fe_xyz_shape_2D.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
 
 
// C++ includes
 
// Local includes
#include "libmesh/fe.h"
 
#include "libmesh/elem.h"
#include "libmesh/int_range.h"
 
 
namespace libMesh
{
 
 
LIBMESH_DEFAULT_VECTORIZED_FE(2,XYZ)
 
 
template <>
Real FE<2,XYZ>::shape(const Elem * elem,
                      const Order libmesh_dbg_var(order),
                      const unsigned int i,
                      const Point & point_in,
                      const bool libmesh_dbg_var(add_p_level))
{
#if LIBMESH_DIM > 1
 
  libmesh_assert(elem);
 
  Point avg = elem->vertex_average();
  Point max_distance = Point(0.,0.,0.);
  for (const Point & p : elem->node_ref_range())
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(avg(d) - p(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }
 
  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = avg(0);
  const Real yc = avg(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
 
#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + add_p_level * elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);
 
 
  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
      // constant
    case 0:
      return 1.;
 
      // linear
    case 1:
      return dx;
 
    case 2:
      return dy;
 
      // quadratics
    case 3:
      return dx*dx;
 
    case 4:
      return dx*dy;
 
    case 5:
      return dy*dy;
 
      // cubics
    case 6:
      return dx*dx*dx;
 
    case 7:
      return dx*dx*dy;
 
    case 8:
      return dx*dy*dy;
 
    case 9:
      return dy*dy*dy;
 
      // quartics
    case 10:
      return dx*dx*dx*dx;
 
    case 11:
      return dx*dx*dx*dy;
 
    case 12:
      return dx*dx*dy*dy;
 
    case 13:
      return dx*dy*dy*dy;
 
    case 14:
      return dy*dy*dy*dy;
 
    default:
      unsigned int o = 0;
      for (; i >= (o+1)*(o+2)/2; o++) { }
      unsigned int i2 = i - (o*(o+1)/2);
      Real val = 1.;
      for (unsigned int index=i2; index != o; index++)
        val *= dx;
      for (unsigned int index=0; index != i2; index++)
        val *= dy;
      return val;
    }
 
#else // LIBMESH_DIM <= 1
  libmesh_assert(true || order || add_p_level);
  libmesh_ignore(elem, i, point_in);
  libmesh_not_implemented();
#endif
}
 
 
 
 
template <>
Real FE<2,XYZ>::shape(const ElemType,
                      const Order,
                      const unsigned int,
                      const Point &)
{
  libmesh_error_msg("XYZ polynomials require the element.");
  return 0.;
}
 
 
 
template <>
Real FE<2,XYZ>::shape(const FEType fet,
                      const Elem * elem,
                      const unsigned int i,
                      const Point & p,
                      const bool add_p_level)
{
  return FE<2,XYZ>::shape(elem, fet.order, i, p, add_p_level);
}
 
 
 
 
 
template <>
Real FE<2,XYZ>::shape_deriv(const Elem * elem,
                            const Order libmesh_dbg_var(order),
                            const unsigned int i,
                            const unsigned int j,
                            const Point & point_in,
                            const bool libmesh_dbg_var(add_p_level))
{
#if LIBMESH_DIM > 1
 
 
  libmesh_assert_less (j, 2);
  libmesh_assert(elem);
 
  Point avg = elem->vertex_average();
  Point max_distance = Point(0.,0.,0.);
  for (const Point & p : elem->node_ref_range())
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(avg(d) - p(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }
 
  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = avg(0);
  const Real yc = avg(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
 
#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + add_p_level * elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);
 
  // monomials. since they are hierarchic we only need one case block.
 
  switch (j)
    {
      // d()/dx
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;
 
            // linears
          case 1:
            return 1./distx;
 
          case 2:
            return 0.;
 
            // quadratics
          case 3:
            return 2.*dx/distx;
 
          case 4:
            return dy/distx;
 
          case 5:
            return 0.;
 
            // cubics
          case 6:
            return 3.*dx*dx/distx;
 
          case 7:
            return 2.*dx*dy/distx;
 
          case 8:
            return dy*dy/distx;
 
          case 9:
            return 0.;
 
            // quartics
          case 10:
            return 4.*dx*dx*dx/distx;
 
          case 11:
            return 3.*dx*dx*dy/distx;
 
          case 12:
            return 2.*dx*dy*dy/distx;
 
          case 13:
            return dy*dy*dy/distx;
 
          case 14:
            return 0.;
 
          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = o - i2;
            for (unsigned int index=i2+1; index < o; index++)
              val *= dx;
            for (unsigned int index=0; index != i2; index++)
              val *= dy;
            return val/distx;
          }
      }
 
 
      // d()/dy
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;
 
            // linears
          case 1:
            return 0.;
 
          case 2:
            return 1./disty;
 
            // quadratics
          case 3:
            return 0.;
 
          case 4:
            return dx/disty;
 
          case 5:
            return 2.*dy/disty;
 
            // cubics
          case 6:
            return 0.;
 
          case 7:
            return dx*dx/disty;
 
          case 8:
            return 2.*dx*dy/disty;
 
          case 9:
            return 3.*dy*dy/disty;
 
            // quartics
          case 10:
            return 0.;
 
          case 11:
            return dx*dx*dx/disty;
 
          case 12:
            return 2.*dx*dx*dy/disty;
 
          case 13:
            return 3.*dx*dy*dy/disty;
 
          case 14:
            return 4.*dy*dy*dy/disty;
 
          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = i2;
            for (unsigned int index=i2; index != o; index++)
              val *= dx;
            for (unsigned int index=1; index <= i2; index++)
              val *= dy;
            return val/disty;
          }
      }
 
 
    default:
      libmesh_error_msg("Invalid j = " << j);
    }
 
#else // LIBMESH_DIM <= 1
  libmesh_assert(true || order || add_p_level);
  libmesh_ignore(elem, i, j, point_in);
  libmesh_not_implemented();
#endif
}
 
 
template <>
Real FE<2,XYZ>::shape_deriv(const ElemType,
                            const Order,
                            const unsigned int,
                            const unsigned int,
                            const Point &)
{
  libmesh_error_msg("XYZ polynomials require the element.");
  return 0.;
}
 
 
template <>
Real FE<2,XYZ>::shape_deriv(const FEType fet,
                            const Elem * elem,
                            const unsigned int i,
                            const unsigned int j,
                            const Point & p,
                            const bool add_p_level)
{
  return FE<2,XYZ>::shape_deriv(elem, fet.order, i, j, p, add_p_level);
}
 
 
 
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
 
template <>
Real FE<2,XYZ>::shape_second_deriv(const Elem * elem,
                                   const Order libmesh_dbg_var(order),
                                   const unsigned int i,
                                   const unsigned int j,
                                   const Point & point_in,
                                   const bool libmesh_dbg_var(add_p_level))
{
#if LIBMESH_DIM > 1
 
  libmesh_assert_less_equal (j, 2);
  libmesh_assert(elem);
 
  Point avg = elem->vertex_average();
  Point max_distance = Point(0.,0.,0.);
  for (auto p : make_range(elem->n_nodes()))
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(avg(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }
 
  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = avg(0);
  const Real yc = avg(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
  const Real dist2x = pow(distx,2.);
  const Real dist2y = pow(disty,2.);
  const Real distxy = distx * disty;
 
#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + add_p_level * elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);
 
  // monomials. since they are hierarchic we only need one case block.
 
  switch (j)
    {
      // d^2()/dx^2
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            // linears
          case 1:
          case 2:
            return 0.;
 
            // quadratics
          case 3:
            return 2./dist2x;
 
          case 4:
          case 5:
            return 0.;
 
            // cubics
          case 6:
            return 6.*dx/dist2x;
 
          case 7:
            return 2.*dy/dist2x;
 
          case 8:
          case 9:
            return 0.;
 
            // quartics
          case 10:
            return 12.*dx*dx/dist2x;
 
          case 11:
            return 6.*dx*dy/dist2x;
 
          case 12:
            return 2.*dy*dy/dist2x;
 
          case 13:
          case 14:
            return 0.;
 
          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = (o - i2) * (o - i2 - 1);
            for (unsigned int index=i2+2; index < o; index++)
              val *= dx;
            for (unsigned int index=0; index != i2; index++)
              val *= dy;
            return val/dist2x;
          }
      }
 
      // d^2()/dxdy
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:
 
            // linears
          case 1:
          case 2:
            return 0.;
 
            // quadratics
          case 3:
            return 0.;
 
          case 4:
            return 1./distxy;
 
          case 5:
            return 0.;
 
            // cubics
          case 6:
            return 0.;
          case 7:
            return 2.*dx/distxy;
 
          case 8:
            return 2.*dy/distxy;
 
          case 9:
            return 0.;
 
            // quartics
          case 10:
            return 0.;
 
          case 11:
            return 3.*dx*dx/distxy;
 
          case 12:
            return 4.*dx*dy/distxy;
 
          case 13:
            return 3.*dy*dy/distxy;
 
          case 14:
            return 0.;
 
          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = (o - i2) * i2;
            for (unsigned int index=i2+1; index < o; index++)
              val *= dx;
            for (unsigned int index=1; index < i2; index++)
              val *= dy;
            return val/distxy;
          }
      }
 
      // d^2()/dy^2
    case 2:
      {
        switch (i)
          {
            // constants
          case 0:
 
            // linears
          case 1:
          case 2:
            return 0.;
 
            // quadratics
          case 3:
          case 4:
            return 0.;
 
          case 5:
            return 2./dist2y;
 
            // cubics
          case 6:
            return 0.;
 
          case 7:
            return 0.;
 
          case 8:
            return 2.*dx/dist2y;
 
          case 9:
            return 6.*dy/dist2y;
 
            // quartics
          case 10:
          case 11:
            return 0.;
 
          case 12:
            return 2.*dx*dx/dist2y;
 
          case 13:
            return 6.*dx*dy/dist2y;
 
          case 14:
            return 12.*dy*dy/dist2y;
 
          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = i2 * (i2 - 1);
            for (unsigned int index=i2; index != o; index++)
              val *= dx;
            for (unsigned int index=2; index < i2; index++)
              val *= dy;
            return val/dist2y;
          }
      }
 
    default:
      libmesh_error_msg("Invalid shape function derivative j = " << j);
    }
 
#else // LIBMESH_DIM <= 1
  libmesh_assert(true || order || add_p_level);
  libmesh_ignore(elem, i, j, point_in);
  libmesh_not_implemented();
#endif
}
 
 
template <>
Real FE<2,XYZ>::shape_second_deriv(const ElemType,
                                   const Order,
                                   const unsigned int,
                                   const unsigned int,
                                   const Point &)
{
  libmesh_error_msg("XYZ polynomials require the element.");
  return 0.;
}
 
 
 
template <>
Real FE<2,XYZ>::shape_second_deriv(const FEType fet,
                                   const Elem * elem,
                                   const unsigned int i,
                                   const unsigned int j,
                                   const Point & p,
                                   const bool add_p_level)
{
  return FE<2,XYZ>::shape_second_deriv(elem, fet.order, i, j, p, add_p_level);
}
 
 
#endif
 
} // namespace libMesh