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inf_fe_map.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2012 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/libmesh_config.h"
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
#include "libmesh/inf_fe.h"
#include "libmesh/fe.h"
#include "libmesh/elem.h"
#include "libmesh/inf_fe_macro.h"
#include "libmesh/libmesh_logging.h"

namespace libMesh
{



// ------------------------------------------------------------
// InfFE static class members concerned with coordinate
// mapping


template <unsigned int Dim, FEFamily T_radial, InfMapType T_map>
Point InfFE<Dim,T_radial,T_map>::map (const Elem* inf_elem,
                                      const Point& reference_point)
{
  libmesh_assert(inf_elem);
  libmesh_assert_not_equal_to (Dim, 0);

  AutoPtr<Elem>      base_elem (Base::build_elem (inf_elem));

  const Order        radial_mapping_order (Radial::mapping_order());
  const Real         v                    (reference_point(Dim-1));

  // map in the base face
  Point base_point;
  switch (Dim)
    {
    case 1:
      base_point = inf_elem->point(0);
      break;
    case 2:
      base_point = FE<1,LAGRANGE>::map (base_elem.get(), reference_point);
      break;
    case 3:
      base_point = FE<2,LAGRANGE>::map (base_elem.get(), reference_point);
      break;
#ifdef DEBUG
    default:
        libmesh_error();
#endif
    }


  // map in the outer node face not necessary. Simply
  // compute the outer_point = base_point + (base_point-origin)
  const Point outer_point (base_point * 2. - inf_elem->origin());

  Point p;

  // there are only two mapping shapes in radial direction
  p.add_scaled (base_point,  InfFE<Dim,INFINITE_MAP,T_map>::eval (v, radial_mapping_order, 0));
  p.add_scaled (outer_point, InfFE<Dim,INFINITE_MAP,T_map>::eval (v, radial_mapping_order, 1));

  return p;
}





template <unsigned int Dim, FEFamily T_radial, InfMapType T_map>
Point InfFE<Dim,T_radial,T_map>::inverse_map (const Elem* inf_elem,
                                              const Point& physical_point,
                                              const Real tolerance,
                                              const bool secure,
                                              const bool interpolated)
{
  libmesh_assert(inf_elem);
  libmesh_assert_greater_equal (tolerance, 0.);


  // Start logging the map inversion.
  START_LOG("inverse_map()", "InfFE");


  // 1.)
  // build a base element to do the map inversion in the base face
  AutoPtr<Elem> base_elem (Base::build_elem (inf_elem));

  const Order    base_mapping_order     (base_elem->default_order());
  const ElemType base_mapping_elem_type (base_elem->type());
  const unsigned int n_base_mapping_sf = Base::n_base_mapping_sf (base_mapping_elem_type,
                                                                  base_mapping_order);

  const ElemType inf_elem_type = inf_elem->type();
  if (inf_elem_type != INFHEX8 &&
      inf_elem_type != INFPRISM6)
    {
      libMesh::err << "ERROR: InfFE::inverse_map is currently implemented only for\n"
                    << " infinite elments of type InfHex8 and InfPrism6." << std::endl;
      libmesh_error();
    }


  // 2.)
  // just like in FE<Dim-1,LAGRANGE>::inverse_map(): compute
  // the local coordinates, but only in the base element.
  // The radial part can then be computed directly later on.


  // How much did the point on the reference
  // element change by in this Newton step?
  Real inverse_map_error = 0.;


  // The point on the reference element.  This is
  // the "initial guess" for Newton's method.  The
  // centroid seems like a good idea, but computing
  // it is a little more intensive than, say taking
  // the zero point.
  //
  // Convergence should be insensitive of this choice
  // for "good" elements.
  Point p; // the zero point.  No computation required



  // Now find the intersection of a plane represented by the base
  // element nodes and the line given by the origin of the infinite
  // element and the physical point.
  Point intersection;

  // the origin of the infinite lement
  const Point o = inf_elem->origin();

  switch (Dim)
    {

      // unnecessary for 1D
    case 1:
      {
        break;
      }

    case 2:
      {
        libMesh::err << "ERROR: InfFE::inverse_map is not yet implemented"
                      << " in 2d" << std::endl;
        libmesh_error();
        break;
      }


    case 3:
      {
        // references to the nodal points of the base element
        const Point& p0 = base_elem->point(0);
        const Point& p1 = base_elem->point(1);
        const Point& p2 = base_elem->point(2);

        // a reference to the physical point
        const Point& fp = physical_point;

        // The intersection of the plane and the line is given by
        // can be computed solving a linear 3x3 system
        // a*({p1}-{p0})+b*({p2}-{p0})-c*({fp}-{o})={fp}-{p0}.
        const Real c_factor = -(p1(0)*fp(1)*p0(2)-p1(0)*fp(2)*p0(1)
                                +fp(0)*p1(2)*p0(1)-p0(0)*fp(1)*p1(2)
                                +p0(0)*fp(2)*p1(1)+p2(0)*fp(2)*p0(1)
                                -p2(0)*fp(1)*p0(2)-fp(0)*p2(2)*p0(1)
                                +fp(0)*p0(2)*p2(1)+p0(0)*fp(1)*p2(2)
                                -p0(0)*fp(2)*p2(1)-fp(0)*p0(2)*p1(1)
                                +p0(2)*p2(0)*p1(1)-p0(1)*p2(0)*p1(2)
                                -fp(0)*p1(2)*p2(1)+p2(1)*p0(0)*p1(2)
                                -p2(0)*fp(2)*p1(1)-p1(0)*fp(1)*p2(2)
                                +p2(2)*p1(0)*p0(1)+p1(0)*fp(2)*p2(1)
                                -p0(2)*p1(0)*p2(1)-p2(2)*p0(0)*p1(1)
                                +fp(0)*p2(2)*p1(1)+p2(0)*fp(1)*p1(2))/
                                (fp(0)*p1(2)*p0(1)-p1(0)*fp(2)*p0(1)
                                 +p1(0)*fp(1)*p0(2)-p1(0)*o(1)*p0(2)
                                 +o(0)*p2(2)*p0(1)-p0(0)*fp(2)*p2(1)
                                 +p1(0)*o(1)*p2(2)+fp(0)*p0(2)*p2(1)
                                 -fp(0)*p1(2)*p2(1)-p0(0)*o(1)*p2(2)
                                 +p0(0)*fp(1)*p2(2)-o(0)*p0(2)*p2(1)
                                 +o(0)*p1(2)*p2(1)-p2(0)*fp(2)*p1(1)
                                 +fp(0)*p2(2)*p1(1)-p2(0)*fp(1)*p0(2)
                                 -o(2)*p0(0)*p1(1)-fp(0)*p0(2)*p1(1)
                                 +p0(0)*o(1)*p1(2)+p0(0)*fp(2)*p1(1)
                                 -p0(0)*fp(1)*p1(2)-o(0)*p1(2)*p0(1)
                                 -p2(0)*o(1)*p1(2)-o(2)*p2(0)*p0(1)
                                 -o(2)*p1(0)*p2(1)+o(2)*p0(0)*p2(1)
                                 -fp(0)*p2(2)*p0(1)+o(2)*p2(0)*p1(1)
                                 +p2(0)*o(1)*p0(2)+p2(0)*fp(1)*p1(2)
                                 +p2(0)*fp(2)*p0(1)-p1(0)*fp(1)*p2(2)
                                 +p1(0)*fp(2)*p2(1)-o(0)*p2(2)*p1(1)
                                 +o(2)*p1(0)*p0(1)+o(0)*p0(2)*p1(1));


        // Compute the intersection with
        // {intersection} = {fp} + c*({fp}-{o}).
        intersection.add_scaled(fp,1.+c_factor);
        intersection.add_scaled(o,-c_factor);

        break;
      }
    }

  unsigned int cnt = 0;


  do
    {

      // Increment in current iterate \p p, will be computed.
      // Automatically initialized to all zero.  Note that
      // in 3D, actually only the first two entries are
      // filled by the inverse map, and in 2D only the first.
      Point dp;

      // The form of the map and how we invert it depends
      // on the dimension that we are in.
      switch (Dim)
        {

          //------------------------------------------------------------------
          // 1D infinite element - no map inversion necessary
        case 1:
          {
            break;
          }


          //------------------------------------------------------------------
          // 2D infinite element - 1D map inversion
          //
          // In this iteration scheme only search for the local coordinate
          // in xi direction.  Once xi is determined, the radial coordinate eta is
          // uniquely determined, and there is no need to iterate in that direction.
        case 2:
          {

            // Where our current iterate \p p maps to.
            const Point physical_guess = FE<1,LAGRANGE>::map (base_elem.get(), p);


            // How far our current iterate is from the actual point.
            const Point delta = physical_point - physical_guess;


            const Point dxi = FE<1,LAGRANGE>::map_xi (base_elem.get(), p);


            // For details on Newton's method see fe_map.C
            const Real G = dxi*dxi;

            if (secure)
              libmesh_assert_greater (G, 0.);

            const Real Ginv = 1./G;

            const Real  dxidelta = dxi*delta;

            // compute only the first coordinate
            dp(0) = Ginv*dxidelta;

            break;
          }



          //------------------------------------------------------------------
          // 3D infinite element - 2D map inversion
          //
          // In this iteration scheme only search for the local coordinates
          // in xi and eta direction.  Once xi, eta are determined, the radial
          // coordinate zeta may directly computed.
        case 3:
          {


            // Where our current iterate \p p maps to.
            const Point physical_guess = FE<2,LAGRANGE>::map (base_elem.get(), p);

            // How far our current iterate is from the actual point.
            // const Point delta = physical_point - physical_guess;
            const Point delta = intersection - physical_guess;

            const Point dxi  = FE<2,LAGRANGE>::map_xi  (base_elem.get(), p);
            const Point deta = FE<2,LAGRANGE>::map_eta (base_elem.get(), p);


            // For details on Newton's method see fe_map.C
            const Real
              G11 = dxi*dxi,  G12 = dxi*deta,
              G21 = dxi*deta, G22 = deta*deta;


            const Real det = (G11*G22 - G12*G21);

            if (secure)
              {
                libmesh_assert_greater (det, 0.);
                libmesh_assert_greater (std::abs(det), 1.e-10);
              }

            const Real inv_det = 1./det;

            const Real
              Ginv11 =  G22*inv_det,
              Ginv12 = -G12*inv_det,

              Ginv21 = -G21*inv_det,
              Ginv22 =  G11*inv_det;


            const Real  dxidelta  = dxi*delta;
            const Real  detadelta = deta*delta;

            // compute only the first two coordinates.
            dp(0) = (Ginv11*dxidelta + Ginv12*detadelta);
            dp(1) = (Ginv21*dxidelta + Ginv22*detadelta);

            break;
          }



          // Some other dimension?
        default:
          libmesh_error();
        } // end switch(Dim), dp now computed



      // determine the error in computing the local coordinates
      // in the base: ||P_n+1 - P_n||
      inverse_map_error = dp.size();


      // P_n+1 = P_n + dp
      p.add (dp);


      // Increment the iteration count.
      cnt++;


      // Watch for divergence of Newton's
      // method.
      if (cnt > 10)
        {
          if (secure || !secure)
            {
              libmesh_here();
              {
                libMesh::err << "WARNING: Newton scheme has not converged in "
                              << cnt << " iterations:\n"
                              << "   physical_point="
                              << physical_point
                              << "   dp="
                              << dp
                              << "   p="
                              << p
                              << "   error=" << inverse_map_error
                              << std::endl;
              }
            }

          if (cnt > 20)
            {
              libMesh::err << "ERROR: Newton scheme FAILED to converge in "
                            << cnt << " iterations!" << std::endl;
              libmesh_error();
            }

          // else
          //  {
          //    break;
          //  }
        }
    }
  while (inverse_map_error > tolerance);



  // 4.
  //
  // Now that we have the local coordinates in the base,
  // compute the interpolated radial distance a(s,t) \p a_interpolated
  if (interpolated)
    switch (Dim)
      {
      case 1:
        {
          Real a_interpolated = Point( inf_elem->point(0)
                                       - inf_elem->point(n_base_mapping_sf) ).size();

          p(0) = 1. - 2*a_interpolated/physical_point(0);

#ifdef DEBUG
          // the radial distance should always be >= -1.

          if (p(0)+1 < tolerance)
            {
              libmesh_here();
              libMesh::err << "WARNING: radial distance p(0) is "
                            << p(0)
                            << std::endl;
            }
#endif

          break;
        }


      case 2:
        {
          Real a_interpolated = 0.;

          // the distance between the origin and the physical point
          const Real fp_o_dist = Point(o-physical_point).size();

          for (unsigned int i=0; i<n_base_mapping_sf; i++)
            {
              // the radial distance of the i-th base mapping point
              const Real dist_i = Point( inf_elem->point(i)
                                         - inf_elem->point(i+n_base_mapping_sf) ).size();
              // weight with the corresponding shape function
              a_interpolated += dist_i * FE<1,LAGRANGE>::shape(base_mapping_elem_type,
                                                               base_mapping_order,
                                                               i,
                                                               p);
            }

          p(1) = 1. - 2*a_interpolated/fp_o_dist;

#ifdef DEBUG
          // the radial distance should always be >= -1.

          // if (p(1)+1 < tolerance)
          //  {
          //    libmesh_here();
          //    libMesh::err << "WARNING: radial distance p(1) is "
          //          << p(1)
          //          << std::endl;
          //  }
#endif

          break;
        }


      case 3:
        {
          Real a_interpolated = 0.;


          // the distance between the origin and the physical point
          const Real fp_o_dist = Point(o-physical_point).size();

          for (unsigned int i=0; i<n_base_mapping_sf; i++)
            {
              // the radial distance of the i-th base mapping point
              const Real dist_i = Point( inf_elem->point(i)
                                         - inf_elem->point(i+n_base_mapping_sf) ).size();

              // weight with the corresponding shape function
              a_interpolated += dist_i * FE<2,LAGRANGE>::shape(base_mapping_elem_type,
                                                               base_mapping_order,
                                                               i,
                                                               p);

            }

          p(2) = 1. - 2*a_interpolated/fp_o_dist;

#ifdef DEBUG


          // the radial distance should always be >= -1.

          // if (p(2)+1 < tolerance)
          //  {
          // libmesh_here();
          // libMesh::err << "WARNING: radial distance p(2) is "
          //          << p(2)
          //          << std::endl;
          //  }
#endif

          break;
        }

      default:
        libmesh_error();
      } // end switch(Dim), p fully computed, including radial part

  // if we do not want the interpolated distance, then
  // use newton iteration to get the actual distance
  else
    {
      // distance from the physical point to the ifem origin
      const Real fp_o_dist = Point(o-physical_point).size();

      // the distance from the intersection on the
      // base to the origin
      const Real a_dist = intersection.size();

      // element coordinate in radial direction
      // here our first guess is 0.
      Real v = 0.;

      // the order of the radial mapping
      const Order radial_mapping_order (Radial::mapping_order());

      unsigned int cnt = 0;
      inverse_map_error = 0.;

      // Newton iteration in 1-D
      do
        {
          // the mapping in radial direction
          // note that we only have two mapping functions in
          // radial direction
          const Real r = a_dist * InfFE<Dim,INFINITE_MAP,T_map>::eval (v, radial_mapping_order, 0)
            + 2. * a_dist * InfFE<Dim,INFINITE_MAP,T_map>::eval (v, radial_mapping_order, 1);

          const Real dr = a_dist * InfFE<Dim,INFINITE_MAP,T_map>::eval_deriv (v, radial_mapping_order, 0)
            + 2. * a_dist * InfFE<Dim,INFINITE_MAP,T_map>::eval_deriv (v, radial_mapping_order, 1);

          const Real G = dr*dr;
          const Real Ginv = 1./G;

          const Real delta = fp_o_dist - r;
          const Real drdelta = dr*delta;

          Real dp = Ginv*drdelta;

          // update the radial coordinate
          v += dp;

          // note that v should be smaller than 1,
          // since radial mapping function tends to infinity
          if (v >= 1.)
            v = .9999;

          inverse_map_error = std::fabs(dp);

          // increment iteration count
          cnt ++;
          if (cnt > 20)
            {

              libMesh::err << "ERROR: 1D Newton scheme FAILED to converge"
                            << std::endl;

              libmesh_error();
            }


        }
      while (inverse_map_error > tolerance);

      switch (Dim)
        {
        case 1:
          {
            p(0) = v;
            break;
          }
        case 2:
          {
            p(1) = v;
            break;
          }
        case 3:
          {
            p(2) = v;
            break;
          }
        default:
          libmesh_error();
        }
    }

  // If we are in debug mode do a sanity check.  Make sure
  // the point \p p on the reference element actually does
  // map to the point \p physical_point within a tolerance.
#ifdef DEBUG
  /*
        const Point check = InfFE<Dim,T_radial,T_map>::map (inf_elem, p);
        const Point diff  = physical_point - check;

        if (diff.size() > tolerance)
        {
        libmesh_here();
        libMesh::err << "WARNING:  diff is "
        << diff.size()
        << std::endl;
        }
  */
#endif




  // Stop logging the map inversion.
  STOP_LOG("inverse_map()", "InfFE");

  return p;
}

template <unsigned int Dim, FEFamily T_radial, InfMapType T_map>
void InfFE<Dim,T_radial,T_map>::inverse_map (const Elem* elem,
                                             const std::vector<Point>& physical_points,
                                             std::vector<Point>&       reference_points,
                                             const Real tolerance,
                                             const bool secure)
{
  // The number of points to find the
  // inverse map of
  const std::size_t n_points = physical_points.size();

  // Resize the vector to hold the points
  // on the reference element
  reference_points.resize(n_points);

  // Find the coordinates on the reference
  // element of each point in physical space
  for (unsigned int p=0; p<n_points; p++)
    reference_points[p] =
      InfFE<Dim,T_radial,T_map>::inverse_map (elem, physical_points[p], tolerance, secure, false);
}




//--------------------------------------------------------------
// Explicit instantiations using the macro from inf_fe_macro.h
//INSTANTIATE_INF_FE(1,CARTESIAN);

//INSTANTIATE_INF_FE(2,CARTESIAN);

//INSTANTIATE_INF_FE(3,CARTESIAN);

INSTANTIATE_INF_FE_MBRF(1,CARTESIAN,Point,inverse_map(const Elem*,const Point&,const Real,const bool,const bool));
INSTANTIATE_INF_FE_MBRF(2,CARTESIAN,Point,inverse_map(const Elem*,const Point&,const Real,const bool,const bool));
INSTANTIATE_INF_FE_MBRF(3,CARTESIAN,Point,inverse_map(const Elem*,const Point&,const Real,const bool,const bool));

INSTANTIATE_INF_FE_MBRF(1,CARTESIAN,void,inverse_map(const Elem*,const std::vector<Point>&,std::vector<Point>&,const Real, const bool));
INSTANTIATE_INF_FE_MBRF(2,CARTESIAN,void,inverse_map(const Elem*,const std::vector<Point>&,std::vector<Point>&,const Real, const bool));
INSTANTIATE_INF_FE_MBRF(3,CARTESIAN,void,inverse_map(const Elem*,const std::vector<Point>&,std::vector<Point>&,const Real, const bool));


} // namespace libMesh


#endif //ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS

---

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Last modified: February 05 2013 19:54:47 UTC

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