cell_hex27.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
 
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
 
// Local includes
#include "libmesh/cell_hex27.h"
#include "libmesh/edge_edge3.h"
#include "libmesh/face_quad9.h"
#include "libmesh/enum_io_package.h"
#include "libmesh/enum_order.h"
 
namespace libMesh
{
 
 
 
// ------------------------------------------------------------
// Hex27 class static member initializations
const int Hex27::num_nodes;
const int Hex27::nodes_per_side;
const int Hex27::nodes_per_edge;
 
const unsigned int Hex27::side_nodes_map[Hex27::num_sides][Hex27::nodes_per_side] =
  {
    {0, 3, 2, 1, 11, 10,  9,  8, 20}, // Side 0
    {0, 1, 5, 4,  8, 13, 16, 12, 21}, // Side 1
    {1, 2, 6, 5,  9, 14, 17, 13, 22}, // Side 2
    {2, 3, 7, 6, 10, 15, 18, 14, 23}, // Side 3
    {3, 0, 4, 7, 11, 12, 19, 15, 24}, // Side 4
    {4, 5, 6, 7, 16, 17, 18, 19, 25}  // Side 5
  };
 
const unsigned int Hex27::edge_nodes_map[Hex27::num_edges][Hex27::nodes_per_edge] =
  {
    {0, 1, 8},  // Edge 0
    {1, 2, 9},  // Edge 1
    {2, 3, 10}, // Edge 2
    {0, 3, 11}, // Edge 3
    {0, 4, 12}, // Edge 4
    {1, 5, 13}, // Edge 5
    {2, 6, 14}, // Edge 6
    {3, 7, 15}, // Edge 7
    {4, 5, 16}, // Edge 8
    {5, 6, 17}, // Edge 9
    {6, 7, 18}, // Edge 10
    {4, 7, 19}  // Edge 11
  };
 
// ------------------------------------------------------------
// Hex27 class member functions
 
bool Hex27::is_vertex(const unsigned int i) const
{
  if (i < 8)
    return true;
  return false;
}
 
bool Hex27::is_edge(const unsigned int i) const
{
  if (i < 8)
    return false;
  if (i > 19)
    return false;
  return true;
}
 
bool Hex27::is_face(const unsigned int i) const
{
  if (i == 26)
    return false;
  if (i > 19)
    return true;
  return false;
}
 
bool Hex27::is_node_on_side(const unsigned int n,
                            const unsigned int s) const
{
  libmesh_assert_less (s, n_sides());
  return std::find(std::begin(side_nodes_map[s]),
                   std::end(side_nodes_map[s]),
                   n) != std::end(side_nodes_map[s]);
}
 
std::vector<unsigned>
Hex27::nodes_on_side(const unsigned int s) const
{
  libmesh_assert_less(s, n_sides());
  return {std::begin(side_nodes_map[s]), std::end(side_nodes_map[s])};
}
 
std::vector<unsigned>
Hex27::nodes_on_edge(const unsigned int e) const
{
  libmesh_assert_less(e, n_edges());
  return {std::begin(edge_nodes_map[e]), std::end(edge_nodes_map[e])};
}
 
bool Hex27::is_node_on_edge(const unsigned int n,
                            const unsigned int e) const
{
  libmesh_assert_less (e, n_edges());
  return std::find(std::begin(edge_nodes_map[e]),
                   std::end(edge_nodes_map[e]),
                   n) != std::end(edge_nodes_map[e]);
}
 
 
 
bool Hex27::has_affine_map() const
{
  // Make sure x-edge endpoints are affine
  Point v = this->point(1) - this->point(0);
  if (!v.relative_fuzzy_equals(this->point(2) - this->point(3), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(5) - this->point(4), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(6) - this->point(7), affine_tol))
    return false;
  // Make sure x-edges are straight
  // and x-face and center points are centered
  v /= 2;
  if (!v.relative_fuzzy_equals(this->point(8) - this->point(0), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(10) - this->point(3), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(16) - this->point(4), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(18) - this->point(7), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(20) - this->point(11), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(21) - this->point(12), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(23) - this->point(15), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(25) - this->point(19), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(26) - this->point(24), affine_tol))
    return false;
  // Make sure xz-faces are identical parallelograms
  v = this->point(4) - this->point(0);
  if (!v.relative_fuzzy_equals(this->point(7) - this->point(3), affine_tol))
    return false;
  v /= 2;
  if (!v.relative_fuzzy_equals(this->point(12) - this->point(0), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(13) - this->point(1), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(14) - this->point(2), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(15) - this->point(3), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(22) - this->point(9), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(24) - this->point(11), affine_tol))
    return false;
  // Make sure y-edges are straight
  v = (this->point(3) - this->point(0))/2;
  if (!v.relative_fuzzy_equals(this->point(11) - this->point(0), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(9) - this->point(1), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(17) - this->point(5), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(19) - this->point(4), affine_tol))
    return false;
  // If all the above checks out, the map is affine
  return true;
}
 
 
 
Order Hex27::default_order() const
{
  return SECOND;
}
 
 
 
dof_id_type Hex27::key (const unsigned int s) const
{
  libmesh_assert_less (s, this->n_sides());
 
  // Think of a unit cube: (-1,1) x (-1,1) x (1,1)
  switch (s)
    {
    case 0:  // the face at z=0
 
      return
        this->compute_key (this->node_id(20));
 
    case 1:  // the face at y = 0
 
      return
        this->compute_key (this->node_id(21));
 
    case 2:  // the face at x=1
 
      return
        this->compute_key (this->node_id(22));
 
    case 3: // the face at y=1
 
      return
        this->compute_key (this->node_id(23));
 
    case 4: // the face at x=0
 
      return
        this->compute_key (this->node_id(24));
 
    case 5: // the face at z=1
 
      return
        this->compute_key (this->node_id(25));
 
    default:
      libmesh_error_msg("Invalid side " << s);
    }
}
 
 
 
unsigned int Hex27::local_side_node(unsigned int side,
                                    unsigned int side_node) const
{
  libmesh_assert_less (side, this->n_sides());
  libmesh_assert_less (side_node, Hex27::nodes_per_side);
 
  return Hex27::side_nodes_map[side][side_node];
}
 
 
 
unsigned int Hex27::local_edge_node(unsigned int edge,
                                    unsigned int edge_node) const
{
  libmesh_assert_less (edge, this->n_edges());
  libmesh_assert_less (edge_node, Hex27::nodes_per_edge);
 
  return Hex27::edge_nodes_map[edge][edge_node];
}
 
 
 
std::unique_ptr<Elem> Hex27::build_side_ptr (const unsigned int i)
{
  return this->simple_build_side_ptr<Quad9, Hex27>(i);
}
 
 
 
void Hex27::build_side_ptr (std::unique_ptr<Elem> & side,
                            const unsigned int i)
{
  this->simple_build_side_ptr<Hex27>(side, i, QUAD9);
}
 
 
 
std::unique_ptr<Elem> Hex27::build_edge_ptr (const unsigned int i)
{
  return this->simple_build_edge_ptr<Edge3,Hex27>(i);
}
 
 
 
void Hex27::build_edge_ptr (std::unique_ptr<Elem> & edge, const unsigned int i)
{
  this->simple_build_edge_ptr<Hex27>(edge, i, EDGE3);
}
 
 
 
void Hex27::connectivity(const unsigned int sc,
                         const IOPackage iop,
                         std::vector<dof_id_type> & conn) const
{
  libmesh_assert(_nodes);
  libmesh_assert_less (sc, this->n_sub_elem());
  libmesh_assert_not_equal_to (iop, INVALID_IO_PACKAGE);
 
  conn.resize(8);
 
  switch (iop)
    {
    case TECPLOT:
      {
        switch (sc)
          {
          case 0:
 
            conn[0] = this->node_id(0)+1;
            conn[1] = this->node_id(8)+1;
            conn[2] = this->node_id(20)+1;
            conn[3] = this->node_id(11)+1;
            conn[4] = this->node_id(12)+1;
            conn[5] = this->node_id(21)+1;
            conn[6] = this->node_id(26)+1;
            conn[7] = this->node_id(24)+1;
 
            return;
 
          case 1:
 
            conn[0] = this->node_id(8)+1;
            conn[1] = this->node_id(1)+1;
            conn[2] = this->node_id(9)+1;
            conn[3] = this->node_id(20)+1;
            conn[4] = this->node_id(21)+1;
            conn[5] = this->node_id(13)+1;
            conn[6] = this->node_id(22)+1;
            conn[7] = this->node_id(26)+1;
 
            return;
 
          case 2:
 
            conn[0] = this->node_id(11)+1;
            conn[1] = this->node_id(20)+1;
            conn[2] = this->node_id(10)+1;
            conn[3] = this->node_id(3)+1;
            conn[4] = this->node_id(24)+1;
            conn[5] = this->node_id(26)+1;
            conn[6] = this->node_id(23)+1;
            conn[7] = this->node_id(15)+1;
 
            return;
 
          case 3:
 
            conn[0] = this->node_id(20)+1;
            conn[1] = this->node_id(9)+1;
            conn[2] = this->node_id(2)+1;
            conn[3] = this->node_id(10)+1;
            conn[4] = this->node_id(26)+1;
            conn[5] = this->node_id(22)+1;
            conn[6] = this->node_id(14)+1;
            conn[7] = this->node_id(23)+1;
 
            return;
 
          case 4:
 
            conn[0] = this->node_id(12)+1;
            conn[1] = this->node_id(21)+1;
            conn[2] = this->node_id(26)+1;
            conn[3] = this->node_id(24)+1;
            conn[4] = this->node_id(4)+1;
            conn[5] = this->node_id(16)+1;
            conn[6] = this->node_id(25)+1;
            conn[7] = this->node_id(19)+1;
 
            return;
 
          case 5:
 
            conn[0] = this->node_id(21)+1;
            conn[1] = this->node_id(13)+1;
            conn[2] = this->node_id(22)+1;
            conn[3] = this->node_id(26)+1;
            conn[4] = this->node_id(16)+1;
            conn[5] = this->node_id(5)+1;
            conn[6] = this->node_id(17)+1;
            conn[7] = this->node_id(25)+1;
 
            return;
 
          case 6:
 
            conn[0] = this->node_id(24)+1;
            conn[1] = this->node_id(26)+1;
            conn[2] = this->node_id(23)+1;
            conn[3] = this->node_id(15)+1;
            conn[4] = this->node_id(19)+1;
            conn[5] = this->node_id(25)+1;
            conn[6] = this->node_id(18)+1;
            conn[7] = this->node_id(7)+1;
 
            return;
 
          case 7:
 
            conn[0] = this->node_id(26)+1;
            conn[1] = this->node_id(22)+1;
            conn[2] = this->node_id(14)+1;
            conn[3] = this->node_id(23)+1;
            conn[4] = this->node_id(25)+1;
            conn[5] = this->node_id(17)+1;
            conn[6] = this->node_id(6)+1;
            conn[7] = this->node_id(18)+1;
 
            return;
 
          default:
            libmesh_error_msg("Invalid sc = " << sc);
          }
      }
 
    case VTK:
      {
        // VTK now supports VTK_TRIQUADRATIC_HEXAHEDRON directly
        conn.resize(27);
 
        conn[0] = this->node_id(0);
        conn[1] = this->node_id(1);
        conn[2] = this->node_id(2);
        conn[3] = this->node_id(3);
        conn[4] = this->node_id(4);
        conn[5] = this->node_id(5);
        conn[6] = this->node_id(6);
        conn[7] = this->node_id(7);
        conn[8] = this->node_id(8);
        conn[9] = this->node_id(9);
        conn[10] = this->node_id(10);
        conn[11] = this->node_id(11);
        conn[12] = this->node_id(16);
        conn[13] = this->node_id(17);
        conn[14] = this->node_id(18);
        conn[15] = this->node_id(19);
        conn[16] = this->node_id(12);
        conn[17] = this->node_id(13);
        conn[18] = this->node_id(14);
        conn[19] = this->node_id(15);
        conn[20] = this->node_id(24);
        conn[21] = this->node_id(22);
        conn[22] = this->node_id(21);
        conn[23] = this->node_id(23);
        conn[24] = this->node_id(20);
        conn[25] = this->node_id(25);
        conn[26] = this->node_id(26);
 
        return;
      }
 
    default:
      libmesh_error_msg("Unsupported IO package " << iop);
    }
}
 
 
 
 
 
unsigned int Hex27::n_second_order_adjacent_vertices (const unsigned int n) const
{
  switch (n)
    {
    case 8:
    case 9:
    case 10:
    case 11:
    case 12:
    case 13:
    case 14:
    case 15:
    case 16:
    case 17:
    case 18:
    case 19:
      return 2;
 
    case 20:
    case 21:
    case 22:
    case 23:
    case 24:
    case 25:
      return 4;
 
    case 26:
      return 8;
 
    default:
      libmesh_error_msg("Invalid node number n = " << n);
    }
}
 
 
 
unsigned short int Hex27::second_order_adjacent_vertex (const unsigned int n,
                                                        const unsigned int v) const
{
  libmesh_assert_greater_equal (n, this->n_vertices());
  libmesh_assert_less (n, this->n_nodes());
 
  switch (n)
    {
      /*
       * these are all nodes that are unique to Hex27,
       * use our _remaining.... matrix
       */
    case 20:
    case 21:
    case 22:
    case 23:
    case 24:
    case 25:
      {
        libmesh_assert_less (v, 4);
        return _remaining_second_order_adjacent_vertices[n-20][v];
      }
 
      /*
       * for the bubble node the return value is simply v.
       * Why? -- the user asks for the v-th adjacent vertex,
       * from \p n_second_order_adjacent_vertices() there
       * are 8 adjacent vertices, and these happen to be
       * 0..7
       */
    case 26:
      {
        libmesh_assert_less (v, 8);
        return static_cast<unsigned short int>(v);
      }
 
      /*
       * nodes 8..19:
       * these are all nodes that are identical for
       * Hex20 and Hex27.  Therefore use the
       * matrix stored in cell_hex.C
       */
    default:
      {
        libmesh_assert_less (v, 2);
        return _second_order_adjacent_vertices[n-this->n_vertices()][v];
      }
    }
}
 
 
 
const unsigned short int Hex27::_remaining_second_order_adjacent_vertices[6][4] =
  {
    { 0,  1,  2,  3}, // vertices adjacent to node 20   face nodes
    { 0,  1,  4,  5}, // vertices adjacent to node 21
    { 1,  2,  5,  6}, // vertices adjacent to node 22
    { 2,  3,  6,  7}, // vertices adjacent to node 23
    { 0,  3,  4,  7}, // vertices adjacent to node 24
    { 4,  5,  6,  7}, // vertices adjacent to node 25
  };
 
 
 
std::pair<unsigned short int, unsigned short int>
Hex27::second_order_child_vertex (const unsigned int n) const
{
  libmesh_assert_greater_equal (n, this->n_vertices());
  libmesh_assert_less (n, this->n_nodes());
  /*
   * the _second_order_vertex_child_* vectors are
   * stored in cell_hex.C, since they are identical
   * for Hex20 and Hex27 (for the first 12 higher-order nodes)
   */
  return std::pair<unsigned short int, unsigned short int>
    (_second_order_vertex_child_number[n],
     _second_order_vertex_child_index[n]);
}
 
 
 
Real Hex27::volume () const
{
  // This specialization is good for Lagrange mappings only in general
  if (this->mapping_type() != LAGRANGE_MAP)
    return this->Elem::volume();
 
  // Make copies of our points.  It makes the subsequent calculations a bit
  // shorter and avoids dereferencing the same pointer multiple times.
  Point
    x0 = point(0),   x1 = point(1),   x2 = point(2),   x3 = point(3),   x4 = point(4),   x5 = point(5),   x6 = point(6),   x7 = point(7),   x8 = point(8),
    x9 = point(9),   x10 = point(10), x11 = point(11), x12 = point(12), x13 = point(13), x14 = point(14), x15 = point(15), x16 = point(16), x17 = point(17),
    x18 = point(18), x19 = point(19), x20 = point(20), x21 = point(21), x22 = point(22), x23 = point(23), x24 = point(24), x25 = point(25), x26 = point(26);
 
  // The constant components of the dx/dxi vector,
  // dx/dxi = \vec{a000} + \vec{a001}*zeta + \vec{a002}*zeta^2 + ...
  // All of the xi^2 terms are zero.
  // These were copied directly from the output of a Python script.
  Point dx_dxi[3][3][3] =
    {
      {
        {
          x22/2 - x24/2, // 0, 0, 0
          x11/4 + x17/4 - x19/4 - x9/4, // 0, 0, 1
          -x11/4 + x17/4 - x19/4 - x22/2 + x24/2 + x9/4 // 0, 0, 2
        },
 
        {
          x12/4 - x13/4 + x14/4 - x15/4, // 0, 1, 0
          -x0/8 + x1/8 - x2/8 + x3/8 + x4/8 - x5/8 + x6/8 - x7/8, // 0, 1, 1
          x0/8 - x1/8 - x12/4 + x13/4 - x14/4 + x15/4 + x2/8 - x3/8 + x4/8 - x5/8 + x6/8 - x7/8 // 0, 1, 2
        },
 
        {
          -x12/4 + x13/4 + x14/4 - x15/4 - x22/2 + x24/2, // 0, 2, 0
          x0/8 - x1/8 - x11/4 - x17/4 + x19/4 - x2/8 + x3/8 - x4/8 + x5/8 + x6/8 - x7/8 + x9/4, // 0, 2, 1
          -x0/8 + x1/8 + x11/4 + x12/4 - x13/4 - x14/4 + x15/4 - x17/4 + x19/4 + x2/8 + x22/2 - x24/2 - x3/8 - x4/8 + x5/8 + x6/8 - x7/8 - x9/4 // 0, 2, 2
        }
      },
      {
        {
          x22 + x24 - 2*x26, // 1, 0, 0
          -x11/2 + x17/2 + x19/2 + x20 - x25 - x9/2, // 1, 0, 1
          x11/2 + x17/2 + x19/2 - x20 - x22 - x24 - x25 + 2*x26 + x9/2 // 1, 0, 2
        },
 
        {
          -x12/2 - x13/2 + x14/2 + x15/2 + x21 - x23, // 1, 1, 0
          x0/4 + x1/4 + x10/2 + x16/2 - x18/2 - x2/4 - x3/4 - x4/4 - x5/4 + x6/4 + x7/4 - x8/2, // 1, 1, 1
          -x0/4 - x1/4 - x10/2 + x12/2 + x13/2 - x14/2 - x15/2 + x16/2 - x18/2 + x2/4 - x21 + x23 + x3/4 - x4/4 - x5/4 + x6/4 + x7/4 + x8/2 // 1, 1, 2
        },
 
        {
          x12/2 + x13/2 + x14/2 + x15/2 - x21 - x22 - x23 - x24 + 2*x26, // 1, 2, 0
          -x0/4 - x1/4 + x10/2 + x11/2 - x16/2 - x17/2 - x18/2 - x19/2 - x2/4 - x20 + x25 - x3/4 + x4/4 + x5/4 + x6/4 + x7/4 + x8/2 + x9/2, // 1, 2, 1
          x0/4 + x1/4 - x10/2 - x11/2 - x12/2 - x13/2 - x14/2 - x15/2 - x16/2 - x17/2 - x18/2 - x19/2 + x2/4 + x20 + x21 + x22 + x23 + x24 + x25 - 2*x26 + x3/4 + x4/4 + x5/4 + x6/4 + x7/4 - x8/2 - x9/2 // 1, 2, 2
        }
      },
      {
        {Point(0,0,0), Point(0,0,0), Point(0,0,0)},
        {Point(0,0,0), Point(0,0,0), Point(0,0,0)},
        {Point(0,0,0), Point(0,0,0), Point(0,0,0)}
      }
    };
 
 
 
  // The constant components of the dx/deta vector, all of the eta^2
  // terms are zero. These were copied directly from the output of a
  // Python script.
  Point dx_deta[3][3][3] =
    {
      {
        {
          -x21/2 + x23/2, // 0, 0, 0
          -x10/4 - x16/4 + x18/4 + x8/4, // 0, 0, 1
          x10/4 - x16/4 + x18/4 + x21/2 - x23/2 - x8/4 // 0, 0, 2
        },
        {
          x21 + x23 - 2*x26, // 0, 1, 0
          -x10/2 + x16/2 + x18/2 + x20 - x25 - x8/2, // 0, 1, 1
          x10/2 + x16/2 + x18/2 - x20 - x21 - x23 - x25 + 2*x26 + x8/2 // 0, 1, 2
        },
        {
          Point(0,0,0), // 0, 2, 0
          Point(0,0,0), // 0, 2, 1
          Point(0,0,0)  // 0, 2, 2
        }
      },
 
      {
        {
          x12/4 - x13/4 + x14/4 - x15/4, // 1, 0, 0
          -x0/8 + x1/8 - x2/8 + x3/8 + x4/8 - x5/8 + x6/8 - x7/8, // 1, 0, 1
          x0/8 - x1/8 - x12/4 + x13/4 - x14/4 + x15/4 + x2/8 - x3/8 + x4/8 - x5/8 + x6/8 - x7/8 // 1, 0, 2
        },
        {
          -x12/2 + x13/2 + x14/2 - x15/2 - x22 + x24, // 1, 1, 0
          x0/4 - x1/4 - x11/2 - x17/2 + x19/2 - x2/4 + x3/4 - x4/4 + x5/4 + x6/4 - x7/4 + x9/2, // 1, 1, 1
          -x0/4 + x1/4 + x11/2 + x12/2 - x13/2 - x14/2 + x15/2 - x17/2 + x19/2 + x2/4 + x22 - x24 - x3/4 - x4/4 + x5/4 + x6/4 - x7/4 - x9/2 // 1, 1, 2
        },
        {
          Point(0,0,0), // 1, 2, 0
          Point(0,0,0), // 1, 2, 1
          Point(0,0,0)  // 1, 2, 2
        }
      },
 
      {
        {
          -x12/4 - x13/4 + x14/4 + x15/4 + x21/2 - x23/2, // 2, 0, 0
          x0/8 + x1/8 + x10/4 + x16/4 - x18/4 - x2/8 - x3/8 - x4/8 - x5/8 + x6/8 + x7/8 - x8/4, // 2, 0, 1
          -x0/8 - x1/8 - x10/4 + x12/4 + x13/4 - x14/4 - x15/4 + x16/4 - x18/4 + x2/8 - x21/2 + x23/2 + x3/8 - x4/8 - x5/8 + x6/8 + x7/8 + x8/4, // 2, 0, 2
        },
        {
          x12/2 + x13/2 + x14/2 + x15/2 - x21 - x22 - x23 - x24 + 2*x26, // 2, 1, 0
          -x0/4 - x1/4 + x10/2 + x11/2 - x16/2 - x17/2 - x18/2 - x19/2 - x2/4 - x20 + x25 - x3/4 + x4/4 + x5/4 + x6/4 + x7/4 + x8/2 + x9/2, // 2, 1, 1
          x0/4 + x1/4 - x10/2 - x11/2 - x12/2 - x13/2 - x14/2 - x15/2 - x16/2 - x17/2 - x18/2 - x19/2 + x2/4 + x20 + x21 + x22 + x23 + x24 + x25 - 2*x26 + x3/4 + x4/4 + x5/4 + x6/4 + x7/4 - x8/2 - x9/2 // 2, 1, 2
        },
        {
          Point(0,0,0), // 2, 2, 0
          Point(0,0,0), // 2, 2, 1
          Point(0,0,0)  // 2, 2, 2
        }
      }
    };
 
 
 
  // The constant components of the dx/dzeta vector, all of the zeta^2
  // terms are zero. These were copied directly from the output of a
  // Python script.
  Point dx_dzeta[3][3][3] =
    {
      {
        {
          -x20/2 + x25/2, // 0, 0, 0
          x20 + x25 - 2*x26, // 0, 0, 1
          Point(0,0,0) // 0, 0, 2
        },
        {
          -x10/4 - x16/4 + x18/4 + x8/4, // 0, 1, 0
          x10/2 - x16/2 + x18/2 + x21 - x23 - x8/2, // 0, 1, 1
          Point(0,0,0) // 0, 1, 2
        },
        {
          -x10/4 + x16/4 + x18/4 + x20/2 - x25/2 - x8/4, // 0, 2, 0
          x10/2 + x16/2 + x18/2 - x20 - x21 - x23 - x25 + 2*x26 + x8/2, // 0, 2, 1
          Point(0,0,0) // 0, 2, 2
        }
      },
      {
        {
          x11/4 + x17/4 - x19/4 - x9/4, // 1, 0, 0
          -x11/2 + x17/2 - x19/2 - x22 + x24 + x9/2, // 1, 0, 1
          Point(0,0,0) // 1, 0, 2
        },
        {
          -x0/8 + x1/8 - x2/8 + x3/8 + x4/8 - x5/8 + x6/8 - x7/8, // 1, 1, 0
          x0/4 - x1/4 - x12/2 + x13/2 - x14/2 + x15/2 + x2/4 - x3/4 + x4/4 - x5/4 + x6/4 - x7/4, // 1, 1, 1
          Point(0,0,0) // 1, 1, 2
        },
        {
          x0/8 - x1/8 - x11/4 - x17/4 + x19/4 - x2/8 + x3/8 - x4/8 + x5/8 + x6/8 - x7/8 + x9/4, // 1, 2, 0
          -x0/4 + x1/4 + x11/2 + x12/2 - x13/2 - x14/2 + x15/2 - x17/2 + x19/2 + x2/4 + x22 - x24 - x3/4 - x4/4 + x5/4 + x6/4 - x7/4 - x9/2, // 1, 2, 1
          Point(0,0,0) // 1, 2, 2
        }
      },
      {
        {
          -x11/4 + x17/4 + x19/4 + x20/2 - x25/2 - x9/4, // 2, 0, 0
          x11/2 + x17/2 + x19/2 - x20 - x22 - x24 - x25 + 2*x26 + x9/2, // 2, 0, 1
          Point(0,0,0) // 2, 0, 2
        },
        {
          x0/8 + x1/8 + x10/4 + x16/4 - x18/4 - x2/8 - x3/8 - x4/8 - x5/8 + x6/8 + x7/8 - x8/4, // 2, 1, 0
          -x0/4 - x1/4 - x10/2 + x12/2 + x13/2 - x14/2 - x15/2 + x16/2 - x18/2 + x2/4 - x21 + x23 + x3/4 - x4/4 - x5/4 + x6/4 + x7/4 + x8/2, // 2, 1, 1
          Point(0,0,0)  // 2, 1, 2
        },
        {
          -x0/8 - x1/8 + x10/4 + x11/4 - x16/4 - x17/4 - x18/4 - x19/4 - x2/8 - x20/2 + x25/2 - x3/8 + x4/8 + x5/8 + x6/8 + x7/8 + x8/4 + x9/4, // 2, 2, 0
          x0/4 + x1/4 - x10/2 - x11/2 - x12/2 - x13/2 - x14/2 - x15/2 - x16/2 - x17/2 - x18/2 - x19/2 + x2/4 + x20 + x21 + x22 + x23 + x24 + x25 - 2*x26 + x3/4 + x4/4 + x5/4 + x6/4 + x7/4 - x8/2 - x9/2, // 2, 2, 1
          Point(0,0,0) // 2, 2, 2
        }
      }
    };
 
  // 3x3 quadrature, exact for bi-quintics
  const int N = 3;
  const Real w[N] = {5./9, 8./9, 5./9};
 
  // Quadrature point locations raised to powers.  q[0][2] is
  // quadrature point 0, squared, q[1][1] is quadrature point 1 to the
  // first power, etc.
  const Real q[N][N] =
    {
      //^0   ^1                 ^2
      {  1., -std::sqrt(15)/5., 15./25},
      {  1., 0.,                0.},
      {  1., std::sqrt(15)/5.,  15./25}
    };
 
  Real vol = 0.;
  for (int i=0; i<N; ++i)
    for (int j=0; j<N; ++j)
      for (int k=0; k<N; ++k)
        {
          // Compute dx_dxi, dx_deta, dx_dzeta at the current quadrature point.
          Point dx_dxi_q, dx_deta_q, dx_dzeta_q;
          for (int ii=0; ii<N; ++ii)
            for (int jj=0; jj<N; ++jj)
              for (int kk=0; kk<N; ++kk)
                {
                  Real coeff = q[i][ii] * q[j][jj] * q[k][kk];
 
                  dx_dxi_q   += coeff * dx_dxi[ii][jj][kk];
                  dx_deta_q  += coeff * dx_deta[ii][jj][kk];
                  dx_dzeta_q += coeff * dx_dzeta[ii][jj][kk];
                }
 
          // Compute scalar triple product, multiply by weight, and accumulate volume.
          vol += w[i] * w[j] * w[k] * triple_product(dx_dxi_q, dx_deta_q, dx_dzeta_q);
        }
 
  return vol;
}
 
 
 
 
 
 
 
#ifdef LIBMESH_ENABLE_AMR
 
const Real Hex27::_embedding_matrix[Hex27::num_children][Hex27::num_nodes][Hex27::num_nodes] =
  {
    // embedding matrix for child 0
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000 }, // 7
      {   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 10
      {   0.375000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 11
      {   0.375000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.750000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.750000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 19
      {   0.140625, -0.0468750,  0.0156250, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500, -0.0937500,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 20
      {   0.140625, -0.0468750,    0.00000,    0.00000, -0.0468750,  0.0156250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 21
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,   0.281250,   0.281250,    0.00000, -0.0937500,    0.00000, -0.0937500,   0.562500 }, // 22
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,   0.281250,    0.00000, -0.0937500,    0.00000,   0.281250, -0.0937500,   0.562500 }, // 23
      {   0.140625,    0.00000,    0.00000, -0.0468750, -0.0468750,    0.00000,    0.00000,  0.0156250,    0.00000,    0.00000,    0.00000,   0.281250,   0.281250,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625, -0.0468750,  0.0156250, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500, -0.0937500,   0.281250,    0.00000,   0.562500 }, // 25
      { 0.052734375, -0.017578125, 0.005859375, -0.017578125, -0.017578125, 0.005859375, -0.001953125, 0.005859375, 0.10546875, -0.03515625, -0.03515625, 0.10546875, 0.10546875, -0.03515625, 0.01171875, -0.03515625, -0.03515625, 0.01171875, 0.01171875, -0.03515625, 0.2109375, 0.2109375, -0.0703125, -0.0703125, 0.2109375, -0.0703125, 0.421875 } // 26
    },
 
    // embedding matrix for child 1
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 7
      {  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 12
      {    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.750000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.750000 }, // 19
      { -0.0468750,   0.140625, -0.0468750,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250,   0.281250, -0.0937500, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 20
      { -0.0468750,   0.140625,    0.00000,    0.00000,  0.0156250, -0.0468750,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 21
      {    0.00000,   0.140625, -0.0468750,    0.00000,    0.00000, -0.0468750,  0.0156250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000 }, // 22
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,   0.281250,    0.00000,   0.281250,    0.00000, -0.0937500, -0.0937500,   0.562500 }, // 23
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,   0.281250,   0.281250,    0.00000, -0.0937500,    0.00000, -0.0937500,   0.562500 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,   0.140625, -0.0468750,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250,   0.281250, -0.0937500, -0.0937500,    0.00000,   0.562500 }, // 25
      { -0.017578125, 0.052734375, -0.017578125, 0.005859375, 0.005859375, -0.017578125, 0.005859375, -0.001953125, 0.10546875, 0.10546875, -0.03515625, -0.03515625, -0.03515625, 0.10546875, -0.03515625, 0.01171875, -0.03515625, -0.03515625, 0.01171875, 0.01171875, 0.2109375, 0.2109375, 0.2109375, -0.0703125, -0.0703125, -0.0703125, 0.421875 } // 26
 
    },
 
    // embedding matrix for child 2
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 10
      {  -0.125000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.750000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,   0.750000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 19
      { -0.0468750,  0.0156250, -0.0468750,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500, -0.0937500,   0.281250,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 20
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,   0.281250,    0.00000, -0.0937500,    0.00000,   0.281250, -0.0937500,   0.562500 }, // 21
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,   0.281250, -0.0937500,    0.00000,   0.281250,    0.00000, -0.0937500,   0.562500 }, // 22
      {    0.00000,    0.00000, -0.0468750,   0.140625,    0.00000,    0.00000,  0.0156250, -0.0468750,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000 }, // 23
      { -0.0468750,    0.00000,    0.00000,   0.140625,  0.0156250,    0.00000,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,  0.0156250, -0.0468750,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500, -0.0937500,   0.281250,   0.281250,    0.00000,   0.562500 }, // 25
      { -0.017578125, 0.005859375, -0.017578125, 0.052734375, 0.005859375, -0.001953125, 0.005859375, -0.017578125, -0.03515625, -0.03515625, 0.10546875, 0.10546875, -0.03515625, 0.01171875, -0.03515625, 0.10546875, 0.01171875, 0.01171875, -0.03515625, -0.03515625, 0.2109375, -0.0703125, -0.0703125, 0.2109375, 0.2109375, -0.0703125, 0.421875 } // 26
    },
 
    // embedding matrix for child 3
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.750000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,   0.750000 }, // 19
      {  0.0156250, -0.0468750,   0.140625, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,   0.281250, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 20
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,   0.281250,    0.00000,   0.281250,    0.00000, -0.0937500, -0.0937500,   0.562500 }, // 21
      {    0.00000, -0.0468750,   0.140625,    0.00000,    0.00000,  0.0156250, -0.0468750,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000 }, // 22
      {    0.00000,    0.00000,   0.140625, -0.0468750,    0.00000,    0.00000, -0.0468750,  0.0156250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000 }, // 23
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,   0.281250, -0.0937500,    0.00000,   0.281250,    0.00000, -0.0937500,   0.562500 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250, -0.0468750,   0.140625, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,   0.281250, -0.0937500,    0.00000,   0.562500 }, // 25
      { 0.005859375, -0.017578125, 0.052734375, -0.017578125, -0.001953125, 0.005859375, -0.017578125, 0.005859375, -0.03515625, 0.10546875, 0.10546875, -0.03515625, 0.01171875, -0.03515625, 0.10546875, -0.03515625, 0.01171875, -0.03515625, -0.03515625, 0.01171875, 0.2109375, -0.0703125, 0.2109375, 0.2109375, -0.0703125, -0.0703125, 0.421875 } // 26
    },
 
    // embedding matrix for child 4
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.750000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 11
      {  -0.125000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,   0.750000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 19
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625, -0.0468750,  0.0156250, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500, -0.0937500,   0.281250,    0.00000,   0.562500 }, // 20
      { -0.0468750,  0.0156250,    0.00000,    0.00000,   0.140625, -0.0468750,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 21
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000, -0.0937500,   0.281250,    0.00000, -0.0937500,    0.00000,   0.281250,   0.562500 }, // 22
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625, -0.0937500,    0.00000, -0.0937500,    0.00000,   0.281250,   0.281250,   0.562500 }, // 23
      { -0.0468750,    0.00000,    0.00000,  0.0156250,   0.140625,    0.00000,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,   0.140625, -0.0468750,  0.0156250, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500, -0.0937500,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000 }, // 25
      { -0.017578125, 0.005859375, -0.001953125, 0.005859375, 0.052734375, -0.017578125, 0.005859375, -0.017578125, -0.03515625, 0.01171875, 0.01171875, -0.03515625, 0.10546875, -0.03515625, 0.01171875, -0.03515625, 0.10546875, -0.03515625, -0.03515625, 0.10546875, -0.0703125, 0.2109375, -0.0703125, -0.0703125, 0.2109375, 0.2109375, 0.421875 } // 26
    },
 
    // embedding matrix for child 5
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.750000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 12
      {    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,   0.750000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 19
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,   0.140625, -0.0468750,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250,   0.281250, -0.0937500, -0.0937500,    0.00000,   0.562500 }, // 20
      {  0.0156250, -0.0468750,    0.00000,    0.00000, -0.0468750,   0.140625,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 21
      {    0.00000, -0.0468750,  0.0156250,    0.00000,    0.00000,   0.140625, -0.0468750,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000 }, // 22
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750, -0.0937500,    0.00000,   0.281250,    0.00000, -0.0937500,   0.281250,   0.562500 }, // 23
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750,    0.00000, -0.0937500,   0.281250,    0.00000, -0.0937500,    0.00000,   0.281250,   0.562500 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,   0.140625, -0.0468750,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.281250,   0.281250, -0.0937500, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000 }, // 25
      { 0.005859375, -0.017578125, 0.005859375, -0.001953125, -0.017578125, 0.052734375, -0.017578125, 0.005859375, -0.03515625, -0.03515625, 0.01171875, 0.01171875, -0.03515625, 0.10546875, -0.03515625, 0.01171875, 0.10546875, 0.10546875, -0.03515625, -0.03515625, -0.0703125, 0.2109375, 0.2109375, -0.0703125, -0.0703125, 0.2109375, 0.421875 } // 26
    },
 
    // embedding matrix for child 6
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,   0.750000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,   0.750000 }, // 13
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 19
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,  0.0156250, -0.0468750,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500, -0.0937500,   0.281250,   0.281250,    0.00000,   0.562500 }, // 20
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625, -0.0937500,    0.00000, -0.0937500,    0.00000,   0.281250,   0.281250,   0.562500 }, // 21
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000, -0.0937500, -0.0937500,    0.00000,   0.281250,    0.00000,   0.281250,   0.562500 }, // 22
      {    0.00000,    0.00000,  0.0156250, -0.0468750,    0.00000,    0.00000, -0.0468750,   0.140625,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000 }, // 23
      {  0.0156250,    0.00000,    0.00000, -0.0468750, -0.0468750,    0.00000,    0.00000,   0.140625,    0.00000,    0.00000,    0.00000, -0.0937500, -0.0937500,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,  0.0156250, -0.0468750,   0.140625,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500, -0.0937500,   0.281250,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000 }, // 25
      { 0.005859375, -0.001953125, 0.005859375, -0.017578125, -0.017578125, 0.005859375, -0.017578125, 0.052734375, 0.01171875, 0.01171875, -0.03515625, -0.03515625, -0.03515625, 0.01171875, -0.03515625, 0.10546875, -0.03515625, -0.03515625, 0.10546875, 0.10546875, -0.0703125, -0.0703125, -0.0703125, 0.2109375, 0.2109375, 0.2109375, 0.421875 } // 26
    },
 
    // embedding matrix for child 7
    {
      //         0           1           2           3           4           5           6           7           8           9          10          11          12          13          14          15          16          17          18          19          20          21          22          23          24          25          26
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000 }, // 0
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 1
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 7
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 9
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 10
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,   0.750000 }, // 11
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,   0.750000 }, // 12
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 13
      {    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 14
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 15
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 16
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 17
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 18
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 19
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250, -0.0468750,   0.140625, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,   0.281250, -0.0937500,    0.00000,   0.562500 }, // 20
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,  0.0156250,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.140625,    0.00000, -0.0468750, -0.0937500,    0.00000,   0.281250,    0.00000, -0.0937500,   0.281250,   0.562500 }, // 21
      {    0.00000,  0.0156250, -0.0468750,    0.00000,    0.00000, -0.0468750,   0.140625,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000,    0.00000 }, // 22
      {    0.00000,    0.00000, -0.0468750,  0.0156250,    0.00000,    0.00000,   0.140625, -0.0468750,    0.00000,    0.00000, -0.0937500,    0.00000,    0.00000,    0.00000,   0.281250, -0.0937500,    0.00000,    0.00000,   0.281250,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000,    0.00000,    0.00000 }, // 23
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250,    0.00000, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0468750,    0.00000,   0.140625,    0.00000, -0.0937500, -0.0937500,    0.00000,   0.281250,    0.00000,   0.281250,   0.562500 }, // 24
      {    0.00000,    0.00000,    0.00000,    0.00000,  0.0156250, -0.0468750,   0.140625, -0.0468750,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000, -0.0937500,   0.281250,   0.281250, -0.0937500,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.562500,    0.00000 }, // 25
      { -0.001953125, 0.005859375, -0.017578125, 0.005859375, 0.005859375, -0.017578125, 0.052734375, -0.017578125, 0.01171875, -0.03515625, -0.03515625, 0.01171875, 0.01171875, -0.03515625, 0.10546875, -0.03515625, -0.03515625, 0.10546875, 0.10546875, -0.03515625, -0.0703125, -0.0703125, 0.2109375, 0.2109375, -0.0703125, 0.2109375, 0.421875 } // 26
    }
  };
 
#endif
 
 
void
Hex27::permute(unsigned int perm_num)
{
  libmesh_assert_less (perm_num, 24);
  const unsigned int side = perm_num % 6;
  const unsigned int rotate = perm_num / 6;
 
  for (unsigned int i = 0; i != rotate; ++i)
    {
      swap4nodes(0,1,2,3);
      swap4nodes(4,5,6,7);
      swap4nodes(8,9,10,11);
      swap4nodes(12,13,14,15);
      swap4nodes(16,17,18,19);
      swap4nodes(21,22,23,24);
      swap4neighbors(1,2,3,4);
    }
 
  switch (side) {
  case 0:
    break;
  case 1:
    swap4nodes(3,7,4,0);
    swap4nodes(11,15,19,12);
    swap4nodes(10,18,16,8);
    swap4nodes(2,6,5,1);
    swap4nodes(9,14,17,13);
    swap4nodes(20,23,25,21);
    swap4neighbors(0,3,5,1);
    break;
  case 2:
    swap4nodes(0,4,5,1);
    swap4nodes(8,12,16,13);
    swap4nodes(3,7,6,2);
    swap4nodes(10,15,18,14);
    swap4nodes(11,19,17,9);
    swap4nodes(20,24,25,22);
    swap4neighbors(0,4,5,2);
    break;
  case 3:
    swap4nodes(0,4,7,3);
    swap4nodes(12,19,15,11);
    swap4nodes(8,16,18,10);
    swap4nodes(1,5,6,2);
    swap4nodes(13,17,14,9);
    swap4nodes(20,21,25,23);
    swap4neighbors(0,1,5,3);
    break;
  case 4:
    swap4nodes(1,5,4,0);
    swap4nodes(8,13,16,12);
    swap4nodes(9,17,19,11);
    swap4nodes(2,6,7,3);
    swap4nodes(10,14,18,15);
    swap4nodes(20,22,25,24);
    swap4neighbors(0,2,5,4);
    break;
  case 5:
    swap2nodes(0,7);
    swap2nodes(8,18);
    swap2nodes(1,6);
    swap2nodes(2,5);
    swap2nodes(10,16);
    swap2nodes(3,4);
    swap2nodes(11,19);
    swap2nodes(12,15);
    swap2nodes(9,17);
    swap2nodes(13,14);
    swap2nodes(20,25);
    swap2nodes(21,23);
    swap2neighbors(0,5);
    swap2neighbors(1,3);
    break;
  default:
    libmesh_error();
  }
}
 
 
void
Hex27::flip(BoundaryInfo * boundary_info)
{
  libmesh_assert(boundary_info);
 
  swap2nodes(0,1);
  swap2nodes(2,3);
  swap2nodes(4,5);
  swap2nodes(6,7);
  swap2nodes(9,11);
  swap2nodes(12,13);
  swap2nodes(14,15);
  swap2nodes(17,19);
  swap2nodes(22,24);
  swap2neighbors(2,4);
  swap2boundarysides(2,4,boundary_info);
  swap2boundaryedges(1,3,boundary_info);
  swap2boundaryedges(4,5,boundary_info);
  swap2boundaryedges(6,7,boundary_info);
  swap2boundaryedges(9,11,boundary_info);
}
 
 
unsigned int Hex27::center_node_on_side(const unsigned short side) const
{
  libmesh_assert_less (side, Hex27::num_sides);
  return side + 20;
}
 
 
ElemType
Hex27::side_type (const unsigned int libmesh_dbg_var(s)) const
{
  libmesh_assert_less (s, 6);
  return QUAD9;
}
 
 
} // namespace libMesh