QualityMetric/DFT/I_DFT.cpp

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/* ***************************************************************** 
    MESQUITE -- The Mesh Quality Improvement Toolkit

    Copyright 2004 Sandia Corporation and Argonne National
    Laboratory.  Under the terms of Contract DE-AC04-94AL85000 
    with Sandia Corporation, the U.S. Government retains certain 
    rights in this software.

    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 
    (lgpl.txt) along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
    diachin2@llnl.gov, djmelan@sandia.gov, mbrewer@sandia.gov, 
    pknupp@sandia.gov, tleurent@mcs.anl.gov, tmunson@mcs.anl.gov      
   
  ***************************************************************** */
#include "I_DFT.hpp"
#include "I_DFTFamilyFunctions.hpp"
#include "TargetMatrix.hpp"

using namespace Mesquite;
   

bool I_DFT::evaluate_element(PatchData& pd,
                             MsqMeshEntity* e,
                             double& m, 
                             MsqError &err)
{
  // Only works with the weighted average

  MsqError  mErr;
  MsqVertex *vertices = pd.get_vertex_array(err); MSQ_ERRZERO(err);

  EntityTopology topo = e->get_element_type();

  const size_t nv = e->vertex_count();
  const size_t *v_i = e->get_vertex_index_array();

  size_t idx = pd.get_element_index(e);
  const TargetMatrix *W = pd.targetMatrices.get_element_corner_tags(&pd, idx, err );
  MSQ_ERRZERO(err);

  // Initialize constants for the metric
  const double delta = useBarrierDelta ? pd.get_barrier_delta(err) :
    (mGamma ? 0 : 1);
  MSQ_ERRZERO(err);

  const int tetInd[4][4] = {{0, 1, 2, 3}, {1, 2, 0, 3},
                            {2, 0, 1, 3}, {3, 2, 1, 0}};
  const int hexInd[8][4] = {{0, 1, 3, 4}, {1, 2, 0, 5},
                            {2, 3, 1, 6}, {3, 0, 2, 7},
                            {4, 7, 5, 0}, {5, 4, 6, 1},
                            {6, 5, 7, 2}, {7, 6, 4, 3}};

  // Variables used for computing the metric
  double   mMetric;             // Metric value
  bool     mValid;              // Validity of the metric
  int      i, j;

  m = 0.0;
  switch(topo) {
  case TRIANGLE:
    assert(3 == nv);

    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    for (i = 0; i < 3; ++i) {
      for (j = 0; j < 3; ++j) {
        mCoords[j] = vertices[v_i[tetInd[i][j]]];
      }
      
      mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;

      QR(mQ, mR, W[i]);
      inv(invR, mR);
      mValid = m_gdft_2(mMetric, mCoords, mNormals[i], invR, mQ, 
                        mAlpha, mGamma, delta, mBeta);
      if (!mValid) return false;
      m += W[i].get_cK() * mMetric;
      
    }

    m *= MSQ_ONE_THIRD;
    break;

  case QUADRILATERAL:
    assert(4 == nv);

    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    for (i = 0; i < 4; ++i) {
      for (j = 0; j < 3; ++j) {
        mCoords[j] = vertices[v_i[hexInd[i][j]]];
      }

      QR(mQ, mR, W[i]);
      inv(invR, mR);
      mValid = m_gdft_2(mMetric, mCoords, mNormals[i], invR, mQ, 
                        mAlpha, mGamma, delta, mBeta);
      if (!mValid) return false;
      m += W[i].get_cK() * mMetric;
    }

    m *= 0.25;
    break;

  case TETRAHEDRON:
    assert(4 == nv);

    for (i = 0; i < 4; ++i) {
      for (j = 0; j < 4; ++j) {
        mCoords[j] = vertices[v_i[tetInd[i][j]]];
      }

      QR(mQ, mR, W[i]);
      inv(invR, mR);
      mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                        mAlpha, mGamma, delta, mBeta);
      
      if (!mValid) return false;
      m += W[i].get_cK() * mMetric;
    }

    m *= 0.25;
    break;

  case HEXAHEDRON:
    assert(8 == nv);

    for (i = 0; i < 8; ++i) {
      for (j = 0; j < 4; ++j) {
        mCoords[j] = vertices[v_i[hexInd[i][j]]];
      }

      QR(mQ, mR, W[i]);
      inv(invR, mR);
      mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                        mAlpha, mGamma, delta, mBeta);
      
      if (!mValid) return false;
      m += W[i].get_cK() * mMetric;
    }

    m *= 0.125;
    break;

  default:
    MSQ_SETERR(err)("element type not implemented.",MsqError::NOT_IMPLEMENTED);
    return false;
  }

  return true;
}

bool I_DFT::compute_element_analytical_gradient(PatchData &pd,
                                                MsqMeshEntity *e,
                                                MsqVertex *fv[], 
                                                Vector3D g[],
                                                int nfv, 
                                                double &m,
                                                MsqError &err)
{
  // Only works with the weighted average

  MsqVertex *vertices = pd.get_vertex_array(err); MSQ_ERRZERO(err);
  EntityTopology topo = e->get_element_type();

  const size_t nv = e->vertex_count();
  const size_t *v_i = e->get_vertex_index_array();

  size_t idx = pd.get_element_index(e);
  const TargetMatrix *W = pd.targetMatrices.get_element_corner_tags(&pd, idx, err );
  MSQ_ERRZERO(err);

  // Initialize constants for the metric
  const double delta = useBarrierDelta ? pd.get_barrier_delta(err) :
    (mGamma ? 0 : 1);
    //const double delta = useBarrierDelta ? pd.get_barrier_delta(err) : 0;
  MSQ_ERRZERO(err);

  const int tetInd[4][4] = {{0, 1, 2, 3}, {1, 2, 0, 3},
                            {2, 0, 1, 3}, {3, 2, 1, 0}};
  const int hexInd[8][4] = {{0, 1, 3, 4}, {1, 2, 0, 5},
                            {2, 3, 1, 6}, {3, 0, 2, 7},
                            {4, 7, 5, 0}, {5, 4, 6, 1},
                            {6, 5, 7, 2}, {7, 6, 4, 3}};

  // Variables used for computing the metric
  double   mMetric;             // Metric value
  bool     mValid;              // Validity of the metric
  int      i, j, mVert;

  m = 0.0;
  switch(topo) {
  case TRIANGLE:
    assert(3 == nv);
    
    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    // The following analytic calculation only works correctly if the
    // normal is constant.  If the normal is not constant, you need
    // to get the gradient of the normal with respect to the vertex
    // positions to obtain the correct values.
    
    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the gradient.

      g[0] = 0.0;
      for (i = 0; i < 3; ++i) {
        mVert = -1;
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
          if (vertices + v_i[tetInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }

        if (mVert >= 0) {
          mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          switch(mVert) {
          case 0:
            mValid = g_gdft_2_v0(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = g_gdft_2_v1(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = g_gdft_2_v2(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          g[0] += W[i].get_cK() * mGrads[0];
        }
        else {
          // For triangles, the free vertex must appear in every element.
          // Therefore, there these accumulations should not get used.

          mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          mValid = m_gdft_2(mMetric, mCoords, mNormals[i],
                            invR, mQ, mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= MSQ_ONE_THIRD;
      g[0] *= MSQ_ONE_THIRD;
    }
    else {
      for (i = 0; i < 3; ++i) {
        mAccGrads[i] = 0.0;
      }
    
      for (i = 0; i < 3; ++i) {
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
        }
      
        mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
      
        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = g_gdft_2(mMetric, mGrads, mCoords, mNormals[i], invR, mQ, 
                          mAlpha, mGamma, delta, mBeta);

        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;
        for (j = 0; j < 3; ++j) {
          mAccGrads[tetInd[i][j]] += W[i].get_cK() * mGrads[j];
        }
      }

      m *= MSQ_ONE_THIRD;
      for (i = 0; i < 3; ++i) {
        mAccGrads[i] *= MSQ_ONE_THIRD;
      }

      // This is not very efficient, but is one way to select correct 
      // gradients.  For gradients, info is returned only for free 
      // vertices, in the order of fv[].

      for (i = 0; i < 3; ++i) {
        for (j = 0; j < nfv; ++j) {
          if (vertices + v_i[i] == fv[j]) {
            g[j] = mAccGrads[i];
          }
        }
      }
    }
    break;

  case QUADRILATERAL:
    assert(4 == nv);
    
    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    // The following analytic calculation only works correctly if the
    // normal is constant.  If the normal is not constant, you need
    // to get the gradient of the normal with respect to the vertex
    // positions to obtain the correct values.

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the gradient.

      g[0] = 0.0;
      for (i = 0; i < 4; ++i) {
        mVert = -1;
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
          if (vertices + v_i[hexInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }

        if (mVert >= 0) {
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          switch(mVert) {
          case 0:
            mValid = g_gdft_2_v0(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = g_gdft_2_v1(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = g_gdft_2_v2(mMetric, mGrads[0], mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          g[0] += W[i].get_cK() * mGrads[0];
        }
        else {
          // For quadrilaterals, the free vertex only appears in three 
          // elements.  Therefore, there these accumulations are needed 
          // to get the true local objective function.  Note: this code 
          // can be commented out for local codes to improve performance 
          // because you are unable to change the contributions from the 
          // elements where the free vertex does not appear.  (If the 
          // weight matrices change, then the code needs to be modified.)
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          mValid = m_gdft_2(mMetric, mCoords, mNormals[i],
                            invR, mQ, mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= 0.25;
      g[0] *= 0.25;
    }
    else {
      for (i = 0; i < 4; ++i) {
        mAccGrads[i] = 0.0;
      }

      for (i = 0; i < 4; ++i) {
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
        }

        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = g_gdft_2(mMetric, mGrads, mCoords, mNormals[i], invR, mQ, 
                          mAlpha, mGamma, delta, mBeta);

        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;
        for (j = 0; j < 3; ++j) {
          mAccGrads[hexInd[i][j]] += W[i].get_cK() * mGrads[j];
        }
      }
    
      m *= 0.25;
      for (i = 0; i < 4; ++i) {
        mAccGrads[i] *= 0.25;
      }

      // This is not very efficient, but is one way to select correct gradients
      // For gradients, info is returned only for free vertices, in the order 
      // of fv[].

      for (i = 0; i < 4; ++i) {
        for (j = 0; j < nfv; ++j) {
          if (vertices + v_i[i] == fv[j]) {
            g[j] = mAccGrads[i];
          }
        }
      }
    }
    break;

  case TETRAHEDRON:
    assert(4 == nv);

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the gradient.

      g[0] = 0.0;
      for (i = 0; i < 4; ++i) {
        mVert = -1;
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
          if (vertices + v_i[tetInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }
        
        if (mVert >= 0) {
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = g_gdft_3_v0(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = g_gdft_3_v1(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 2:
            mValid = g_gdft_3_v2(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = g_gdft_3_v3(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          g[0] += W[i].get_cK() * mGrads[0];
        }
        else {
          // For tetrahedrons, the free vertex must appear in every element.
          // Therefore, there these accumulations should not get used.

          QR(mQ, mR, W[i]);
          inv(invR, mR);
          mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                            mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= 0.25;
      g[0] *= 0.25;
    }
    else {
      for (i = 0; i < 4; ++i) {
        mAccGrads[i] = 0.0;
      }
      
      for (i = 0; i < 4; ++i) {
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
        }
        
        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = g_gdft_3(mMetric, mGrads, mCoords, invR, mQ, 
                          mAlpha, mGamma, delta, mBeta);
        
        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;
        
        for (j = 0; j < 4; ++j) {
          mAccGrads[tetInd[i][j]] += W[i].get_cK() * mGrads[j];
        }
      }
      
      m *= 0.25;
      for (i = 0; i < 4; ++i) {
        mAccGrads[i] *= 0.25;
      }
      
      // This is not very efficient, but is one way to select correct 
      // gradients.  For gradients, info is returned only for free vertices, 
      // in the order of fv[].

      for (i = 0; i < 4; ++i) {
        for (j = 0; j < nfv; ++j) {
          if (vertices + v_i[i] == fv[j]) {
            g[j] = mAccGrads[i];
          }
        }
      }
    }
    break;

  case HEXAHEDRON:
    assert(8 == nv);

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the gradient.

      g[0] = 0.0;
      for (i = 0; i < 8; ++i) {
        mVert = -1;
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
          if (vertices + v_i[hexInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }

        if (mVert >= 0) {
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = g_gdft_3_v0(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = g_gdft_3_v1(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 2:
            mValid = g_gdft_3_v2(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = g_gdft_3_v3(mMetric, mGrads[0], mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
          }
          
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          g[0] += W[i].get_cK() * mGrads[0];
        }
        else {
          // For hexahedrons, the free vertex only appears in four elements.
          // Therefore, there these accumulations are needed to get the
          // true local objective function.  Note: this code can be commented 
          // out for local codes to improve performance because you are 
          // unable to change the contributions from the elements where the 
          // free vertex does not appear.  (If the weight matrices change, 
          // then the code needs to be modified.)

          QR(mQ, mR, W[i]);
          inv(invR, mR);
          mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                            mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }
      
      m *= 0.125;
      g[0] *= 0.125;
    }
    else {
      for (i = 0; i < 8; ++i) {
        mAccGrads[i] = 0.0;
      }
      
      for (i = 0; i < 8; ++i) {
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
        }
        
        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = g_gdft_3(mMetric, mGrads, mCoords, invR, mQ, 
                          mAlpha, mGamma, delta, mBeta);
        
        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;
        
        for (j = 0; j < 4; ++j) {
          mAccGrads[hexInd[i][j]] += W[i].get_cK() * mGrads[j];
        }
      }
      
      m *= 0.125;
      for (i = 0; i < 8; ++i) {
        mAccGrads[i] *= 0.125;
      }
      
      // This is not very efficient, but is one way to select correct 
      // gradients.  For gradients, info is returned only for free 
      // vertices, in the order of fv[].
      
      for (i = 0; i < 8; ++i) {
        for (j = 0; j < nfv; ++j) {
          if (vertices + v_i[i] == fv[j]) {
            g[j] = mAccGrads[i];
          }
        }
      }
    }
    break;

  default:
    MSQ_SETERR(err)("element type not implemented.",MsqError::NOT_IMPLEMENTED);
    return false;
  }

  return true;
}

bool I_DFT::compute_element_analytical_hessian(PatchData &pd,
                                               MsqMeshEntity *e,
                                               MsqVertex *fv[], 
                                               Vector3D g[],
                                               Matrix3D h[],
                                               int nfv, 
                                               double &m,
                                               MsqError &err)
{
  // Only works with the weighted average

  MsqVertex *vertices = pd.get_vertex_array(err);  MSQ_ERRZERO(err);
  EntityTopology topo = e->get_element_type();

  const size_t nv = e->vertex_count();
  const size_t *v_i = e->get_vertex_index_array();

  size_t idx = pd.get_element_index(e);
  const TargetMatrix *W = pd.targetMatrices.get_element_corner_tags(&pd, idx, err );
  MSQ_ERRZERO(err);

  // Initialize constants for the metric
  const double delta = useBarrierDelta ? pd.get_barrier_delta(err) :
    (mGamma ? 0 : 1);  
    //const double delta = useBarrierDelta ? pd.get_barrier_delta(err) : 0;
  MSQ_ERRZERO(err);

  const int tetInd[4][4] = {{0, 1, 2, 3}, {1, 2, 0, 3},
                            {2, 0, 1, 3}, {3, 2, 1, 0}};
  const int hexInd[8][4] = {{0, 1, 3, 4}, {1, 2, 0, 5},
                            {2, 3, 1, 6}, {3, 0, 2, 7},
                            {4, 7, 5, 0}, {5, 4, 6, 1},
                            {6, 5, 7, 2}, {7, 6, 4, 3}};

  // Variables used for computing the metric
  double   mMetric;             // Metric value
  bool     mValid;              // Validity of the metric
  int      i, j, k, l, mVert;
  int      row, col, loc;

  m = 0.0;
  switch(topo) {
  case TRIANGLE:
    assert(3 == nv);
    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    // The following analytic calculation only works correctly if the
    // normal is constant.  If the normal is not constant, you need
    // to get the gradient of the normal with respect to the vertex
    // positions to obtain the correct values.

    // Zero out the hessian and gradient vector
    for (i = 0; i < 3; ++i) {
      g[i] = 0.0;
    }

    for (i = 0; i < 6; ++i) {
      h[i].zero();
    }

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the 
      // gradient and Hessian.

      Vector3D mG;
      Matrix3D mH;

      mG = 0.0;
      mH.zero();

      for (i = 0; i < 3; ++i) {
        mVert = -1;
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
          if (vertices + v_i[tetInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }
        
        if (mVert >= 0) {
          mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = h_gdft_2_v0(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = h_gdft_2_v1(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = h_gdft_2_v2(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          mG += W[i].get_cK() * mGrads[0];
          mH += W[i].get_cK() * mHessians[0];
        }
        else {
          // For triangles, the free vertex must appear in every element.
          // Therefore, there these accumulations should not get used.

          mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          mValid = m_gdft_2(mMetric, mCoords, mNormals[i],
                            invR, mQ, mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= MSQ_ONE_THIRD;
      mG *= MSQ_ONE_THIRD;
      mH *= MSQ_ONE_THIRD;

      for (i = 0; i < 3; ++i) {
        if (vertices + v_i[i] == fv[0]) {
          // free vertex, see next
          g[i] = mG;
          switch(i) {
          case 0:
            h[0] = mH;
            break;

          case 1:
            h[3] = mH;
            break;

          default:
            h[5] = mH;
            break;
          }
          break;
        }
      }
    }
    else {
      // Compute the metric and sum them together
      for (i = 0; i < 3; ++i) {
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
        }

        mNormals[i] *= MSQ_3RT_2_OVER_6RT_3;
      
        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = h_gdft_2(mMetric, mGrads, mHessians, mCoords, mNormals[i], 
                          invR, mQ, mAlpha, mGamma, delta, mBeta);

        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;

        for (j = 0; j < 3; ++j) {
          g[tetInd[i][j]] += W[i].get_cK() * mGrads[j];
        }

        l = 0;
        for (j = 0; j < 3; ++j) {
          for (k = j; k < 3; ++k) {
            row = tetInd[i][j];
            col = tetInd[i][k];

            if (row <= col) {
              loc = 3*row - (row*(row+1)/2) + col;
              h[loc] += W[i].get_cK() * mHessians[l];
            }
            else {
              loc = 3*col - (col*(col+1)/2) + row;
              h[loc] += W[i].get_cK() * transpose(mHessians[l]);
            }
            ++l;
          }
        }
      }

      m *= MSQ_ONE_THIRD;
      for (i = 0; i < 3; ++i) {
        g[i] *= MSQ_ONE_THIRD;
      }

      for (i = 0; i < 6; ++i) {
        h[i] *= MSQ_ONE_THIRD;
      }

      // zero out fixed elements of g
      j = 0;
      for (i = 0; i < 3; ++i) {
        if (vertices + v_i[i] == fv[j]) {
          // if free vertex, see next
          ++j;
        }
        else {
          // else zero gradient entry and hessian entries.
          g[i] = 0.;

          switch(i) {
          case 0:
            h[0].zero(); h[1].zero(); h[2].zero();
            break;
          
          case 1:
            h[1].zero(); h[3].zero(); h[4].zero();
            break;
          
          case 2:
            h[2].zero(); h[4].zero(); h[5].zero();
            break;
          }
        }
      }
    }
    break;

  case QUADRILATERAL:
    assert(4 == nv);
    e->compute_corner_normals( mNormals, pd, err ); MSQ_ERRZERO(err);

    // The following analytic calculation only works correctly if the
    // normal is constant.  If the normal is not constant, you need
    // to get the gradient of the normal with respect to the vertex
    // positions to obtain the correct values.

    // Zero out the hessian and gradient vector
    for (i = 0; i < 4; ++i) {
      g[i] = 0.0;
    }

    for (i = 0; i < 10; ++i) {
      h[i].zero();
    }

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the 
      // gradient and Hessian.

      Vector3D mG;
      Matrix3D mH;

      mG = 0.0;
      mH.zero();

      for (i = 0; i < 4; ++i) {
        mVert = -1;
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
          if (vertices + v_i[hexInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }
        
        if (mVert >= 0) {
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = h_gdft_2_v0(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = h_gdft_2_v1(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = h_gdft_2_v2(mMetric, mGrads[0], mHessians[0],
                                 mCoords, mNormals[i],
                                 invR, mQ, mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          mG += W[i].get_cK() * mGrads[0];
          mH += W[i].get_cK() * mHessians[0];
        }
        else {
          // For quadrilaterals, the free vertex only appears in three 
          // elements.  Therefore, there these accumulations are needed 
          // to get the true local objective function.  Note: this code 
          // can be commented out for local codes to improve performance 
          // because you are unable to change the contributions from the 
          // elements where the free vertex does not appear.  (If the 
          // weight matrices change, then the code needs to be modified.)
          
          QR(mQ, mR, W[i]);
          inv(invR, mR);

          mValid = m_gdft_2(mMetric, mCoords, mNormals[i],
                            invR, mQ, mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= 0.25;
      mG *= 0.25;
      mH *= 0.25;

      for (i = 0; i < 4; ++i) {
        if (vertices + v_i[i] == fv[0]) {
          // free vertex, see next
          g[i] = mG;
          switch(i) {
          case 0:
            h[0] = mH;
            break;

          case 1:
            h[4] = mH;
            break;

          case 2:
            h[7] = mH;
            break;

          default:
            h[9] = mH;
            break;
          }
          break;
        }
      }
    }
    else {
      // Compute the metric and sum them together
      for (i = 0; i < 4; ++i) {
        for (j = 0; j < 3; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
        }

        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = h_gdft_2(mMetric, mGrads, mHessians, mCoords, mNormals[i], 
                          invR, mQ, mAlpha, mGamma, delta, mBeta);

        if (!mValid) return false;
        m += W[i].get_cK() * mMetric;

        for (j = 0; j < 3; ++j) {
          g[hexInd[i][j]] += W[i].get_cK() * mGrads[j];
        }

        l = 0;
        for (j = 0; j < 3; ++j) {
          for (k = j; k < 3; ++k) {
            row = hexInd[i][j];
            col = hexInd[i][k];

            if (row <= col) {
              loc = 4*row - (row*(row+1)/2) + col;
              h[loc] += W[i].get_cK() * mHessians[l];
            }
            else {
              loc = 4*col - (col*(col+1)/2) + row;
              h[loc] += W[i].get_cK() * transpose(mHessians[l]);
            }
            ++l;
          }
        }
      }

      m *= 0.25;
      for (i = 0; i < 4; ++i) {
        g[i] *= 0.25;
      }

      for (i = 0; i < 10; ++i) {
        h[i] *= 0.25;
      }

      // zero out fixed elements of gradient and Hessian
      j = 0;
      for (i = 0; i < 4; ++i) {
        if (vertices + v_i[i] == fv[j]) {
          // if free vertex, see next
          ++j;
        }
        else {
          // else zero gradient entry and hessian entries.
          g[i] = 0.;

          switch(i) {
          case 0:
            h[0].zero();   h[1].zero();   h[2].zero();   h[3].zero();
            break;
          
          case 1:
            h[1].zero();   h[4].zero();   h[5].zero();   h[6].zero();
            break;
          
          case 2:
            h[2].zero();   h[5].zero();   h[7].zero();  h[8].zero();
            break;
          
          case 3:
            h[3].zero();   h[6].zero();   h[8].zero();  h[9].zero();
            break;
          }
        }
      }
    }
    break;

  case TETRAHEDRON:
    assert(4 == nv);

    // Zero out the hessian and gradient vector
    for (i = 0; i < 4; ++i) {
      g[i] = 0.0;
    }

    for (i = 0; i < 10; ++i) {
      h[i].zero();
    }

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the 
      // gradient and Hessian.

      Vector3D mG;
      Matrix3D mH;

      mG = 0.0;
      mH.zero();

      for (i = 0; i < 4; ++i) {
        mVert = -1;
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
          if (vertices + v_i[tetInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }
        
        if (mVert >= 0) {
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = h_gdft_3_v0(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = h_gdft_3_v1(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 2:
            mValid = h_gdft_3_v2(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = h_gdft_3_v3(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          mG += W[i].get_cK() * mGrads[0];
          mH += W[i].get_cK() * mHessians[0];
        }
        else {
          // For tetrahedrons, the free vertex must appear in every element.
          // Therefore, there these accumulations should not get used.

          QR(mQ, mR, W[i]);
          inv(invR, mR);
          mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                            mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= 0.25;
      mG *= 0.25;
      mH *= 0.25;

      for (i = 0; i < 4; ++i) {
        if (vertices + v_i[i] == fv[0]) {
          // free vertex, see next
          g[i] = mG;
          switch(i) {
          case 0:
            h[0] = mH;
            break;

          case 1:
            h[4] = mH;
            break;

          case 2:
            h[7] = mH;
            break;

          default:
            h[9] = mH;
            break;
          }
          break;
        }
      }
    }
    else {
      // Compute the metric and sum them together
      for (i = 0; i < 4; ++i) {
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[tetInd[i][j]]];
        }
        
        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = h_gdft_3(mMetric, mGrads, mHessians, mCoords, invR, mQ,
                          mAlpha, mGamma, delta, mBeta);
        
        if (!mValid) return false;
        
        m += W[i].get_cK() * mMetric;
        
        for (j = 0; j < 4; ++j) {
          g[tetInd[i][j]] += W[i].get_cK() * mGrads[j];
        }
        
        l = 0;
        for (j = 0; j < 4; ++j) {
          for (k = j; k < 4; ++k) {
            row = tetInd[i][j];
            col = tetInd[i][k];
            
            if (row <= col) {
              loc = 4*row - (row*(row+1)/2) + col;
              h[loc] += W[i].get_cK() * mHessians[l];
            }
            else {
              loc = 4*col - (col*(col+1)/2) + row;
              h[loc] += W[i].get_cK() * transpose(mHessians[l]);
            }
            ++l;
          }
        }
      }
      
      m *= 0.25;
      for (i = 0; i < 4; ++i) {
        g[i] *= 0.25;
      }
      
      for (i = 0; i < 10; ++i) {
        h[i] *= 0.25;
      }
      
      // zero out fixed elements of g
      j = 0;
      for (i = 0; i < 4; ++i) {
        if (vertices + v_i[i] == fv[j]) {
          // if free vertex, see next
          ++j;
        }
        else {
          // else zero gradient entry and hessian entries.
          g[i] = 0.;
          
          switch(i) {
          case 0:
            h[0].zero(); h[1].zero(); h[2].zero(); h[3].zero();
            break;
            
          case 1:
            h[1].zero(); h[4].zero(); h[5].zero(); h[6].zero();
            break;
            
          case 2:
            h[2].zero(); h[5].zero(); h[7].zero(); h[8].zero();
            break;

          case 3:
            h[3].zero(); h[6].zero(); h[8].zero(); h[9].zero();
            break;
          }
        }
      }
    }
    break;

  case HEXAHEDRON:
    assert(8 == nv);

    // Zero out the hessian and gradient vector
    for (i = 0; i < 8; ++i) {
      g[i] = 0.0;
    }

    for (i = 0; i < 36; ++i) {
      h[i].zero();
    }

    if (1 == nfv) {
      // One free vertex; use the specialized code for computing the 
      // gradient and Hessian.

      Vector3D mG;
      Matrix3D mH;

      mG = 0.0;
      mH.zero();

      for (i = 0; i < 8; ++i) {
        mVert = -1;
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
          if (vertices + v_i[hexInd[i][j]] == fv[0]) {
            mVert = j;
          }
        }
        
        if (mVert >= 0) {
          QR(mQ, mR, W[i]);
          inv(invR, mR);
          
          switch(mVert) {
          case 0:
            mValid = h_gdft_3_v0(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 1:
            mValid = h_gdft_3_v1(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          case 2:
            mValid = h_gdft_3_v2(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
            
          default:
            mValid = h_gdft_3_v3(mMetric, mGrads[0], mHessians[0],
                                 mCoords, invR, mQ, 
                                 mAlpha, mGamma, delta, mBeta);
            break;
          }

          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
          mG += W[i].get_cK() * mGrads[0];
          mH += W[i].get_cK() * mHessians[0];
        }
        else {
          // For hexahedrons, the free vertex only appears in four elements.
          // Therefore, there these accumulations are needed to get the
          // true local objective function.  Note: this code can be commented 
          // out for local codes to improve performance because you are 
          // unable to change the contributions from the elements where the 
          // free vertex does not appear.  (If the weight matrices change, 
          // then the code needs to be modified.)

          QR(mQ, mR, W[i]);
          inv(invR, mR);
          mValid = m_gdft_3(mMetric, mCoords, invR, mQ, 
                            mAlpha, mGamma, delta, mBeta);
          if (!mValid) return false;
          m += W[i].get_cK() * mMetric;
        }
      }

      m *= 0.125;
      mG *= 0.125;
      mH *= 0.125;

      for (i = 0; i < 8; ++i) {
        if (vertices + v_i[i] == fv[0]) {
          // free vertex, see next
          g[i] = mG;
          switch(i) {
          case 0:
            h[0] = mH;
            break;

          case 1:
            h[8] = mH;
            break;

          case 2:
            h[15] = mH;
            break;

          case 3:
            h[21] = mH;
            break;

          case 4:
            h[26] = mH;
            break;

          case 5:
            h[30] = mH;
            break;

          case 6:
            h[33] = mH;
            break;

          default:
            h[35] = mH;
            break;
          }
          break;
        }
      }
    }
    else {
      // Compute the metric and sum them together
      for (i = 0; i < 8; ++i) {
        for (j = 0; j < 4; ++j) {
          mCoords[j] = vertices[v_i[hexInd[i][j]]];
        }

        QR(mQ, mR, W[i]);
        inv(invR, mR);
        mValid = h_gdft_3(mMetric, mGrads, mHessians, mCoords, invR, mQ,
                          mAlpha, mGamma, delta, mBeta);
      
        if (!mValid) return false;

        m += W[i].get_cK() * mMetric;

        for (j = 0; j < 4; ++j) {
          g[hexInd[i][j]] += W[i].get_cK() * mGrads[j];
        }

        l = 0;
        for (j = 0; j < 4; ++j) {
          for (k = j; k < 4; ++k) {
            row = hexInd[i][j];
            col = hexInd[i][k];

            if (row <= col) {
              loc = 8*row - (row*(row+1)/2) + col;
              h[loc] += W[i].get_cK() * mHessians[l];
            }
            else {
              loc = 8*col - (col*(col+1)/2) + row;
              h[loc] += W[i].get_cK() * transpose(mHessians[l]);
            }
            ++l;
          }
        }
      }

      m *= 0.125;
      for (i = 0; i < 8; ++i) {
        g[i] *= 0.125;
      }

      for (i = 0; i < 36; ++i) {
        h[i] *= 0.125;
      }

      // zero out fixed elements of gradient and Hessian
      j = 0;
      for (i = 0; i < 8; ++i) {
        if (vertices + v_i[i] == fv[j]) {
          // if free vertex, see next
          ++j;
        }
        else {
          // else zero gradient entry and hessian entries.
          g[i] = 0.;

          switch(i) {
          case 0:
            h[0].zero();   h[1].zero();   h[2].zero();   h[3].zero();
            h[4].zero();   h[5].zero();   h[6].zero();   h[7].zero();
            break;
          
          case 1:
            h[1].zero();   h[8].zero();   h[9].zero();   h[10].zero();
            h[11].zero();  h[12].zero();  h[13].zero();  h[14].zero();
            break;
          
          case 2:
            h[2].zero();   h[9].zero();   h[15].zero();  h[16].zero();
            h[17].zero();  h[18].zero();  h[19].zero();  h[20].zero();
            break;
          
          case 3:
            h[3].zero();   h[10].zero();  h[16].zero();  h[21].zero();
            h[22].zero();  h[23].zero();  h[24].zero();  h[25].zero();
            break;
          
          case 4:
            h[4].zero();   h[11].zero();  h[17].zero();  h[22].zero();
            h[26].zero();  h[27].zero();  h[28].zero();  h[29].zero();
            break;
          
          case 5:
            h[5].zero();   h[12].zero();  h[18].zero();  h[23].zero();
            h[27].zero();  h[30].zero();  h[31].zero();  h[32].zero();
            break;
          
          case 6:
            h[6].zero();   h[13].zero();  h[19].zero();  h[24].zero();
            h[28].zero();  h[31].zero();  h[33].zero();  h[34].zero();
            break;
          
          case 7:
            h[7].zero();   h[14].zero();  h[20].zero();  h[25].zero();
            h[29].zero();  h[32].zero();  h[34].zero();  h[35].zero();
            break;
          }
        }
      }
    }
    break;

  default:
    MSQ_SETERR(err)("element type not implemented.",MsqError::NOT_IMPLEMENTED);
    return false;
  }
  return true;
}