                          // data.aux fields (because it has only dim-1
                          // components), but we can still compute the
                          // boundary form by simply looking at the number of
                          // the face
                          output_data.boundary_forms[i][0] =
                            (face_no == 0 ? -1 : +1);
                          break;
                        case 2:
                          output_data.boundary_forms[i] =
                            cross_product_2d(data.aux[0][i]);
                          break;
                        case 3:
                          output_data.boundary_forms[i] =
                            cross_product_3d(data.aux[0][i], data.aux[1][i]);
                          break;
                        default:
                          DEAL_II_NOT_IMPLEMENTED();
                      }
                }
              else //(dim < spacedim)
                {
                  // in the codim-one case, the boundary form results from the
                  // cross product of all the face tangential vectors and the
                  // cell normal vector
                  //
                  // to compute the cell normal, use the same method used in
                  // fill_fe_values for cells above
                  AssertDimension(data.output_data->jacobians.size(),
                                  n_q_points);
 
                  for (unsigned int point = 0; point < n_q_points; ++point)
                    {
                      const DerivativeForm<1, dim, spacedim> contravariant =
                        data.output_data->jacobians[point];
                      if (dim == 1)
                        {
                          // J is a tangent vector
                          output_data.boundary_forms[point] =
                            contravariant.transpose()[0];
                          output_data.boundary_forms[point] /=
                            (face_no == 0 ? -1. : +1.) *
                            output_data.boundary_forms[point].norm();
                        }
 
                      if (dim == 2)
                        {
                          const DerivativeForm<1, spacedim, dim> DX_t =
                            contravariant.transpose();
 
                          Tensor<1, spacedim> cell_normal =
                            cross_product_3d(DX_t[0], DX_t[1]);
                          cell_normal /= cell_normal.norm();
 
                          // then compute the face normal from the face
                          // tangent and the cell normal:
                          output_data.boundary_forms[point] =
                            cross_product_3d(data.aux[0][point], cell_normal);
                        }
                    }
                }
            }
 
          if (update_flags & update_JxW_values)
            for (unsigned int i = 0; i < output_data.boundary_forms.size(); ++i)
              {
                output_data.JxW_values[i] =
                  output_data.boundary_forms[i].norm() * weights[i];
 
                if (subface_no != numbers::invalid_unsigned_int)
                  {
                    const double area_ratio = GeometryInfo<dim>::subface_ratio(
                      cell->subface_case(face_no), subface_no);
                    output_data.JxW_values[i] *= area_ratio;
                  }
              }
 
          if (update_flags & update_normal_vectors)
            for (unsigned int i = 0; i < output_data.normal_vectors.size(); ++i)
              output_data.normal_vectors[i] =
                Point<spacedim>(output_data.boundary_forms[i] /
                                output_data.boundary_forms[i].norm());
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    do_fill_fe_face_values(
      const ::MappingQ<dim, spacedim> &mapping,
      const typename ::Triangulation<dim, spacedim>::cell_iterator &cell,
      const unsigned int                                            face_no,
      const unsigned int                                            subface_no,
      const typename QProjector<dim>::DataSetDescriptor             data_set,
      const Quadrature<dim - 1>                                    &quadrature,
      const typename ::MappingQ<dim, spacedim>::InternalData &data,
      const std::vector<Polynomials::Polynomial<double>> &polynomials_1d,
      const std::vector<unsigned int> &renumber_lexicographic_to_hierarchic,
      internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
        &output_data)
    {
      const ArrayView<const Point<dim>> quadrature_points(
        &data.quadrature_points[data_set], quadrature.size());
      if (dim > 1 && data.tensor_product_quadrature)
        {
          maybe_update_q_points_Jacobians_and_grads_tensor<dim, spacedim>(
            CellSimilarity::none,
            data,
            output_data.quadrature_points,
            output_data.jacobians,
            output_data.inverse_jacobians,
            output_data.jacobian_grads);
        }
      else
        {
          internal::MappingQImplementation::
            maybe_update_q_points_Jacobians_generic(
              CellSimilarity::none,
              data,
              quadrature_points,
              polynomials_1d,
              renumber_lexicographic_to_hierarchic,
              output_data.quadrature_points,
              output_data.jacobians,
              output_data.inverse_jacobians);
          maybe_update_jacobian_grads<dim, spacedim>(
            CellSimilarity::none,
            data,
            quadrature_points,
            polynomials_1d,
            renumber_lexicographic_to_hierarchic,
            output_data.jacobian_grads);
        }
      maybe_update_jacobian_pushed_forward_grads<dim, spacedim>(
        CellSimilarity::none,
        data,
        quadrature_points,
        polynomials_1d,
        renumber_lexicographic_to_hierarchic,
        output_data.jacobian_pushed_forward_grads);
      maybe_update_jacobian_2nd_derivatives<dim, spacedim>(
        CellSimilarity::none,
        data,
        quadrature_points,
        polynomials_1d,
        renumber_lexicographic_to_hierarchic,
        output_data.jacobian_2nd_derivatives);
      maybe_update_jacobian_pushed_forward_2nd_derivatives<dim, spacedim>(
        CellSimilarity::none,
        data,
        quadrature_points,
        polynomials_1d,
        renumber_lexicographic_to_hierarchic,
        output_data.jacobian_pushed_forward_2nd_derivatives);
      maybe_update_jacobian_3rd_derivatives<dim, spacedim>(
        CellSimilarity::none,
        data,
        quadrature_points,
        polynomials_1d,
        renumber_lexicographic_to_hierarchic,
        output_data.jacobian_3rd_derivatives);
      maybe_update_jacobian_pushed_forward_3rd_derivatives<dim, spacedim>(
        CellSimilarity::none,
        data,
        quadrature_points,
        polynomials_1d,
        renumber_lexicographic_to_hierarchic,
        output_data.jacobian_pushed_forward_3rd_derivatives);
 
      maybe_compute_face_data(mapping,
                              cell,
                              face_no,
                              subface_no,
                              quadrature.size(),
                              quadrature.get_weights(),
                              data,
                              output_data);
    }
 
 
 
    template <int dim, int spacedim, int rank>
    inline void
    transform_fields(
      const ArrayView<const Tensor<rank, dim>>                &input,
      const MappingKind                                        mapping_kind,
      const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
      const ArrayView<Tensor<rank, spacedim>>                 &output)
    {
      AssertDimension(input.size(), output.size());
      Assert((dynamic_cast<
                const typename ::MappingQ<dim, spacedim>::InternalData *>(
                &mapping_data) != nullptr),
             ExcInternalError());
      const typename ::MappingQ<dim, spacedim>::InternalData &data =
        static_cast<
          const typename ::MappingQ<dim, spacedim>::InternalData &>(
          mapping_data);
 
      switch (mapping_kind)
        {
          case mapping_contravariant:
            {
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                output[i] = apply_transformation(data.output_data->jacobians[i],
                                                 input[i]);
 
              return;
            }
 
          case mapping_piola:
            {
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
              Assert(data.update_each & update_volume_elements,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_volume_elements"));
              Assert(rank == 1, ExcMessage("Only for rank 1"));
              if (rank != 1)
                return;
 
              for (unsigned int i = 0; i < output.size(); ++i)
                {
                  output[i] =
                    apply_transformation(data.output_data->jacobians[i],
                                         input[i]);
                  output[i] /= data.volume_elements[i];
                }
              return;
            }
          // We still allow this operation as in the
          // reference cell Derivatives are Tensor
          // rather than DerivativeForm
          case mapping_covariant:
            {
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                output[i] = apply_transformation(
                  data.output_data->inverse_jacobians[i].transpose(), input[i]);
 
              return;
            }
 
          default:
            DEAL_II_NOT_IMPLEMENTED();
        }
    }
 
 
 
    template <int dim, int spacedim, int rank>
    inline void
    transform_gradients(
      const ArrayView<const Tensor<rank, dim>>                &input,
      const MappingKind                                        mapping_kind,
      const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
      const ArrayView<Tensor<rank, spacedim>>                 &output)
    {
      AssertDimension(input.size(), output.size());
      Assert((dynamic_cast<
                const typename ::MappingQ<dim, spacedim>::InternalData *>(
                &mapping_data) != nullptr),
             ExcInternalError());
      const typename ::MappingQ<dim, spacedim>::InternalData &data =
        static_cast<
          const typename ::MappingQ<dim, spacedim>::InternalData &>(
          mapping_data);
 
      switch (mapping_kind)
        {
          case mapping_contravariant_gradient:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
              Assert(rank == 2, ExcMessage("Only for rank 2"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                {
                  const DerivativeForm<1, spacedim, dim> A =
                    apply_transformation(data.output_data->jacobians[i],
                                         transpose(input[i]));
                  output[i] = apply_transformation(
                    data.output_data->inverse_jacobians[i].transpose(),
                    A.transpose());
                }
 
              return;
            }
 
          case mapping_covariant_gradient:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
              Assert(rank == 2, ExcMessage("Only for rank 2"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                {
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[i].transpose();
                  const DerivativeForm<1, spacedim, dim> A =
                    apply_transformation(covariant, transpose(input[i]));
                  output[i] = apply_transformation(covariant, A.transpose());
                }
 
              return;
            }
 
          case mapping_piola_gradient:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
              Assert(data.update_each & update_volume_elements,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_volume_elements"));
              Assert(rank == 2, ExcMessage("Only for rank 2"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                {
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[i].transpose();
                  const DerivativeForm<1, spacedim, dim> A =
                    apply_transformation(covariant, input[i]);
                  const Tensor<2, spacedim> T =
                    apply_transformation(data.output_data->jacobians[i],
                                         A.transpose());
 
                  output[i] = transpose(T);
                  output[i] /= data.volume_elements[i];
                }
 
              return;
            }
 
          default:
            DEAL_II_NOT_IMPLEMENTED();
        }
    }
 
 
 
    template <int dim, int spacedim>
    inline void
    transform_hessians(
      const ArrayView<const Tensor<3, dim>>                   &input,
      const MappingKind                                        mapping_kind,
      const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
      const ArrayView<Tensor<3, spacedim>>                    &output)
    {
      AssertDimension(input.size(), output.size());
      Assert((dynamic_cast<
                const typename ::MappingQ<dim, spacedim>::InternalData *>(
                &mapping_data) != nullptr),
             ExcInternalError());
      const typename ::MappingQ<dim, spacedim>::InternalData &data =
        static_cast<
          const typename ::MappingQ<dim, spacedim>::InternalData &>(
          mapping_data);
 
      switch (mapping_kind)
        {
          case mapping_contravariant_hessian:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
 
              for (unsigned int q = 0; q < output.size(); ++q)
                {
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[q].transpose();
                  const DerivativeForm<1, dim, spacedim> contravariant =
                    data.output_data->jacobians[q];
 
                  for (unsigned int i = 0; i < spacedim; ++i)
                    {
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] =
                              contravariant[i][0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] +=
                                contravariant[i][I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = covariant[j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += covariant[j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] = covariant[k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] += covariant[k][K] * tmp2[K];
                            }
                        }
                    }
                }
              return;
            }
 
          case mapping_covariant_hessian:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
 
              for (unsigned int q = 0; q < output.size(); ++q)
                {
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[q].transpose();
 
                  for (unsigned int i = 0; i < spacedim; ++i)
                    {
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] = covariant[i][0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] += covariant[i][I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = covariant[j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += covariant[j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] = covariant[k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] += covariant[k][K] * tmp2[K];
                            }
                        }
                    }
                }
 
              return;
            }
 
          case mapping_piola_hessian:
            {
              Assert(data.update_each & update_covariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_contravariant_transformation"));
              Assert(data.update_each & update_volume_elements,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_volume_elements"));
 
              for (unsigned int q = 0; q < output.size(); ++q)
                {
                  const DerivativeForm<1, dim, spacedim> covariant =
                    data.output_data->inverse_jacobians[q].transpose();
                  const DerivativeForm<1, dim, spacedim> contravariant =
                    data.output_data->jacobians[q];
                  for (unsigned int i = 0; i < spacedim; ++i)
                    {
                      double factor[dim];
                      for (unsigned int I = 0; I < dim; ++I)
                        factor[I] =
                          contravariant[i][I] * (1. / data.volume_elements[q]);
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] = factor[0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] += factor[I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = covariant[j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += covariant[j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] = covariant[k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] += covariant[k][K] * tmp2[K];
                            }
                        }
                    }
                }
 
              return;
            }
 
          default:
            DEAL_II_NOT_IMPLEMENTED();
        }
    }
 
 
 
    template <int dim, int spacedim, int rank>
    inline void
    transform_differential_forms(
      const ArrayView<const DerivativeForm<rank, dim, spacedim>> &input,
      const MappingKind                                           mapping_kind,
      const typename Mapping<dim, spacedim>::InternalDataBase    &mapping_data,
      const ArrayView<Tensor<rank + 1, spacedim>>                &output)
    {
      AssertDimension(input.size(), output.size());
      Assert((dynamic_cast<
                const typename ::MappingQ<dim, spacedim>::InternalData *>(
                &mapping_data) != nullptr),
             ExcInternalError());
      const typename ::MappingQ<dim, spacedim>::InternalData &data =
        static_cast<
          const typename ::MappingQ<dim, spacedim>::InternalData &>(
          mapping_data);
 
      switch (mapping_kind)
        {
          case mapping_covariant:
            {
              Assert(data.update_each & update_contravariant_transformation,
                     typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                       "update_covariant_transformation"));
 
              for (unsigned int i = 0; i < output.size(); ++i)
                output[i] = apply_transformation(
                  data.output_data->inverse_jacobians[i].transpose(), input[i]);
 
              return;
            }
          default:
            DEAL_II_NOT_IMPLEMENTED();
        }
    }
  } // namespace MappingQImplementation
} // namespace internal
 
DEAL_II_NAMESPACE_CLOSE
 
#endif