fe_h1.cpp

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// Copyright (c) 2010-2023, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.

// H1 Finite Element classes

#include "fe_h1.hpp"

namespace mfem
{

using namespace std;

H1_SegmentElement::H1_SegmentElement(const int p, const int btype)
   : NodalTensorFiniteElement(1, p, VerifyClosed(btype), H1_DOF_MAP)
{
   const double *cp = poly1d.ClosedPoints(p, b_type);

#ifndef MFEM_THREAD_SAFE
   shape_x.SetSize(p+1);
   dshape_x.SetSize(p+1);
   d2shape_x.SetSize(p+1);
#endif

   Nodes.IntPoint(0).x = cp[0];
   Nodes.IntPoint(1).x = cp[p];
   for (int i = 1; i < p; i++)
   {
      Nodes.IntPoint(i+1).x = cp[i];
   }
}

void H1_SegmentElement::CalcShape(const IntegrationPoint &ip,
                                  Vector &shape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1);
#endif

   basis1d.Eval(ip.x, shape_x);

   shape(0) = shape_x(0);
   shape(1) = shape_x(p);
   for (int i = 1; i < p; i++)
   {
      shape(i+1) = shape_x(i);
   }
}

void H1_SegmentElement::CalcDShape(const IntegrationPoint &ip,
                                   DenseMatrix &dshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), dshape_x(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x);

   dshape(0,0) = dshape_x(0);
   dshape(1,0) = dshape_x(p);
   for (int i = 1; i < p; i++)
   {
      dshape(i+1,0) = dshape_x(i);
   }
}

void H1_SegmentElement::CalcHessian(const IntegrationPoint &ip,
                                    DenseMatrix &Hessian) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), dshape_x(p+1), d2shape_x(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);

   Hessian(0,0) = d2shape_x(0);
   Hessian(1,0) = d2shape_x(p);
   for (int i = 1; i < p; i++)
   {
      Hessian(i+1,0) = d2shape_x(i);
   }
}

void H1_SegmentElement::ProjectDelta(int vertex, Vector &dofs) const
{
   const int p = order;
   const double *cp = poly1d.ClosedPoints(p, b_type);

   switch (vertex)
   {
      case 0:
         dofs(0) = poly1d.CalcDelta(p, (1.0 - cp[0]));
         dofs(1) = poly1d.CalcDelta(p, (1.0 - cp[p]));
         for (int i = 1; i < p; i++)
         {
            dofs(i+1) = poly1d.CalcDelta(p, (1.0 - cp[i]));
         }
         break;

      case 1:
         dofs(0) = poly1d.CalcDelta(p, cp[0]);
         dofs(1) = poly1d.CalcDelta(p, cp[p]);
         for (int i = 1; i < p; i++)
         {
            dofs(i+1) = poly1d.CalcDelta(p, cp[i]);
         }
         break;
   }
}


H1_QuadrilateralElement::H1_QuadrilateralElement(const int p, const int btype)
   : NodalTensorFiniteElement(2, p, VerifyClosed(btype), H1_DOF_MAP)
{
   const double *cp = poly1d.ClosedPoints(p, b_type);

#ifndef MFEM_THREAD_SAFE
   const int p1 = p + 1;

   shape_x.SetSize(p1);
   shape_y.SetSize(p1);
   dshape_x.SetSize(p1);
   dshape_y.SetSize(p1);
   d2shape_x.SetSize(p1);
   d2shape_y.SetSize(p1);
#endif

   int o = 0;
   for (int j = 0; j <= p; j++)
   {
      for (int i = 0; i <= p; i++)
      {
         Nodes.IntPoint(dof_map[o++]).Set2(cp[i], cp[j]);
      }
   }
}

void H1_QuadrilateralElement::CalcShape(const IntegrationPoint &ip,
                                        Vector &shape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1);
#endif

   basis1d.Eval(ip.x, shape_x);
   basis1d.Eval(ip.y, shape_y);

   for (int o = 0, j = 0; j <= p; j++)
      for (int i = 0; i <= p; i++)
      {
         shape(dof_map[o++]) = shape_x(i)*shape_y(j);
      }
}

void H1_QuadrilateralElement::CalcDShape(const IntegrationPoint &ip,
                                         DenseMatrix &dshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1), dshape_x(p+1), dshape_y(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x);
   basis1d.Eval(ip.y, shape_y, dshape_y);

   for (int o = 0, j = 0; j <= p; j++)
   {
      for (int i = 0; i <= p; i++)
      {
         dshape(dof_map[o],0) = dshape_x(i)* shape_y(j);
         dshape(dof_map[o],1) =  shape_x(i)*dshape_y(j);  o++;
      }
   }
}

void H1_QuadrilateralElement::CalcHessian(const IntegrationPoint &ip,
                                          DenseMatrix &Hessian) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1), dshape_x(p+1), dshape_y(p+1),
          d2shape_x(p+1), d2shape_y(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);
   basis1d.Eval(ip.y, shape_y, dshape_y, d2shape_y);

   for (int o = 0, j = 0; j <= p; j++)
   {
      for (int i = 0; i <= p; i++)
      {
         Hessian(dof_map[o],0) = d2shape_x(i)*  shape_y(j);
         Hessian(dof_map[o],1) =  dshape_x(i)* dshape_y(j);
         Hessian(dof_map[o],2) =   shape_x(i)*d2shape_y(j);  o++;
      }
   }
}

void H1_QuadrilateralElement::ProjectDelta(int vertex, Vector &dofs) const
{
   const int p = order;
   const double *cp = poly1d.ClosedPoints(p, b_type);

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1);
#endif

   for (int i = 0; i <= p; i++)
   {
      shape_x(i) = poly1d.CalcDelta(p, (1.0 - cp[i]));
      shape_y(i) = poly1d.CalcDelta(p, cp[i]);
   }

   switch (vertex)
   {
      case 0:
         for (int o = 0, j = 0; j <= p; j++)
            for (int i = 0; i <= p; i++)
            {
               dofs(dof_map[o++]) = shape_x(i)*shape_x(j);
            }
         break;
      case 1:
         for (int o = 0, j = 0; j <= p; j++)
            for (int i = 0; i <= p; i++)
            {
               dofs(dof_map[o++]) = shape_y(i)*shape_x(j);
            }
         break;
      case 2:
         for (int o = 0, j = 0; j <= p; j++)
            for (int i = 0; i <= p; i++)
            {
               dofs(dof_map[o++]) = shape_y(i)*shape_y(j);
            }
         break;
      case 3:
         for (int o = 0, j = 0; j <= p; j++)
            for (int i = 0; i <= p; i++)
            {
               dofs(dof_map[o++]) = shape_x(i)*shape_y(j);
            }
         break;
   }
}


H1_HexahedronElement::H1_HexahedronElement(const int p, const int btype)
   : NodalTensorFiniteElement(3, p, VerifyClosed(btype), H1_DOF_MAP)
{
   const double *cp = poly1d.ClosedPoints(p, b_type);

#ifndef MFEM_THREAD_SAFE
   const int p1 = p + 1;

   shape_x.SetSize(p1);
   shape_y.SetSize(p1);
   shape_z.SetSize(p1);
   dshape_x.SetSize(p1);
   dshape_y.SetSize(p1);
   dshape_z.SetSize(p1);
   d2shape_x.SetSize(p1);
   d2shape_y.SetSize(p1);
   d2shape_z.SetSize(p1);
#endif

   int o = 0;
   for (int k = 0; k <= p; k++)
      for (int j = 0; j <= p; j++)
         for (int i = 0; i <= p; i++)
         {
            Nodes.IntPoint(dof_map[o++]).Set3(cp[i], cp[j], cp[k]);
         }
}

void H1_HexahedronElement::CalcShape(const IntegrationPoint &ip,
                                     Vector &shape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1), shape_z(p+1);
#endif

   basis1d.Eval(ip.x, shape_x);
   basis1d.Eval(ip.y, shape_y);
   basis1d.Eval(ip.z, shape_z);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j <= p; j++)
         for (int i = 0; i <= p; i++)
         {
            shape(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_z(k);
         }
}

void H1_HexahedronElement::CalcDShape(const IntegrationPoint &ip,
                                      DenseMatrix &dshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1),  shape_y(p+1),  shape_z(p+1);
   Vector dshape_x(p+1), dshape_y(p+1), dshape_z(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x);
   basis1d.Eval(ip.y, shape_y, dshape_y);
   basis1d.Eval(ip.z, shape_z, dshape_z);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j <= p; j++)
         for (int i = 0; i <= p; i++)
         {
            dshape(dof_map[o],0) = dshape_x(i)* shape_y(j)* shape_z(k);
            dshape(dof_map[o],1) =  shape_x(i)*dshape_y(j)* shape_z(k);
            dshape(dof_map[o],2) =  shape_x(i)* shape_y(j)*dshape_z(k);  o++;
         }
}

void H1_HexahedronElement::CalcHessian(const IntegrationPoint &ip,
                                       DenseMatrix &Hessian) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1),  shape_y(p+1),  shape_z(p+1);
   Vector dshape_x(p+1), dshape_y(p+1), dshape_z(p+1);
   Vector d2shape_x(p+1), d2shape_y(p+1), d2shape_z(p+1);
#endif

   basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);
   basis1d.Eval(ip.y, shape_y, dshape_y, d2shape_y);
   basis1d.Eval(ip.z, shape_z, dshape_z, d2shape_z);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j <= p; j++)
         for (int i = 0; i <= p; i++)
         {
            Hessian(dof_map[o],0) = d2shape_x(i)*  shape_y(j)*  shape_z(k);
            Hessian(dof_map[o],1) =  dshape_x(i)* dshape_y(j)*  shape_z(k);
            Hessian(dof_map[o],2) =  dshape_x(i)*  shape_y(j)* dshape_z(k);
            Hessian(dof_map[o],3) =   shape_x(i)*d2shape_y(j)*  shape_z(k);
            Hessian(dof_map[o],4) =   shape_x(i)* dshape_y(j)* dshape_z(k);
            Hessian(dof_map[o],5) =   shape_x(i)*  shape_y(j)*d2shape_z(k);
            o++;
         }
}

void H1_HexahedronElement::ProjectDelta(int vertex, Vector &dofs) const
{
   const int p = order;
   const double *cp = poly1d.ClosedPoints(p,b_type);

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p+1), shape_y(p+1);
#endif

   for (int i = 0; i <= p; i++)
   {
      shape_x(i) = poly1d.CalcDelta(p, (1.0 - cp[i]));
      shape_y(i) = poly1d.CalcDelta(p, cp[i]);
   }

   switch (vertex)
   {
      case 0:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_x(i)*shape_x(j)*shape_x(k);
               }
         break;
      case 1:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_y(i)*shape_x(j)*shape_x(k);
               }
         break;
      case 2:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_y(i)*shape_y(j)*shape_x(k);
               }
         break;
      case 3:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_x(k);
               }
         break;
      case 4:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_x(i)*shape_x(j)*shape_y(k);
               }
         break;
      case 5:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_y(i)*shape_x(j)*shape_y(k);
               }
         break;
      case 6:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_y(i)*shape_y(j)*shape_y(k);
               }
         break;
      case 7:
         for (int o = 0, k = 0; k <= p; k++)
            for (int j = 0; j <= p; j++)
               for (int i = 0; i <= p; i++)
               {
                  dofs(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_y(k);
               }
         break;
   }
}

H1_TriangleElement::H1_TriangleElement(const int p, const int btype)
   : NodalFiniteElement(2, Geometry::TRIANGLE, ((p + 1)*(p + 2))/2, p,
                        FunctionSpace::Pk)
{
   const double *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));

#ifndef MFEM_THREAD_SAFE
   shape_x.SetSize(p + 1);
   shape_y.SetSize(p + 1);
   shape_l.SetSize(p + 1);
   dshape_x.SetSize(p + 1);
   dshape_y.SetSize(p + 1);
   dshape_l.SetSize(p + 1);
   ddshape_x.SetSize(p + 1);
   ddshape_y.SetSize(p + 1);
   ddshape_l.SetSize(p + 1);
   u.SetSize(dof);
   du.SetSize(dof, dim);
   ddu.SetSize(dof, (dim * (dim + 1)) / 2 );
#else
   Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1);
#endif

   int p2p3 = 2*p + 3;
   auto idx = [p2p3](int i, int j) { return ((p2p3-j)*j)/2+i; };
   lex_ordering.SetSize(dof);

   // vertices
   lex_ordering[idx(0,0)] = 0;
   Nodes.IntPoint(0).Set2(cp[0], cp[0]);
   lex_ordering[idx(p,0)] = 1;
   Nodes.IntPoint(1).Set2(cp[p], cp[0]);
   lex_ordering[idx(0,p)] = 2;
   Nodes.IntPoint(2).Set2(cp[0], cp[p]);

   // edges
   int o = 3;
   for (int i = 1; i < p; i++)
   {
      lex_ordering[idx(i,0)] = o;
      Nodes.IntPoint(o++).Set2(cp[i], cp[0]);
   }
   for (int i = 1; i < p; i++)
   {
      lex_ordering[idx(p-i,i)] = o;
      Nodes.IntPoint(o++).Set2(cp[p-i], cp[i]);
   }
   for (int i = 1; i < p; i++)
   {
      lex_ordering[idx(0,p-i)] = o;
      Nodes.IntPoint(o++).Set2(cp[0], cp[p-i]);
   }

   // interior
   for (int j = 1; j < p; j++)
      for (int i = 1; i + j < p; i++)
      {
         const double w = cp[i] + cp[j] + cp[p-i-j];
         lex_ordering[idx(i,j)] = o;
         Nodes.IntPoint(o++).Set2(cp[i]/w, cp[j]/w);
      }

   DenseMatrix T(dof);
   for (int k = 0; k < dof; k++)
   {
      IntegrationPoint &ip = Nodes.IntPoint(k);
      poly1d.CalcBasis(p, ip.x, shape_x);
      poly1d.CalcBasis(p, ip.y, shape_y);
      poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l);

      o = 0;
      for (int j = 0; j <= p; j++)
         for (int i = 0; i + j <= p; i++)
         {
            T(o++, k) = shape_x(i)*shape_y(j)*shape_l(p-i-j);
         }
   }

   Ti.Factor(T);
   // mfem::out << "H1_TriangleElement(" << p << ") : "; Ti.TestInversion();
}

void H1_TriangleElement::CalcShape(const IntegrationPoint &ip,
                                   Vector &shape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1), u(dof);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x);
   poly1d.CalcBasis(p, ip.y, shape_y);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l);

   for (int o = 0, j = 0; j <= p; j++)
      for (int i = 0; i + j <= p; i++)
      {
         u(o++) = shape_x(i)*shape_y(j)*shape_l(p-i-j);
      }

   Ti.Mult(u, shape);
}

void H1_TriangleElement::CalcDShape(const IntegrationPoint &ip,
                                    DenseMatrix &dshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector  shape_x(p + 1),  shape_y(p + 1),  shape_l(p + 1);
   Vector dshape_x(p + 1), dshape_y(p + 1), dshape_l(p + 1);
   DenseMatrix du(dof, dim);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x, dshape_x);
   poly1d.CalcBasis(p, ip.y, shape_y, dshape_y);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l, dshape_l);

   for (int o = 0, j = 0; j <= p; j++)
      for (int i = 0; i + j <= p; i++)
      {
         int k = p - i - j;
         du(o,0) = ((dshape_x(i)* shape_l(k)) -
                    ( shape_x(i)*dshape_l(k)))*shape_y(j);
         du(o,1) = ((dshape_y(j)* shape_l(k)) -
                    ( shape_y(j)*dshape_l(k)))*shape_x(i);
         o++;
      }

   Ti.Mult(du, dshape);
}

void H1_TriangleElement::CalcHessian(const IntegrationPoint &ip,
                                     DenseMatrix &ddshape) const
{
   const int p = order;
#ifdef MFEM_THREAD_SAFE
   Vector   shape_x(p + 1),   shape_y(p + 1),   shape_l(p + 1);
   Vector  dshape_x(p + 1),  dshape_y(p + 1),  dshape_l(p + 1);
   Vector ddshape_x(p + 1), ddshape_y(p + 1), ddshape_l(p + 1);
   DenseMatrix ddu(dof, dim);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x, dshape_x, ddshape_x);
   poly1d.CalcBasis(p, ip.y, shape_y, dshape_y, ddshape_y);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l, dshape_l, ddshape_l);

   for (int o = 0, j = 0; j <= p; j++)
      for (int i = 0; i + j <= p; i++)
      {
         int k = p - i - j;
         // u_xx, u_xy, u_yy
         ddu(o,0) = ((ddshape_x(i) * shape_l(k)) - 2. * (dshape_x(i) * dshape_l(k)) +
                     (shape_x(i) * ddshape_l(k))) * shape_y(j);
         ddu(o,1) = (((shape_x(i) * ddshape_l(k)) - dshape_x(i) * dshape_l(k)) * shape_y(
                        j)) + (((dshape_x(i) * shape_l(k)) - (shape_x(i) * dshape_l(k))) * dshape_y(j));
         ddu(o,2) = ((ddshape_y(j) * shape_l(k)) - 2. * (dshape_y(j) * dshape_l(k)) +
                     (shape_y(j) * ddshape_l(k))) * shape_x(i);
         o++;
      }

   Ti.Mult(ddu, ddshape);
}


H1_TetrahedronElement::H1_TetrahedronElement(const int p, const int btype)
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, ((p + 1)*(p + 2)*(p + 3))/6,
                        p, FunctionSpace::Pk)
{
   const double *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));

#ifndef MFEM_THREAD_SAFE
   shape_x.SetSize(p + 1);
   shape_y.SetSize(p + 1);
   shape_z.SetSize(p + 1);
   shape_l.SetSize(p + 1);
   dshape_x.SetSize(p + 1);
   dshape_y.SetSize(p + 1);
   dshape_z.SetSize(p + 1);
   dshape_l.SetSize(p + 1);
   ddshape_x.SetSize(p + 1);
   ddshape_y.SetSize(p + 1);
   ddshape_z.SetSize(p + 1);
   ddshape_l.SetSize(p + 1);
   u.SetSize(dof);
   du.SetSize(dof, dim);
   ddu.SetSize(dof, (dim * (dim + 1)) / 2);
#else
   Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
#endif

   auto tri = [](int k) { return (k*(k + 1))/2; };
   auto tet = [](int k) { return (k*(k + 1)*(k + 2))/6; };
   int ndof = tet(p+1);
   auto idx = [tri, tet, p, ndof](int i, int j, int k)
   {
      return ndof - tet(p - k) - tri(p + 1 - k - j) + i;
   };

   lex_ordering.SetSize(dof);

   // vertices
   lex_ordering[idx(0,0,0)] = 0;
   Nodes.IntPoint(0).Set3(cp[0], cp[0], cp[0]);
   lex_ordering[idx(p,0,0)] = 1;
   Nodes.IntPoint(1).Set3(cp[p], cp[0], cp[0]);
   lex_ordering[idx(0,p,0)] = 2;
   Nodes.IntPoint(2).Set3(cp[0], cp[p], cp[0]);
   lex_ordering[idx(0,0,p)] = 3;
   Nodes.IntPoint(3).Set3(cp[0], cp[0], cp[p]);

   // edges (see Tetrahedron::edges in mesh/tetrahedron.cpp)
   int o = 4;
   for (int i = 1; i < p; i++)  // (0,1)
   {
      lex_ordering[idx(i,0,0)] = o;
      Nodes.IntPoint(o++).Set3(cp[i], cp[0], cp[0]);
   }
   for (int i = 1; i < p; i++)  // (0,2)
   {
      lex_ordering[idx(0,i,0)] = o;
      Nodes.IntPoint(o++).Set3(cp[0], cp[i], cp[0]);
   }
   for (int i = 1; i < p; i++)  // (0,3)
   {
      lex_ordering[idx(0,0,i)] = o;
      Nodes.IntPoint(o++).Set3(cp[0], cp[0], cp[i]);
   }
   for (int i = 1; i < p; i++)  // (1,2)
   {
      lex_ordering[idx(p-i,i,0)] = o;
      Nodes.IntPoint(o++).Set3(cp[p-i], cp[i], cp[0]);
   }
   for (int i = 1; i < p; i++)  // (1,3)
   {
      lex_ordering[idx(p-i,0,i)] = o;
      Nodes.IntPoint(o++).Set3(cp[p-i], cp[0], cp[i]);
   }
   for (int i = 1; i < p; i++)  // (2,3)
   {
      lex_ordering[idx(0,p-i,i)] = o;
      Nodes.IntPoint(o++).Set3(cp[0], cp[p-i], cp[i]);
   }

   // faces (see Mesh::GenerateFaces in mesh/mesh.cpp)
   for (int j = 1; j < p; j++)
      for (int i = 1; i + j < p; i++)  // (1,2,3)
      {
         lex_ordering[idx(p-i-j,i,j)] = o;
         double w = cp[i] + cp[j] + cp[p-i-j];
         Nodes.IntPoint(o++).Set3(cp[p-i-j]/w, cp[i]/w, cp[j]/w);
      }
   for (int j = 1; j < p; j++)
      for (int i = 1; i + j < p; i++)  // (0,3,2)
      {
         lex_ordering[idx(0,j,i)] = o;
         double w = cp[i] + cp[j] + cp[p-i-j];
         Nodes.IntPoint(o++).Set3(cp[0], cp[j]/w, cp[i]/w);
      }
   for (int j = 1; j < p; j++)
      for (int i = 1; i + j < p; i++)  // (0,1,3)
      {
         lex_ordering[idx(i,0,j)] = o;
         double w = cp[i] + cp[j] + cp[p-i-j];
         Nodes.IntPoint(o++).Set3(cp[i]/w, cp[0], cp[j]/w);
      }
   for (int j = 1; j < p; j++)
      for (int i = 1; i + j < p; i++)  // (0,2,1)
      {
         lex_ordering[idx(j,i,0)] = o;
         double w = cp[i] + cp[j] + cp[p-i-j];
         Nodes.IntPoint(o++).Set3(cp[j]/w, cp[i]/w, cp[0]);
      }

   // interior
   for (int k = 1; k < p; k++)
      for (int j = 1; j + k < p; j++)
         for (int i = 1; i + j + k < p; i++)
         {
            lex_ordering[idx(i,j,k)] = o;
            double w = cp[i] + cp[j] + cp[k] + cp[p-i-j-k];
            Nodes.IntPoint(o++).Set3(cp[i]/w, cp[j]/w, cp[k]/w);
         }

   DenseMatrix T(dof);
   for (int m = 0; m < dof; m++)
   {
      IntegrationPoint &ip = Nodes.IntPoint(m);
      poly1d.CalcBasis(p, ip.x, shape_x);
      poly1d.CalcBasis(p, ip.y, shape_y);
      poly1d.CalcBasis(p, ip.z, shape_z);
      poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l);

      o = 0;
      for (int k = 0; k <= p; k++)
         for (int j = 0; j + k <= p; j++)
            for (int i = 0; i + j + k <= p; i++)
            {
               T(o++, m) = shape_x(i)*shape_y(j)*shape_z(k)*shape_l(p-i-j-k);
            }
   }

   Ti.Factor(T);
   // mfem::out << "H1_TetrahedronElement(" << p << ") : "; Ti.TestInversion();
}

void H1_TetrahedronElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
   Vector u(dof);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x);
   poly1d.CalcBasis(p, ip.y, shape_y);
   poly1d.CalcBasis(p, ip.z, shape_z);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j + k <= p; j++)
         for (int i = 0; i + j + k <= p; i++)
         {
            u(o++) = shape_x(i)*shape_y(j)*shape_z(k)*shape_l(p-i-j-k);
         }

   Ti.Mult(u, shape);
}

void H1_TetrahedronElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector  shape_x(p + 1),  shape_y(p + 1),  shape_z(p + 1),  shape_l(p + 1);
   Vector dshape_x(p + 1), dshape_y(p + 1), dshape_z(p + 1), dshape_l(p + 1);
   DenseMatrix du(dof, dim);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x, dshape_x);
   poly1d.CalcBasis(p, ip.y, shape_y, dshape_y);
   poly1d.CalcBasis(p, ip.z, shape_z, dshape_z);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l, dshape_l);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j + k <= p; j++)
         for (int i = 0; i + j + k <= p; i++)
         {
            int l = p - i - j - k;
            du(o,0) = ((dshape_x(i)* shape_l(l)) -
                       ( shape_x(i)*dshape_l(l)))*shape_y(j)*shape_z(k);
            du(o,1) = ((dshape_y(j)* shape_l(l)) -
                       ( shape_y(j)*dshape_l(l)))*shape_x(i)*shape_z(k);
            du(o,2) = ((dshape_z(k)* shape_l(l)) -
                       ( shape_z(k)*dshape_l(l)))*shape_x(i)*shape_y(j);
            o++;
         }

   Ti.Mult(du, dshape);
}

void H1_TetrahedronElement::CalcHessian(const IntegrationPoint &ip,
                                        DenseMatrix &ddshape) const
{
   const int p = order;

#ifdef MFEM_THREAD_SAFE
   Vector   shape_x(p + 1),   shape_y(p + 1),   shape_z(p + 1),   shape_l(p + 1);
   Vector  dshape_x(p + 1),  dshape_y(p + 1),  dshape_z(p + 1),  dshape_l(p + 1);
   Vector ddshape_x(p + 1), ddshape_y(p + 1), ddshape_z(p + 1), ddshape_l(p + 1);
   DenseMatrix ddu(dof, ((dim + 1) * dim) / 2);
#endif

   poly1d.CalcBasis(p, ip.x, shape_x, dshape_x, ddshape_x);
   poly1d.CalcBasis(p, ip.y, shape_y, dshape_y, ddshape_y);
   poly1d.CalcBasis(p, ip.z, shape_z, dshape_z, ddshape_z);
   poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l, dshape_l, ddshape_l);

   for (int o = 0, k = 0; k <= p; k++)
      for (int j = 0; j + k <= p; j++)
         for (int i = 0; i + j + k <= p; i++)
         {
            // u_xx, u_xy, u_xz, u_yy, u_yz, u_zz
            int l = p - i - j - k;
            ddu(o,0) = ((ddshape_x(i) * shape_l(l)) - 2. * (dshape_x(i) * dshape_l(l)) +
                        (shape_x(i) * ddshape_l(l))) * shape_y(j) * shape_z(k);
            ddu(o,1) = ((dshape_y(j) * ((dshape_x(i) * shape_l(l)) -
                                        (shape_x(i) * dshape_l(l)))) +
                        (shape_y(j) * ((ddshape_l(l) * shape_x(i)) -
                                       (dshape_x(i) * dshape_l(l)))))* shape_z(k);
            ddu(o,2) = ((dshape_z(k) * ((dshape_x(i) * shape_l(l)) -
                                        (shape_x(i) * dshape_l(l)))) +
                        (shape_z(k) * ((ddshape_l(l) * shape_x(i)) -
                                       (dshape_x(i) * dshape_l(l)))))* shape_y(j);
            ddu(o,3) = ((ddshape_y(j) * shape_l(l)) - 2. * (dshape_y(j) * dshape_l(l)) +
                        (shape_y(j) * ddshape_l(l))) * shape_x(i) * shape_z(k);
            ddu(o,4) = ((dshape_z(k) * ((dshape_y(j) * shape_l(l)) -
                                        (shape_y(j)*dshape_l(l))) ) +
                        (shape_z(k)* ((ddshape_l(l)*shape_y(j)) -
                                      (dshape_y(j) * dshape_l(l)) ) ) )* shape_x(i);
            ddu(o,5) = ((ddshape_z(k) * shape_l(l)) - 2. * (dshape_z(k) * dshape_l(l)) +
                        (shape_z(k) * ddshape_l(l))) * shape_y(j) * shape_x(i);
            o++;
         }
   Ti.Mult(ddu, ddshape);
}

// TODO: use a FunctionSpace specific to wedges instead of Qk.
H1_WedgeElement::H1_WedgeElement(const int p,
                                 const int btype)
   : NodalFiniteElement(3, Geometry::PRISM, ((p + 1)*(p + 1)*(p + 2))/2,
                        p, FunctionSpace::Qk),
     TriangleFE(p, btype),
     SegmentFE(p, btype)
{
#ifndef MFEM_THREAD_SAFE
   t_shape.SetSize(TriangleFE.GetDof());
   s_shape.SetSize(SegmentFE.GetDof());
   t_dshape.SetSize(TriangleFE.GetDof(), 2);
   s_dshape.SetSize(SegmentFE.GetDof(), 1);
#endif

   t_dof.SetSize(dof);
   s_dof.SetSize(dof);

   int p2p3 = 2*p + 3, ntri = ((p + 1)*(p + 2))/2;
   auto idx = [p2p3,ntri](int i, int j, int k)
   {
      return k*ntri + ((p2p3-j)*j)/2+i;
   };

   lex_ordering.SetSize(dof);
   int o = 0;

   // Nodal DoFs
   lex_ordering[idx(0,0,0)] = o++;
   lex_ordering[idx(p,0,0)] = o++;
   lex_ordering[idx(0,p,0)] = o++;
   lex_ordering[idx(0,0,p)] = o++;
   lex_ordering[idx(p,0,p)] = o++;
   lex_ordering[idx(0,p,p)] = o++;
   t_dof[0] = 0; s_dof[0] = 0;
   t_dof[1] = 1; s_dof[1] = 0;
   t_dof[2] = 2; s_dof[2] = 0;
   t_dof[3] = 0; s_dof[3] = 1;
   t_dof[4] = 1; s_dof[4] = 1;
   t_dof[5] = 2; s_dof[5] = 1;

   // Edge DoFs
   int k = 0;
   int ne = p-1;
   for (int i=1; i<p; i++)
   {
      lex_ordering[idx(i,0,0)] = o + 0*ne + k;
      lex_ordering[idx(p-i,i,0)] = o + 1*ne + k;
      lex_ordering[idx(0,p-i,0)] = o + 2*ne + k;
      lex_ordering[idx(i,0,p)] = o + 3*ne + k;
      lex_ordering[idx(p-i,i,p)] = o + 4*ne + k;
      lex_ordering[idx(0,p-i,p)] = o + 5*ne + k;
      lex_ordering[idx(0,0,i)] = o + 6*ne + k;
      lex_ordering[idx(p,0,i)] = o + 7*ne + k;
      lex_ordering[idx(0,p,i)] = o + 8*ne + k;
      t_dof[5 + 0 * ne + i] = 2 + 0 * ne + i; s_dof[5 + 0 * ne + i] = 0;
      t_dof[5 + 1 * ne + i] = 2 + 1 * ne + i; s_dof[5 + 1 * ne + i] = 0;
      t_dof[5 + 2 * ne + i] = 2 + 2 * ne + i; s_dof[5 + 2 * ne + i] = 0;
      t_dof[5 + 3 * ne + i] = 2 + 0 * ne + i; s_dof[5 + 3 * ne + i] = 1;
      t_dof[5 + 4 * ne + i] = 2 + 1 * ne + i; s_dof[5 + 4 * ne + i] = 1;
      t_dof[5 + 5 * ne + i] = 2 + 2 * ne + i; s_dof[5 + 5 * ne + i] = 1;
      t_dof[5 + 6 * ne + i] = 0;              s_dof[5 + 6 * ne + i] = i + 1;
      t_dof[5 + 7 * ne + i] = 1;              s_dof[5 + 7 * ne + i] = i + 1;
      t_dof[5 + 8 * ne + i] = 2;              s_dof[5 + 8 * ne + i] = i + 1;
      ++k;
   }
   o += 9*ne;

   // Triangular Face DoFs
   k=0;
   int nt = (p-1)*(p-2)/2;
   for (int j=1; j<p; j++)
   {
      for (int i=1; i<p-j; i++)
      {
         int l = j - p + (((2 * p - 1) - i) * i) / 2;
         lex_ordering[idx(i,j,0)] = o+l;
         lex_ordering[idx(i,j,p)] = o+nt+k;
         t_dof[6 + 9 * ne + k]      = 3 * p + l; s_dof[6 + 9 * ne + k]      = 0;
         t_dof[6 + 9 * ne + nt + k] = 3 * p + k; s_dof[6 + 9 * ne + nt + k] = 1;
         k++;
      }
   }
   o += 2*nt;

   // Quadrilateral Face DoFs
   k=0;
   int nq = (p-1)*(p-1);
   for (int j=1; j<p; j++)
   {
      for (int i=1; i<p; i++)
      {
         lex_ordering[idx(i,0,j)] = o+k;
         lex_ordering[idx(p-i,i,j)] = o+nq+k;
         lex_ordering[idx(0,p-i,j)] = o+2*nq+k;

         t_dof[6 + 9 * ne + 2 * nt + 0 * nq + k] = 2 + 0 * ne + i;
         t_dof[6 + 9 * ne + 2 * nt + 1 * nq + k] = 2 + 1 * ne + i;
         t_dof[6 + 9 * ne + 2 * nt + 2 * nq + k] = 2 + 2 * ne + i;

         s_dof[6 + 9 * ne + 2 * nt + 0 * nq + k] = 1 + j;
         s_dof[6 + 9 * ne + 2 * nt + 1 * nq + k] = 1 + j;
         s_dof[6 + 9 * ne + 2 * nt + 2 * nq + k] = 1 + j;

         k++;
      }
   }
   o += 3*nq;

   // Interior DoFs
   int m=0;
   for (k=1; k<p; k++)
   {
      int l=0;
      for (int j=1; j<p; j++)
      {
         for (int i=1; i+j<p; i++)
         {
            lex_ordering[idx(i,j,k)] = o++;
            t_dof[6 + 9 * ne + 2 * nt + 3 * nq + m] = 3 * p + l;
            s_dof[6 + 9 * ne + 2 * nt + 3 * nq + m] = 1 + k;
            l++; m++;
         }
      }
   }

   // Define Nodes
   const IntegrationRule & t_Nodes = TriangleFE.GetNodes();
   const IntegrationRule & s_Nodes = SegmentFE.GetNodes();
   for (int i=0; i<dof; i++)
   {
      Nodes.IntPoint(i).x = t_Nodes.IntPoint(t_dof[i]).x;
      Nodes.IntPoint(i).y = t_Nodes.IntPoint(t_dof[i]).y;
      Nodes.IntPoint(i).z = s_Nodes.IntPoint(s_dof[i]).x;
   }
}

void H1_WedgeElement::CalcShape(const IntegrationPoint &ip,
                                Vector &shape) const
{
#ifdef MFEM_THREAD_SAFE
   Vector t_shape(TriangleFE.GetDof());
   Vector s_shape(SegmentFE.GetDof());
#endif

   IntegrationPoint ipz; ipz.x = ip.z; ipz.y = 0.0; ipz.z = 0.0;

   TriangleFE.CalcShape(ip, t_shape);
   SegmentFE.CalcShape(ipz, s_shape);

   for (int i=0; i<dof; i++)
   {
      shape[i] = t_shape[t_dof[i]] * s_shape[s_dof[i]];
   }
}

void H1_WedgeElement::CalcDShape(const IntegrationPoint &ip,
                                 DenseMatrix &dshape) const
{
#ifdef MFEM_THREAD_SAFE
   Vector      t_shape(TriangleFE.GetDof());
   DenseMatrix t_dshape(TriangleFE.GetDof(), 2);
   Vector      s_shape(SegmentFE.GetDof());
   DenseMatrix s_dshape(SegmentFE.GetDof(), 1);
#endif

   IntegrationPoint ipz; ipz.x = ip.z; ipz.y = 0.0; ipz.z = 0.0;

   TriangleFE.CalcShape(ip, t_shape);
   TriangleFE.CalcDShape(ip, t_dshape);
   SegmentFE.CalcShape(ipz, s_shape);
   SegmentFE.CalcDShape(ipz, s_dshape);

   for (int i=0; i<dof; i++)
   {
      dshape(i, 0) = t_dshape(t_dof[i],0) * s_shape[s_dof[i]];
      dshape(i, 1) = t_dshape(t_dof[i],1) * s_shape[s_dof[i]];
      dshape(i, 2) = t_shape[t_dof[i]] * s_dshape(s_dof[i],0);
   }
}

}