face_quad9.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/edge_edge3.h"
#include "libmesh/face_quad9.h"
#include "libmesh/enum_io_package.h"
#include "libmesh/enum_order.h"
 
namespace libMesh
{
 
 
 
 
// ------------------------------------------------------------
// Quad9 class static member initializations
const int Quad9::num_nodes;
const int Quad9::nodes_per_side;
 
const unsigned int Quad9::side_nodes_map[Quad9::num_sides][Quad9::nodes_per_side] =
  {
    {0, 1, 4}, // Side 0
    {1, 2, 5}, // Side 1
    {2, 3, 6}, // Side 2
    {3, 0, 7}  // Side 3
  };
 
 
#ifdef LIBMESH_ENABLE_AMR
 
const Real Quad9::_embedding_matrix[Quad9::num_children][Quad9::num_nodes][Quad9::num_nodes] =
  {
    // embedding matrix for child 0
    {
      //         0           1           2           3           4           5           6           7           8
      {    1.00000,    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,    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,    1.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 3
      {   0.375000,  -0.125000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,   0.750000 }, // 6
      {   0.375000,    0.00000,    0.00000,  -0.125000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 7
      {   0.140625, -0.0468750,  0.0156250, -0.0468750,   0.281250, -0.0937500, -0.0937500,   0.281250,   0.562500 }  // 8
    },
 
    // embedding matrix for child 1
    {
      //         0           1           2           3           4           5           6           7           8
      {    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    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 }, // 1
      {    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,    1.00000 }, // 3
      {  -0.125000,   0.375000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 4
      {    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,   0.750000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,   0.375000,    0.00000,  -0.125000,    0.00000,   0.750000 }, // 7
      { -0.0468750,   0.140625, -0.0468750,  0.0156250,   0.281250,   0.281250, -0.0937500, -0.0937500,   0.562500 }  // 8
    },
 
    // embedding matrix for child 2
    {
      //         0           1           2           3           4           5           6           7           8
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,    1.00000,    0.00000 }, // 0
      {    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,    1.00000,    0.00000,    0.00000 }, // 2
      {    0.00000,    0.00000,    0.00000,    1.00000,    0.00000,    0.00000,    0.00000,    0.00000,    0.00000 }, // 3
      {    0.00000,    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,   0.750000 }, // 4
      {    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 5
      {    0.00000,    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 6
      {  -0.125000,    0.00000,    0.00000,   0.375000,    0.00000,    0.00000,    0.00000,   0.750000,    0.00000 }, // 7
      { -0.0468750,  0.0156250, -0.0468750,   0.140625, -0.0937500, -0.0937500,   0.281250,   0.281250,   0.562500 }  // 8
    },
 
    // embedding matrix for child 3
    {
      //         0           1           2           3           4           5           6           7           8
      {    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,    1.00000,    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 }, // 2
      {    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,    0.00000,   0.375000,    0.00000,  -0.125000,   0.750000 }, // 4
      {    0.00000,  -0.125000,   0.375000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000,    0.00000 }, // 5
      {    0.00000,    0.00000,   0.375000,  -0.125000,    0.00000,    0.00000,   0.750000,    0.00000,    0.00000 }, // 6
      {    0.00000,    0.00000,    0.00000,    0.00000,  -0.125000,    0.00000,   0.375000,    0.00000,   0.750000 }, // 7
      {  0.0156250, -0.0468750,   0.140625, -0.0468750, -0.0937500,   0.281250,   0.281250, -0.0937500,   0.562500 }  // 8
    }
  };
 
#endif
 
 
 
// ------------------------------------------------------------
// Quad9 class member functions
 
bool Quad9::is_vertex(const unsigned int i) const
{
  if (i < 4)
    return true;
  return false;
}
 
bool Quad9::is_edge(const unsigned int i) const
{
  if (i < 4)
    return false;
  if (i > 7)
    return false;
  return true;
}
 
bool Quad9::is_face(const unsigned int i) const
{
  if (i > 7)
    return true;
  return false;
}
 
bool Quad9::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>
Quad9::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>
Quad9::nodes_on_edge(const unsigned int e) const
{
  return nodes_on_side(e);
}
 
bool Quad9::has_affine_map() const
{
  // make sure corners form a parallelogram
  Point v = this->point(1) - this->point(0);
  if (!v.relative_fuzzy_equals(this->point(2) - this->point(3), affine_tol))
    return false;
  // make sure "horizontal" sides are straight
  v /= 2;
  if (!v.relative_fuzzy_equals(this->point(4) - this->point(0), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(6) - this->point(3), affine_tol))
    return false;
  // make sure "vertical" sides are straight
  // and the center node is centered
  v = (this->point(3) - this->point(0))/2;
  if (!v.relative_fuzzy_equals(this->point(7) - this->point(0), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(5) - this->point(1), affine_tol) ||
      !v.relative_fuzzy_equals(this->point(8) - this->point(4), affine_tol))
    return false;
  return true;
}
 
 
 
Order Quad9::default_order() const
{
  return SECOND;
}
 
 
 
dof_id_type Quad9::key (const unsigned int s) const
{
  libmesh_assert_less (s, this->n_sides());
 
  switch (s)
    {
    case 0:
 
      return
        this->compute_key (this->node_id(4));
 
    case 1:
 
      return
        this->compute_key (this->node_id(5));
 
    case 2:
 
      return
        this->compute_key (this->node_id(6));
 
    case 3:
 
      return
        this->compute_key (this->node_id(7));
 
    default:
      libmesh_error_msg("Invalid side s = " << s);
    }
}
 
 
 
dof_id_type Quad9::key () const
{
  return this->compute_key(this->node_id(8));
}
 
 
 
unsigned int Quad9::local_side_node(unsigned int side,
                                    unsigned int side_node) const
{
  libmesh_assert_less (side, this->n_sides());
  libmesh_assert_less (side_node, Quad9::nodes_per_side);
 
  return Quad9::side_nodes_map[side][side_node];
}
 
 
 
std::unique_ptr<Elem> Quad9::build_side_ptr (const unsigned int i)
{
  return this->simple_build_side_ptr<Edge3, Quad9>(i);
}
 
 
 
void Quad9::build_side_ptr (std::unique_ptr<Elem> & side,
                            const unsigned int i)
{
  this->simple_build_side_ptr<Quad9>(side, i, EDGE3);
}
 
 
 
void Quad9::connectivity(const unsigned int sf,
                         const IOPackage iop,
                         std::vector<dof_id_type> & conn) const
{
  libmesh_assert_less (sf, this->n_sub_elem());
  libmesh_assert_not_equal_to (iop, INVALID_IO_PACKAGE);
 
  conn.resize(4);
 
  switch (iop)
    {
    case TECPLOT:
      {
        switch(sf)
          {
          case 0:
            // linear sub-quad 0
            conn[0] = this->node_id(0)+1;
            conn[1] = this->node_id(4)+1;
            conn[2] = this->node_id(8)+1;
            conn[3] = this->node_id(7)+1;
            return;
 
          case 1:
            // linear sub-quad 1
            conn[0] = this->node_id(4)+1;
            conn[1] = this->node_id(1)+1;
            conn[2] = this->node_id(5)+1;
            conn[3] = this->node_id(8)+1;
            return;
 
          case 2:
            // linear sub-quad 2
            conn[0] = this->node_id(7)+1;
            conn[1] = this->node_id(8)+1;
            conn[2] = this->node_id(6)+1;
            conn[3] = this->node_id(3)+1;
            return;
 
          case 3:
            // linear sub-quad 3
            conn[0] = this->node_id(8)+1;
            conn[1] = this->node_id(5)+1;
            conn[2] = this->node_id(2)+1;
            conn[3] = this->node_id(6)+1;
            return;
 
          default:
            libmesh_error_msg("Invalid sf = " << sf);
          }
      }
 
    case VTK:
      {
        conn.resize(Quad9::num_nodes);
        for (auto i : index_range(conn))
          conn[i] = this->node_id(i);
        return;
      }
 
    default:
      libmesh_error_msg("Unsupported IO package " << iop);
    }
}
 
 
 
BoundingBox Quad9::loose_bounding_box () const
{
  // This might have curved edges, or might be a curved surface in
  // 3-space, in which case the full bounding box can be larger than
  // the bounding box of just the nodes.
  //
  //
  // FIXME - I haven't yet proven the formula below to be correct for
  // biquadratics - RHS
  Point pmin, pmax;
 
  for (unsigned d=0; d<LIBMESH_DIM; ++d)
    {
      const Real center = this->point(8)(d);
      Real hd = std::abs(center - this->point(0)(d));
      for (unsigned int p=0; p != 8; ++p)
        hd = std::max(hd, std::abs(center - this->point(p)(d)));
 
      pmin(d) = center - hd;
      pmax(d) = center + hd;
    }
 
  return BoundingBox(pmin, pmax);
}
 
 
 
Real Quad9::volume () const
{
  // This specialization is good for Lagrange mappings only
  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);
 
  // Construct constant data vectors.
  // \vec{x}_{\xi}  = \vec{a1}*xi*eta^2 + \vec{b1}*eta**2 + \vec{c1}*xi*eta + \vec{d1}*xi + \vec{e1}*eta + \vec{f1}
  // \vec{x}_{\eta} = \vec{a2}*xi^2*eta + \vec{b2}*xi**2  + \vec{c2}*xi*eta + \vec{d2}*xi + \vec{e2}*eta + \vec{f2}
  // This is copy-pasted directly from the output of a Python script.
  Point
    a1 = x0/2 + x1/2 + x2/2 + x3/2 - x4 - x5 - x6 - x7 + 2*x8,
    b1 = -x0/4 + x1/4 + x2/4 - x3/4 - x5/2 + x7/2,
    c1 = -x0/2 - x1/2 + x2/2 + x3/2 + x4 - x6,
    d1 = x5 + x7 - 2*x8,
    e1 = x0/4 - x1/4 + x2/4 - x3/4,
    f1 = x5/2 - x7/2,
    a2 = a1,
    b2 = -x0/4 - x1/4 + x2/4 + x3/4 + x4/2 - x6/2,
    c2 = -x0/2 + x1/2 + x2/2 - x3/2 - x5 + x7,
    d2 = x0/4 - x1/4 + x2/4 - x3/4,
    e2 = x4 + x6 - 2*x8,
    f2 = -x4/2 + x6/2;
 
  // 3x3 quadrature, exact for bi-quintics
  const unsigned int N = 3;
  const Real q[N] = {-std::sqrt(15)/5., 0., std::sqrt(15)/5.};
  const Real w[N] = {5./9, 8./9, 5./9};
 
  Real vol=0.;
  for (unsigned int i=0; i<N; ++i)
    for (unsigned int j=0; j<N; ++j)
      vol += w[i] * w[j] *
        cross_norm(q[i]*q[j]*q[j]*a1 + q[j]*q[j]*b1 + q[j]*q[i]*c1 + q[i]*d1 + q[j]*e1 + f1,
                   q[i]*q[i]*q[j]*a2 + q[i]*q[i]*b2 + q[j]*q[i]*c2 + q[i]*d2 + q[j]*e2 + f2);
 
  return vol;
}
 
 
 
 
unsigned int Quad9::n_second_order_adjacent_vertices (const unsigned int n) const
{
  switch (n)
    {
    case 4:
    case 5:
    case 6:
    case 7:
      return 2;
 
    case 8:
      return 4;
 
    default:
      libmesh_error_msg("Invalid n = " << n);
    }
}
 
 
 
unsigned short int Quad9::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)
    {
    case 8:
      {
        libmesh_assert_less (v, 4);
        return static_cast<unsigned short int>(v);
      }
 
    default:
      {
        libmesh_assert_less (v, 2);
        // use the matrix that we inherited from \p Quad
        return _second_order_adjacent_vertices[n-this->n_vertices()][v];
      }
    }
}
 
 
 
std::pair<unsigned short int, unsigned short int>
Quad9::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 face_quad.C, since they are identical
   * for Quad8 and Quad9 (for the first 4 higher-order nodes)
   */
  return std::pair<unsigned short int, unsigned short int>
    (_second_order_vertex_child_number[n],
     _second_order_vertex_child_index[n]);
}
 
 
void Quad9::permute(unsigned int perm_num)
{
  libmesh_assert_less (perm_num, 4);
 
  for (unsigned int i = 0; i != perm_num; ++i)
    {
      swap4nodes(0,1,2,3);
      swap4nodes(4,5,6,7);
      swap4neighbors(0,1,2,3);
    }
}
 
 
void Quad9::flip(BoundaryInfo * boundary_info)
{
  libmesh_assert(boundary_info);
 
  swap2nodes(0,1);
  swap2nodes(2,3);
  swap2nodes(5,7);
  swap2neighbors(1,3);
  swap2boundarysides(1,3,boundary_info);
  swap2boundaryedges(1,3,boundary_info);
}
 
 
unsigned int Quad9::center_node_on_side(const unsigned short side) const
{
  libmesh_assert_less (side, Quad9::num_sides);
  return side + 4;
}
 
 
ElemType Quad9::side_type (const unsigned int libmesh_dbg_var(s)) const
{
  libmesh_assert_less (s, 4);
  return EDGE3;
}
 
 
} // namespace libMesh