TetExp.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: TetExp.cpp
//
// For more information, please see: http://www.nektar.info
//
// The MIT License
//
// Copyright (c) 2006 Division of Applied Mathematics, Brown University (USA),
// Department of Aeronautics, Imperial College London (UK), and Scientific
// Computing and Imaging Institute, University of Utah (USA).
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
//
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//
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// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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// DEALINGS IN THE SOFTWARE.
//
// Description:
//
///////////////////////////////////////////////////////////////////////////////
 
#include <boost/core/ignore_unused.hpp>
 
#include <LibUtilities/Foundations/Interp.h>
#include <LibUtilities/Foundations/InterpCoeff.h>
#include <LocalRegions/TetExp.h>
#include <SpatialDomains/SegGeom.h>
 
using namespace std;
 
namespace Nektar
{
namespace LocalRegions
{
/**
 * @class TetExp
 * Defines a Tetrahedral local expansion.
 */
 
/**
 * \brief Constructor using BasisKey class for quadrature points and
 * order definition
 *
 * @param   Ba          Basis key for first coordinate.
 * @param   Bb          Basis key for second coordinate.
 * @param   Bc          Basis key for third coordinate.
 */
TetExp::TetExp(const LibUtilities::BasisKey &Ba,
               const LibUtilities::BasisKey &Bb,
               const LibUtilities::BasisKey &Bc,
               const SpatialDomains::TetGeomSharedPtr &geom)
    : StdExpansion(LibUtilities::StdTetData::getNumberOfCoefficients(
                       Ba.GetNumModes(), Bb.GetNumModes(), Bc.GetNumModes()),
                   3, Ba, Bb, Bc),
      StdExpansion3D(LibUtilities::StdTetData::getNumberOfCoefficients(
                         Ba.GetNumModes(), Bb.GetNumModes(), Bc.GetNumModes()),
                     Ba, Bb, Bc),
      StdTetExp(Ba, Bb, Bc), Expansion(geom), Expansion3D(geom),
      m_matrixManager(
          std::bind(&Expansion3D::CreateMatrix, this, std::placeholders::_1),
          std::string("TetExpMatrix")),
      m_staticCondMatrixManager(std::bind(&Expansion::CreateStaticCondMatrix,
                                          this, std::placeholders::_1),
                                std::string("TetExpStaticCondMatrix"))
{
}
 
/**
 * \brief Copy Constructor
 */
TetExp::TetExp(const TetExp &T)
    : StdRegions::StdExpansion(T), StdRegions::StdExpansion3D(T),
      StdRegions::StdTetExp(T), Expansion(T), Expansion3D(T),
      m_matrixManager(T.m_matrixManager),
      m_staticCondMatrixManager(T.m_staticCondMatrixManager)
{
}
 
//-----------------------------
// Integration Methods
//-----------------------------
/**
 * \brief Integrate the physical point list \a inarray over region
 *
 * @param   inarray     Definition of function to be returned at
 *                      quadrature point of expansion.
 * @returns \f$\int^1_{-1}\int^1_{-1} \int^1_{-1}
 *   u(\eta_1, \eta_2, \eta_3) J[i,j,k] d \eta_1 d \eta_2 d \eta_3 \f$
 * where \f$inarray[i,j,k] = u(\eta_{1i},\eta_{2j},\eta_{3k})
 * \f$ and \f$ J[i,j,k] \f$ is the Jacobian evaluated at the quadrature
 * point.
 */
NekDouble TetExp::v_Integral(const Array<OneD, const NekDouble> &inarray)
{
    int nquad0                       = m_base[0]->GetNumPoints();
    int nquad1                       = m_base[1]->GetNumPoints();
    int nquad2                       = m_base[2]->GetNumPoints();
    Array<OneD, const NekDouble> jac = m_metricinfo->GetJac(GetPointsKeys());
    NekDouble retrunVal;
    Array<OneD, NekDouble> tmp(nquad0 * nquad1 * nquad2);
 
    // multiply inarray with Jacobian
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        Vmath::Vmul(nquad0 * nquad1 * nquad2, &jac[0], 1,
                    (NekDouble *)&inarray[0], 1, &tmp[0], 1);
    }
    else
    {
        Vmath::Smul(nquad0 * nquad1 * nquad2, (NekDouble)jac[0],
                    (NekDouble *)&inarray[0], 1, &tmp[0], 1);
    }
 
    // call StdTetExp version;
    retrunVal = StdTetExp::v_Integral(tmp);
 
    return retrunVal;
}
 
//-----------------------------
// Differentiation Methods
//-----------------------------
/**
 * \brief Differentiate \a inarray in the three coordinate directions.
 *
 * @param   inarray     Input array of values at quadrature points to
 *                      be differentiated.
 * @param   out_d0      Derivative in first coordinate direction.
 * @param   out_d1      Derivative in second coordinate direction.
 * @param   out_d2      Derivative in third coordinate direction.
 */
void TetExp::v_PhysDeriv(const Array<OneD, const NekDouble> &inarray,
                         Array<OneD, NekDouble> &out_d0,
                         Array<OneD, NekDouble> &out_d1,
                         Array<OneD, NekDouble> &out_d2)
{
    int TotPts = m_base[0]->GetNumPoints() * m_base[1]->GetNumPoints() *
                 m_base[2]->GetNumPoints();
 
    Array<TwoD, const NekDouble> df =
        m_metricinfo->GetDerivFactors(GetPointsKeys());
    Array<OneD, NekDouble> Diff0 = Array<OneD, NekDouble>(3 * TotPts);
    Array<OneD, NekDouble> Diff1 = Diff0 + TotPts;
    Array<OneD, NekDouble> Diff2 = Diff1 + TotPts;
 
    StdTetExp::v_PhysDeriv(inarray, Diff0, Diff1, Diff2);
 
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        if (out_d0.size())
        {
            Vmath::Vmul(TotPts, &df[0][0], 1, &Diff0[0], 1, &out_d0[0], 1);
            Vmath::Vvtvp(TotPts, &df[1][0], 1, &Diff1[0], 1, &out_d0[0], 1,
                         &out_d0[0], 1);
            Vmath::Vvtvp(TotPts, &df[2][0], 1, &Diff2[0], 1, &out_d0[0], 1,
                         &out_d0[0], 1);
        }
 
        if (out_d1.size())
        {
            Vmath::Vmul(TotPts, &df[3][0], 1, &Diff0[0], 1, &out_d1[0], 1);
            Vmath::Vvtvp(TotPts, &df[4][0], 1, &Diff1[0], 1, &out_d1[0], 1,
                         &out_d1[0], 1);
            Vmath::Vvtvp(TotPts, &df[5][0], 1, &Diff2[0], 1, &out_d1[0], 1,
                         &out_d1[0], 1);
        }
 
        if (out_d2.size())
        {
            Vmath::Vmul(TotPts, &df[6][0], 1, &Diff0[0], 1, &out_d2[0], 1);
            Vmath::Vvtvp(TotPts, &df[7][0], 1, &Diff1[0], 1, &out_d2[0], 1,
                         &out_d2[0], 1);
            Vmath::Vvtvp(TotPts, &df[8][0], 1, &Diff2[0], 1, &out_d2[0], 1,
                         &out_d2[0], 1);
        }
    }
    else // regular geometry
    {
        if (out_d0.size())
        {
            Vmath::Smul(TotPts, df[0][0], &Diff0[0], 1, &out_d0[0], 1);
            Blas::Daxpy(TotPts, df[1][0], &Diff1[0], 1, &out_d0[0], 1);
            Blas::Daxpy(TotPts, df[2][0], &Diff2[0], 1, &out_d0[0], 1);
        }
 
        if (out_d1.size())
        {
            Vmath::Smul(TotPts, df[3][0], &Diff0[0], 1, &out_d1[0], 1);
            Blas::Daxpy(TotPts, df[4][0], &Diff1[0], 1, &out_d1[0], 1);
            Blas::Daxpy(TotPts, df[5][0], &Diff2[0], 1, &out_d1[0], 1);
        }
 
        if (out_d2.size())
        {
            Vmath::Smul(TotPts, df[6][0], &Diff0[0], 1, &out_d2[0], 1);
            Blas::Daxpy(TotPts, df[7][0], &Diff1[0], 1, &out_d2[0], 1);
            Blas::Daxpy(TotPts, df[8][0], &Diff2[0], 1, &out_d2[0], 1);
        }
    }
}
 
//-----------------------------
// Transforms
//-----------------------------
/**
 * \brief Forward transform from physical quadrature space stored in
 * \a inarray and evaluate the expansion coefficients and store
 * in \a (this)->_coeffs
 *
 * @param   inarray     Array of physical quadrature points to be
 *                      transformed.
 * @param   outarray    Array of coefficients to update.
 */
void TetExp::v_FwdTrans(const Array<OneD, const NekDouble> &inarray,
                        Array<OneD, NekDouble> &outarray)
{
    if ((m_base[0]->Collocation()) && (m_base[1]->Collocation()) &&
        (m_base[2]->Collocation()))
    {
        Vmath::Vcopy(GetNcoeffs(), &inarray[0], 1, &outarray[0], 1);
    }
    else
    {
        IProductWRTBase(inarray, outarray);
 
        // get Mass matrix inverse
        MatrixKey masskey(StdRegions::eInvMass, DetShapeType(), *this);
        DNekScalMatSharedPtr matsys = m_matrixManager[masskey];
 
        // copy inarray in case inarray == outarray
        DNekVec in(m_ncoeffs, outarray);
        DNekVec out(m_ncoeffs, outarray, eWrapper);
 
        out = (*matsys) * in;
    }
}
 
//-----------------------------
// Inner product functions
//-----------------------------
/**
 * \brief Calculate the inner product of inarray with respect to the
 * basis B=m_base0*m_base1*m_base2 and put into outarray:
 *
 * \f$ \begin{array}{rcl} I_{pqr} = (\phi_{pqr}, u)_{\delta}
 *   & = & \sum_{i=0}^{nq_0} \sum_{j=0}^{nq_1} \sum_{k=0}^{nq_2}
 *     \psi_{p}^{a} (\eta_{1i}) \psi_{pq}^{b} (\eta_{2j}) \psi_{pqr}^{c}
 *     (\eta_{3k}) w_i w_j w_k u(\eta_{1,i} \eta_{2,j} \eta_{3,k})
 * J_{i,j,k}\\ & = & \sum_{i=0}^{nq_0} \psi_p^a(\eta_{1,i})
 *   \sum_{j=0}^{nq_1} \psi_{pq}^b(\eta_{2,j}) \sum_{k=0}^{nq_2}
 *   \psi_{pqr}^c u(\eta_{1i},\eta_{2j},\eta_{3k}) J_{i,j,k}
 * \end{array} \f$ \n
 * where
 * \f$ \phi_{pqr} (\xi_1 , \xi_2 , \xi_3)
 *   = \psi_p^a (\eta_1) \psi_{pq}^b (\eta_2) \psi_{pqr}^c (\eta_3) \f$
 * which can be implemented as \n
 * \f$f_{pqr} (\xi_{3k})
 *   = \sum_{k=0}^{nq_3} \psi_{pqr}^c u(\eta_{1i},\eta_{2j},\eta_{3k})
 * J_{i,j,k} = {\bf B_3 U}   \f$ \n
 * \f$ g_{pq} (\xi_{3k})
 *   = \sum_{j=0}^{nq_1} \psi_{pq}^b (\xi_{2j}) f_{pqr} (\xi_{3k})
 *   = {\bf B_2 F}  \f$ \n
 * \f$ (\phi_{pqr}, u)_{\delta}
 *   = \sum_{k=0}^{nq_0} \psi_{p}^a (\xi_{3k}) g_{pq} (\xi_{3k})
 *   = {\bf B_1 G} \f$
 */
void TetExp::v_IProductWRTBase(const Array<OneD, const NekDouble> &inarray,
                               Array<OneD, NekDouble> &outarray)
{
    v_IProductWRTBase_SumFac(inarray, outarray);
}
 
void TetExp::v_IProductWRTBase_SumFac(
    const Array<OneD, const NekDouble> &inarray,
    Array<OneD, NekDouble> &outarray, bool multiplybyweights)
{
    const int nquad0 = m_base[0]->GetNumPoints();
    const int nquad1 = m_base[1]->GetNumPoints();
    const int nquad2 = m_base[2]->GetNumPoints();
    const int order0 = m_base[0]->GetNumModes();
    const int order1 = m_base[1]->GetNumModes();
    Array<OneD, NekDouble> wsp(nquad1 * nquad2 * order0 +
                               nquad2 * order0 * (order1 + 1) / 2);
 
    if (multiplybyweights)
    {
        Array<OneD, NekDouble> tmp(nquad0 * nquad1 * nquad2);
 
        MultiplyByQuadratureMetric(inarray, tmp);
        IProductWRTBase_SumFacKernel(
            m_base[0]->GetBdata(), m_base[1]->GetBdata(), m_base[2]->GetBdata(),
            tmp, outarray, wsp, true, true, true);
    }
    else
    {
        IProductWRTBase_SumFacKernel(
            m_base[0]->GetBdata(), m_base[1]->GetBdata(), m_base[2]->GetBdata(),
            inarray, outarray, wsp, true, true, true);
    }
}
 
/**
 * @brief Calculates the inner product \f$ I_{pqr} = (u,
 * \partial_{x_i} \phi_{pqr}) \f$.
 *
 * The derivative of the basis functions is performed using the chain
 * rule in order to incorporate the geometric factors. Assuming that
 * the basis functions are a tensor product
 * \f$\phi_{pqr}(\eta_1,\eta_2,\eta_3) =
 * \phi_1(\eta_1)\phi_2(\eta_2)\phi_3(\eta_3)\f$, this yields the
 * result
 *
 * \f[
 * I_{pqr} = \sum_{j=1}^3 \left(u, \frac{\partial u}{\partial \eta_j}
 * \frac{\partial \eta_j}{\partial x_i}\right)
 * \f]
 *
 * In the prismatic element, we must also incorporate a second set of
 * geometric factors which incorporate the collapsed co-ordinate
 * system, so that
 *
 * \f[ \frac{\partial\eta_j}{\partial x_i} = \sum_{k=1}^3
 * \frac{\partial\eta_j}{\partial\xi_k}\frac{\partial\xi_k}{\partial
 * x_i} \f]
 *
 * These derivatives can be found on p152 of Sherwin & Karniadakis.
 *
 * @param dir       Direction in which to take the derivative.
 * @param inarray   The function \f$ u \f$.
 * @param outarray  Value of the inner product.
 */
void TetExp::v_IProductWRTDerivBase(const int dir,
                                    const Array<OneD, const NekDouble> &inarray,
                                    Array<OneD, NekDouble> &outarray)
{
    const int nquad0 = m_base[0]->GetNumPoints();
    const int nquad1 = m_base[1]->GetNumPoints();
    const int nquad2 = m_base[2]->GetNumPoints();
    const int order0 = m_base[0]->GetNumModes();
    const int order1 = m_base[1]->GetNumModes();
    const int nqtot  = nquad0 * nquad1 * nquad2;
 
    Array<OneD, NekDouble> tmp1(nqtot);
    Array<OneD, NekDouble> tmp2(nqtot);
    Array<OneD, NekDouble> tmp3(nqtot);
    Array<OneD, NekDouble> tmp4(nqtot);
    Array<OneD, NekDouble> tmp6(m_ncoeffs);
    Array<OneD, NekDouble> wsp(nquad1 * nquad2 * order0 +
                               nquad2 * order0 * (order1 + 1) / 2);
 
    MultiplyByQuadratureMetric(inarray, tmp1);
 
    Array<OneD, Array<OneD, NekDouble>> tmp2D{3};
    tmp2D[0] = tmp2;
    tmp2D[1] = tmp3;
    tmp2D[2] = tmp4;
 
    TetExp::v_AlignVectorToCollapsedDir(dir, tmp1, tmp2D);
 
    IProductWRTBase_SumFacKernel(m_base[0]->GetDbdata(), m_base[1]->GetBdata(),
                                 m_base[2]->GetBdata(), tmp2, outarray, wsp,
                                 false, true, true);
 
    IProductWRTBase_SumFacKernel(m_base[0]->GetBdata(), m_base[1]->GetDbdata(),
                                 m_base[2]->GetBdata(), tmp3, tmp6, wsp, true,
                                 false, true);
 
    Vmath::Vadd(m_ncoeffs, tmp6, 1, outarray, 1, outarray, 1);
 
    IProductWRTBase_SumFacKernel(m_base[0]->GetBdata(), m_base[1]->GetBdata(),
                                 m_base[2]->GetDbdata(), tmp4, tmp6, wsp, true,
                                 true, false);
 
    Vmath::Vadd(m_ncoeffs, tmp6, 1, outarray, 1, outarray, 1);
}
 
void TetExp::v_AlignVectorToCollapsedDir(
    const int dir, const Array<OneD, const NekDouble> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray)
{
    int i, j;
 
    const int nquad0 = m_base[0]->GetNumPoints();
    const int nquad1 = m_base[1]->GetNumPoints();
    const int nquad2 = m_base[2]->GetNumPoints();
    const int nqtot  = nquad0 * nquad1 * nquad2;
 
    const Array<OneD, const NekDouble> &z0 = m_base[0]->GetZ();
    const Array<OneD, const NekDouble> &z1 = m_base[1]->GetZ();
    const Array<OneD, const NekDouble> &z2 = m_base[2]->GetZ();
 
    Array<OneD, NekDouble> tmp2(nqtot);
    Array<OneD, NekDouble> tmp3(nqtot);
 
    const Array<TwoD, const NekDouble> &df =
        m_metricinfo->GetDerivFactors(GetPointsKeys());
 
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        Vmath::Vmul(nqtot, &df[3 * dir][0], 1, inarray.get(), 1, tmp2.get(), 1);
        Vmath::Vmul(nqtot, &df[3 * dir + 1][0], 1, inarray.get(), 1, tmp3.get(),
                    1);
        Vmath::Vmul(nqtot, &df[3 * dir + 2][0], 1, inarray.get(), 1,
                    outarray[2].get(), 1);
    }
    else
    {
        Vmath::Smul(nqtot, df[3 * dir][0], inarray.get(), 1, tmp2.get(), 1);
        Vmath::Smul(nqtot, df[3 * dir + 1][0], inarray.get(), 1, tmp3.get(), 1);
        Vmath::Smul(nqtot, df[3 * dir + 2][0], inarray.get(), 1,
                    outarray[2].get(), 1);
    }
 
    NekDouble g0, g1, g1a, g2, g3;
    int k, cnt;
 
    for (cnt = 0, k = 0; k < nquad2; ++k)
    {
        for (j = 0; j < nquad1; ++j)
        {
            g2 = 2.0 / (1.0 - z2[k]);
            g1 = g2 / (1.0 - z1[j]);
            g0 = 2.0 * g1;
            g3 = (1.0 + z1[j]) * g2 * 0.5;
 
            for (i = 0; i < nquad0; ++i, ++cnt)
            {
                g1a = g1 * (1 + z0[i]);
 
                outarray[0][cnt] =
                    g0 * tmp2[cnt] + g1a * (tmp3[cnt] + outarray[2][cnt]);
 
                outarray[1][cnt] = g2 * tmp3[cnt] + g3 * outarray[2][cnt];
            }
        }
    }
}
 
//-----------------------------
// Evaluation functions
//-----------------------------
 
/**
 * Given the local cartesian coordinate \a Lcoord evaluate the
 * value of physvals at this point by calling through to the
 * StdExpansion method
 */
NekDouble TetExp::v_StdPhysEvaluate(
    const Array<OneD, const NekDouble> &Lcoord,
    const Array<OneD, const NekDouble> &physvals)
{
    // Evaluate point in local (eta) coordinates.
    return StdExpansion3D::v_PhysEvaluate(Lcoord, physvals);
}
 
/**
 * @param   coord       Physical space coordinate
 * @returns Evaluation of expansion at given coordinate.
 */
NekDouble TetExp::v_PhysEvaluate(const Array<OneD, const NekDouble> &coord,
                                 const Array<OneD, const NekDouble> &physvals)
{
    ASSERTL0(m_geom, "m_geom not defined");
 
    Array<OneD, NekDouble> Lcoord = Array<OneD, NekDouble>(3);
 
    // Get the local (eta) coordinates of the point
    m_geom->GetLocCoords(coord, Lcoord);
 
    // Evaluate point in local (eta) coordinates.
    return StdExpansion3D::v_PhysEvaluate(Lcoord, physvals);
}
 
NekDouble TetExp::v_PhysEvaluate(const Array<OneD, NekDouble> &coord,
                                 const Array<OneD, const NekDouble> &inarray,
                                 std::array<NekDouble, 3> &firstOrderDerivs)
{
    Array<OneD, NekDouble> Lcoord(3);
    ASSERTL0(m_geom, "m_geom not defined");
    m_geom->GetLocCoords(coord, Lcoord);
    return StdTetExp::v_PhysEvaluate(Lcoord, inarray, firstOrderDerivs);
}
 
/**
 * \brief Get the coordinates "coords" at the local coordinates "Lcoords"
 */
void TetExp::v_GetCoord(const Array<OneD, const NekDouble> &Lcoords,
                        Array<OneD, NekDouble> &coords)
{
    int i;
 
    ASSERTL1(Lcoords[0] <= -1.0 && Lcoords[0] >= 1.0 && Lcoords[1] <= -1.0 &&
                 Lcoords[1] >= 1.0 && Lcoords[2] <= -1.0 && Lcoords[2] >= 1.0,
             "Local coordinates are not in region [-1,1]");
 
    // m_geom->FillGeom(); // TODO: implement FillGeom()
 
    for (i = 0; i < m_geom->GetCoordim(); ++i)
    {
        coords[i] = m_geom->GetCoord(i, Lcoords);
    }
}
 
void TetExp::v_GetCoords(Array<OneD, NekDouble> &coords_0,
                         Array<OneD, NekDouble> &coords_1,
                         Array<OneD, NekDouble> &coords_2)
{
    Expansion::v_GetCoords(coords_0, coords_1, coords_2);
}
 
//-----------------------------
// Helper functions
//-----------------------------
 
/**
 * \brief Return Shape of region, using  ShapeType enum list.
 */
LibUtilities::ShapeType TetExp::v_DetShapeType() const
{
    return LibUtilities::eTetrahedron;
}
 
StdRegions::StdExpansionSharedPtr TetExp::v_GetStdExp(void) const
{
    return MemoryManager<StdRegions::StdTetExp>::AllocateSharedPtr(
        m_base[0]->GetBasisKey(), m_base[1]->GetBasisKey(),
        m_base[2]->GetBasisKey());
}
 
StdRegions::StdExpansionSharedPtr TetExp::v_GetLinStdExp(void) const
{
    LibUtilities::BasisKey bkey0(m_base[0]->GetBasisType(), 2,
                                 m_base[0]->GetPointsKey());
    LibUtilities::BasisKey bkey1(m_base[1]->GetBasisType(), 2,
                                 m_base[1]->GetPointsKey());
    LibUtilities::BasisKey bkey2(m_base[2]->GetBasisType(), 2,
                                 m_base[2]->GetPointsKey());
 
    return MemoryManager<StdRegions::StdTetExp>::AllocateSharedPtr(bkey0, bkey1,
                                                                   bkey2);
}
 
void TetExp::v_ExtractDataToCoeffs(
    const NekDouble *data, const std::vector<unsigned int> &nummodes,
    const int mode_offset, NekDouble *coeffs,
    std::vector<LibUtilities::BasisType> &fromType)
{
    boost::ignore_unused(fromType);
 
    int data_order0 = nummodes[mode_offset];
    int fillorder0  = min(m_base[0]->GetNumModes(), data_order0);
    int data_order1 = nummodes[mode_offset + 1];
    int order1      = m_base[1]->GetNumModes();
    int fillorder1  = min(order1, data_order1);
    int data_order2 = nummodes[mode_offset + 2];
    int order2      = m_base[2]->GetNumModes();
    int fillorder2  = min(order2, data_order2);
 
    switch (m_base[0]->GetBasisType())
    {
        case LibUtilities::eModified_A:
        {
            int i, j;
            int cnt  = 0;
            int cnt1 = 0;
 
            ASSERTL1(m_base[1]->GetBasisType() == LibUtilities::eModified_B,
                     "Extraction routine not set up for this basis");
            ASSERTL1(m_base[2]->GetBasisType() == LibUtilities::eModified_C,
                     "Extraction routine not set up for this basis");
 
            Vmath::Zero(m_ncoeffs, coeffs, 1);
            for (j = 0; j < fillorder0; ++j)
            {
                for (i = 0; i < fillorder1 - j; ++i)
                {
                    Vmath::Vcopy(fillorder2 - j - i, &data[cnt], 1,
                                 &coeffs[cnt1], 1);
                    cnt += data_order2 - j - i;
                    cnt1 += order2 - j - i;
                }
 
                // count out data for j iteration
                for (i = fillorder1 - j; i < data_order1 - j; ++i)
                {
                    cnt += data_order2 - j - i;
                }
 
                for (i = fillorder1 - j; i < order1 - j; ++i)
                {
                    cnt1 += order2 - j - i;
                }
            }
        }
        break;
        default:
            ASSERTL0(false, "basis is either not set up or not "
                            "hierarchicial");
    }
}
 
/**
 * \brief Returns the physical values at the quadrature points of a face
 */
void TetExp::v_GetTracePhysMap(const int face, Array<OneD, int> &outarray)
{
    int nquad0 = m_base[0]->GetNumPoints();
    int nquad1 = m_base[1]->GetNumPoints();
    int nquad2 = m_base[2]->GetNumPoints();
 
    int nq0 = 0;
    int nq1 = 0;
 
    // get forward aligned faces.
    switch (face)
    {
        case 0:
        {
            nq0 = nquad0;
            nq1 = nquad1;
            if (outarray.size() != nq0 * nq1)
            {
                outarray = Array<OneD, int>(nq0 * nq1);
            }
 
            for (int i = 0; i < nquad0 * nquad1; ++i)
            {
                outarray[i] = i;
            }
 
            break;
        }
        case 1:
        {
            nq0 = nquad0;
            nq1 = nquad2;
            if (outarray.size() != nq0 * nq1)
            {
                outarray = Array<OneD, int>(nq0 * nq1);
            }
 
            // Direction A and B positive
            for (int k = 0; k < nquad2; k++)
            {
                for (int i = 0; i < nquad0; ++i)
                {
                    outarray[k * nquad0 + i] = (nquad0 * nquad1 * k) + i;
                }
            }
            break;
        }
        case 2:
        {
            nq0 = nquad1;
            nq1 = nquad2;
            if (outarray.size() != nq0 * nq1)
            {
                outarray = Array<OneD, int>(nq0 * nq1);
            }
 
            // Directions A and B positive
            for (int j = 0; j < nquad1 * nquad2; ++j)
            {
                outarray[j] = nquad0 - 1 + j * nquad0;
            }
            break;
        }
        case 3:
        {
            nq0 = nquad1;
            nq1 = nquad2;
            if (outarray.size() != nq0 * nq1)
            {
                outarray = Array<OneD, int>(nq0 * nq1);
            }
 
            // Directions A and B positive
            for (int j = 0; j < nquad1 * nquad2; ++j)
            {
                outarray[j] = j * nquad0;
            }
        }
        break;
        default:
            ASSERTL0(false, "face value (> 3) is out of range");
            break;
    }
}
 
/**
 * \brief Compute the normal of a triangular face
 */
void TetExp::v_ComputeTraceNormal(const int face)
{
    int i;
    const SpatialDomains::GeomFactorsSharedPtr &geomFactors =
        GetGeom()->GetMetricInfo();
 
    LibUtilities::PointsKeyVector ptsKeys = GetPointsKeys();
    for (int i = 0; i < ptsKeys.size(); ++i)
    {
        // Need at least 2 points for computing normals
        if (ptsKeys[i].GetNumPoints() == 1)
        {
            LibUtilities::PointsKey pKey(2, ptsKeys[i].GetPointsType());
            ptsKeys[i] = pKey;
        }
    }
 
    SpatialDomains::GeomType type = geomFactors->GetGtype();
    const Array<TwoD, const NekDouble> &df =
        geomFactors->GetDerivFactors(ptsKeys);
    const Array<OneD, const NekDouble> &jac = geomFactors->GetJac(ptsKeys);
 
    LibUtilities::BasisKey tobasis0 = GetTraceBasisKey(face, 0);
    LibUtilities::BasisKey tobasis1 = GetTraceBasisKey(face, 1);
 
    // number of face quadrature points
    int nq_face = tobasis0.GetNumPoints() * tobasis1.GetNumPoints();
 
    int vCoordDim = GetCoordim();
 
    m_traceNormals[face] = Array<OneD, Array<OneD, NekDouble>>(vCoordDim);
    Array<OneD, Array<OneD, NekDouble>> &normal = m_traceNormals[face];
    for (i = 0; i < vCoordDim; ++i)
    {
        normal[i] = Array<OneD, NekDouble>(nq_face);
    }
 
    size_t nqb                     = nq_face;
    size_t nbnd                    = face;
    m_elmtBndNormDirElmtLen[nbnd]  = Array<OneD, NekDouble>{nqb, 0.0};
    Array<OneD, NekDouble> &length = m_elmtBndNormDirElmtLen[nbnd];
 
    // Regular geometry case
    if (type == SpatialDomains::eRegular ||
        type == SpatialDomains::eMovingRegular)
    {
        NekDouble fac;
 
        // Set up normals
        switch (face)
        {
            case 0:
            {
                for (i = 0; i < vCoordDim; ++i)
                {
                    normal[i][0] = -df[3 * i + 2][0];
                }
 
                break;
            }
            case 1:
            {
                for (i = 0; i < vCoordDim; ++i)
                {
                    normal[i][0] = -df[3 * i + 1][0];
                }
 
                break;
            }
            case 2:
            {
                for (i = 0; i < vCoordDim; ++i)
                {
                    normal[i][0] =
                        df[3 * i][0] + df[3 * i + 1][0] + df[3 * i + 2][0];
                }
 
                break;
            }
            case 3:
            {
                for (i = 0; i < vCoordDim; ++i)
                {
                    normal[i][0] = -df[3 * i][0];
                }
                break;
            }
            default:
                ASSERTL0(false, "face is out of range (edge < 3)");
        }
 
        // normalise
        fac = 0.0;
        for (i = 0; i < vCoordDim; ++i)
        {
            fac += normal[i][0] * normal[i][0];
        }
        fac = 1.0 / sqrt(fac);
        Vmath::Fill(nqb, fac, length, 1);
 
        for (i = 0; i < vCoordDim; ++i)
        {
            Vmath::Fill(nq_face, fac * normal[i][0], normal[i], 1);
        }
    }
    else
    {
        // Set up deformed normals
        int j, k;
 
        int nq0 = ptsKeys[0].GetNumPoints();
        int nq1 = ptsKeys[1].GetNumPoints();
        int nq2 = ptsKeys[2].GetNumPoints();
        int nqtot;
        int nq01 = nq0 * nq1;
 
        // number of elemental quad points
        if (face == 0)
        {
            nqtot = nq01;
        }
        else if (face == 1)
        {
            nqtot = nq0 * nq2;
        }
        else
        {
            nqtot = nq1 * nq2;
        }
 
        LibUtilities::PointsKey points0;
        LibUtilities::PointsKey points1;
 
        Array<OneD, NekDouble> faceJac(nqtot);
        Array<OneD, NekDouble> normals(vCoordDim * nqtot, 0.0);
 
        // Extract Jacobian along face and recover local derivates
        // (dx/dr) for polynomial interpolation by multiplying m_gmat by
        // jacobian
        switch (face)
        {
            case 0:
            {
                for (j = 0; j < nq01; ++j)
                {
                    normals[j]             = -df[2][j] * jac[j];
                    normals[nqtot + j]     = -df[5][j] * jac[j];
                    normals[2 * nqtot + j] = -df[8][j] * jac[j];
                    faceJac[j]             = jac[j];
                }
 
                points0 = ptsKeys[0];
                points1 = ptsKeys[1];
                break;
            }
 
            case 1:
            {
                for (j = 0; j < nq0; ++j)
                {
                    for (k = 0; k < nq2; ++k)
                    {
                        int tmp                      = j + nq01 * k;
                        normals[j + k * nq0]         = -df[1][tmp] * jac[tmp];
                        normals[nqtot + j + k * nq0] = -df[4][tmp] * jac[tmp];
                        normals[2 * nqtot + j + k * nq0] =
                            -df[7][tmp] * jac[tmp];
                        faceJac[j + k * nq0] = jac[tmp];
                    }
                }
 
                points0 = ptsKeys[0];
                points1 = ptsKeys[2];
                break;
            }
 
            case 2:
            {
                for (j = 0; j < nq1; ++j)
                {
                    for (k = 0; k < nq2; ++k)
                    {
                        int tmp = nq0 - 1 + nq0 * j + nq01 * k;
                        normals[j + k * nq1] =
                            (df[0][tmp] + df[1][tmp] + df[2][tmp]) * jac[tmp];
                        normals[nqtot + j + k * nq1] =
                            (df[3][tmp] + df[4][tmp] + df[5][tmp]) * jac[tmp];
                        normals[2 * nqtot + j + k * nq1] =
                            (df[6][tmp] + df[7][tmp] + df[8][tmp]) * jac[tmp];
                        faceJac[j + k * nq1] = jac[tmp];
                    }
                }
 
                points0 = ptsKeys[1];
                points1 = ptsKeys[2];
                break;
            }
 
            case 3:
            {
                for (j = 0; j < nq1; ++j)
                {
                    for (k = 0; k < nq2; ++k)
                    {
                        int tmp                      = j * nq0 + nq01 * k;
                        normals[j + k * nq1]         = -df[0][tmp] * jac[tmp];
                        normals[nqtot + j + k * nq1] = -df[3][tmp] * jac[tmp];
                        normals[2 * nqtot + j + k * nq1] =
                            -df[6][tmp] * jac[tmp];
                        faceJac[j + k * nq1] = jac[tmp];
                    }
                }
 
                points0 = ptsKeys[1];
                points1 = ptsKeys[2];
                break;
            }
 
            default:
                ASSERTL0(false, "face is out of range (face < 3)");
        }
 
        Array<OneD, NekDouble> work(nq_face, 0.0);
        // Interpolate Jacobian and invert
        LibUtilities::Interp2D(points0, points1, faceJac,
                               tobasis0.GetPointsKey(), tobasis1.GetPointsKey(),
                               work);
        Vmath::Sdiv(nq_face, 1.0, &work[0], 1, &work[0], 1);
 
        // Interpolate normal and multiply by inverse Jacobian.
        for (i = 0; i < vCoordDim; ++i)
        {
            LibUtilities::Interp2D(points0, points1, &normals[i * nqtot],
                                   tobasis0.GetPointsKey(),
                                   tobasis1.GetPointsKey(), &normal[i][0]);
            Vmath::Vmul(nq_face, work, 1, normal[i], 1, normal[i], 1);
        }
 
        // Normalise to obtain unit normals.
        Vmath::Zero(nq_face, work, 1);
        for (i = 0; i < GetCoordim(); ++i)
        {
            Vmath::Vvtvp(nq_face, normal[i], 1, normal[i], 1, work, 1, work, 1);
        }
 
        Vmath::Vsqrt(nq_face, work, 1, work, 1);
        Vmath::Sdiv(nq_face, 1.0, work, 1, work, 1);
 
        Vmath::Vcopy(nqb, work, 1, length, 1);
 
        for (i = 0; i < GetCoordim(); ++i)
        {
            Vmath::Vmul(nq_face, normal[i], 1, work, 1, normal[i], 1);
        }
    }
}
 
//-----------------------------
// Operator creation functions
//-----------------------------
void TetExp::v_HelmholtzMatrixOp(const Array<OneD, const NekDouble> &inarray,
                                 Array<OneD, NekDouble> &outarray,
                                 const StdRegions::StdMatrixKey &mkey)
{
    TetExp::v_HelmholtzMatrixOp_MatFree(inarray, outarray, mkey);
}
 
void TetExp::v_LaplacianMatrixOp(const Array<OneD, const NekDouble> &inarray,
                                 Array<OneD, NekDouble> &outarray,
                                 const StdRegions::StdMatrixKey &mkey)
{
    TetExp::v_LaplacianMatrixOp_MatFree(inarray, outarray, mkey);
}
 
void TetExp::v_LaplacianMatrixOp(const int k1, const int k2,
                                 const Array<OneD, const NekDouble> &inarray,
                                 Array<OneD, NekDouble> &outarray,
                                 const StdRegions::StdMatrixKey &mkey)
{
    StdExpansion::LaplacianMatrixOp_MatFree(k1, k2, inarray, outarray, mkey);
}
 
void TetExp::v_SVVLaplacianFilter(Array<OneD, NekDouble> &array,
                                  const StdRegions::StdMatrixKey &mkey)
{
    int nq = GetTotPoints();
 
    // Calculate sqrt of the Jacobian
    Array<OneD, const NekDouble> jac = m_metricinfo->GetJac(GetPointsKeys());
    Array<OneD, NekDouble> sqrt_jac(nq);
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        Vmath::Vsqrt(nq, jac, 1, sqrt_jac, 1);
    }
    else
    {
        Vmath::Fill(nq, sqrt(jac[0]), sqrt_jac, 1);
    }
 
    // Multiply array by sqrt(Jac)
    Vmath::Vmul(nq, sqrt_jac, 1, array, 1, array, 1);
 
    // Apply std region filter
    StdTetExp::v_SVVLaplacianFilter(array, mkey);
 
    // Divide by sqrt(Jac)
    Vmath::Vdiv(nq, array, 1, sqrt_jac, 1, array, 1);
}
 
//-----------------------------
// Matrix creation functions
//-----------------------------
DNekMatSharedPtr TetExp::v_GenMatrix(const StdRegions::StdMatrixKey &mkey)
{
    DNekMatSharedPtr returnval;
 
    switch (mkey.GetMatrixType())
    {
        case StdRegions::eHybridDGHelmholtz:
        case StdRegions::eHybridDGLamToU:
        case StdRegions::eHybridDGLamToQ0:
        case StdRegions::eHybridDGLamToQ1:
        case StdRegions::eHybridDGLamToQ2:
        case StdRegions::eHybridDGHelmBndLam:
        case StdRegions::eInvLaplacianWithUnityMean:
            returnval = Expansion3D::v_GenMatrix(mkey);
            break;
        default:
            returnval = StdTetExp::v_GenMatrix(mkey);
    }
 
    return returnval;
}
 
DNekMatSharedPtr TetExp::v_CreateStdMatrix(const StdRegions::StdMatrixKey &mkey)
{
    LibUtilities::BasisKey bkey0 = m_base[0]->GetBasisKey();
    LibUtilities::BasisKey bkey1 = m_base[1]->GetBasisKey();
    LibUtilities::BasisKey bkey2 = m_base[2]->GetBasisKey();
    StdRegions::StdTetExpSharedPtr tmp =
        MemoryManager<StdTetExp>::AllocateSharedPtr(bkey0, bkey1, bkey2);
 
    return tmp->GetStdMatrix(mkey);
}
 
DNekScalMatSharedPtr TetExp::v_GetLocMatrix(const MatrixKey &mkey)
{
    return m_matrixManager[mkey];
}
 
void TetExp::v_DropLocMatrix(const MatrixKey &mkey)
{
    m_matrixManager.DeleteObject(mkey);
}
 
DNekScalBlkMatSharedPtr TetExp::v_GetLocStaticCondMatrix(const MatrixKey &mkey)
{
    return m_staticCondMatrixManager[mkey];
}
 
void TetExp::v_DropLocStaticCondMatrix(const MatrixKey &mkey)
{
    m_staticCondMatrixManager.DeleteObject(mkey);
}
 
void TetExp::GeneralMatrixOp_MatOp(const Array<OneD, const NekDouble> &inarray,
                                   Array<OneD, NekDouble> &outarray,
                                   const StdRegions::StdMatrixKey &mkey)
{
    DNekScalMatSharedPtr mat = GetLocMatrix(mkey);
 
    if (inarray.get() == outarray.get())
    {
        Array<OneD, NekDouble> tmp(m_ncoeffs);
        Vmath::Vcopy(m_ncoeffs, inarray.get(), 1, tmp.get(), 1);
 
        Blas::Dgemv('N', m_ncoeffs, m_ncoeffs, mat->Scale(),
                    (mat->GetOwnedMatrix())->GetPtr().get(), m_ncoeffs,
                    tmp.get(), 1, 0.0, outarray.get(), 1);
    }
    else
    {
        Blas::Dgemv('N', m_ncoeffs, m_ncoeffs, mat->Scale(),
                    (mat->GetOwnedMatrix())->GetPtr().get(), m_ncoeffs,
                    inarray.get(), 1, 0.0, outarray.get(), 1);
    }
}
 
void TetExp::v_LaplacianMatrixOp_MatFree_Kernel(
    const Array<OneD, const NekDouble> &inarray,
    Array<OneD, NekDouble> &outarray, Array<OneD, NekDouble> &wsp)
{
    // This implementation is only valid when there are no
    // coefficients associated to the Laplacian operator
    if (m_metrics.count(eMetricLaplacian00) == 0)
    {
        ComputeLaplacianMetric();
    }
 
    int nquad0 = m_base[0]->GetNumPoints();
    int nquad1 = m_base[1]->GetNumPoints();
    int nquad2 = m_base[2]->GetNumPoints();
    int nqtot  = nquad0 * nquad1 * nquad2;
 
    ASSERTL1(wsp.size() >= 6 * nqtot, "Insufficient workspace size.");
    ASSERTL1(m_ncoeffs <= nqtot, "Workspace not set up for ncoeffs > nqtot");
 
    const Array<OneD, const NekDouble> &base0  = m_base[0]->GetBdata();
    const Array<OneD, const NekDouble> &base1  = m_base[1]->GetBdata();
    const Array<OneD, const NekDouble> &base2  = m_base[2]->GetBdata();
    const Array<OneD, const NekDouble> &dbase0 = m_base[0]->GetDbdata();
    const Array<OneD, const NekDouble> &dbase1 = m_base[1]->GetDbdata();
    const Array<OneD, const NekDouble> &dbase2 = m_base[2]->GetDbdata();
    const Array<OneD, const NekDouble> &metric00 =
        m_metrics[eMetricLaplacian00];
    const Array<OneD, const NekDouble> &metric01 =
        m_metrics[eMetricLaplacian01];
    const Array<OneD, const NekDouble> &metric02 =
        m_metrics[eMetricLaplacian02];
    const Array<OneD, const NekDouble> &metric11 =
        m_metrics[eMetricLaplacian11];
    const Array<OneD, const NekDouble> &metric12 =
        m_metrics[eMetricLaplacian12];
    const Array<OneD, const NekDouble> &metric22 =
        m_metrics[eMetricLaplacian22];
 
    // Allocate temporary storage
    Array<OneD, NekDouble> wsp0(2 * nqtot, wsp);
    Array<OneD, NekDouble> wsp1(nqtot, wsp + 1 * nqtot);
    Array<OneD, NekDouble> wsp2(nqtot, wsp + 2 * nqtot);
    Array<OneD, NekDouble> wsp3(nqtot, wsp + 3 * nqtot);
    Array<OneD, NekDouble> wsp4(nqtot, wsp + 4 * nqtot);
    Array<OneD, NekDouble> wsp5(nqtot, wsp + 5 * nqtot);
 
    // LAPLACIAN MATRIX OPERATION
    // wsp1 = du_dxi1 = D_xi1 * inarray = D_xi1 * u
    // wsp2 = du_dxi2 = D_xi2 * inarray = D_xi2 * u
    // wsp2 = du_dxi3 = D_xi3 * inarray = D_xi3 * u
    StdExpansion3D::PhysTensorDeriv(inarray, wsp0, wsp1, wsp2);
 
    // wsp0 = k = g0 * wsp1 + g1 * wsp2 = g0 * du_dxi1 + g1 * du_dxi2
    // wsp2 = l = g1 * wsp1 + g2 * wsp2 = g0 * du_dxi1 + g1 * du_dxi2
    // where g0, g1 and g2 are the metric terms set up in the GeomFactors class
    // especially for this purpose
    Vmath::Vvtvvtp(nqtot, &metric00[0], 1, &wsp0[0], 1, &metric01[0], 1,
                   &wsp1[0], 1, &wsp3[0], 1);
    Vmath::Vvtvp(nqtot, &metric02[0], 1, &wsp2[0], 1, &wsp3[0], 1, &wsp3[0], 1);
    Vmath::Vvtvvtp(nqtot, &metric01[0], 1, &wsp0[0], 1, &metric11[0], 1,
                   &wsp1[0], 1, &wsp4[0], 1);
    Vmath::Vvtvp(nqtot, &metric12[0], 1, &wsp2[0], 1, &wsp4[0], 1, &wsp4[0], 1);
    Vmath::Vvtvvtp(nqtot, &metric02[0], 1, &wsp0[0], 1, &metric12[0], 1,
                   &wsp1[0], 1, &wsp5[0], 1);
    Vmath::Vvtvp(nqtot, &metric22[0], 1, &wsp2[0], 1, &wsp5[0], 1, &wsp5[0], 1);
 
    // outarray = m = (D_xi1 * B)^T * k
    // wsp1     = n = (D_xi2 * B)^T * l
    IProductWRTBase_SumFacKernel(dbase0, base1, base2, wsp3, outarray, wsp0,
                                 false, true, true);
    IProductWRTBase_SumFacKernel(base0, dbase1, base2, wsp4, wsp2, wsp0, true,
                                 false, true);
    Vmath::Vadd(m_ncoeffs, wsp2.get(), 1, outarray.get(), 1, outarray.get(), 1);
    IProductWRTBase_SumFacKernel(base0, base1, dbase2, wsp5, wsp2, wsp0, true,
                                 true, false);
    Vmath::Vadd(m_ncoeffs, wsp2.get(), 1, outarray.get(), 1, outarray.get(), 1);
}
 
void TetExp::v_ComputeLaplacianMetric()
{
    if (m_metrics.count(eMetricQuadrature) == 0)
    {
        ComputeQuadratureMetric();
    }
 
    int i, j;
    const unsigned int nqtot = GetTotPoints();
    const unsigned int dim   = 3;
    const MetricType m[3][3] = {
        {eMetricLaplacian00, eMetricLaplacian01, eMetricLaplacian02},
        {eMetricLaplacian01, eMetricLaplacian11, eMetricLaplacian12},
        {eMetricLaplacian02, eMetricLaplacian12, eMetricLaplacian22}};
 
    for (unsigned int i = 0; i < dim; ++i)
    {
        for (unsigned int j = i; j < dim; ++j)
        {
            m_metrics[m[i][j]] = Array<OneD, NekDouble>(nqtot);
        }
    }
 
    // Define shorthand synonyms for m_metrics storage
    Array<OneD, NekDouble> g0(m_metrics[m[0][0]]);
    Array<OneD, NekDouble> g1(m_metrics[m[1][1]]);
    Array<OneD, NekDouble> g2(m_metrics[m[2][2]]);
    Array<OneD, NekDouble> g3(m_metrics[m[0][1]]);
    Array<OneD, NekDouble> g4(m_metrics[m[0][2]]);
    Array<OneD, NekDouble> g5(m_metrics[m[1][2]]);
 
    // Allocate temporary storage
    Array<OneD, NekDouble> alloc(7 * nqtot, 0.0);
    Array<OneD, NekDouble> h0(alloc);               // h0
    Array<OneD, NekDouble> h1(alloc + 1 * nqtot);   // h1
    Array<OneD, NekDouble> h2(alloc + 2 * nqtot);   // h2
    Array<OneD, NekDouble> h3(alloc + 3 * nqtot);   // h3
    Array<OneD, NekDouble> wsp4(alloc + 4 * nqtot); // wsp4
    Array<OneD, NekDouble> wsp5(alloc + 5 * nqtot); // wsp5
    Array<OneD, NekDouble> wsp6(alloc + 6 * nqtot); // wsp6
    // Reuse some of the storage as workspace
    Array<OneD, NekDouble> wsp7(alloc);             // wsp7
    Array<OneD, NekDouble> wsp8(alloc + 1 * nqtot); // wsp8
    Array<OneD, NekDouble> wsp9(alloc + 2 * nqtot); // wsp9
 
    const Array<TwoD, const NekDouble> &df =
        m_metricinfo->GetDerivFactors(GetPointsKeys());
    const Array<OneD, const NekDouble> &z0 = m_base[0]->GetZ();
    const Array<OneD, const NekDouble> &z1 = m_base[1]->GetZ();
    const Array<OneD, const NekDouble> &z2 = m_base[2]->GetZ();
    const unsigned int nquad0              = m_base[0]->GetNumPoints();
    const unsigned int nquad1              = m_base[1]->GetNumPoints();
    const unsigned int nquad2              = m_base[2]->GetNumPoints();
 
    for (j = 0; j < nquad2; ++j)
    {
        for (i = 0; i < nquad1; ++i)
        {
            Vmath::Fill(nquad0, 4.0 / (1.0 - z1[i]) / (1.0 - z2[j]),
                        &h0[0] + i * nquad0 + j * nquad0 * nquad1, 1);
            Vmath::Fill(nquad0, 2.0 / (1.0 - z1[i]) / (1.0 - z2[j]),
                        &h1[0] + i * nquad0 + j * nquad0 * nquad1, 1);
            Vmath::Fill(nquad0, 2.0 / (1.0 - z2[j]),
                        &h2[0] + i * nquad0 + j * nquad0 * nquad1, 1);
            Vmath::Fill(nquad0, (1.0 + z1[i]) / (1.0 - z2[j]),
                        &h3[0] + i * nquad0 + j * nquad0 * nquad1, 1);
        }
    }
    for (i = 0; i < nquad0; i++)
    {
        Blas::Dscal(nquad1 * nquad2, 1 + z0[i], &h1[0] + i, nquad0);
    }
 
    // Step 3. Construct combined metric terms for physical space to
    // collapsed coordinate system.
    // Order of construction optimised to minimise temporary storage
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        // wsp4
        Vmath::Vadd(nqtot, &df[1][0], 1, &df[2][0], 1, &wsp4[0], 1);
        Vmath::Vvtvvtp(nqtot, &df[0][0], 1, &h0[0], 1, &wsp4[0], 1, &h1[0], 1,
                       &wsp4[0], 1);
        // wsp5
        Vmath::Vadd(nqtot, &df[4][0], 1, &df[5][0], 1, &wsp5[0], 1);
        Vmath::Vvtvvtp(nqtot, &df[3][0], 1, &h0[0], 1, &wsp5[0], 1, &h1[0], 1,
                       &wsp5[0], 1);
        // wsp6
        Vmath::Vadd(nqtot, &df[7][0], 1, &df[8][0], 1, &wsp6[0], 1);
        Vmath::Vvtvvtp(nqtot, &df[6][0], 1, &h0[0], 1, &wsp6[0], 1, &h1[0], 1,
                       &wsp6[0], 1);
 
        // g0
        Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp4[0], 1, &wsp5[0], 1, &wsp5[0],
                       1, &g0[0], 1);
        Vmath::Vvtvp(nqtot, &wsp6[0], 1, &wsp6[0], 1, &g0[0], 1, &g0[0], 1);
 
        // g4
        Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &wsp4[0], 1, &df[5][0], 1, &wsp5[0],
                       1, &g4[0], 1);
        Vmath::Vvtvp(nqtot, &df[8][0], 1, &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
 
        // overwrite h0, h1, h2
        // wsp7 (h2f1 + h3f2)
        Vmath::Vvtvvtp(nqtot, &df[1][0], 1, &h2[0], 1, &df[2][0], 1, &h3[0], 1,
                       &wsp7[0], 1);
        // wsp8 (h2f4 + h3f5)
        Vmath::Vvtvvtp(nqtot, &df[4][0], 1, &h2[0], 1, &df[5][0], 1, &h3[0], 1,
                       &wsp8[0], 1);
        // wsp9 (h2f7 + h3f8)
        Vmath::Vvtvvtp(nqtot, &df[7][0], 1, &h2[0], 1, &df[8][0], 1, &h3[0], 1,
                       &wsp9[0], 1);
 
        // g3
        Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp7[0], 1, &wsp5[0], 1, &wsp8[0],
                       1, &g3[0], 1);
        Vmath::Vvtvp(nqtot, &wsp6[0], 1, &wsp9[0], 1, &g3[0], 1, &g3[0], 1);
 
        // overwrite wsp4, wsp5, wsp6
        // g1
        Vmath::Vvtvvtp(nqtot, &wsp7[0], 1, &wsp7[0], 1, &wsp8[0], 1, &wsp8[0],
                       1, &g1[0], 1);
        Vmath::Vvtvp(nqtot, &wsp9[0], 1, &wsp9[0], 1, &g1[0], 1, &g1[0], 1);
 
        // g5
        Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &wsp7[0], 1, &df[5][0], 1, &wsp8[0],
                       1, &g5[0], 1);
        Vmath::Vvtvp(nqtot, &df[8][0], 1, &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
 
        // g2
        Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &df[2][0], 1, &df[5][0], 1,
                       &df[5][0], 1, &g2[0], 1);
        Vmath::Vvtvp(nqtot, &df[8][0], 1, &df[8][0], 1, &g2[0], 1, &g2[0], 1);
    }
    else
    {
        // wsp4
        Vmath::Svtsvtp(nqtot, df[0][0], &h0[0], 1, df[1][0] + df[2][0], &h1[0],
                       1, &wsp4[0], 1);
        // wsp5
        Vmath::Svtsvtp(nqtot, df[3][0], &h0[0], 1, df[4][0] + df[5][0], &h1[0],
                       1, &wsp5[0], 1);
        // wsp6
        Vmath::Svtsvtp(nqtot, df[6][0], &h0[0], 1, df[7][0] + df[8][0], &h1[0],
                       1, &wsp6[0], 1);
 
        // g0
        Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp4[0], 1, &wsp5[0], 1, &wsp5[0],
                       1, &g0[0], 1);
        Vmath::Vvtvp(nqtot, &wsp6[0], 1, &wsp6[0], 1, &g0[0], 1, &g0[0], 1);
 
        // g4
        Vmath::Svtsvtp(nqtot, df[2][0], &wsp4[0], 1, df[5][0], &wsp5[0], 1,
                       &g4[0], 1);
        Vmath::Svtvp(nqtot, df[8][0], &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
 
        // overwrite h0, h1, h2
        // wsp7 (h2f1 + h3f2)
        Vmath::Svtsvtp(nqtot, df[1][0], &h2[0], 1, df[2][0], &h3[0], 1,
                       &wsp7[0], 1);
        // wsp8 (h2f4 + h3f5)
        Vmath::Svtsvtp(nqtot, df[4][0], &h2[0], 1, df[5][0], &h3[0], 1,
                       &wsp8[0], 1);
        // wsp9 (h2f7 + h3f8)
        Vmath::Svtsvtp(nqtot, df[7][0], &h2[0], 1, df[8][0], &h3[0], 1,
                       &wsp9[0], 1);
 
        // g3
        Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp7[0], 1, &wsp5[0], 1, &wsp8[0],
                       1, &g3[0], 1);
        Vmath::Vvtvp(nqtot, &wsp6[0], 1, &wsp9[0], 1, &g3[0], 1, &g3[0], 1);
 
        // overwrite wsp4, wsp5, wsp6
        // g1
        Vmath::Vvtvvtp(nqtot, &wsp7[0], 1, &wsp7[0], 1, &wsp8[0], 1, &wsp8[0],
                       1, &g1[0], 1);
        Vmath::Vvtvp(nqtot, &wsp9[0], 1, &wsp9[0], 1, &g1[0], 1, &g1[0], 1);
 
        // g5
        Vmath::Svtsvtp(nqtot, df[2][0], &wsp7[0], 1, df[5][0], &wsp8[0], 1,
                       &g5[0], 1);
        Vmath::Svtvp(nqtot, df[8][0], &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
 
        // g2
        Vmath::Fill(nqtot,
                    df[2][0] * df[2][0] + df[5][0] * df[5][0] +
                        df[8][0] * df[8][0],
                    &g2[0], 1);
    }
 
    for (unsigned int i = 0; i < dim; ++i)
    {
        for (unsigned int j = i; j < dim; ++j)
        {
            MultiplyByQuadratureMetric(m_metrics[m[i][j]], m_metrics[m[i][j]]);
        }
    }
}
 
/** @brief: This method gets all of the factors which are
    required as part of the Gradient Jump Penalty
    stabilisation and involves the product of the normal and
    geometric factors along the element trace.
*/
void TetExp::v_NormalTraceDerivFactors(
    Array<OneD, Array<OneD, NekDouble>> &d0factors,
    Array<OneD, Array<OneD, NekDouble>> &d1factors,
    Array<OneD, Array<OneD, NekDouble>> &d2factors)
{
    int nquad0 = GetNumPoints(0);
    int nquad1 = GetNumPoints(1);
    int nquad2 = GetNumPoints(2);
 
    const Array<TwoD, const NekDouble> &df =
        m_metricinfo->GetDerivFactors(GetPointsKeys());
 
    if (d0factors.size() != 4)
    {
        d0factors = Array<OneD, Array<OneD, NekDouble>>(4);
        d1factors = Array<OneD, Array<OneD, NekDouble>>(4);
        d2factors = Array<OneD, Array<OneD, NekDouble>>(4);
    }
 
    if (d0factors[0].size() != nquad0 * nquad1)
    {
        d0factors[0] = Array<OneD, NekDouble>(nquad0 * nquad1);
        d1factors[0] = Array<OneD, NekDouble>(nquad0 * nquad1);
        d2factors[0] = Array<OneD, NekDouble>(nquad0 * nquad1);
    }
 
    if (d0factors[1].size() != nquad0 * nquad2)
    {
        d0factors[1] = Array<OneD, NekDouble>(nquad0 * nquad2);
        d1factors[1] = Array<OneD, NekDouble>(nquad0 * nquad2);
        d2factors[1] = Array<OneD, NekDouble>(nquad0 * nquad2);
    }
 
    if (d0factors[2].size() != nquad1 * nquad2)
    {
        d0factors[2] = Array<OneD, NekDouble>(nquad1 * nquad2);
        d0factors[3] = Array<OneD, NekDouble>(nquad1 * nquad2);
        d1factors[2] = Array<OneD, NekDouble>(nquad1 * nquad2);
        d1factors[3] = Array<OneD, NekDouble>(nquad1 * nquad2);
        d2factors[2] = Array<OneD, NekDouble>(nquad1 * nquad2);
        d2factors[3] = Array<OneD, NekDouble>(nquad1 * nquad2);
    }
 
    // Outwards normals
    const Array<OneD, const Array<OneD, NekDouble>> &normal_0 =
        GetTraceNormal(0);
    const Array<OneD, const Array<OneD, NekDouble>> &normal_1 =
        GetTraceNormal(1);
    const Array<OneD, const Array<OneD, NekDouble>> &normal_2 =
        GetTraceNormal(2);
    const Array<OneD, const Array<OneD, NekDouble>> &normal_3 =
        GetTraceNormal(3);
 
    int ncoords = normal_0.size();
 
    // first gather together standard cartesian inner products
    if (m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
    {
        // face 0
        for (int i = 0; i < nquad0 * nquad1; ++i)
        {
            d0factors[0][i] = df[0][i] * normal_0[0][i];
            d1factors[0][i] = df[1][i] * normal_0[0][i];
            d2factors[0][i] = df[2][i] * normal_0[0][i];
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int i = 0; i < nquad0 * nquad1; ++i)
            {
                d0factors[0][i] += df[3 * n][i] * normal_0[n][i];
                d1factors[0][i] += df[3 * n + 1][i] * normal_0[n][i];
                d2factors[0][i] += df[3 * n + 2][i] * normal_0[n][i];
            }
        }
 
        // face 1
        for (int j = 0; j < nquad2; ++j)
        {
            for (int i = 0; i < nquad0; ++i)
            {
                d0factors[1][j * nquad0 + i] = df[0][j * nquad0 * nquad1 + i] *
                                               normal_1[0][j * nquad0 + i];
                d1factors[1][j * nquad0 + i] = df[1][j * nquad0 * nquad1 + i] *
                                               normal_1[0][j * nquad0 + i];
                d2factors[1][j * nquad0 + i] = df[2][j * nquad0 * nquad1 + i] *
                                               normal_1[0][j * nquad0 + i];
            }
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int j = 0; j < nquad2; ++j)
            {
                for (int i = 0; i < nquad0; ++i)
                {
                    d0factors[1][j * nquad0 + i] +=
                        df[3 * n][j * nquad0 * nquad1 + i] *
                        normal_1[0][j * nquad0 + i];
                    d1factors[1][j * nquad0 + i] +=
                        df[3 * n + 1][j * nquad0 * nquad1 + i] *
                        normal_1[0][j * nquad0 + i];
                    d2factors[1][j * nquad0 + i] +=
                        df[3 * n + 2][j * nquad0 * nquad1 + i] *
                        normal_1[0][j * nquad0 + i];
                }
            }
        }
 
        // faces 2 and 3
        for (int j = 0; j < nquad2; ++j)
        {
            for (int i = 0; i < nquad1; ++i)
            {
                d0factors[2][j * nquad1 + i] =
                    df[0][(j * nquad1 + i + 1) * nquad0 - 1] *
                    normal_2[0][j * nquad1 + i];
                d1factors[2][j * nquad1 + i] =
                    df[1][(j * nquad1 + i + 1) * nquad0 - 1] *
                    normal_2[0][j * nquad1 + i];
                d2factors[2][j * nquad1 + i] =
                    df[2][(j * nquad1 + i + 1) * nquad0 - 1] *
                    normal_2[0][j * nquad1 + i];
 
                d0factors[3][j * nquad1 + i] =
                    df[0][(j * nquad1 + i) * nquad0] *
                    normal_3[0][j * nquad1 + i];
                d1factors[3][j * nquad1 + i] =
                    df[1][(j * nquad1 + i) * nquad0] *
                    normal_3[0][j * nquad1 + i];
                d2factors[3][j * nquad1 + i] =
                    df[2][(j * nquad1 + i) * nquad0] *
                    normal_3[0][j * nquad1 + i];
            }
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int j = 0; j < nquad2; ++j)
            {
                for (int i = 0; i < nquad1; ++i)
                {
                    d0factors[2][j * nquad1 + i] +=
                        df[3 * n][(j * nquad1 + i + 1) * nquad0 - 1] *
                        normal_2[n][j * nquad1 + i];
                    d1factors[2][j * nquad1 + i] +=
                        df[3 * n + 1][(j * nquad1 + i + 1) * nquad0 - 1] *
                        normal_2[n][j * nquad1 + i];
                    d2factors[2][j * nquad1 + i] +=
                        df[3 * n + 2][(j * nquad1 + i + 1) * nquad0 - 1] *
                        normal_2[n][j * nquad1 + i];
 
                    d0factors[3][j * nquad1 + i] +=
                        df[3 * n][(j * nquad1 + i) * nquad0] *
                        normal_3[n][j * nquad1 + i];
                    d1factors[3][j * nquad1 + i] +=
                        df[3 * n + 1][(j * nquad1 + i) * nquad0] *
                        normal_3[n][j * nquad1 + i];
                    d2factors[3][j * nquad1 + i] +=
                        df[3 * n + 2][(j * nquad1 + i) * nquad0] *
                        normal_3[n][j * nquad1 + i];
                }
            }
        }
    }
    else
    {
        // Face 0
        for (int i = 0; i < nquad0 * nquad1; ++i)
        {
            d0factors[0][i] = df[0][0] * normal_0[0][i];
            d1factors[0][i] = df[1][0] * normal_0[0][i];
            d2factors[0][i] = df[2][0] * normal_0[0][i];
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int i = 0; i < nquad0 * nquad1; ++i)
            {
                d0factors[0][i] += df[3 * n][0] * normal_0[n][i];
                d1factors[0][i] += df[3 * n + 1][0] * normal_0[n][i];
                d2factors[0][i] += df[3 * n + 2][0] * normal_0[n][i];
            }
        }
 
        // face 1
        for (int i = 0; i < nquad0 * nquad2; ++i)
        {
            d0factors[1][i] = df[0][0] * normal_1[0][i];
            d1factors[1][i] = df[1][0] * normal_1[0][i];
            d2factors[1][i] = df[2][0] * normal_1[0][i];
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int i = 0; i < nquad0 * nquad2; ++i)
            {
                d0factors[1][i] += df[3 * n][0] * normal_1[n][i];
                d1factors[1][i] += df[3 * n + 1][0] * normal_1[n][i];
                d2factors[1][i] += df[3 * n + 2][0] * normal_1[n][i];
            }
        }
 
        // faces 2 and 3
        for (int i = 0; i < nquad1 * nquad2; ++i)
        {
            d0factors[2][i] = df[0][0] * normal_2[0][i];
            d0factors[3][i] = df[0][0] * normal_3[0][i];
 
            d1factors[2][i] = df[1][0] * normal_2[0][i];
            d1factors[3][i] = df[1][0] * normal_3[0][i];
 
            d2factors[2][i] = df[2][0] * normal_2[0][i];
            d2factors[3][i] = df[2][0] * normal_3[0][i];
        }
 
        for (int n = 1; n < ncoords; ++n)
        {
            for (int i = 0; i < nquad1 * nquad2; ++i)
            {
                d0factors[2][i] += df[3 * n][0] * normal_2[n][i];
                d0factors[3][i] += df[3 * n][0] * normal_3[n][i];
 
                d1factors[2][i] += df[3 * n + 1][0] * normal_2[n][i];
                d1factors[3][i] += df[3 * n + 1][0] * normal_3[n][i];
 
                d2factors[2][i] += df[3 * n + 2][0] * normal_2[n][i];
                d2factors[3][i] += df[3 * n + 2][0] * normal_3[n][i];
            }
        }
    }
}
} // namespace LocalRegions
} // namespace Nektar