Extrapolate.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: Extrapolate.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:
//
// The above copyright notice and this permission notice shall be included
// in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
// THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.
//
// Description: Abstract base class for Extrapolate.
//
///////////////////////////////////////////////////////////////////////////////
 
#include <IncNavierStokesSolver/EquationSystems/Extrapolate.h>
#include <LibUtilities/Communication/Comm.h>
 
using namespace std;
 
namespace Nektar
{
NekDouble Extrapolate::StifflyStable_Betaq_Coeffs[3][3] = {
    {1.0, 0.0, 0.0}, {2.0, -1.0, 0.0}, {3.0, -3.0, 1.0}};
NekDouble Extrapolate::StifflyStable_Alpha_Coeffs[3][3] = {
    {1.0, 0.0, 0.0}, {2.0, -0.5, 0.0}, {3.0, -1.5, 1.0 / 3.0}};
NekDouble Extrapolate::StifflyStable_Gamma0_Coeffs[3] = {1.0, 1.5, 11.0 / 6.0};
 
ExtrapolateFactory &GetExtrapolateFactory()
{
    static ExtrapolateFactory instance;
    return instance;
}
 
Extrapolate::Extrapolate(const LibUtilities::SessionReaderSharedPtr pSession,
                         Array<OneD, MultiRegions::ExpListSharedPtr> pFields,
                         MultiRegions::ExpListSharedPtr pPressure,
                         const Array<OneD, int> pVel,
                         const SolverUtils::AdvectionSharedPtr advObject)
    : m_session(pSession), m_fields(pFields), m_pressure(pPressure),
      m_velocity(pVel), m_advObject(advObject)
{
    m_session->LoadParameter("TimeStep", m_timestep, 0.01);
    m_comm = m_session->GetComm();
}
 
Extrapolate::~Extrapolate()
{
}
 
std::string Extrapolate::def =
    LibUtilities::SessionReader::RegisterDefaultSolverInfo(
        "StandardExtrapolate", "StandardExtrapolate");
 
/**
 *
 */
void Extrapolate::AddDuDt(void)
{
    if (m_numHBCDof)
    {
        // Update velocity BF at n+1 (actually only needs doing if
        // velocity is time dependent on HBCs)
        IProductNormVelocityBCOnHBC(m_iprodnormvel[m_intSteps]);
 
        // Calculate acceleration term at level n based on previous steps
        v_AccelerationBDF(m_iprodnormvel);
 
        // Subtract acceleration term off m_pressureHBCs[nlevels-1]
        Vmath::Svtvp(m_numHBCDof, -1.0 / m_timestep, m_iprodnormvel[m_intSteps],
                     1, m_pressureHBCs[m_intSteps - 1], 1,
                     m_pressureHBCs[m_intSteps - 1], 1);
    }
}
 
/**
 *
 */
void Extrapolate::AddVelBC(void)
{
    if (m_numHBCDof)
    {
        int order = std::min(m_pressureCalls, m_intSteps);
 
        // Update velocity BF at n+1 (actually only needs doing if
        // velocity is time dependent on HBCs)
        IProductNormVelocityBCOnHBC(m_iprodnormvel[0]);
 
        // Subtract acceleration term off m_pressureHBCs[nlevels-1]
        Vmath::Svtvp(m_numHBCDof,
                     -1.0 * StifflyStable_Gamma0_Coeffs[order - 1] / m_timestep,
                     m_iprodnormvel[0], 1, m_pressureHBCs[m_intSteps - 1], 1,
                     m_pressureHBCs[m_intSteps - 1], 1);
    }
}
 
/**
 * Unified routine for calculation high-oder terms
 */
void Extrapolate::v_CalcNeumannPressureBCs(
    const Array<OneD, const Array<OneD, NekDouble>> &fields,
    const Array<OneD, const Array<OneD, NekDouble>> &N, NekDouble kinvis)
{
    size_t n, cnt;
 
    Array<OneD, NekDouble> Pvals;
 
    Array<OneD, Array<OneD, NekDouble>> Velocity(m_curl_dim);
    Array<OneD, Array<OneD, NekDouble>> Advection(m_bnd_dim);
 
    Array<OneD, Array<OneD, NekDouble>> BndValues(m_bnd_dim);
    Array<OneD, Array<OneD, NekDouble>> Q(m_curl_dim);
 
    MultiRegions::ExpListSharedPtr BndElmtExp;
    for (n = cnt = 0; n < m_PBndConds.size(); ++n)
    {
        // High order boundary condition;
        if ((m_hbcType[n] == eHBCNeumann) || (m_hbcType[n] == eConvectiveOBC))
        {
            m_fields[0]->GetBndElmtExpansion(n, BndElmtExp, false);
            int nqb = m_PBndExp[n]->GetTotPoints();
            int nq  = BndElmtExp->GetTotPoints();
 
            for (int i = 0; i < m_bnd_dim; i++)
            {
                BndValues[i] = Array<OneD, NekDouble>(nqb, 0.0);
            }
 
            for (int i = 0; i < m_curl_dim; i++)
            {
                Q[i] = Array<OneD, NekDouble>(nq, 0.0);
            }
 
            // Obtaining fields on BndElmtExp
            for (int i = 0; i < m_curl_dim; i++)
            {
                m_fields[0]->ExtractPhysToBndElmt(n, fields[i], Velocity[i]);
            }
 
            if (N.size()) // not required for some extrapolation
            {
                for (int i = 0; i < m_bnd_dim; i++)
                {
                    m_fields[0]->ExtractPhysToBndElmt(n, N[i], Advection[i]);
                }
            }
 
            // CurlCurl
            BndElmtExp->CurlCurl(Velocity, Q);
 
            // Mounting advection component into the high-order condition
            for (int i = 0; i < m_bnd_dim; i++)
            {
                MountHOPBCs(nq, kinvis, Q[i], Advection[i]);
            }
 
            Pvals = (m_pressureHBCs[m_intSteps - 1]) + cnt;
 
            // Getting values on the boundary and filling the pressure bnd
            // expansion. Multiplication by the normal is required
            for (int i = 0; i < m_bnd_dim; i++)
            {
                m_fields[0]->ExtractElmtToBndPhys(n, Q[i], BndValues[i]);
            }
 
            m_PBndExp[n]->NormVectorIProductWRTBase(BndValues, Pvals);
 
            // Get offset for next terms
            cnt += m_PBndExp[n]->GetNcoeffs();
        }
    }
}
 
// do nothing unless otherwise defined.
void Extrapolate::v_CorrectPressureBCs(const Array<OneD, NekDouble> &pressure)
{
    boost::ignore_unused(pressure);
}
 
// do nothing unless otherwise defined.
void Extrapolate::v_AddNormVelOnOBC(const int nbcoeffs, const int nreg,
                                    Array<OneD, Array<OneD, NekDouble>> &u)
{
    boost::ignore_unused(nbcoeffs, nreg, u);
}
 
void Extrapolate::CalcOutflowBCs(
    const Array<OneD, const Array<OneD, NekDouble>> &fields, NekDouble kinvis)
{
    if (!m_houtflow.get())
    {
        return;
    }
 
    Array<OneD, Array<OneD, NekDouble>> Velocity(m_curl_dim);
 
    MultiRegions::ExpListSharedPtr BndElmtExp;
    size_t cnt = 0;
 
    // Evaluate robin primitive coefficient here so they can be
    // updated whem m_int > 1 Currently not using this update
    // since we only using u^n at outflow instead of BDF rule.
    UpdateRobinPrimCoeff();
 
    for (size_t n = 0; n < m_PBndConds.size(); ++n)
    {
        if ((m_hbcType[n] == eOBC) || (m_hbcType[n] == eConvectiveOBC))
        {
            // Get expansion with element on this boundary
            m_fields[0]->GetBndElmtExpansion(n, BndElmtExp, false);
            int nqb = m_PBndExp[n]->GetTotPoints();
            int nq  = BndElmtExp->GetTotPoints();
 
            // Get velocity and extrapolate
            for (int i = 0; i < m_curl_dim; i++)
            {
                m_fields[0]->ExtractPhysToBndElmt(
                    n, fields[i],
                    m_houtflow->m_outflowVel[cnt][i][m_intSteps - 1]);
                ExtrapolateArray(m_houtflow->m_outflowVel[cnt][i]);
                Velocity[i] = m_houtflow->m_outflowVel[cnt][i][m_intSteps - 1];
            }
 
            // Homogeneous case needs conversion to physical space
            if (m_fields[0]->GetWaveSpace())
            {
                for (int i = 0; i < m_curl_dim; i++)
                {
                    BndElmtExp->HomogeneousBwdTrans(Velocity[i].size(),
                                                    Velocity[i], Velocity[i]);
                }
                BndElmtExp->SetWaveSpace(false);
            }
 
            // Get normal vector
            Array<OneD, Array<OneD, NekDouble>> normals;
            m_fields[0]->GetBoundaryNormals(n, normals);
 
            // Calculate n.gradU.n, div(U)
            Array<OneD, NekDouble> nGradUn(nqb, 0.0);
            Array<OneD, NekDouble> divU(nqb, 0.0);
            Array<OneD, Array<OneD, NekDouble>> grad(m_curl_dim);
            Array<OneD, NekDouble> bndVal(nqb, 0.0);
            for (int i = 0; i < m_curl_dim; i++)
            {
                grad[i] = Array<OneD, NekDouble>(nq, 0.0);
            }
            for (int i = 0; i < m_curl_dim; i++)
            {
                if (m_curl_dim == 2)
                {
                    BndElmtExp->PhysDeriv(Velocity[i], grad[0], grad[1]);
                }
                else
                {
                    BndElmtExp->PhysDeriv(Velocity[i], grad[0], grad[1],
                                          grad[2]);
                }
 
                for (int j = 0; j < m_curl_dim; j++)
                {
                    m_fields[0]->ExtractElmtToBndPhys(n, grad[j], bndVal);
                    // div(U) = gradU_ii
                    if (i == j)
                    {
                        Vmath::Vadd(nqb, divU, 1, bndVal, 1, divU, 1);
                    }
                    // n.gradU.n = gradU_ij n_i n_j
                    Vmath::Vmul(nqb, normals[i], 1, bndVal, 1, bndVal, 1);
                    Vmath::Vvtvp(nqb, normals[j], 1, bndVal, 1, nGradUn, 1,
                                 nGradUn, 1);
                }
            }
 
            // Obtain u at the boundary
            Array<OneD, Array<OneD, NekDouble>> u(m_curl_dim);
            for (int i = 0; i < m_curl_dim; i++)
            {
                u[i] = Array<OneD, NekDouble>(nqb, 0.0);
                m_fields[0]->ExtractElmtToBndPhys(n, Velocity[i], u[i]);
            }
 
            // Calculate u.n and u^2
            Array<OneD, NekDouble> un(nqb, 0.0);
            Array<OneD, NekDouble> u2(nqb, 0.0);
            for (int i = 0; i < m_curl_dim; i++)
            {
                Vmath::Vvtvp(nqb, normals[i], 1, u[i], 1, un, 1, un, 1);
                Vmath::Vvtvp(nqb, u[i], 1, u[i], 1, u2, 1, u2, 1);
            }
 
            // Calculate S_0(u.n) = 0.5*(1-tanh(u.n/*U0*delta))
            Array<OneD, NekDouble> S0(nqb, 0.0);
            for (int i = 0; i < nqb; i++)
            {
                S0[i] = 0.5 * (1.0 - tanh(un[i] / (m_houtflow->m_U0 *
                                                   m_houtflow->m_delta)));
            }
 
            // Calculate E(n,u) = ((theta+alpha2)*0.5*(u^2)n +
            //                    (1-theta+alpha1)*0.5*(n.u)u ) * S_0(u.n)
            NekDouble k1 =
                0.5 * (m_houtflow->m_obcTheta + m_houtflow->m_obcAlpha2);
            NekDouble k2 =
                0.5 * (1 - m_houtflow->m_obcTheta + m_houtflow->m_obcAlpha1);
 
            Array<OneD, Array<OneD, NekDouble>> E(m_curl_dim);
            for (int i = 0; i < m_curl_dim; i++)
            {
                E[i] = Array<OneD, NekDouble>(nqb, 0.0);
 
                Vmath::Smul(nqb, k1, u2, 1, E[i], 1);
                Vmath::Vmul(nqb, E[i], 1, normals[i], 1, E[i], 1);
                // Use bndVal as a temporary storage
                Vmath::Smul(nqb, k2, un, 1, bndVal, 1);
                Vmath::Vvtvp(nqb, u[i], 1, bndVal, 1, E[i], 1, E[i], 1);
                Vmath::Vmul(nqb, E[i], 1, S0, 1, E[i], 1);
            }
 
            // if non-zero forcing is provided we want to subtract
            // value if we want to interpret values as being the
            // desired pressure value. This is now precribed from
            // the velocity forcing to be consistent with the
            // paper except f_b = -f_b
 
            // Calculate (E(n,u) + f_b).n
            Array<OneD, NekDouble> En(nqb, 0.0);
            for (int i = 0; i < m_bnd_dim; i++)
            {
                // Use bndVal as temporary
                Vmath::Vsub(nqb, E[i], 1,
                            m_houtflow->m_UBndExp[i][n]->GetPhys(), 1, bndVal,
                            1);
 
                Vmath::Vvtvp(nqb, normals[i], 1, bndVal, 1, En, 1, En, 1);
            }
 
            // Calculate pressure bc = kinvis*n.gradU.n - E.n + f_b.n
            Array<OneD, NekDouble> pbc(nqb, 0.0);
            Vmath::Svtvm(nqb, kinvis, nGradUn, 1, En, 1, pbc, 1);
 
            if (m_hbcType[n] == eOBC)
            {
 
                if (m_PBndExp[n]->GetWaveSpace())
                {
                    m_PBndExp[n]->HomogeneousFwdTrans(nqb, pbc, bndVal);
                    m_PBndExp[n]->FwdTrans(bndVal,
                                           m_PBndExp[n]->UpdateCoeffs());
                }
                else
                {
                    m_PBndExp[n]->FwdTrans(pbc, m_PBndExp[n]->UpdateCoeffs());
                }
            }
            else if (m_hbcType[n] == eConvectiveOBC) // add outflow values to
                                                     // calculation from HBC
            {
                int nbcoeffs = m_PBndExp[n]->GetNcoeffs();
                Array<OneD, NekDouble> bndCoeffs(nbcoeffs, 0.0);
                if (m_PBndExp[n]->GetWaveSpace())
                {
                    m_PBndExp[n]->HomogeneousFwdTrans(nqb, pbc, bndVal);
                    m_PBndExp[n]->IProductWRTBase(bndVal, bndCoeffs);
                }
                else
                {
                    m_PBndExp[n]->IProductWRTBase(pbc, bndCoeffs);
                }
                // Note we have the negative of what is in the Dong paper in
                // bndVal
                Vmath::Svtvp(nbcoeffs, m_houtflow->m_pressurePrimCoeff[n],
                             bndCoeffs, 1, m_PBndExp[n]->UpdateCoeffs(), 1,
                             m_PBndExp[n]->UpdateCoeffs(), 1);
 
                // evaluate u^n at outflow boundary for velocity BC
                for (int i = 0; i < m_curl_dim; i++)
                {
                    m_fields[0]->ExtractElmtToBndPhys(
                        n, m_houtflow->m_outflowVel[cnt][i][0],
                        m_houtflow->m_outflowVelBnd[cnt][i][m_intSteps - 1]);
 
                    EvaluateBDFArray(m_houtflow->m_outflowVelBnd[cnt][i]);
 
                    // point u[i] to BDF evalauted value \hat{u}
                    u[i] = m_houtflow->m_outflowVelBnd[cnt][i][m_intSteps - 1];
                }
 
                // Add normal velocity if weak pressure
                // formulation. In this case there is an
                // additional \int \hat{u}.n ds on the outflow
                // boundary since we use the inner product wrt
                // deriv of basis in pressure solve.
                AddNormVelOnOBC(cnt, n, u);
            }
 
            // Calculate velocity boundary conditions
            if (m_hbcType[n] == eOBC)
            {
                //    = (pbc n - kinvis divU n)
                Vmath::Smul(nqb, kinvis, divU, 1, divU, 1);
                Vmath::Vsub(nqb, pbc, 1, divU, 1, bndVal, 1);
            }
            else if (m_hbcType[n] == eConvectiveOBC)
            {
                //    = (-kinvis divU n)
                Vmath::Smul(nqb, -1.0 * kinvis, divU, 1, bndVal, 1);
 
                // pbc  needs to be added after pressure solve
            }
 
            for (int i = 0; i < m_curl_dim; ++i)
            {
                // Reuse divU  -> En
                Vmath::Vvtvp(nqb, normals[i], 1, bndVal, 1, E[i], 1, divU, 1);
                // - f_b
                Vmath::Vsub(nqb, divU, 1,
                            m_houtflow->m_UBndExp[i][n]->GetPhys(), 1, divU, 1);
                // * 1/kinvis
                Vmath::Smul(nqb, 1.0 / kinvis, divU, 1, divU, 1);
 
                if (m_hbcType[n] == eConvectiveOBC)
                {
                    Vmath::Svtvp(nqb, m_houtflow->m_velocityPrimCoeff[i][n],
                                 u[i], 1, divU, 1, divU, 1);
                }
 
                if (m_houtflow->m_UBndExp[i][n]->GetWaveSpace())
                {
                    m_houtflow->m_UBndExp[i][n]->HomogeneousFwdTrans(nqb, divU,
                                                                     divU);
                }
 
                m_houtflow->m_UBndExp[i][n]->IProductWRTBase(
                    divU, m_houtflow->m_UBndExp[i][n]->UpdateCoeffs());
            }
 
            // Get offset for next terms
            cnt++;
        }
    }
}
 
void Extrapolate::AddPressureToOutflowBCs(NekDouble kinvis)
{
    if (!m_houtflow.get())
    {
        return;
    }
 
    for (size_t n = 0; n < m_PBndConds.size(); ++n)
    {
        if (m_hbcType[n] == eConvectiveOBC)
        {
            int nqb = m_PBndExp[n]->GetTotPoints();
            int ncb = m_PBndExp[n]->GetNcoeffs();
 
            m_pressure->FillBndCondFromField(n, m_pressure->GetCoeffs());
            Array<OneD, NekDouble> pbc(nqb);
 
            m_PBndExp[n]->BwdTrans(m_PBndExp[n]->GetCoeffs(), pbc);
 
            if (m_PBndExp[n]->GetWaveSpace())
            {
                m_PBndExp[n]->HomogeneousBwdTrans(nqb, pbc, pbc);
            }
 
            Array<OneD, NekDouble> wk(nqb);
            Array<OneD, NekDouble> wk1(ncb);
 
            // Get normal vector
            Array<OneD, Array<OneD, NekDouble>> normals;
            m_fields[0]->GetBoundaryNormals(n, normals);
 
            //  Add  1/kinvis * (pbc n )
            for (int i = 0; i < m_curl_dim; ++i)
            {
                Vmath::Vmul(nqb, normals[i], 1, pbc, 1, wk, 1);
 
                Vmath::Smul(nqb, 1.0 / kinvis, wk, 1, wk, 1);
 
                if (m_houtflow->m_UBndExp[i][n]->GetWaveSpace())
                {
                    m_houtflow->m_UBndExp[i][n]->HomogeneousFwdTrans(nqb, wk,
                                                                     wk);
                }
                m_houtflow->m_UBndExp[i][n]->IProductWRTBase(wk, wk1);
 
                Vmath::Vadd(ncb, wk1, 1,
                            m_houtflow->m_UBndExp[i][n]->GetCoeffs(), 1,
                            m_houtflow->m_UBndExp[i][n]->UpdateCoeffs(), 1);
            }
        }
    }
}
 
void Extrapolate::IProductNormVelocityOnHBC(
    const Array<OneD, const Array<OneD, NekDouble>> &Vel,
    Array<OneD, NekDouble> &IProdVn)
{
    int i;
    size_t n, cnt;
    Array<OneD, NekDouble> IProdVnTmp;
    Array<OneD, Array<OneD, NekDouble>> velbc(m_bnd_dim);
 
    for (n = cnt = 0; n < m_PBndConds.size(); ++n)
    {
        // High order boundary condition;
        if (m_hbcType[n] == eHBCNeumann)
        {
            for (i = 0; i < m_bnd_dim; ++i)
            {
                m_fields[0]->ExtractPhysToBnd(n, Vel[i], velbc[i]);
            }
            IProdVnTmp = IProdVn + cnt;
            m_PBndExp[n]->NormVectorIProductWRTBase(velbc, IProdVnTmp);
            cnt += m_PBndExp[n]->GetNcoeffs();
        }
        else if (m_hbcType[n] == eConvectiveOBC) // skip over conective OBC
        {
            cnt += m_PBndExp[n]->GetNcoeffs();
        }
    }
}
 
void Extrapolate::IProductNormVelocityBCOnHBC(Array<OneD, NekDouble> &IProdVn)
{
 
    if (!m_HBCnumber)
    {
        return;
    }
    int i;
    size_t n, cnt;
    Array<OneD, NekDouble> IProdVnTmp;
    Array<OneD, Array<OneD, NekDouble>> velbc(m_bnd_dim);
    Array<OneD, Array<OneD, MultiRegions::ExpListSharedPtr>> VelBndExp(
        m_bnd_dim);
    for (i = 0; i < m_bnd_dim; ++i)
    {
        VelBndExp[i] = m_fields[m_velocity[i]]->GetBndCondExpansions();
    }
 
    for (n = cnt = 0; n < m_PBndConds.size(); ++n)
    {
        // High order boundary condition;
        if (m_hbcType[n] == eHBCNeumann)
        {
            for (i = 0; i < m_bnd_dim; ++i)
            {
                velbc[i] = Array<OneD, NekDouble>(
                    VelBndExp[i][n]->GetTotPoints(), 0.0);
                VelBndExp[i][n]->SetWaveSpace(
                    m_fields[m_velocity[i]]->GetWaveSpace());
                VelBndExp[i][n]->BwdTrans(VelBndExp[i][n]->GetCoeffs(),
                                          velbc[i]);
            }
            IProdVnTmp = IProdVn + cnt;
            m_PBndExp[n]->NormVectorIProductWRTBase(velbc, IProdVnTmp);
            cnt += m_PBndExp[n]->GetNcoeffs();
        }
        else if (m_hbcType[n] == eConvectiveOBC)
        {
            // skip over convective OBC
            cnt += m_PBndExp[n]->GetNcoeffs();
        }
    }
}
 
/**
 * Function to roll time-level storages to the next step layout.
 * The stored data associated with the oldest time-level
 * (not required anymore) are moved to the top, where they will
 * be overwritten as the solution process progresses.
 */
void Extrapolate::RollOver(Array<OneD, Array<OneD, NekDouble>> &input)
{
    int nlevels = input.size();
 
    Array<OneD, NekDouble> tmp;
 
    tmp = input[nlevels - 1];
 
    for (int n = nlevels - 1; n > 0; --n)
    {
        input[n] = input[n - 1];
    }
 
    input[0] = tmp;
}
 
/**
 * Initialize HOBCs
 */
void Extrapolate::GenerateHOPBCMap(
    const LibUtilities::SessionReaderSharedPtr &pSession)
{
    m_PBndConds = m_pressure->GetBndConditions();
    m_PBndExp   = m_pressure->GetBndCondExpansions();
 
    size_t cnt, n;
 
    // Storage array for high order pressure BCs
    m_pressureHBCs = Array<OneD, Array<OneD, NekDouble>>(m_intSteps);
    m_iprodnormvel = Array<OneD, Array<OneD, NekDouble>>(m_intSteps + 1);
 
    // Get useful values for HOBCs
    m_HBCnumber = 0;
    m_numHBCDof = 0;
 
    int outHBCnumber = 0;
    int numOutHBCPts = 0;
 
    m_hbcType = Array<OneD, HBCType>(m_PBndConds.size(), eNOHBC);
    for (n = 0; n < m_PBndConds.size(); ++n)
    {
        // High order boundary Neumann Condiiton
        if (boost::iequals(m_PBndConds[n]->GetUserDefined(), "H"))
        {
            m_hbcType[n] = eHBCNeumann;
            m_numHBCDof += m_PBndExp[n]->GetNcoeffs();
            m_HBCnumber += m_PBndExp[n]->GetExpSize();
        }
 
        // High order outflow convective condition
        if (m_PBndConds[n]->GetBoundaryConditionType() ==
                SpatialDomains::eRobin &&
            boost::iequals(m_PBndConds[n]->GetUserDefined(), "HOutflow"))
        {
            m_hbcType[n] = eConvectiveOBC;
            m_numHBCDof += m_PBndExp[n]->GetNcoeffs();
            m_HBCnumber += m_PBndExp[n]->GetExpSize();
            numOutHBCPts += m_PBndExp[n]->GetTotPoints();
            outHBCnumber++;
        }
        // High order outflow boundary condition;
        else if (boost::iequals(m_PBndConds[n]->GetUserDefined(), "HOutflow"))
        {
            m_hbcType[n] = eOBC;
            numOutHBCPts += m_PBndExp[n]->GetTotPoints();
            outHBCnumber++;
        }
    }
 
    m_iprodnormvel[0] = Array<OneD, NekDouble>(m_numHBCDof, 0.0);
    for (int n = 0; n < m_intSteps; ++n)
    {
        m_pressureHBCs[n]     = Array<OneD, NekDouble>(m_numHBCDof, 0.0);
        m_iprodnormvel[n + 1] = Array<OneD, NekDouble>(m_numHBCDof, 0.0);
    }
 
    m_pressureCalls = 0;
 
    switch (m_pressure->GetExpType())
    {
        case MultiRegions::e2D:
        {
            m_curl_dim = 2;
            m_bnd_dim  = 2;
        }
        break;
        case MultiRegions::e3DH1D:
        {
            m_curl_dim = 3;
            m_bnd_dim  = 2;
        }
        break;
        case MultiRegions::e3DH2D:
        {
            m_curl_dim = 3;
            m_bnd_dim  = 1;
        }
        break;
        case MultiRegions::e3D:
        {
            m_curl_dim = 3;
            m_bnd_dim  = 3;
        }
        break;
        default:
            ASSERTL0(0, "Dimension not supported");
            break;
    }
 
    // Initialise storage for outflow HOBCs
    if (numOutHBCPts > 0)
    {
        m_houtflow = MemoryManager<HighOrderOutflow>::AllocateSharedPtr(
            numOutHBCPts, outHBCnumber, m_curl_dim, pSession);
 
        MultiRegions::ExpListSharedPtr BndElmtExp;
 
        // set up boundary expansions link
        for (int i = 0; i < m_curl_dim; ++i)
        {
            m_houtflow->m_UBndExp[i] =
                m_fields[m_velocity[i]]->GetBndCondExpansions();
        }
 
        for (n = 0, cnt = 0; n < m_PBndConds.size(); ++n)
        {
            if (boost::iequals(m_PBndConds[n]->GetUserDefined(), "HOutflow"))
            {
                m_houtflow->m_outflowVel[cnt] =
                    Array<OneD, Array<OneD, Array<OneD, NekDouble>>>(
                        m_curl_dim);
 
                m_houtflow->m_outflowVelBnd[cnt] =
                    Array<OneD, Array<OneD, Array<OneD, NekDouble>>>(
                        m_curl_dim);
 
                m_fields[0]->GetBndElmtExpansion(n, BndElmtExp, false);
                int nqb = m_PBndExp[n]->GetTotPoints();
                int nq  = BndElmtExp->GetTotPoints();
                for (int j = 0; j < m_curl_dim; ++j)
                {
                    m_houtflow->m_outflowVel[cnt][j] =
                        Array<OneD, Array<OneD, NekDouble>>(m_intSteps);
 
                    m_houtflow->m_outflowVelBnd[cnt][j] =
                        Array<OneD, Array<OneD, NekDouble>>(m_intSteps);
 
                    for (int k = 0; k < m_intSteps; ++k)
                    {
                        m_houtflow->m_outflowVel[cnt][j][k] =
                            Array<OneD, NekDouble>(nq, 0.0);
                        m_houtflow->m_outflowVelBnd[cnt][j][k] =
                            Array<OneD, NekDouble>(nqb, 0.0);
                    }
                }
                cnt++;
            }
 
            // evaluate convective primitive coefficient if
            // convective OBCs are used
            if (m_hbcType[n] == eConvectiveOBC)
            {
                // initialise convective members of
                // HighOrderOutflow struct
                if (m_houtflow->m_pressurePrimCoeff.size() == 0)
                {
                    m_houtflow->m_pressurePrimCoeff =
                        Array<OneD, NekDouble>(m_PBndConds.size(), 0.0);
                    m_houtflow->m_velocityPrimCoeff =
                        Array<OneD, Array<OneD, NekDouble>>(m_curl_dim);
 
                    for (int i = 0; i < m_curl_dim; ++i)
                    {
                        m_houtflow->m_velocityPrimCoeff[i] =
                            Array<OneD, NekDouble>(m_PBndConds.size(), 0.0);
                    }
                }
 
                LibUtilities::Equation coeff =
                    std::static_pointer_cast<
                        SpatialDomains::RobinBoundaryCondition>(m_PBndConds[n])
                        ->m_robinPrimitiveCoeff;
 
                // checkout equation evaluation options!!
                m_houtflow->m_pressurePrimCoeff[n] = coeff.Evaluate();
 
                for (int i = 0; i < m_curl_dim; ++i)
                {
                    Array<OneD, const SpatialDomains::BoundaryConditionShPtr>
                        UBndConds = m_fields[m_velocity[i]]->GetBndConditions();
 
                    LibUtilities::Equation coeff1 =
                        std::static_pointer_cast<
                            SpatialDomains::RobinBoundaryCondition>(
                            UBndConds[n])
                            ->m_robinPrimitiveCoeff;
 
                    m_houtflow->m_defVelPrimCoeff[i] = coeff1.GetExpression();
 
                    ASSERTL1(UBndConds[n]->GetBoundaryConditionType() ==
                                 SpatialDomains::eRobin,
                             "Require Velocity "
                             "conditions to be of Robin type when pressure"
                             "outflow is specticied as Robin Boundary type");
 
                    // checkout equation evaluation options!!
                    m_houtflow->m_velocityPrimCoeff[i][n] = coeff1.Evaluate();
                }
            }
        }
    }
}
 
void Extrapolate::UpdateRobinPrimCoeff(void)
{
 
    if ((m_pressureCalls == 1) || (m_pressureCalls > m_intSteps))
    {
        return;
    }
 
    for (size_t n = 0; n < m_PBndConds.size(); ++n)
    {
        // Get expansion with element on this boundary
        if (m_hbcType[n] == eConvectiveOBC)
        {
            for (int i = 0; i < m_curl_dim; ++i)
            {
                Array<OneD, SpatialDomains::BoundaryConditionShPtr> UBndConds =
                    m_fields[m_velocity[i]]->UpdateBndConditions();
 
                std::string primcoeff =
                    m_houtflow->m_defVelPrimCoeff[i] + "*" +
                    boost::lexical_cast<std::string>(
                        StifflyStable_Gamma0_Coeffs[m_pressureCalls - 1]);
 
                SpatialDomains::RobinBCShPtr rcond = std::dynamic_pointer_cast<
                    SpatialDomains::RobinBoundaryCondition>(UBndConds[n]);
 
                SpatialDomains::BoundaryConditionShPtr bcond =
                    MemoryManager<SpatialDomains::RobinBoundaryCondition>::
                        AllocateSharedPtr(
                            m_session, rcond->m_robinFunction.GetExpression(),
                            primcoeff, rcond->GetUserDefined(),
                            rcond->m_filename);
 
                UBndConds[n] = bcond;
            }
        }
    }
}
 
/**
 *
 */
Array<OneD, NekDouble> Extrapolate::GetMaxStdVelocity(
    const Array<OneD, Array<OneD, NekDouble>> inarray)
{
    // Checking if the problem is 2D
    ASSERTL0(m_curl_dim >= 2, "Method not implemented for 1D");
 
    size_t n_points_0 = m_fields[0]->GetExp(0)->GetTotPoints();
    size_t n_element  = m_fields[0]->GetExpSize();
    size_t nvel       = inarray.size();
    size_t cnt;
 
    NekDouble pntVelocity;
 
    // Getting the standard velocity vector
    Array<OneD, Array<OneD, NekDouble>> stdVelocity(nvel);
    Array<OneD, NekDouble> tmp;
    Array<OneD, NekDouble> maxV(n_element, 0.0);
    LibUtilities::PointsKeyVector ptsKeys;
 
    for (size_t i = 0; i < nvel; ++i)
    {
        stdVelocity[i] = Array<OneD, NekDouble>(n_points_0);
    }
 
    cnt = 0.0;
    for (size_t el = 0; el < n_element; ++el)
    {
        size_t n_points = m_fields[0]->GetExp(el)->GetTotPoints();
        ptsKeys         = m_fields[0]->GetExp(el)->GetPointsKeys();
 
        // reset local space
        if (n_points != n_points_0)
        {
            for (size_t j = 0; j < nvel; ++j)
            {
                stdVelocity[j] = Array<OneD, NekDouble>(n_points, 0.0);
            }
            n_points_0 = n_points;
        }
        else
        {
            for (size_t j = 0; j < nvel; ++j)
            {
                Vmath::Zero(n_points, stdVelocity[j], 1);
            }
        }
 
        Array<TwoD, const NekDouble> gmat = m_fields[0]
                                                ->GetExp(el)
                                                ->GetGeom()
                                                ->GetMetricInfo()
                                                ->GetDerivFactors(ptsKeys);
 
        if (m_fields[0]->GetExp(el)->GetGeom()->GetMetricInfo()->GetGtype() ==
            SpatialDomains::eDeformed)
        {
            for (size_t j = 0; j < nvel; ++j)
            {
                for (size_t k = 0; k < nvel; ++k)
                {
                    Vmath::Vvtvp(n_points, gmat[k * nvel + j], 1,
                                 tmp = inarray[k] + cnt, 1, stdVelocity[j], 1,
                                 stdVelocity[j], 1);
                }
            }
        }
        else
        {
            for (size_t j = 0; j < nvel; ++j)
            {
                for (size_t k = 0; k < nvel; ++k)
                {
                    Vmath::Svtvp(n_points, gmat[k * nvel + j][0],
                                 tmp = inarray[k] + cnt, 1, stdVelocity[j], 1,
                                 stdVelocity[j], 1);
                }
            }
        }
        cnt += n_points;
 
        // Calculate total velocity in stdVelocity[0]
        Vmath::Vmul(n_points, stdVelocity[0], 1, stdVelocity[0], 1,
                    stdVelocity[0], 1);
        for (size_t k = 1; k < nvel; ++k)
        {
            Vmath::Vvtvp(n_points, stdVelocity[k], 1, stdVelocity[k], 1,
                         stdVelocity[0], 1, stdVelocity[0], 1);
        }
        pntVelocity = Vmath::Vmax(n_points, stdVelocity[0], 1);
        maxV[el]    = sqrt(pntVelocity);
    }
 
    return maxV;
}
 
std::string Extrapolate::v_GetSubStepName(void)
{
    return "";
}
 
/**
 *    At the start, the newest value is stored in array[nlevels-1]
 *        and the previous values in the first positions
 *    At the end, the extrapolated value is stored in array[nlevels-1]
 *        and the storage has been updated to included the new value
 */
void Extrapolate::ExtrapolateArray(Array<OneD, Array<OneD, NekDouble>> &array)
{
    int nint    = min(m_pressureCalls, m_intSteps);
    int nlevels = array.size();
    int nPts    = array[0].size();
 
    // Update array
    RollOver(array);
 
    // Extrapolate to outarray
    Vmath::Smul(nPts, StifflyStable_Betaq_Coeffs[nint - 1][nint - 1],
                array[nint - 1], 1, array[nlevels - 1], 1);
 
    for (int n = 0; n < nint - 1; ++n)
    {
        Vmath::Svtvp(nPts, StifflyStable_Betaq_Coeffs[nint - 1][n], array[n], 1,
                     array[nlevels - 1], 1, array[nlevels - 1], 1);
    }
}
 
/**
 *    At the start, the newest value is stored in array[nlevels-1]
 *        and the previous values in the first positions
 *    At the end, the value of the bdf explicit part is stored in
 * array[nlevels-1] and the storage has been updated to included the new value
 */
void Extrapolate::EvaluateBDFArray(Array<OneD, Array<OneD, NekDouble>> &array)
{
    int nint    = min(m_pressureCalls, m_intSteps);
    int nlevels = array.size();
    int nPts    = array[0].size();
 
    // Update array
    RollOver(array);
 
    // Extrapolate to outarray
    Vmath::Smul(nPts, StifflyStable_Alpha_Coeffs[nint - 1][nint - 1],
                array[nint - 1], 1, array[nlevels - 1], 1);
 
    for (int n = 0; n < nint - 1; ++n)
    {
        Vmath::Svtvp(nPts, StifflyStable_Alpha_Coeffs[nint - 1][n], array[n], 1,
                     array[nlevels - 1], 1, array[nlevels - 1], 1);
    }
}
 
/**
 *    At the start, the newest value is stored in array[nlevels-1]
 *        and the previous values in the first positions
 *    At the end, the acceleration from BDF is stored in array[nlevels-1]
 *        and the storage has been updated to included the new value
 */
void Extrapolate::v_AccelerationBDF(Array<OneD, Array<OneD, NekDouble>> &array)
{
    int nlevels = array.size();
    int nPts    = array[0].size();
 
    if (nPts)
    {
        // Update array
        RollOver(array);
 
        // Calculate acceleration using Backward Differentiation Formula
        Array<OneD, NekDouble> accelerationTerm(nPts, 0.0);
        if (m_pressureCalls > 2)
        {
            int acc_order = min(m_pressureCalls - 2, m_intSteps);
            Vmath::Smul(nPts, StifflyStable_Gamma0_Coeffs[acc_order - 1],
                        array[0], 1, accelerationTerm, 1);
 
            for (int i = 0; i < acc_order; i++)
            {
                Vmath::Svtvp(
                    nPts, -1 * StifflyStable_Alpha_Coeffs[acc_order - 1][i],
                    array[i + 1], 1, accelerationTerm, 1, accelerationTerm, 1);
            }
        }
        array[nlevels - 1] = accelerationTerm;
    }
}
 
void Extrapolate::CopyPressureHBCsToPbndExp(void)
{
    size_t n, cnt;
    for (cnt = n = 0; n < m_PBndConds.size(); ++n)
    {
        if ((m_hbcType[n] == eHBCNeumann) || (m_hbcType[n] == eConvectiveOBC))
        {
            int nq = m_PBndExp[n]->GetNcoeffs();
            Vmath::Vcopy(nq, &(m_pressureHBCs[m_intSteps - 1])[cnt], 1,
                         &(m_PBndExp[n]->UpdateCoeffs()[0]), 1);
            cnt += nq;
        }
    }
}
} // namespace Nektar