Intrepid_PointToolsDef.hpp

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#ifdef _MSC_VER
#include "winmath.h"
#endif


namespace Intrepid {

  template<class Scalar, class ArrayType>
  void PointTools::getLattice(ArrayType &pts ,
                              const shards::CellTopology& cellType ,
                              const int order ,
                              const int offset ,
                              const EPointType pointType ) 
  {
    switch (pointType) {
    case POINTTYPE_EQUISPACED:
      getEquispacedLattice<Scalar,ArrayType>( cellType , pts , order , offset );
      break;
    case POINTTYPE_WARPBLEND:

      getWarpBlendLattice<Scalar,ArrayType>( cellType , pts , order , offset );
      break;
    default:
      TEUCHOS_TEST_FOR_EXCEPTION( true ,
                          std::invalid_argument ,
                          "PointTools::getLattice: invalid EPointType" );
    }
    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::getGaussPoints( ArrayType &pts ,
                                   const int order )
  {

    Scalar *z = new Scalar[order+1];
    Scalar *w = new Scalar[order+1];

    IntrepidPolylib::zwgj( z , w , order + 1 , 0.0 , 0.0 );
    for (int i=0;i<order+1;i++) {
      pts(i,0) = z[i];
    }

    delete []z;
    delete []w;
  }

  
  template<class Scalar, class ArrayType>
  void PointTools::getEquispacedLattice(const shards::CellTopology& cellType ,
                                        ArrayType &points ,
                                        const int order ,
                                        const int offset )

  {
    switch (cellType.getKey()) {
    case shards::Tetrahedron<4>::key:
    case shards::Tetrahedron<8>::key:
    case shards::Tetrahedron<10>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 3 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getEquispacedLattice): points argument is ill-sized." );
      getEquispacedLatticeTetrahedron<Scalar,ArrayType>( points , order , offset );
      break;
    case shards::Triangle<3>::key:
    case shards::Triangle<4>::key:
    case shards::Triangle<6>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 2 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getEquispacedLattice): points argument is ill-sized." );
      getEquispacedLatticeTriangle<Scalar,ArrayType>( points , order , offset );
      break;
    case shards::Line<2>::key:
    case shards::Line<3>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 1 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getEquispacedLattice): points argument is ill-sized." );
      getEquispacedLatticeLine<Scalar,ArrayType>( points , order , offset );
      break;
    default:
      TEUCHOS_TEST_FOR_EXCEPTION( true , std::invalid_argument ,
                          ">>> ERROR (Intrepid::PointTools::getEquispacedLattice): Illegal cell type" );
    }
    
  }

  template<class Scalar, class ArrayType>
  void PointTools::getWarpBlendLattice( const shards::CellTopology& cellType ,
                                        ArrayType &points ,
                                        const int order ,
                                        const int offset )

  {

    switch (cellType.getKey()) {
    case shards::Tetrahedron<4>::key:
    case shards::Tetrahedron<8>::key:
    case shards::Tetrahedron<10>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 3 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getWarpBlendLattice): points argument is ill-sized." );

      getWarpBlendLatticeTetrahedron<Scalar,ArrayType>( points , order , offset );
      break;
    case shards::Triangle<3>::key:
    case shards::Triangle<4>::key:
    case shards::Triangle<6>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 2 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getWarpBlendLattice): points argument is ill-sized." );                    

      getWarpBlendLatticeTriangle<Scalar,ArrayType>( points , order , offset );
      break;
    case shards::Line<2>::key:
    case shards::Line<3>::key:
      TEUCHOS_TEST_FOR_EXCEPTION( ( points.dimension(0) != getLatticeSize( cellType , order , offset ) ) 
                          || points.dimension(1) != 1 ,
                          std::invalid_argument ,
                          ">>> ERROR(PointTools::getWarpBlendLattice): points argument is ill-sized." );

      getWarpBlendLatticeLine<Scalar,ArrayType>( points , order , offset );
      break;
    default:
      TEUCHOS_TEST_FOR_EXCEPTION( true , std::invalid_argument ,
                          ">>> ERROR (Intrepid::PointTools::getWarpBlendLattice): Illegal cell type" );
    }
    
  }

  template<class Scalar, class ArrayType>
  void PointTools::getEquispacedLatticeLine( ArrayType &points ,
                                             const int order ,
                                             const int offset )
  {
    TEUCHOS_TEST_FOR_EXCEPTION( order < 0 ,
                        std::invalid_argument ,
                        ">>> ERROR (Intrepid::PointTools::getEquispacedLatticeLine): order must be positive" );
    if (order == 0) {
      points(0,0) = 0.0;
    }
    else {
      const Scalar h = 2.0 / order;
      
      for (int i=offset;i<=order-offset;i++) {
        points(i-offset,0) = -1.0 + h * (Scalar) i;
      }
    }

    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::getEquispacedLatticeTriangle( ArrayType &points ,
                                                 const int order ,
                                                 const int offset )
  {
    TEUCHOS_TEST_FOR_EXCEPTION( order <= 0 ,
                        std::invalid_argument ,
                        ">>> ERROR (Intrepid::PointTools::getEquispacedLatticeLine): order must be positive" );

    const Scalar h = 1.0 / order;
    int cur = 0;

    for (int i=offset;i<=order-offset;i++) {
      for (int j=offset;j<=order-i-offset;j++) {
        points(cur,0) = (Scalar)0.0 + (Scalar) j * h ;
        points(cur,1) = (Scalar)0.0 + (Scalar) i * h;
        cur++;
      }
    }

    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::getEquispacedLatticeTetrahedron( ArrayType &points ,
                                                    const int order ,
                                                    const int offset )
  {
    TEUCHOS_TEST_FOR_EXCEPTION( (order <= 0) ,
                        std::invalid_argument ,
                        ">>> ERROR (Intrepid::PointTools::getEquispacedLatticeTetrahedron): order must be positive" );

    const Scalar h = 1.0 / order;
    int cur = 0;

    for (int i=offset;i<=order-offset;i++) {
      for (int j=offset;j<=order-i-offset;j++) {
        for (int k=offset;k<=order-i-j-offset;k++) {
          points(cur,0) = (Scalar) k * h;
          points(cur,1) = (Scalar) j * h;
          points(cur,2) = (Scalar) i * h;
          cur++;
        }
      }
    }

    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::getWarpBlendLatticeLine( ArrayType &points ,
                                            const int order ,
                                            const int offset )
  {
    Scalar *z = new Scalar[order+1];
    Scalar *w = new Scalar[order+1];
    
    // order is order of polynomial degree.  The Gauss-Lobatto points are accurate
    // up to degree 2*i-1
    IntrepidPolylib::zwglj( z , w , order+1 , 0.0 , 0.0 );

    for (int i=offset;i<=order-offset;i++) {
      points(i-offset,0) = z[i];
    }

    delete []z;
    delete []w;

    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::warpFactor( const int order , 
                              const ArrayType &xnodes ,
                              const ArrayType &xout ,
                              ArrayType &warp)
  {
    TEUCHOS_TEST_FOR_EXCEPTION( ( warp.dimension(0) != xout.dimension(0) ) ,
                        std::invalid_argument ,
                        ">>> ERROR (PointTools::warpFactor): xout and warp must be same size." );

    warp.initialize();

    FieldContainer<Scalar> d( xout.dimension(0) );
    d.initialize();

    FieldContainer<Scalar> xeq( order + 1 ,1);
    PointTools::getEquispacedLatticeLine<Scalar,ArrayType>( xeq , order , 0 );
    xeq.resize( order + 1 );

    TEUCHOS_TEST_FOR_EXCEPTION( ( xeq.dimension(0) != xnodes.dimension(0) ) ,
                        std::invalid_argument ,
                        ">>> ERROR (PointTools::warpFactor): xeq and xnodes must be same size." );
    
    for (int i=0;i<=order;i++) {

      for (int k=0;k<d.dimension(0);k++) {
        d(k) = xnodes(i) - xeq(i);
      }

      for (int j=1;j<order;j++) {
        if (i != j) {
          for (int k=0;k<d.dimension(0);k++) {
            d(k) = d(k) * ( (xout(k)-xeq(j)) / (xeq(i)-xeq(j)) );
          }
        }
      }
      
      // deflate end roots
      if ( i != 0 ) {
        for (int k=0;k<d.dimension(0);k++) {
          d(k) = -d(k) / (xeq(i) - xeq(0));
        }
      }

      if (i != order ) {
        for (int k=0;k<d.dimension(0);k++) {
          d(k) = d(k) / (xeq(i) - xeq(order));
        }
      }

      for (int k=0;k<d.dimension(0);k++) {
        warp(k) += d(k);
      }

    }


    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::getWarpBlendLatticeTriangle( ArrayType &points ,
                                                const int order ,
                                                const int offset  )
  {
    /* get Gauss-Lobatto points */

    Intrepid::FieldContainer<Scalar> gaussX( order + 1 , 1 );
    
    PointTools::getWarpBlendLatticeLine<Scalar,FieldContainer<Scalar> >( gaussX , order , 0 );
    
    gaussX.resize(gaussX.dimension(0));

    Scalar alpopt[] = {0.0000,0.0000,1.4152,0.1001,0.2751,0.9800,1.0999,
                        1.2832,1.3648, 1.4773, 1.4959, 1.5743, 1.5770, 1.6223, 1.6258};

    Scalar alpha;

    if (order >= 1 && order < 16) {
      alpha = alpopt[order-1];
    }
    else {
      alpha = 5.0 / 3.0;
    }

    const int p = order; /* switch to Warburton's notation */
    int N = (p+1)*(p+2)/2;
    
    /* equidistributed nodes on equilateral triangle */
    Intrepid::FieldContainer<Scalar> L1( N );
    Intrepid::FieldContainer<Scalar> L2( N );
    Intrepid::FieldContainer<Scalar> L3( N );     
    Intrepid::FieldContainer<Scalar> X(N);
    Intrepid::FieldContainer<Scalar> Y(N);

    int sk = 0;
    for (int n=1;n<=p+1;n++) {
      for (int m=1;m<=p+2-n;m++) {
        L1(sk) = (n-1) / (Scalar)p;
        L3(sk) = (m-1) / (Scalar)p;
        L2(sk) = 1.0 - L1(sk) - L3(sk);
        sk++;
      }
    }
    
    for (int n=0;n<N;n++) {
      X(n) = -L2(n) + L3(n);
      Y(n) = (-L2(n) - L3(n) + 2*L1(n))/1.7320508075688772;
    }

    /* get blending function for each node at each edge */
    Intrepid::FieldContainer<Scalar> blend1(N);
    Intrepid::FieldContainer<Scalar> blend2(N);
    Intrepid::FieldContainer<Scalar> blend3(N);
    
    for (int n=0;n<N;n++) {
      blend1(n) = 4.0 * L2(n) * L3(n);
      blend2(n) = 4.0 * L1(n) * L3(n);
      blend3(n) = 4.0 * L1(n) * L2(n);
    }
    
    /* get difference of each barycentric coordinate */
    Intrepid::FieldContainer<Scalar> L3mL2(N);
    Intrepid::FieldContainer<Scalar> L1mL3(N);
    Intrepid::FieldContainer<Scalar> L2mL1(N);

    for (int k=0;k<N;k++) {
      L3mL2(k) = L3(k)-L2(k);
      L1mL3(k) = L1(k)-L3(k);
      L2mL1(k) = L2(k)-L1(k);
    }

    FieldContainer<Scalar> warpfactor1(N);
    FieldContainer<Scalar> warpfactor2(N);
    FieldContainer<Scalar> warpfactor3(N);
    
    warpFactor<Scalar,FieldContainer<Scalar> >( order , gaussX , L3mL2 , warpfactor1 );
    warpFactor<Scalar,FieldContainer<Scalar> >( order , gaussX , L1mL3 , warpfactor2 );
    warpFactor<Scalar,FieldContainer<Scalar> >( order , gaussX , L2mL1 , warpfactor3 );

    FieldContainer<Scalar> warp1(N);
    FieldContainer<Scalar> warp2(N);
    FieldContainer<Scalar> warp3(N);

    for (int k=0;k<N;k++) {
      warp1(k) = blend1(k) * warpfactor1(k) *
        ( 1.0 + alpha * alpha * L1(k) * L1(k) );
      warp2(k) = blend2(k) * warpfactor2(k) *
        ( 1.0 + alpha * alpha * L2(k) * L2(k) );
      warp3(k) = blend3(k) * warpfactor3(k) *
        ( 1.0 + alpha * alpha * L3(k) * L3(k) );
    }

    for (int k=0;k<N;k++) {
      X(k) += 1.0 * warp1(k) + cos( 2.0 * M_PI / 3.0 ) * warp2(k) + cos(4*M_PI/3.0) * warp3(k);
      Y(k) += 0.0 * warp1(k) + sin( 2.0 * M_PI / 3.0 ) * warp2(k) + sin( 4*M_PI/3.0) * warp3(k);
    }

    FieldContainer<Scalar> warXY(N,2);
    
    for (int k=0;k<N;k++) {
      warXY(k,0) = X(k);
      warXY(k,1) = Y(k);
    }


    // finally, convert the warp-blend points to the correct triangle
    FieldContainer<Scalar> warburtonVerts(1,3,2);
    warburtonVerts(0,0,0) = -1.0;
    warburtonVerts(0,0,1) = -1.0/sqrt(3.0);
    warburtonVerts(0,1,0) = 1.0;
    warburtonVerts(0,1,1) = -1.0/sqrt(3.0);
    warburtonVerts(0,2,0) = 0.0;
    warburtonVerts(0,2,1) = 2.0/sqrt(3.0);

    FieldContainer<Scalar> refPts(N,2);

    Intrepid::CellTools<Scalar>::mapToReferenceFrame( refPts ,
                                                      warXY ,
                                                      warburtonVerts ,
                                                      shards::getCellTopologyData< shards::Triangle<3> >(),
                                                      0 );

    // now write from refPts into points, taking care of offset
    int noffcur = 0;  // index into refPts
    int offcur = 0;   // index int points
    for (int i=0;i<=order;i++) {
      for (int j=0;j<=order-i;j++) {
        if ( (i >= offset) && (i <= order-offset) &&
              (j >= offset) && (j <= order-i-offset) ) {
          points(offcur,0) = refPts(noffcur,0);
          points(offcur,1) = refPts(noffcur,1);
          offcur++;
        }
        noffcur++;
      }
    }

    return;
  }
  

  template<class Scalar, class ArrayType>
  void PointTools::warpShiftFace3D( const int order ,
                                    const Scalar pval ,
                                    const ArrayType &L1,
                                    const ArrayType &L2,
                                    const ArrayType &L3,
                                    const ArrayType &L4,
                                    ArrayType &dxy)
  {
    evalshift<Scalar,ArrayType>(order,pval,L2,L3,L4,dxy);
    return;
  }

  template<class Scalar, class ArrayType>
  void PointTools::evalshift( const int order ,
                              const Scalar pval ,
                              const ArrayType &L1 ,
                              const ArrayType &L2 ,
                              const ArrayType &L3 ,
                              ArrayType &dxy )
  {
    // get Gauss-Lobatto-nodes
    FieldContainer<Scalar> gaussX(order+1,1);
    PointTools::getWarpBlendLatticeLine<Scalar,FieldContainer<Scalar> >( gaussX , order , 0 );
    gaussX.resize(order+1);
    const int N = L1.dimension(0);
    
    // Warburton code reverses them
    for (int k=0;k<=order;k++) {
      gaussX(k) *= -1.0;
    }

    // blending function at each node for each edge
    FieldContainer<Scalar> blend1(N);
    FieldContainer<Scalar> blend2(N);
    FieldContainer<Scalar> blend3(N);

    for (int i=0;i<N;i++) {
      blend1(i) = L2(i) * L3(i);
      blend2(i) = L1(i) * L3(i);
      blend3(i) = L1(i) * L2(i);
    }

    // amount of warp for each node for each edge
    FieldContainer<Scalar> warpfactor1(N);
    FieldContainer<Scalar> warpfactor2(N);
    FieldContainer<Scalar> warpfactor3(N);

    // differences of barycentric coordinates 
    FieldContainer<Scalar> L3mL2(N);
    FieldContainer<Scalar> L1mL3(N);
    FieldContainer<Scalar> L2mL1(N);
    
    for (int k=0;k<N;k++) {
      L3mL2(k) = L3(k)-L2(k);
      L1mL3(k) = L1(k)-L3(k);
      L2mL1(k) = L2(k)-L1(k);
    }
    
    evalwarp<Scalar,FieldContainer<Scalar> >( warpfactor1 , order , gaussX , L3mL2 );
    evalwarp<Scalar,FieldContainer<Scalar> >( warpfactor2 , order , gaussX , L1mL3 );
    evalwarp<Scalar,FieldContainer<Scalar> >( warpfactor3 , order , gaussX , L2mL1 );
    
    for (int k=0;k<N;k++) {
      warpfactor1(k) *= 4.0;
      warpfactor2(k) *= 4.0;
      warpfactor3(k) *= 4.0;      
    }

    FieldContainer<Scalar> warp1(N);
    FieldContainer<Scalar> warp2(N);
    FieldContainer<Scalar> warp3(N);
    
    for (int k=0;k<N;k++) {
      warp1(k) = blend1(k) * warpfactor1(k) *
        ( 1.0 + pval * pval * L1(k) * L1(k) );
      warp2(k) = blend2(k) * warpfactor2(k) *
        ( 1.0 + pval * pval * L2(k) * L2(k) );
      warp3(k) = blend3(k) * warpfactor3(k) *
        ( 1.0 + pval * pval * L3(k) * L3(k) );
    }

    for (int k=0;k<N;k++) {
      dxy(k,0) = 1.0 * warp1(k) + cos( 2.0 * M_PI / 3.0 ) * warp2(k) + cos( 4.0*M_PI/3.0 ) * warp3(k);
      dxy(k,1) = 0.0 * warp1(k) + sin( 2.0 * M_PI / 3.0 ) * warp2(k) + sin( 4.0*M_PI/3.0 ) * warp3(k);
    }

    return;

  }

  /* one-d edge warping function */
  template<class Scalar, class ArrayType>
  void PointTools::evalwarp( ArrayType &warp ,
                            const int order ,
                            const ArrayType &xnodes ,
                            const ArrayType &xout )
  {
    FieldContainer<Scalar> xeq(order+1);
    FieldContainer<Scalar> d(xout.dimension(0));

    d.initialize();

    for (int i=0;i<=order;i++) {
      xeq(i) = -1.0 + 2.0 * ( order - i ) / order;
    }



    for (int i=0;i<=order;i++) {
      d.initialize( xnodes(i) - xeq(i) );
      for (int j=1;j<order;j++) {
        if (i!=j) {
          for (int k=0;k<d.dimension(0);k++) {
            d(k) = d(k) * (xout(k)-xeq(j))/(xeq(i)-xeq(j));
          }
        }
      }
      if (i!=0) {
        for (int k=0;k<d.dimension(0);k++) {
          d(k) = -d(k)/(xeq(i)-xeq(0));
        }
      }
      if (i!=order) {
        for (int k=0;k<d.dimension(0);k++) {
          d(k) = d(k)/(xeq(i)-xeq(order));
        }
      }
      
      for (int k=0;k<d.dimension(0);k++) {
        warp(k) += d(k);
      } 
    }    

    return;
  }


  template<class Scalar, class ArrayType>
  void PointTools::getWarpBlendLatticeTetrahedron(ArrayType &points ,
                                                  const int order ,
                                                  const int offset  )
  {
    Scalar alphastore[] = { 0,0,0,0.1002, 1.1332,1.5608,1.3413,1.2577,1.1603,
                            1.10153,0.6080,0.4523,0.8856,0.8717,0.9655};
    Scalar alpha;

    if (order <= 15) {
      alpha = alphastore[order-1]; 
    }
    else {
      alpha = 1.0;
    }

    const int N = (order+1)*(order+2)*(order+3)/6;
    Scalar tol = 1.e-10;

    FieldContainer<Scalar> shift(N,3);
    shift.initialize();

    /* create 3d equidistributed nodes on Warburton tet */
    FieldContainer<Scalar> equipoints(N,3);
    int sk = 0;
    for (int n=0;n<=order;n++) {
      for (int m=0;m<=order-n;m++) {
        for (int q=0;q<=order-n-m;q++) {
          equipoints(sk,0) = -1.0 + (q * 2.0 ) / order;
          equipoints(sk,1) = -1.0 + (m * 2.0 ) / order;
          equipoints(sk,2) = -1.0 + (n * 2.0 ) / order;
          sk++;
        }
      }
    }
    

    /* construct barycentric coordinates for equispaced lattice */
    FieldContainer<Scalar> L1(N);
    FieldContainer<Scalar> L2(N);
    FieldContainer<Scalar> L3(N);
    FieldContainer<Scalar> L4(N);
    for (int i=0;i<N;i++) {
      L1(i) = (1.0 + equipoints(i,2)) / 2.0;
      L2(i) = (1.0 + equipoints(i,1)) / 2.0;
      L3(i) = -(1.0 + equipoints(i,0) + equipoints(i,1) + equipoints(i,2)) / 2.0;
      L4(i) = (1.0 + equipoints(i,0)) / 2.0;
    }
    
    
    /* vertices of Warburton tet */
    FieldContainer<Scalar> warVerts(4,3);
    warVerts(0,0) = -1.0;
    warVerts(0,1) = -1.0/sqrt(3.0);
    warVerts(0,2) = -1.0/sqrt(6.0);
    warVerts(1,0) = 1.0;
    warVerts(1,1) = -1.0/sqrt(3.0);
    warVerts(1,2) = -1.0/sqrt(6.0);
    warVerts(2,0) = 0.0;
    warVerts(2,1) = 2.0 / sqrt(3.0);
    warVerts(2,2) = -1.0/sqrt(6.0);
    warVerts(3,0) = 0.0;
    warVerts(3,1) = 0.0;
    warVerts(3,2) = 3.0 / sqrt(6.0);


    /* tangents to faces */
    FieldContainer<Scalar> t1(4,3);
    FieldContainer<Scalar> t2(4,3);
    for (int i=0;i<3;i++) {
      t1(0,i) = warVerts(1,i) - warVerts(0,i);
      t1(1,i) = warVerts(1,i) - warVerts(0,i);
      t1(2,i) = warVerts(2,i) - warVerts(1,i);
      t1(3,i) = warVerts(2,i) - warVerts(0,i);
      t2(0,i) = warVerts(2,i) - 0.5 * ( warVerts(0,i) + warVerts(1,i) );
      t2(1,i) = warVerts(3,i) - 0.5 * ( warVerts(0,i) + warVerts(1,i) );
      t2(2,i) = warVerts(3,i) - 0.5 * ( warVerts(1,i) + warVerts(2,i) );
      t2(3,i) = warVerts(3,i) - 0.5 * ( warVerts(0,i) + warVerts(2,i) );
    }

    /* normalize tangents */
    for (int n=0;n<4;n++) {
      /* Compute norm of t1(n) and t2(n) */
      Scalar normt1n = 0.0;
      Scalar normt2n = 0.0;
      for (int i=0;i<3;i++) {
        normt1n += (t1(n,i) * t1(n,i));
        normt2n += (t2(n,i) * t2(n,i));
      }
      normt1n = sqrt(normt1n);
      normt2n = sqrt(normt2n);
      /* normalize each tangent now */
      for (int i=0;i<3;i++) {
        t1(n,i) /= normt1n;
        t2(n,i) /= normt2n;
      }
    }

    /* undeformed coordinates */
    FieldContainer<Scalar> XYZ(N,3);
    for (int i=0;i<N;i++) {
      for (int j=0;j<3;j++) {
        XYZ(i,j) = L3(i)*warVerts(0,j) + L4(i)*warVerts(1,j) + L2(i)*warVerts(2,j) + L1(i)*warVerts(3,j);
      }
    }

    for (int face=1;face<=4;face++) {
      FieldContainer<Scalar> La, Lb, Lc, Ld;
      FieldContainer<Scalar> warp(N,2);
      FieldContainer<Scalar> blend(N);
      FieldContainer<Scalar> denom(N);
      switch (face) {
      case 1:
        La = L1; Lb = L2; Lc = L3; Ld = L4; break;
      case 2:
        La = L2; Lb = L1; Lc = L3; Ld = L4; break;
      case 3:
        La = L3; Lb = L1; Lc = L4; Ld = L2; break;
      case 4:
        La = L4; Lb = L1; Lc = L3; Ld = L2; break;
      }
      
      /* get warp tangential to face */
      warpShiftFace3D<Scalar,ArrayType>(order,alpha,La,Lb,Lc,Ld,warp);
      
      for (int k=0;k<N;k++) {
        blend(k) = Lb(k) * Lc(k) * Ld(k);
      }

      for (int k=0;k<N;k++) {
        denom(k) = (Lb(k) + 0.5 * La(k)) * (Lc(k) + 0.5*La(k)) * (Ld(k) + 0.5 * La(k));
      }

      for (int k=0;k<N;k++) {
        if (denom(k) > tol) {
          blend(k) *= ( 1.0 + alpha * alpha * La(k) * La(k) ) / denom(k);
        }
      }  


      // compute warp and blend
      for (int k=0;k<N;k++) {
        for (int j=0;j<3;j++) {
          shift(k,j) = shift(k,j) + blend(k) * warp(k,0) * t1(face-1,j)
            + blend(k) * warp(k,1) * t2(face-1,j);
        }
      }

      for (int k=0;k<N;k++) {
        if (La(k) < tol && ( Lb(k) < tol || Lc(k) < tol || Ld(k) < tol )) {
          for (int j=0;j<3;j++) {
            shift(k,j) = warp(k,0) * t1(face-1,j) + warp(k,1) * t2(face-1,j);
          }
        }
      }
      
    }

    FieldContainer<Scalar> updatedPoints(N,3);
    for (int k=0;k<N;k++) {
      for (int j=0;j<3;j++) {
        updatedPoints(k,j) = XYZ(k,j) + shift(k,j);
      }
    }

    warVerts.resize( 1 , 4 , 3 );

    // now we convert to Pavel's reference triangle!
    FieldContainer<Scalar> refPts(N,3);
    CellTools<Scalar>::mapToReferenceFrame( refPts ,updatedPoints ,
                                            warVerts ,
                                            shards::getCellTopologyData<shards::Tetrahedron<4> >() ,
                                            0 );

    // now write from refPts into points, taking offset into account
    int noffcur = 0;
    int offcur = 0;
    for (int i=0;i<=order;i++) {
      for (int j=0;j<=order-i;j++) {
        for (int k=0;k<=order-i-j;k++) {
          if ( (i >= offset) && (i <= order-offset) &&
              (j >= offset) && (j <= order-i-offset) &&
              (k >= offset) && (k <= order-i-j-offset) ) {
            points(offcur,0) = refPts(noffcur,0);
            points(offcur,1) = refPts(noffcur,1);
            points(offcur,2) = refPts(noffcur,2);
            offcur++;
          }
          noffcur++;
        }
      }
    }
                                            


  }
  

} // face Intrepid