vtkQuadraticHexahedron.cxx

/*=========================================================================

  Program:   Visualization Toolkit
  Module:    vtkQuadraticHexahedron.cxx

  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/
#include "vtkQuadraticHexahedron.h"

#include "vtkObjectFactory.h"
#include "vtkCellData.h"
#include "vtkDoubleArray.h"
#include "vtkHexahedron.h"
#include "vtkMath.h"
#include "vtkPointData.h"
#include "vtkQuadraticEdge.h"
#include "vtkQuadraticQuad.h"
#include "vtkPoints.h"

vtkStandardNewMacro(vtkQuadraticHexahedron);

//----------------------------------------------------------------------------
// Construct the hex with 20 points + 7 extra points for internal
// computation.
vtkQuadraticHexahedron::vtkQuadraticHexahedron()
{
  // At times the cell looks like it has 27 points (during interpolation)
  // We initially allocate for 27.
  this->Points->SetNumberOfPoints(27);
  this->PointIds->SetNumberOfIds(27);
  for (int i = 0; i < 27; i++)
  {
    this->Points->SetPoint(i, 0.0, 0.0, 0.0);
    this->PointIds->SetId(i,0);
  }
  this->Points->SetNumberOfPoints(20);
  this->PointIds->SetNumberOfIds(20);

  this->Edge = vtkQuadraticEdge::New();
  this->Face = vtkQuadraticQuad::New();
  this->Hex = vtkHexahedron::New();

  this->PointData = vtkPointData::New();
  this->CellData = vtkCellData::New();
  this->CellScalars = vtkDoubleArray::New();
  this->CellScalars->SetNumberOfTuples(27);
  this->Scalars = vtkDoubleArray::New();
  this->Scalars->SetNumberOfTuples(8);
}

//----------------------------------------------------------------------------
vtkQuadraticHexahedron::~vtkQuadraticHexahedron()
{
  this->Edge->Delete();
  this->Face->Delete();
  this->Hex->Delete();

  this->PointData->Delete();
  this->CellData->Delete();
  this->Scalars->Delete();
  this->CellScalars->Delete();
}


static int LinearHexs[8][8] = {
    { 0,  8, 24, 11, 16, 22, 26, 20},
    { 8,  1,  9, 24, 22, 17, 21, 26},
    {11, 24, 10,  3, 20, 26, 23, 19},
    {24,  9,  2, 10, 26, 21, 18, 23},
    {16, 22, 26, 20,  4, 12, 25, 15},
    {22, 17, 21, 26, 12,  5, 13, 25},
    {20, 26, 23, 19, 15, 25, 14,  7},
    {26, 21, 18, 23, 25, 13,  6, 14} };

static int HexFaces[6][8] = {
    {0, 4, 7, 3, 16, 15, 19, 11},
    {1, 2, 6, 5,  9, 18, 13, 17},
    {0, 1, 5, 4,  8, 17, 12, 16},
    {3, 7, 6, 2, 19, 14, 18, 10},
    {0, 3, 2, 1, 11, 10,  9,  8},
    {4, 5, 6, 7, 12, 13, 14, 15} };

static int HexEdges[12][3] = {
    {0, 1,  8}, {1, 2,  9}, {3, 2, 10}, {0, 3, 11},
    {4, 5, 12}, {5, 6, 13}, {7, 6, 14}, {4, 7, 15},
    {0, 4, 16}, {1, 5, 17}, {3, 7, 19}, {2, 6, 18}
};

static double MidPoints[7][3] = { {0.0,0.5,0.5}, // 20
                                  {1.0,0.5,0.5}, // 21
                                  {0.5,0.0,0.5}, // 22
                                  {0.5,1.0,0.5}, // 23
                                  {0.5,0.5,0.0}, // 24
                                  {0.5,0.5,1.0}, // 25
                                  {0.5,0.5,0.5} }; // 26

//----------------------------------------------------------------------------
int *vtkQuadraticHexahedron::GetEdgeArray(int edgeId)
{
  return HexEdges[edgeId];
}
//----------------------------------------------------------------------------
int *vtkQuadraticHexahedron::GetFaceArray(int faceId)
{
  return HexFaces[faceId];
}

//----------------------------------------------------------------------------
vtkCell *vtkQuadraticHexahedron::GetEdge(int edgeId)
{
  edgeId = (edgeId < 0 ? 0 : (edgeId > 11 ? 11 : edgeId ));

  for (int i=0; i<3; i++)
  {
    this->Edge->PointIds->SetId(i,this->PointIds->GetId(HexEdges[edgeId][i]));
    this->Edge->Points->SetPoint(i,this->Points->GetPoint(HexEdges[edgeId][i]));
  }

  return this->Edge;
}

//----------------------------------------------------------------------------
vtkCell *vtkQuadraticHexahedron::GetFace(int faceId)
{
  faceId = (faceId < 0 ? 0 : (faceId > 5 ? 5 : faceId ));

  for (int i=0; i<8; i++)
  {
    this->Face->PointIds->SetId(i,this->PointIds->GetId(HexFaces[faceId][i]));
    this->Face->Points->SetPoint(i,this->Points->GetPoint(HexFaces[faceId][i]));
  }

  return this->Face;
}

//----------------------------------------------------------------------------
void vtkQuadraticHexahedron::Subdivide(vtkPointData *inPd, vtkCellData *inCd,
                                       vtkIdType cellId, vtkDataArray *cellScalars)
{
  int numMidPts, i, j;
  double weights[20];
  double x[3];
  double s;

  //Copy point and cell attribute data, first make sure it's empty:
  this->PointData->Initialize();
  this->CellData->Initialize();
  // Make sure to copy ALL arrays. These field data have to be
  // identical to the input field data. Otherwise, CopyData
  // that occurs later may not work because the output field
  // data was initialized (CopyAllocate) with the input field
  // data.
  this->PointData->CopyAllOn();
  this->CellData->CopyAllOn();
  this->PointData->CopyAllocate(inPd,27);
  this->CellData->CopyAllocate(inCd,8);
  for (i=0; i<20; i++)
  {
    this->PointData->CopyData(inPd,this->PointIds->GetId(i),i);
    this->CellScalars->SetValue( i, cellScalars->GetTuple1(i));
  }
  for (i=0; i<8; i++)
  {
    this->CellData->CopyData(inCd,cellId,i);
  }

  //Interpolate new values
  double p[3];
  this->Points->Resize(27);
  this->CellScalars->Resize(27);
  for ( numMidPts=0; numMidPts < 7; numMidPts++ )
  {
    this->InterpolationFunctions(MidPoints[numMidPts], weights);

    x[0] = x[1] = x[2] = 0.0;
    s = 0.0;
    for (i=0; i<20; i++)
    {
      this->Points->GetPoint(i, p);
      for (j=0; j<3; j++)
      {
        x[j] += p[j] * weights[i];
      }
      s += cellScalars->GetTuple1(i) * weights[i];
    }
    this->Points->SetPoint(20+numMidPts,x);
    this->CellScalars->SetValue(20+numMidPts,s);
    this->PointData->InterpolatePoint(inPd, 20+numMidPts,
                                      this->PointIds, weights);
  }
}

//----------------------------------------------------------------------------
static const double VTK_DIVERGED = 1.e6;
static const int VTK_HEX_MAX_ITERATION=20;
static const double VTK_HEX_CONVERGED=1.e-04;

int vtkQuadraticHexahedron::EvaluatePosition(const double x[3],
                                             double closestPoint[3],
                                             int& subId, double pcoords[3],
                                             double& dist2, double weights[])
{
  double  params[3] = {0.5, 0.5, 0.5};
  double derivs[60];

  // compute a bound on the volume to get a scale for an acceptable determinant
  vtkIdType diagonals[4][2] = { {0, 6}, {1, 7}, {2, 4}, {3, 5} };
  double longestDiagonal = 0;
  for (int i=0;i<4;i++)
  {
    double pt0[3], pt1[3];
    this->Points->GetPoint(diagonals[i][0], pt0);
    this->Points->GetPoint(diagonals[i][1], pt1);
    double d2 = vtkMath::Distance2BetweenPoints(pt0, pt1);
    if (longestDiagonal < d2)
    {
      longestDiagonal = d2;
    }
  }
  // longestDiagonal value is already squared
  double volumeBound = pow(longestDiagonal, 1.5);
  double determinantTolerance = 1e-20 < .00001*volumeBound ? 1e-20 : .00001*volumeBound;

  //  set initial position for Newton's method
  subId = 0;
  pcoords[0] = pcoords[1] = pcoords[2] = 0.5;

  //  enter iteration loop
  int converged = 0;
  for (int iteration=0;!converged && (iteration < VTK_HEX_MAX_ITERATION);  iteration++)
  {
    //  calculate element interpolation functions and derivatives
    this->InterpolationFunctions(pcoords, weights);
    this->InterpolationDerivs(pcoords, derivs);

    //  calculate newton functions
    double fcol[3] = {0, 0, 0}, rcol[3] = {0, 0, 0}, scol[3] = {0, 0, 0}, tcol[3] = {0, 0, 0};
    for (int i=0; i<20; i++)
    {
      double pt[3];
      this->Points->GetPoint(i, pt);
      for (int j=0; j<3; j++)
      {
        fcol[j] += pt[j] * weights[i];
        rcol[j] += pt[j] * derivs[i];
        scol[j] += pt[j] * derivs[i+20];
        tcol[j] += pt[j] * derivs[i+40];
      }
    }

    for (int i=0; i<3; i++)
    {
      fcol[i] -= x[i];
    }

    //  compute determinants and generate improvements
    double d = vtkMath::Determinant3x3(rcol,scol,tcol);
    if ( fabs(d) < determinantTolerance)
    {
      vtkDebugMacro (<<"Determinant incorrect, iteration " << iteration);
      return -1;
    }

    pcoords[0] = params[0] - 0.5*vtkMath::Determinant3x3 (fcol,scol,tcol) / d;
    pcoords[1] = params[1] - 0.5*vtkMath::Determinant3x3 (rcol,fcol,tcol) / d;
    pcoords[2] = params[2] - 0.5*vtkMath::Determinant3x3 (rcol,scol,fcol) / d;

    //  check for convergence
    if ( ((fabs(pcoords[0]-params[0])) < VTK_HEX_CONVERGED) &&
         ((fabs(pcoords[1]-params[1])) < VTK_HEX_CONVERGED) &&
         ((fabs(pcoords[2]-params[2])) < VTK_HEX_CONVERGED) )
    {
      converged = 1;
    }

    // Test for bad divergence (S.Hirschberg 11.12.2001)
    else if ((fabs(pcoords[0]) > VTK_DIVERGED) ||
             (fabs(pcoords[1]) > VTK_DIVERGED) ||
             (fabs(pcoords[2]) > VTK_DIVERGED))
    {
      return -1;
    }

    //  if not converged, repeat
    else
    {
      params[0] = pcoords[0];
      params[1] = pcoords[1];
      params[2] = pcoords[2];
    }
  }

  //  if not converged, set the parametric coordinates to arbitrary values
  //  outside of element
  if ( !converged )
  {
    return -1;
  }

  this->InterpolationFunctions(pcoords, weights);

  if ( pcoords[0] >= -0.001 && pcoords[0] <= 1.001 &&
  pcoords[1] >= -0.001 && pcoords[1] <= 1.001 &&
  pcoords[2] >= -0.001 && pcoords[2] <= 1.001 )
  {
    if (closestPoint)
    {
      closestPoint[0] = x[0]; closestPoint[1] = x[1]; closestPoint[2] = x[2];
      dist2 = 0.0; //inside hexahedron
    }
    return 1;
  }
  else
  {
    double pc[3], w[20];
    if (closestPoint)
    {
      for (int i=0; i<3; i++) //only approximate, not really true for warped hexa
      {
        if (pcoords[i] < 0.0)
        {
          pc[i] = 0.0;
        }
        else if (pcoords[i] > 1.0)
        {
          pc[i] = 1.0;
        }
        else
        {
          pc[i] = pcoords[i];
        }
      }
      this->EvaluateLocation(subId, pc, closestPoint,
                             static_cast<double *>(w));
      dist2 = vtkMath::Distance2BetweenPoints(closestPoint,x);
    }
    return 0;
  }
}

//----------------------------------------------------------------------------
void vtkQuadraticHexahedron::EvaluateLocation(int& vtkNotUsed(subId),
                                              const double pcoords[3],
                                              double x[3], double *weights)
{
  int i, j;
  double pt[3];

  this->InterpolationFunctions(pcoords, weights);

  x[0] = x[1] = x[2] = 0.0;
  for (i=0; i<20; i++)
  {
    this->Points->GetPoint(i, pt);
    for (j=0; j<3; j++)
    {
      x[j] += pt[j] * weights[i];
    }
  }
}

//----------------------------------------------------------------------------
int vtkQuadraticHexahedron::CellBoundary( int subId,
                                          const double pcoords[3],
                                          vtkIdList *pts )
{
  for ( int i = 0; i < 8; ++ i ) // For each of the eight vertices of the hex
  {
    this->Hex->PointIds->SetId( i, this->PointIds->GetId( i ) );
  }

   return this->Hex->CellBoundary( subId, pcoords, pts );
}

//----------------------------------------------------------------------------
void vtkQuadraticHexahedron::Contour(double value,
                                     vtkDataArray* cellScalars,
                                     vtkIncrementalPointLocator* locator,
                                     vtkCellArray *verts,
                                     vtkCellArray* lines,
                                     vtkCellArray* polys,
                                     vtkPointData* inPd,
                                     vtkPointData* outPd,
                                     vtkCellData* inCd,
                                     vtkIdType cellId,
                                     vtkCellData* outCd)
{
  //subdivide into 8 linear hexs
  this->Subdivide(inPd,inCd,cellId, cellScalars);

  //contour each linear quad separately
  for (int i=0; i<8; i++) // For each subdivided hexahedron
  {
    for (int j=0; j<8; j++) // For each of the eight vertices of the hexhedron
    {
      this->Hex->Points->SetPoint(j,this->Points->GetPoint(LinearHexs[i][j]));
      this->Hex->PointIds->SetId(j,LinearHexs[i][j]);
      this->Scalars->SetValue(j,this->CellScalars->GetValue(LinearHexs[i][j]));
    }
    this->Hex->Contour(value,this->Scalars,locator,verts,lines,polys,
                       this->PointData,outPd,this->CellData,i,outCd);
  }
}

//----------------------------------------------------------------------------
// Line-hex intersection. Intersection has to occur within [0,1] parametric
// coordinates and with specified tolerance.
int vtkQuadraticHexahedron::IntersectWithLine(const double* p1, const double* p2,
                                              double tol, double& t,
                                              double* x, double* pcoords,
                                              int& subId)
{
  int intersection=0;
  double tTemp;
  double pc[3], xTemp[3];
  int faceNum;

  t = VTK_DOUBLE_MAX;
  for (faceNum=0; faceNum<6; faceNum++)
  {
    for (int i=0; i<8; i++)
    {
      this->Face->Points->SetPoint(i,
            this->Points->GetPoint(HexFaces[faceNum][i]));
    }

    if ( this->Face->IntersectWithLine(p1, p2, tol, tTemp,
                                       xTemp, pc, subId) )
    {
      intersection = 1;
      if ( tTemp < t )
      {
        t = tTemp;
        x[0] = xTemp[0]; x[1] = xTemp[1]; x[2] = xTemp[2];
        switch (faceNum)
        {
          case 0:
            pcoords[0] = 0.0; pcoords[1] = pc[1]; pcoords[2] = pc[0];
            break;

          case 1:
            pcoords[0] = 1.0; pcoords[1] = pc[0]; pcoords[2] = pc[1];
            break;

          case 2:
            pcoords[0] = pc[0]; pcoords[1] = 0.0; pcoords[2] = pc[1];
            break;

          case 3:
            pcoords[0] = pc[1]; pcoords[1] = 1.0; pcoords[2] = pc[0];
            break;

          case 4:
            pcoords[0] = pc[1]; pcoords[1] = pc[0]; pcoords[2] = 0.0;
            break;

          case 5:
            pcoords[0] = pc[0]; pcoords[1] = pc[1]; pcoords[2] = 1.0;
            break;
        }
      }
    }
  }
  return intersection;
}

//----------------------------------------------------------------------------
int vtkQuadraticHexahedron::Triangulate(
  int vtkNotUsed(index), vtkIdList *ptIds, vtkPoints *pts)
{
  // results in 22 tets
  ptIds->SetNumberOfIds(22*4);
  pts->SetNumberOfPoints(22*4);

  int p[4];
  p[0] = 8; p[1] = 11; p[2] = 0; p[3] = 16;
  vtkIdType counter = 0;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 1; p[1] = 9; p[2] = 8; p[3] = 17;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 2; p[1] = 10; p[2] = 9; p[3] = 18;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 11; p[1] = 8; p[2] = 10; p[3] = 12;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 10; p[1] = 8; p[2] = 9; p[3] = 12;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 11; p[1] = 10; p[2] = 3; p[3] = 19;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 12; p[1] = 9; p[2] = 10; p[3] = 13;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 13; p[1] = 10; p[2] = 12; p[3] = 14;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 11; p[1] = 12; p[2] = 10; p[3] = 14;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 14; p[1] = 11; p[2] = 12; p[3] = 15;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 12; p[1] = 11; p[2] = 8; p[3] = 16;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 4; p[1] = 15; p[2] = 12; p[3] = 16;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 15; p[1] = 11; p[2] = 12; p[3] = 16;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 9; p[1] = 12; p[2] = 8; p[3] = 17;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 12; p[1] = 13; p[2] = 5; p[3] = 17;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 13; p[1] = 12; p[2] = 9; p[3] = 17;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 13; p[1] = 9; p[2] = 10; p[3] = 18;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 13; p[1] = 14; p[2] = 6; p[3] = 18;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 14; p[1] = 13; p[2] = 10; p[3] = 18;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 11; p[1] = 14; p[2] = 10; p[3] = 19;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 14; p[1] = 15; p[2] = 7; p[3] = 19;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }
  p[0] = 15; p[1] = 14; p[2] = 11; p[3] = 19;
  for (int  i=0; i < 4; i++ )
  {
    ptIds->SetId(counter, this->PointIds->GetId(p[i]));
    pts->SetPoint(counter, this->Points->GetPoint(p[i]));
    counter++;
  }

  return 1;
}

//----------------------------------------------------------------------------
// Given parametric coordinates compute inverse Jacobian transformation
// matrix. Returns 9 elements of 3x3 inverse Jacobian plus interpolation
// function derivatives.
void vtkQuadraticHexahedron::JacobianInverse(const double pcoords[3],
                                             double **inverse,
                                             double derivs[60])
{
  int i, j;
  double *m[3], m0[3], m1[3], m2[3];
  double x[3];

  // compute interpolation function derivatives
  this->InterpolationDerivs(pcoords, derivs);

  // create Jacobian matrix
  m[0] = m0; m[1] = m1; m[2] = m2;
  for (i=0; i < 3; i++) //initialize matrix
  {
    m0[i] = m1[i] = m2[i] = 0.0;
  }

  for ( j=0; j < 20; j++ )
  {
    this->Points->GetPoint(j, x);
    for ( i=0; i < 3; i++ )
    {
      m0[i] += x[i] * derivs[j];
      m1[i] += x[i] * derivs[20 + j];
      m2[i] += x[i] * derivs[40 + j];
    }
  }

  // now find the inverse
  if ( vtkMath::InvertMatrix(m,inverse,3) == 0 )
  {
    vtkErrorMacro(<<"Jacobian inverse not found");
    return;
  }
}

//----------------------------------------------------------------------------
void vtkQuadraticHexahedron::Derivatives(int vtkNotUsed(subId),
                                         const double pcoords[3], const double *values,
                                         int dim, double *derivs)
{
  double *jI[3], j0[3], j1[3], j2[3];
  double functionDerivs[60], sum[3];
  int i, j, k;

  // compute inverse Jacobian and interpolation function derivatives
  jI[0] = j0; jI[1] = j1; jI[2] = j2;
  this->JacobianInverse(pcoords, jI, functionDerivs);

  // now compute derivates of values provided
  for (k=0; k < dim; k++) //loop over values per vertex
  {
    sum[0] = sum[1] = sum[2] = 0.0;
    for ( i=0; i < 20; i++) //loop over interp. function derivatives
    {
      sum[0] += functionDerivs[i] * values[dim*i + k];
      sum[1] += functionDerivs[20 + i] * values[dim*i + k];
      sum[2] += functionDerivs[40 + i] * values[dim*i + k];
    }
    for (j=0; j < 3; j++) //loop over derivative directions
    {
      derivs[3*k + j] = sum[0]*jI[j][0] + sum[1]*jI[j][1] + sum[2]*jI[j][2];
    }
  }
}


//----------------------------------------------------------------------------
// Clip this quadratic hex using scalar value provided. Like contouring,
// except that it cuts the hex to produce tetrahedra.
void vtkQuadraticHexahedron::Clip(double value,
                                  vtkDataArray* cellScalars,
                                  vtkIncrementalPointLocator* locator, vtkCellArray* tets,
                                  vtkPointData* inPd, vtkPointData* outPd,
                                  vtkCellData* inCd, vtkIdType cellId,
                                  vtkCellData* outCd, int insideOut)
{
  //create eight linear hexes
  this->Subdivide(inPd,inCd,cellId,cellScalars);

  //contour each linear hex separately
  for (int i=0; i<8; i++) // For each subdivided hexahedron
  {
    for (int j=0; j<8; j++) // For each of the eight vertices of the hexhedron
    {
      this->Hex->Points->SetPoint(j,this->Points->GetPoint(LinearHexs[i][j]));
      this->Hex->PointIds->SetId(j,LinearHexs[i][j]);
      this->Scalars->SetValue(j,this->CellScalars->GetValue(LinearHexs[i][j]));
    }
    this->Hex->Clip(value,this->Scalars,locator,tets,this->PointData,outPd,
                    this->CellData,i,outCd,insideOut);
  }
}

//----------------------------------------------------------------------------
// Compute interpolation functions for the twenty nodes.
void vtkQuadraticHexahedron::InterpolationFunctions(const double pcoords[3],
                                                    double weights[20])
{
  //VTK needs parametric coordinates to be between (0,1). Isoparametric
  //shape functions are formulated between (-1,1). Here we do a
  //coordinate system conversion from (0,1) to (-1,1).
  double r = 2.0*(pcoords[0]-0.5);
  double s = 2.0*(pcoords[1]-0.5);
  double t = 2.0*(pcoords[2]-0.5);

  double rm = 1.0 - r;
  double rp = 1.0 + r;
  double sm = 1.0 - s;
  double sp = 1.0 + s;
  double tm = 1.0 - t;
  double tp = 1.0 + t;
  double r2 = 1.0 - r*r;
  double s2 = 1.0 - s*s;
  double t2 = 1.0 - t*t;

  //The eight corner points
  weights[0] = 0.125 * rm * sm * tm * (-r - s - t - 2.0);
  weights[1] = 0.125 * rp * sm * tm * ( r - s - t - 2.0);
  weights[2] = 0.125 * rp * sp * tm * ( r + s - t - 2.0);
  weights[3] = 0.125 * rm * sp * tm * (-r + s - t - 2.0);
  weights[4] = 0.125 * rm * sm * tp * (-r - s + t - 2.0);
  weights[5] = 0.125 * rp * sm * tp * ( r - s + t - 2.0);
  weights[6] = 0.125 * rp * sp * tp * ( r + s + t - 2.0);
  weights[7] = 0.125 * rm * sp * tp * (-r + s + t - 2.0);

  //The mid-edge nodes
  weights[8] =  0.25 * r2 * sm * tm;
  weights[9] =  0.25 * s2 * rp * tm;
  weights[10] = 0.25 * r2 * sp * tm;
  weights[11] = 0.25 * s2 * rm * tm;
  weights[12] = 0.25 * r2 * sm * tp;
  weights[13] = 0.25 * s2 * rp * tp;
  weights[14] = 0.25 * r2 * sp * tp;
  weights[15] = 0.25 * s2 * rm * tp;
  weights[16] = 0.25 * t2 * rm * sm;
  weights[17] = 0.25 * t2 * rp * sm;
  weights[18] = 0.25 * t2 * rp * sp;
  weights[19] = 0.25 * t2 * rm * sp;
}

//----------------------------------------------------------------------------
// Derivatives in parametric space.
void vtkQuadraticHexahedron::InterpolationDerivs(const double pcoords[3],
                                                 double derivs[60])
{
  //VTK needs parametric coordinates to be between (0,1). Isoparametric
  //shape functions are formulated between (-1,1). Here we do a
  //coordinate system conversion from (0,1) to (-1,1).
  double r = 2.0*(pcoords[0]-0.5);
  double s = 2.0*(pcoords[1]-0.5);
  double t = 2.0*(pcoords[2]-0.5);

  double rm = 1.0 - r;
  double rp = 1.0 + r;
  double sm = 1.0 - s;
  double sp = 1.0 + s;
  double tm = 1.0 - t;
  double tp = 1.0 + t;

  //r-derivatives
  derivs[0] = -0.125*(sm*tm - 2.0*r*sm*tm - s*sm*tm - t*sm*tm - 2.0*sm*tm);
  derivs[1] =  0.125*(sm*tm + 2.0*r*sm*tm - s*sm*tm - t*sm*tm - 2.0*sm*tm);
  derivs[2] =  0.125*(sp*tm + 2.0*r*sp*tm + s*sp*tm - t*sp*tm - 2.0*sp*tm);
  derivs[3] = -0.125*(sp*tm - 2.0*r*sp*tm + s*sp*tm - t*sp*tm - 2.0*sp*tm);
  derivs[4] = -0.125*(sm*tp - 2.0*r*sm*tp - s*sm*tp + t*sm*tp - 2.0*sm*tp);
  derivs[5] =  0.125*(sm*tp + 2.0*r*sm*tp - s*sm*tp + t*sm*tp - 2.0*sm*tp);
  derivs[6] =  0.125*(sp*tp + 2.0*r*sp*tp + s*sp*tp + t*sp*tp - 2.0*sp*tp);
  derivs[7] = -0.125*(sp*tp - 2.0*r*sp*tp + s*sp*tp + t*sp*tp - 2.0*sp*tp);
  derivs[8] = -0.5*r*sm*tm;
  derivs[9] =  0.25*(tm - s*s*tm);
  derivs[10] = -0.5*r*sp*tm;
  derivs[11] = -0.25*(tm - s*s*tm);
  derivs[12] = -0.5*r*sm*tp;
  derivs[13] =  0.25*(tp - s*s*tp);
  derivs[14] = -0.5*r*sp*tp;
  derivs[15] = -0.25*(tp - s*s*tp);
  derivs[16] = -0.25*(sm - t*t*sm);
  derivs[17] =  0.25*(sm - t*t*sm);
  derivs[18] =  0.25*(sp - t*t*sp);
  derivs[19] = -0.25*(sp - t*t*sp);

  //s-derivatives
  derivs[20] = -0.125*(rm*tm - 2.0*s*rm*tm - r*rm*tm - t*rm*tm - 2.0*rm*tm);
  derivs[21] = -0.125*(rp*tm - 2.0*s*rp*tm + r*rp*tm - t*rp*tm - 2.0*rp*tm);
  derivs[22] =  0.125*(rp*tm + 2.0*s*rp*tm + r*rp*tm - t*rp*tm - 2.0*rp*tm);
  derivs[23] =  0.125*(rm*tm + 2.0*s*rm*tm - r*rm*tm - t*rm*tm - 2.0*rm*tm);
  derivs[24] = -0.125*(rm*tp - 2.0*s*rm*tp - r*rm*tp + t*rm*tp - 2.0*rm*tp);
  derivs[25] = -0.125*(rp*tp - 2.0*s*rp*tp + r*rp*tp + t*rp*tp - 2.0*rp*tp);
  derivs[26] =  0.125*(rp*tp + 2.0*s*rp*tp + r*rp*tp + t*rp*tp - 2.0*rp*tp);
  derivs[27] =  0.125*(rm*tp + 2.0*s*rm*tp - r*rm*tp + t*rm*tp - 2.0*rm*tp);
  derivs[28] = -0.25*(tm - r*r*tm);
  derivs[29] = -0.5*s*rp*tm;
  derivs[30] =  0.25*(tm - r*r*tm);
  derivs[31] = -0.5*s*rm*tm;
  derivs[32] = -0.25*(tp - r*r*tp);
  derivs[33] = -0.5*s*rp*tp;
  derivs[34] =  0.25*(tp - r*r*tp);
  derivs[35] = -0.5*s*rm*tp;
  derivs[36] = -0.25*(rm - t*t*rm);
  derivs[37] = -0.25*(rp - t*t*rp);
  derivs[38] =  0.25*(rp - t*t*rp);
  derivs[39] =  0.25*(rm - t*t*rm);

  //t-derivatives
  derivs[40] = -0.125*(rm*sm - 2.0*t*rm*sm - r*rm*sm - s*rm*sm - 2.0*rm*sm);
  derivs[41] = -0.125*(rp*sm - 2.0*t*rp*sm + r*rp*sm - s*rp*sm - 2.0*rp*sm);
  derivs[42] = -0.125*(rp*sp - 2.0*t*rp*sp + r*rp*sp + s*rp*sp - 2.0*rp*sp);
  derivs[43] = -0.125*(rm*sp - 2.0*t*rm*sp - r*rm*sp + s*rm*sp - 2.0*rm*sp);
  derivs[44] =  0.125*(rm*sm + 2.0*t*rm*sm - r*rm*sm - s*rm*sm - 2.0*rm*sm);
  derivs[45] =  0.125*(rp*sm + 2.0*t*rp*sm + r*rp*sm - s*rp*sm - 2.0*rp*sm);
  derivs[46] =  0.125*(rp*sp + 2.0*t*rp*sp + r*rp*sp + s*rp*sp - 2.0*rp*sp);
  derivs[47] =  0.125*(rm*sp + 2.0*t*rm*sp - r*rm*sp + s*rm*sp - 2.0*rm*sp);
  derivs[48] = -0.25*(sm - r*r*sm);
  derivs[49] = -0.25*(rp - s*s*rp);
  derivs[50] = -0.25*(sp - r*r*sp);
  derivs[51] = -0.25*(rm - s*s*rm);
  derivs[52] =  0.25*(sm - r*r*sm);
  derivs[53] =  0.25*(rp - s*s*rp);
  derivs[54] =  0.25*(sp - r*r*sp);
  derivs[55] =  0.25*(rm - s*s*rm);
  derivs[56] = -0.5*t*rm*sm;
  derivs[57] = -0.5*t*rp*sm;
  derivs[58] = -0.5*t*rp*sp;
  derivs[59] = -0.5*t*rm*sp;

  // we compute derivatives in in [-1; 1] but we need them in [ 0; 1]
  for(int i = 0; i < 60; i++)
    derivs[i] *= 2;

}

//----------------------------------------------------------------------------
static double vtkQHexCellPCoords[60] = {0.0,0.0,0.0, 1.0,0.0,0.0, 1.0,1.0,0.0,
                                        0.0,1.0,0.0, 0.0,0.0,1.0, 1.0,0.0,1.0,
                                        1.0,1.0,1.0, 0.0,1.0,1.0, 0.5,0.0,0.0,
                                        1.0,0.5,0.0, 0.5,1.0,0.0, 0.0,0.5,0.0,
                                        0.5,0.0,1.0, 1.0,0.5,1.0, 0.5,1.0,1.0,
                                        0.0,0.5,1.0, 0.0,0.0,0.5, 1.0,0.0,0.5,
                                        1.0,1.0,0.5, 0.0,1.0,0.5};
double *vtkQuadraticHexahedron::GetParametricCoords()
{
  return vtkQHexCellPCoords;
}

//----------------------------------------------------------------------------
void vtkQuadraticHexahedron::PrintSelf(ostream& os, vtkIndent indent)
{
  this->Superclass::PrintSelf(os,indent);

  os << indent << "Edge:\n";
  this->Edge->PrintSelf(os,indent.GetNextIndent());
  os << indent << "Face:\n";
  this->Face->PrintSelf(os,indent.GetNextIndent());
  os << indent << "Hex:\n";
  this->Hex->PrintSelf(os,indent.GetNextIndent());
  os << indent << "PointData:\n";
  this->PointData->PrintSelf(os,indent.GetNextIndent());
  os << indent << "CellData:\n";
  this->CellData->PrintSelf(os,indent.GetNextIndent());
  os << indent << "Scalars:\n";
  this->Scalars->PrintSelf(os,indent.GetNextIndent());
}