VelocityCorrectionScheme.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: VelocityCorrectionScheme.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
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.
//
// Description: Velocity Correction Scheme for the Incompressible
// Navier Stokes equations
//
///////////////////////////////////////////////////////////////////////////////
 
#include <IncNavierStokesSolver/EquationSystems/VelocityCorrectionScheme.h>
#include <LibUtilities/BasicUtils/Timer.h>
#include <LocalRegions/Expansion2D.h>
#include <LocalRegions/Expansion3D.h>
#include <MultiRegions/ContField.h>
#include <MultiRegions/GJPStabilisation.h>
#include <SolverUtils/Core/Misc.h>
 
#include <boost/algorithm/string.hpp>
 
using namespace std;
 
namespace Nektar
{
using namespace MultiRegions;
 
string VelocityCorrectionScheme::className =
    SolverUtils::GetEquationSystemFactory().RegisterCreatorFunction(
        "VelocityCorrectionScheme", VelocityCorrectionScheme::create);
 
string VelocityCorrectionScheme::solverTypeLookupId =
    LibUtilities::SessionReader::RegisterEnumValue(
        "SolverType", "VelocityCorrectionScheme", eVelocityCorrectionScheme);
 
/**
 * Constructor. Creates ...
 *
 * \param
 * \param
 */
VelocityCorrectionScheme::VelocityCorrectionScheme(
    const LibUtilities::SessionReaderSharedPtr &pSession,
    const SpatialDomains::MeshGraphSharedPtr &pGraph)
    : UnsteadySystem(pSession, pGraph), IncNavierStokes(pSession, pGraph),
      m_varCoeffLap(StdRegions::NullVarCoeffMap)
{
}
 
void VelocityCorrectionScheme::v_InitObject(bool DeclareField)
{
    int n;
 
    IncNavierStokes::v_InitObject(DeclareField);
    m_explicitDiffusion = false;
 
    // Set m_pressure to point to last field of m_fields;
    if (boost::iequals(m_session->GetVariable(m_fields.size() - 1), "p"))
    {
        m_nConvectiveFields = m_fields.size() - 1;
        m_pressure          = m_fields[m_nConvectiveFields];
    }
    else
    {
        ASSERTL0(false, "Need to set up pressure field definition");
    }
 
    // Determine diffusion coefficients for each field
    m_diffCoeff = Array<OneD, NekDouble>(m_nConvectiveFields, m_kinvis);
    for (n = 0; n < m_nConvectiveFields; ++n)
    {
        std::string varName = m_session->GetVariable(n);
        if (m_session->DefinesFunction("DiffusionCoefficient", varName))
        {
            LibUtilities::EquationSharedPtr ffunc =
                m_session->GetFunction("DiffusionCoefficient", varName);
            m_diffCoeff[n] = ffunc->Evaluate();
        }
    }
 
    // Integrate only the convective fields
    for (n = 0; n < m_nConvectiveFields; ++n)
    {
        m_intVariables.push_back(n);
    }
 
    SetUpExtrapolation();
    SetUpSVV();
 
    // check to see if it is explicity turned off
    m_session->MatchSolverInfo("GJPStabilisation", "False",
                               m_useGJPStabilisation, true);
 
    // if GJPStabilisation set to False bool will be true and
    // if not false so negate/revese bool
    m_useGJPStabilisation = !m_useGJPStabilisation;
 
    m_session->MatchSolverInfo("GJPNormalVelocity", "True", m_useGJPNormalVel,
                               false);
 
    if (m_useGJPNormalVel)
    {
        ASSERTL0(boost::iequals(m_session->GetSolverInfo("GJPStabilisation"),
                                "Explicit"),
                 "Can only specify GJPNormalVelocity with"
                 " GJPStabilisation set to Explicit currently");
    }
 
    m_session->LoadParameter("GJPJumpScale", m_GJPJumpScale, 1.0);
 
    m_session->MatchSolverInfo("SmoothAdvection", "True", m_SmoothAdvection,
                               false);
 
    // set explicit time-intregration class operators
    m_ode.DefineOdeRhs(
        &VelocityCorrectionScheme::EvaluateAdvection_SetPressureBCs, this);
 
    // set implicit time-intregration class operators
    m_ode.DefineImplicitSolve(
        &VelocityCorrectionScheme::SolveUnsteadyStokesSystem, this);
 
    // Set up bits for flowrate.
    m_session->LoadParameter("Flowrate", m_flowrate, 0.0);
    m_session->LoadParameter("IO_FlowSteps", m_flowrateSteps, 0);
}
 
void VelocityCorrectionScheme::SetUpExtrapolation()
{
    // creation of the extrapolation object
    if (m_equationType == eUnsteadyNavierStokes ||
        m_equationType == eUnsteadyStokes)
    {
        std::string vExtrapolation = v_GetExtrapolateStr();
        if (m_session->DefinesSolverInfo("Extrapolation"))
        {
            vExtrapolation = v_GetSubSteppingExtrapolateStr(
                m_session->GetSolverInfo("Extrapolation"));
        }
        m_extrapolation = GetExtrapolateFactory().CreateInstance(
            vExtrapolation, m_session, m_fields, m_pressure, m_velocity,
            m_advObject);
 
        m_extrapolation->SetForcing(m_forcing);
        m_extrapolation->SubSteppingTimeIntegration(m_intScheme);
        m_extrapolation->GenerateBndElmtExpansion();
        m_extrapolation->GenerateHOPBCMap(m_session);
        m_IncNavierStokesBCs =
            MemoryManager<IncBoundaryConditions>::AllocateSharedPtr();
        m_IncNavierStokesBCs->Initialize(m_session, m_fields);
    }
}
 
/**
 * @brief Set up the Stokes solution used to impose constant flowrate
 * through a boundary.
 *
 * This routine solves a Stokes equation using a unit forcing direction,
 * specified by the user to be in the desired flow direction. This field can
 * then be used to correct the end of each timestep to impose a constant
 * volumetric flow rate through a user-defined boundary.
 *
 * There are three modes of operation:
 *
 * - Standard two-dimensional or three-dimensional simulations (e.g. pipes
 *   or channels)
 * - 3DH1D simulations where the forcing is not in the homogeneous
 *   direction (e.g. channel flow, where the y-direction of the 2D mesh
 *   is perpendicular to the wall);
 * - 3DH1D simulations where the forcing is in the homogeneous direction
 *   (e.g. pipe flow in the z-direction).
 *
 * In the first two cases, the user should define:
 * - the `Flowrate` parameter, which dictates the volumetric flux through
 *   the reference area
 * - tag a boundary region with the `Flowrate` user-defined type to define
 *   the reference area
 * - define a `FlowrateForce` function with components `ForceX`, `ForceY`
 *   and `ForceZ` that defines a unit forcing in the appropriate direction.
 *
 * In the latter case, the user should define only the `Flowrate`; the
 * reference area is taken to be the homogeneous plane and the force is
 * assumed to be the unit z-vector \f$ \hat{e}_z \f$.
 *
 * This routine solves a single timestep of the Stokes problem
 * (premultiplied by the backwards difference coefficient):
 *
 * \f[ \frac{\partial\mathbf{u}}{\partial t} = -\nabla p +
 * \nu\nabla^2\mathbf{u} + \mathbf{f} \f]
 *
 * with a zero initial condition to obtain a field \f$ \mathbf{u}_s \f$. The
 * flowrate is then corrected at each timestep \f$ n \f$ by adding the
 * correction \f$ \alpha\mathbf{u}_s \f$ where
 *
 * \f[ \alpha = \frac{\overline{Q} - Q(\mathbf{u^n})}{Q(\mathbf{u}_s)} \f]
 *
 * where \f$ Q(\cdot)\f$ is the volumetric flux through the appropriate
 * surface or line, which is implemented in
 * VelocityCorrectionScheme::MeasureFlowrate. For more details, see chapter
 * 3.2 of the thesis of D. Moxey (University of Warwick, 2011).
 */
void VelocityCorrectionScheme::SetupFlowrate(NekDouble aii_dt)
{
    m_flowrateBndID = -1;
    m_flowrateArea  = 0.0;
 
    const Array<OneD, const SpatialDomains::BoundaryConditionShPtr> &bcs =
        m_fields[0]->GetBndConditions();
 
    std::string forces[] = {"X", "Y", "Z"};
    Array<OneD, NekDouble> flowrateForce(m_spacedim, 0.0);
 
    // Set up flowrate forces.
    bool defined = true;
    for (int i = 0; i < m_spacedim; ++i)
    {
        std::string varName = std::string("Force") + forces[i];
        defined = m_session->DefinesFunction("FlowrateForce", varName);
 
        if (!defined && m_HomogeneousType == eHomogeneous1D)
        {
            break;
        }
 
        ASSERTL0(defined,
                 "A 'FlowrateForce' function must defined with components "
                 "[ForceX, ...] to define direction of flowrate forcing");
 
        LibUtilities::EquationSharedPtr ffunc =
            m_session->GetFunction("FlowrateForce", varName);
        flowrateForce[i] = ffunc->Evaluate();
    }
 
    // Define flag for case with homogeneous expansion and forcing not in the
    // z-direction
    m_homd1DFlowinPlane = false;
    if (defined && m_HomogeneousType == eHomogeneous1D)
    {
        m_homd1DFlowinPlane = true;
    }
 
    // For 3DH1D simulations, if force isn't defined then assume in
    // z-direction.
    if (!defined)
    {
        flowrateForce[2] = 1.0;
    }
 
    // Find the boundary condition that is tagged as the flowrate boundary.
    for (size_t i = 0; i < bcs.size(); ++i)
    {
        if (boost::iequals(bcs[i]->GetUserDefined(), "Flowrate"))
        {
            m_flowrateBndID = i;
            break;
        }
    }
 
    int tmpBr = m_flowrateBndID;
    m_comm->AllReduce(tmpBr, LibUtilities::ReduceMax);
    ASSERTL0(tmpBr >= 0 || m_HomogeneousType == eHomogeneous1D,
             "One boundary region must be marked using the 'Flowrate' "
             "user-defined type to monitor the volumetric flowrate.");
 
    // Extract an appropriate expansion list to represents the boundary.
    if (m_flowrateBndID >= 0)
    {
        // For a boundary, extract the boundary itself.
        m_flowrateBnd = m_fields[0]->GetBndCondExpansions()[m_flowrateBndID];
    }
    else if (m_HomogeneousType == eHomogeneous1D && !m_homd1DFlowinPlane)
    {
        // For 3DH1D simulations with no force specified, find the mean
        // (0th) plane.
        Array<OneD, unsigned int> zIDs = m_fields[0]->GetZIDs();
        int tmpId                      = -1;
 
        for (size_t i = 0; i < zIDs.size(); ++i)
        {
            if (zIDs[i] == 0)
            {
                tmpId = i;
                break;
            }
        }
 
        ASSERTL1(tmpId <= 0, "Should be either at location 0 or -1 if not "
                             "found");
 
        if (tmpId != -1)
        {
            m_flowrateBnd = m_fields[0]->GetPlane(tmpId);
        }
    }
 
    // At this point, some processors may not have m_flowrateBnd
    // set if they don't contain the appropriate boundary. To
    // calculate the area, we integrate 1.0 over the boundary
    // (which has been set up with the appropriate subcommunicator
    // to avoid deadlock), and then communicate this to the other
    // processors with an AllReduce.
    if (m_flowrateBnd)
    {
        Array<OneD, NekDouble> inArea(m_flowrateBnd->GetNpoints(), 1.0);
        m_flowrateArea = m_flowrateBnd->Integral(inArea);
    }
    m_comm->AllReduce(m_flowrateArea, LibUtilities::ReduceMax);
 
    // In homogeneous case with forcing not aligned to the z-direction,
    // redefine m_flowrateBnd so it is a 1D expansion
    if (m_HomogeneousType == eHomogeneous1D && m_homd1DFlowinPlane &&
        m_flowrateBnd)
    {
        // For 3DH1D simulations with no force specified, find the mean
        // (0th) plane.
        Array<OneD, unsigned int> zIDs = m_fields[0]->GetZIDs();
        m_planeID                      = -1;
 
        for (size_t i = 0; i < zIDs.size(); ++i)
        {
            if (zIDs[i] == 0)
            {
                m_planeID = i;
                break;
            }
        }
 
        ASSERTL1(m_planeID <= 0, "Should be either at location 0 or -1 "
                                 "if not found");
 
        if (m_planeID != -1)
        {
            m_flowrateBnd =
                m_fields[0]->GetBndCondExpansions()[m_flowrateBndID]->GetPlane(
                    m_planeID);
        }
    }
 
    // Set up some storage for the Stokes solution (to be stored in
    // m_flowrateStokes) and its initial condition (inTmp), which holds the
    // unit forcing.
    int nqTot = m_fields[0]->GetNpoints();
    Array<OneD, Array<OneD, NekDouble>> inTmp(m_spacedim);
    m_flowrateStokes = Array<OneD, Array<OneD, NekDouble>>(m_spacedim);
 
    for (int i = 0; i < m_spacedim; ++i)
    {
        inTmp[i] = Array<OneD, NekDouble>(nqTot, flowrateForce[i] * aii_dt);
        m_flowrateStokes[i] = Array<OneD, NekDouble>(nqTot, 0.0);
 
        if (m_HomogeneousType == eHomogeneous1D)
        {
            Array<OneD, NekDouble> inTmp2(nqTot);
            m_fields[i]->HomogeneousFwdTrans(nqTot, inTmp[i], inTmp2);
            m_fields[i]->SetWaveSpace(true);
            inTmp[i] = inTmp2;
        }
 
        Vmath::Zero(m_fields[i]->GetNcoeffs(), m_fields[i]->UpdateCoeffs(), 1);
    }
 
    // Create temporary extrapolation object to avoid issues with
    // m_extrapolation for HOPBCs using higher order timestepping schemes.
    // Zero pressure BCs in Neumann boundaries that may have been
    // set in the advection step.
    Array<OneD, const SpatialDomains::BoundaryConditionShPtr> PBndConds =
        m_pressure->GetBndConditions();
    Array<OneD, MultiRegions::ExpListSharedPtr> PBndExp =
        m_pressure->GetBndCondExpansions();
    for (size_t n = 0; n < PBndConds.size(); ++n)
    {
        if (PBndConds[n]->GetBoundaryConditionType() ==
            SpatialDomains::eNeumann)
        {
            Vmath::Zero(PBndExp[n]->GetNcoeffs(), PBndExp[n]->UpdateCoeffs(),
                        1);
        }
    }
 
    // Finally, calculate the solution and the flux of the Stokes
    // solution. We set m_greenFlux to maximum numeric limit, which signals
    // to SolveUnsteadyStokesSystem that we don't need to apply a flowrate
    // force.
    m_greenFlux     = numeric_limits<NekDouble>::max();
    m_flowrateAiidt = aii_dt;
 
    //  Save the number of convective field in case it is not set
    //  to spacedim. Only need velocity components for stokes forcing
    int SaveNConvectiveFields = m_nConvectiveFields;
    m_nConvectiveFields       = m_spacedim;
    SolveUnsteadyStokesSystem(inTmp, m_flowrateStokes, 0.0, aii_dt);
    m_nConvectiveFields = SaveNConvectiveFields;
    m_greenFlux         = MeasureFlowrate(m_flowrateStokes);
 
    // If the user specified IO_FlowSteps, open a handle to store output.
    if (m_comm->GetRank() == 0 && m_flowrateSteps &&
        !m_flowrateStream.is_open())
    {
        std::string filename = m_session->GetSessionName();
        filename += ".prs";
        m_flowrateStream.open(filename.c_str());
        m_flowrateStream.setf(ios::scientific, ios::floatfield);
        m_flowrateStream << "# step      time            dP" << endl
                         << "# -------------------------------------------"
                         << endl;
    }
 
    // Replace pressure BCs with those evaluated from advection step
    m_extrapolation->CopyPressureHBCsToPbndExp();
}
 
/**
 * @brief Measure the volumetric flow rate through the volumetric flow rate
 * reference surface.
 *
 * This routine computes the volumetric flow rate
 *
 * \f[
 * Q(\mathbf{u}) = \frac{1}{\mu(R)} \int_R \mathbf{u} \cdot d\mathbf{s}
 * \f]
 *
 * through the boundary region \f$ R \f$.
 */
NekDouble VelocityCorrectionScheme::MeasureFlowrate(
    const Array<OneD, Array<OneD, NekDouble>> &inarray)
{
    NekDouble flowrate = 0.0;
 
    if (m_flowrateBnd && m_flowrateBndID >= 0)
    {
        // If we're an actual boundary, calculate the vector flux through
        // the boundary.
        Array<OneD, Array<OneD, NekDouble>> boundary(m_spacedim);
 
        if (!m_homd1DFlowinPlane)
        {
            // General case
            for (int i = 0; i < m_spacedim; ++i)
            {
                m_fields[i]->ExtractPhysToBnd(m_flowrateBndID, inarray[i],
                                              boundary[i]);
            }
            flowrate = m_flowrateBnd->VectorFlux(boundary);
        }
        else if (m_planeID == 0)
        {
            // Homogeneous with forcing in plane. Calculate flux only on
            // the meanmode - calculateFlux necessary for hybrid
            // parallelisation.
            for (int i = 0; i < m_spacedim; ++i)
            {
                m_fields[i]->GetPlane(m_planeID)->ExtractPhysToBnd(
                    m_flowrateBndID, inarray[i], boundary[i]);
            }
 
            // the flowrate is calculated on the mean mode so it needs to be
            // multiplied by LZ to be consistent with the general case.
            flowrate = m_flowrateBnd->VectorFlux(boundary) *
                       m_session->GetParameter("LZ");
        }
    }
    else if (m_flowrateBnd && !m_homd1DFlowinPlane)
    {
        // 3DH1D case with no Flowrate boundary defined: compute flux
        // through the zero-th (mean) plane.
        flowrate = m_flowrateBnd->Integral(inarray[2]);
    }
 
    // Communication to obtain the total flowrate
    if (!m_homd1DFlowinPlane && m_HomogeneousType == eHomogeneous1D)
    {
        m_comm->GetColumnComm()->AllReduce(flowrate, LibUtilities::ReduceSum);
    }
    else
    {
        m_comm->GetSpaceComm()->AllReduce(flowrate, LibUtilities::ReduceSum);
    }
    return flowrate / m_flowrateArea;
}
 
bool VelocityCorrectionScheme::v_PostIntegrate(int step)
{
    if (m_flowrateSteps > 0)
    {
        if (m_comm->GetRank() == 0 && (step + 1) % m_flowrateSteps == 0)
        {
            m_flowrateStream << setw(8) << step << setw(16) << m_time
                             << setw(16) << m_alpha << endl;
        }
    }
 
    return IncNavierStokes::v_PostIntegrate(step);
}
 
/**
 * Destructor
 */
VelocityCorrectionScheme::~VelocityCorrectionScheme(void)
{
}
 
/**
 *
 */
void VelocityCorrectionScheme::v_GenerateSummary(SolverUtils::SummaryList &s)
{
    AdvectionSystem::v_GenerateSummary(s);
    SolverUtils::AddSummaryItem(s, "Splitting Scheme",
                                "Velocity correction (strong press. form)");
 
    if (m_extrapolation->GetSubStepName().size())
    {
        SolverUtils::AddSummaryItem(s, "Substepping",
                                    m_extrapolation->GetSubStepName());
    }
 
    string dealias = m_homogen_dealiasing ? "Homogeneous1D" : "";
    if (m_specHP_dealiasing)
    {
        dealias += (dealias == "" ? "" : " + ") + string("spectral/hp");
    }
    if (dealias != "")
    {
        SolverUtils::AddSummaryItem(s, "Dealiasing", dealias);
    }
 
    string smoothing = m_useSpecVanVisc ? "spectral/hp" : "";
    if (smoothing != "")
    {
        if (m_svvVarDiffCoeff == NullNekDouble1DArray)
        {
            SolverUtils::AddSummaryItem(
                s, "Smoothing-SpecHP",
                "SVV (" + smoothing + " Exp Kernel(cut-off = " +
                    boost::lexical_cast<string>(m_sVVCutoffRatio) +
                    ", diff coeff = " +
                    boost::lexical_cast<string>(m_sVVDiffCoeff) + "))");
        }
        else
        {
            if (m_IsSVVPowerKernel)
            {
                SolverUtils::AddSummaryItem(
                    s, "Smoothing-SpecHP",
                    "SVV (" + smoothing + " Power Kernel (Power ratio =" +
                        boost::lexical_cast<string>(m_sVVCutoffRatio) +
                        ", diff coeff = " +
                        boost::lexical_cast<string>(m_sVVDiffCoeff) +
                        "*Uh/p))");
            }
            else
            {
                SolverUtils::AddSummaryItem(
                    s, "Smoothing-SpecHP",
                    "SVV (" + smoothing + " DG Kernel (diff coeff = " +
                        boost::lexical_cast<string>(m_sVVDiffCoeff) +
                        "*Uh/p))");
            }
        }
    }
 
    if (m_useHomo1DSpecVanVisc && (m_HomogeneousType == eHomogeneous1D))
    {
        SolverUtils::AddSummaryItem(
            s, "Smoothing-Homo1D",
            "SVV (Homogeneous1D - Exp Kernel(cut-off = " +
                boost::lexical_cast<string>(m_sVVCutoffRatioHomo1D) +
                ", diff coeff = " +
                boost::lexical_cast<string>(m_sVVDiffCoeffHomo1D) + "))");
    }
 
    if (m_useGJPStabilisation)
    {
        SolverUtils::AddSummaryItem(
            s, "GJP Stab. Impl.    ",
            m_session->GetSolverInfo("GJPStabilisation"));
        SolverUtils::AddSummaryItem(s, "GJP Stab. JumpScale", m_GJPJumpScale);
 
        if (boost::iequals(m_session->GetSolverInfo("GJPStabilisation"),
                           "Explicit"))
        {
            SolverUtils::AddSummaryItem(
                s, "GJP Normal Velocity",
                m_session->GetSolverInfo("GJPNormalVelocity"));
        }
    }
}
 
/**
 *
 */
void VelocityCorrectionScheme::v_DoInitialise(bool dumpInitialConditions)
{
    m_F = Array<OneD, Array<OneD, NekDouble>>(m_nConvectiveFields);
 
    for (int i = 0; i < m_nConvectiveFields; ++i)
    {
        m_F[i] = Array<OneD, NekDouble>(m_fields[0]->GetTotPoints(), 0.0);
    }
 
    m_flowrateAiidt = 0.0;
 
    AdvectionSystem::v_DoInitialise(dumpInitialConditions);
 
    // Set up Field Meta Data for output files
    m_fieldMetaDataMap["Kinvis"] = boost::lexical_cast<std::string>(m_kinvis);
    m_fieldMetaDataMap["TimeStep"] =
        boost::lexical_cast<std::string>(m_timestep);
 
    // set boundary conditions here so that any normal component
    // correction are imposed before they are imposed on initial
    // field below
    SetBoundaryConditions(m_time);
    std::map<std::string, NekDouble> params;
    params["Time"] = m_time;
    for (size_t i = 0; i < m_strFrameData.size(); ++i)
    {
        if (std::fabs(m_movingFrameData[i + 21]) != 0.)
        {
            params[m_strFrameData[i]] = m_movingFrameData[i + 21];
        }
    }
    Array<OneD, Array<OneD, NekDouble>> fields;
    Array<OneD, Array<OneD, NekDouble>> Adv;
    m_IncNavierStokesBCs->Update(fields, Adv, params);
 
    // Ensure the initial conditions have correct BCs
    for (size_t i = 0; i < m_fields.size(); ++i)
    {
        m_fields[i]->ImposeDirichletConditions(m_fields[i]->UpdateCoeffs());
        m_fields[i]->LocalToGlobal();
        m_fields[i]->GlobalToLocal();
        m_fields[i]->BwdTrans(m_fields[i]->GetCoeffs(),
                              m_fields[i]->UpdatePhys());
    }
}
 
/**
 *
 */
void VelocityCorrectionScheme::v_TransCoeffToPhys(void)
{
    size_t nfields = m_fields.size() - 1;
    for (size_t k = 0; k < nfields; ++k)
    {
        // Backward Transformation in physical space for time evolution
        m_fields[k]->BwdTrans(m_fields[k]->GetCoeffs(),
                              m_fields[k]->UpdatePhys());
    }
}
 
/**
 *
 */
void VelocityCorrectionScheme::v_TransPhysToCoeff(void)
{
 
    size_t nfields = m_fields.size() - 1;
    for (size_t k = 0; k < nfields; ++k)
    {
        // Forward Transformation in physical space for time evolution
        m_fields[k]->FwdTransLocalElmt(m_fields[k]->GetPhys(),
                                       m_fields[k]->UpdateCoeffs());
    }
}
 
/**
 *
 */
Array<OneD, bool> VelocityCorrectionScheme::v_GetSystemSingularChecks()
{
    int vVar = m_session->GetVariables().size();
    Array<OneD, bool> vChecks(vVar, false);
    vChecks[vVar - 1] = true;
    return vChecks;
}
 
/**
 *
 */
int VelocityCorrectionScheme::v_GetForceDimension()
{
    return m_session->GetVariables().size() - 1;
}
 
/**
 * Explicit part of the method - Advection, Forcing + HOPBCs
 */
void VelocityCorrectionScheme::v_EvaluateAdvection_SetPressureBCs(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble time)
{
    LibUtilities::Timer timer;
    timer.Start();
    EvaluateAdvectionTerms(inarray, outarray, time);
    timer.Stop();
    timer.AccumulateRegion("Advection Terms");
 
    // Smooth advection
    if (m_SmoothAdvection)
    {
        for (int i = 0; i < m_nConvectiveFields; ++i)
        {
            m_pressure->SmoothField(outarray[i]);
        }
    }
 
    // Add forcing terms
    for (auto &x : m_forcing)
    {
        x->Apply(m_fields, inarray, outarray, time);
    }
 
    // Calculate High-Order pressure boundary conditions
    timer.Start();
    std::map<std::string, NekDouble> params;
    params["Kinvis"] = m_kinvis;
    params["Time"]   = time + m_timestep;
    for (size_t i = 0; i < m_strFrameData.size(); ++i)
    {
        if (std::fabs(m_movingFrameData[i + 21]) != 0.)
        {
            params[m_strFrameData[i]] = m_movingFrameData[i + 21];
        }
    }
    m_extrapolation->EvaluatePressureBCs(inarray, outarray, m_kinvis);
    m_IncNavierStokesBCs->Update(inarray, outarray, params);
    timer.Stop();
    timer.AccumulateRegion("Pressure BCs");
}
 
/**
 * Implicit part of the method - Poisson + nConv*Helmholtz
 */
void VelocityCorrectionScheme::v_SolveUnsteadyStokesSystem(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray,
    [[maybe_unused]] const NekDouble time, const NekDouble aii_Dt)
{
    // Set up flowrate if we're starting for the first time or the value of
    // aii_Dt has changed.
    if (m_flowrate > 0.0 && (aii_Dt != m_flowrateAiidt))
    {
        SetupFlowrate(aii_Dt);
    }
 
    size_t physTot = m_fields[0]->GetTotPoints();
 
    // Substep the pressure boundary condition if using substepping
    m_extrapolation->SubStepSetPressureBCs(inarray, aii_Dt, m_kinvis);
 
    // Set up forcing term for pressure Poisson equation
    LibUtilities::Timer timer;
    timer.Start();
    SetUpPressureForcing(inarray, m_F, aii_Dt);
    timer.Stop();
    timer.AccumulateRegion("Pressure Forcing");
 
    // Solve Pressure System
    timer.Start();
    SolvePressure(m_F[0]);
    timer.Stop();
    timer.AccumulateRegion("Pressure Solve");
 
    // Set up forcing term for Helmholtz problems
    timer.Start();
    SetUpViscousForcing(inarray, m_F, aii_Dt);
    timer.Stop();
    timer.AccumulateRegion("Viscous Forcing");
 
    // Solve velocity system
    timer.Start();
    SolveViscous(m_F, inarray, outarray, aii_Dt);
    timer.Stop();
    timer.AccumulateRegion("Viscous Solve");
 
    // Apply flowrate correction
    if (m_flowrate > 0.0 && m_greenFlux != numeric_limits<NekDouble>::max())
    {
        NekDouble currentFlux = MeasureFlowrate(outarray);
        m_alpha               = (m_flowrate - currentFlux) / m_greenFlux;
 
        for (int i = 0; i < m_spacedim; ++i)
        {
            Vmath::Svtvp(physTot, m_alpha, m_flowrateStokes[i], 1, outarray[i],
                         1, outarray[i], 1);
            // Enusre coeff space is updated for next time step
            m_fields[i]->FwdTransLocalElmt(outarray[i],
                                           m_fields[i]->UpdateCoeffs());
            // Impsoe symmetry of flow on coeff space (good to enfore
            // periodicity).
            m_fields[i]->LocalToGlobal();
            m_fields[i]->GlobalToLocal();
        }
    }
}
 
/**
 * Forcing term for Poisson solver solver
 */
void VelocityCorrectionScheme::v_SetUpPressureForcing(
    const Array<OneD, const Array<OneD, NekDouble>> &fields,
    Array<OneD, Array<OneD, NekDouble>> &Forcing, const NekDouble aii_Dt)
{
    size_t i;
    size_t physTot = m_fields[0]->GetTotPoints();
    size_t nvel    = m_velocity.size();
 
    m_fields[0]->PhysDeriv(eX, fields[0], Forcing[0]);
 
    for (i = 1; i < nvel; ++i)
    {
        // Use Forcing[1] as storage since it is not needed for the pressure
        m_fields[i]->PhysDeriv(DirCartesianMap[i], fields[i], Forcing[1]);
        Vmath::Vadd(physTot, Forcing[1], 1, Forcing[0], 1, Forcing[0], 1);
    }
 
    Vmath::Smul(physTot, 1.0 / aii_Dt, Forcing[0], 1, Forcing[0], 1);
}
 
/**
 * Forcing term for Helmholtz solver
 */
void VelocityCorrectionScheme::v_SetUpViscousForcing(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &Forcing, const NekDouble aii_Dt)
{
    NekDouble aii_dtinv = 1.0 / aii_Dt;
    size_t phystot      = m_fields[0]->GetTotPoints();
 
    // Grad p
    m_pressure->BwdTrans(m_pressure->GetCoeffs(), m_pressure->UpdatePhys());
 
    int nvel = m_velocity.size();
    if (nvel == 2)
    {
        m_pressure->PhysDeriv(m_pressure->GetPhys(), Forcing[m_velocity[0]],
                              Forcing[m_velocity[1]]);
    }
    else
    {
        m_pressure->PhysDeriv(m_pressure->GetPhys(), Forcing[m_velocity[0]],
                              Forcing[m_velocity[1]], Forcing[m_velocity[2]]);
    }
 
    // zero convective fields.
    for (int i = nvel; i < m_nConvectiveFields; ++i)
    {
        Vmath::Zero(phystot, Forcing[i], 1);
    }
 
    // Subtract inarray/(aii_dt) and divide by kinvis. Kinvis will
    // need to be updated for the convected fields.
    for (int i = 0; i < m_nConvectiveFields; ++i)
    {
        Blas::Daxpy(phystot, -aii_dtinv, inarray[i], 1, Forcing[i], 1);
        Blas::Dscal(phystot, 1.0 / m_diffCoeff[i], &(Forcing[i])[0], 1);
    }
}
 
/**
 * Solve pressure system
 */
void VelocityCorrectionScheme::v_SolvePressure(
    const Array<OneD, NekDouble> &Forcing)
{
    StdRegions::ConstFactorMap factors;
    // Setup coefficient for equation
    factors[StdRegions::eFactorLambda] = 0.0;
 
    // Solver Pressure Poisson Equation
    m_pressure->HelmSolve(Forcing, m_pressure->UpdateCoeffs(), factors);
 
    // Add presure to outflow bc if using convective like BCs
    m_extrapolation->AddPressureToOutflowBCs(m_kinvis);
}
 
/**
 * Solve velocity system
 */
void VelocityCorrectionScheme::v_SolveViscous(
    const Array<OneD, const Array<OneD, NekDouble>> &Forcing,
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble aii_Dt)
{
    StdRegions::ConstFactorMap factors;
    StdRegions::VarCoeffMap varCoeffMap       = StdRegions::NullVarCoeffMap;
    MultiRegions::VarFactorsMap varFactorsMap = MultiRegions::NullVarFactorsMap;
 
    AppendSVVFactors(factors, varFactorsMap);
    ComputeGJPNormalVelocity(inarray, varCoeffMap);
 
    // Solve Helmholtz system and put in Physical space
    for (int i = 0; i < m_nConvectiveFields; ++i)
    {
        // Add diffusion coefficient to GJP matrix operator (Implicit part)
        if (m_useGJPStabilisation)
        {
            factors[StdRegions::eFactorGJP] = m_GJPJumpScale / m_diffCoeff[i];
        }
 
        // Setup coefficients for equation
        factors[StdRegions::eFactorLambda] = 1.0 / aii_Dt / m_diffCoeff[i];
        m_fields[i]->HelmSolve(Forcing[i], m_fields[i]->UpdateCoeffs(), factors,
                               varCoeffMap, varFactorsMap);
        m_fields[i]->BwdTrans(m_fields[i]->GetCoeffs(), outarray[i]);
    }
}
 
void VelocityCorrectionScheme::SetUpSVV(void)
{
 
    m_session->MatchSolverInfo("SpectralVanishingViscosity", "PowerKernel",
                               m_useSpecVanVisc, false);
 
    if (m_useSpecVanVisc)
    {
        m_useHomo1DSpecVanVisc = true;
    }
    else
    {
        m_useHomo1DSpecVanVisc = false;
 
        m_session->MatchSolverInfo("SpectralVanishingViscositySpectralHP",
                                   "PowerKernel", m_useSpecVanVisc, false);
    }
 
    if (m_useSpecVanVisc)
    {
        m_IsSVVPowerKernel = true;
    }
    else
    {
        m_session->MatchSolverInfo("SpectralVanishingViscosity", "DGKernel",
                                   m_useSpecVanVisc, false);
        if (m_useSpecVanVisc)
        {
            m_useHomo1DSpecVanVisc = true;
        }
        else
        {
            m_session->MatchSolverInfo("SpectralVanishingViscositySpectralHP",
                                       "DGKernel", m_useSpecVanVisc, false);
        }
 
        if (m_useSpecVanVisc)
        {
            m_IsSVVPowerKernel = false;
        }
    }
 
    // set up varcoeff kernel if PowerKernel or DG is specified
    if (m_useSpecVanVisc)
    {
        Array<OneD, Array<OneD, NekDouble>> SVVVelFields =
            NullNekDoubleArrayOfArray;
        if (m_session->DefinesFunction("SVVVelocityMagnitude"))
        {
            if (m_comm->GetRank() == 0)
            {
                cout << "Seting up SVV velocity from "
                        "SVVVelocityMagnitude section in session file"
                     << endl;
            }
            size_t nvel    = m_velocity.size();
            size_t phystot = m_fields[0]->GetTotPoints();
            SVVVelFields   = Array<OneD, Array<OneD, NekDouble>>(nvel);
            vector<string> vars;
            for (size_t i = 0; i < nvel; ++i)
            {
                SVVVelFields[i] = Array<OneD, NekDouble>(phystot);
                vars.push_back(m_session->GetVariable(m_velocity[i]));
            }
 
            // Load up files into  m_fields;
            GetFunction("SVVVelocityMagnitude")->Evaluate(vars, SVVVelFields);
        }
 
        m_svvVarDiffCoeff = Array<OneD, NekDouble>(m_fields[0]->GetNumElmts());
        SVVVarDiffCoeff(1.0, m_svvVarDiffCoeff, SVVVelFields);
        m_session->LoadParameter("SVVDiffCoeff", m_sVVDiffCoeff, 1.0);
    }
    else
    {
        m_svvVarDiffCoeff = NullNekDouble1DArray;
        m_session->LoadParameter("SVVDiffCoeff", m_sVVDiffCoeff, 0.1);
    }
 
    // Load parameters for Spectral Vanishing Viscosity
    if (m_useSpecVanVisc == false)
    {
        m_session->MatchSolverInfo("SpectralVanishingViscosity", "True",
                                   m_useSpecVanVisc, false);
        if (m_useSpecVanVisc == false)
        {
            m_session->MatchSolverInfo("SpectralVanishingViscosity",
                                       "ExpKernel", m_useSpecVanVisc, false);
        }
        m_useHomo1DSpecVanVisc = m_useSpecVanVisc;
 
        if (m_useSpecVanVisc == false)
        {
            m_session->MatchSolverInfo("SpectralVanishingViscositySpectralHP",
                                       "True", m_useSpecVanVisc, false);
            if (m_useSpecVanVisc == false)
            {
                m_session->MatchSolverInfo(
                    "SpectralVanishingViscositySpectralHP", "ExpKernel",
                    m_useSpecVanVisc, false);
            }
        }
    }
 
    // Case of only Homo1D kernel
    if (m_session->DefinesSolverInfo("SpectralVanishingViscosityHomo1D"))
    {
        m_session->MatchSolverInfo("SpectralVanishingViscosityHomo1D", "True",
                                   m_useHomo1DSpecVanVisc, false);
        if (m_useHomo1DSpecVanVisc == false)
        {
            m_session->MatchSolverInfo("SpectralVanishingViscosityHomo1D",
                                       "ExpKernel", m_useHomo1DSpecVanVisc,
                                       false);
        }
    }
 
    m_session->LoadParameter("SVVCutoffRatio", m_sVVCutoffRatio, 0.75);
    m_session->LoadParameter("SVVCutoffRatioHomo1D", m_sVVCutoffRatioHomo1D,
                             m_sVVCutoffRatio);
    m_session->LoadParameter("SVVDiffCoeffHomo1D", m_sVVDiffCoeffHomo1D,
                             m_sVVDiffCoeff);
 
    if (m_HomogeneousType == eHomogeneous1D)
    {
        ASSERTL0(
            m_nConvectiveFields > 2,
            "Expect to have three velocity fields with homogenous expansion");
 
        if (m_useHomo1DSpecVanVisc)
        {
            Array<OneD, unsigned int> planes;
            planes = m_fields[0]->GetZIDs();
 
            size_t num_planes = planes.size();
            Array<OneD, NekDouble> SVV(num_planes, 0.0);
            NekDouble fac;
            size_t kmodes = m_fields[0]->GetHomogeneousBasis()->GetNumModes();
            size_t pstart;
 
            pstart = m_sVVCutoffRatioHomo1D * kmodes;
 
            for (size_t n = 0; n < num_planes; ++n)
            {
                if (planes[n] > pstart)
                {
                    fac = (NekDouble)((planes[n] - kmodes) *
                                      (planes[n] - kmodes)) /
                          ((NekDouble)((planes[n] - pstart) *
                                       (planes[n] - pstart)));
                    SVV[n] = m_sVVDiffCoeffHomo1D * exp(-fac) / m_kinvis;
                }
            }
 
            for (size_t i = 0; i < m_velocity.size(); ++i)
            {
                m_fields[m_velocity[i]]->SetHomo1DSpecVanVisc(SVV);
            }
        }
    }
}
 
void VelocityCorrectionScheme::SVVVarDiffCoeff(
    const NekDouble velmag, Array<OneD, NekDouble> &diffcoeff,
    const Array<OneD, Array<OneD, NekDouble>> &vel)
{
    size_t phystot = m_fields[0]->GetTotPoints();
    size_t nel     = m_fields[0]->GetNumElmts();
    size_t nvel, cnt;
 
    Array<OneD, NekDouble> tmp;
 
    Vmath::Fill(nel, velmag, diffcoeff, 1);
 
    if (vel != NullNekDoubleArrayOfArray)
    {
        Array<OneD, NekDouble> Velmag(phystot);
        nvel = vel.size();
        // calculate magnitude of v
        Vmath::Vmul(phystot, vel[0], 1, vel[0], 1, Velmag, 1);
        for (size_t n = 1; n < nvel; ++n)
        {
            Vmath::Vvtvp(phystot, vel[n], 1, vel[n], 1, Velmag, 1, Velmag, 1);
        }
        Vmath::Vsqrt(phystot, Velmag, 1, Velmag, 1);
 
        cnt = 0;
        Array<OneD, NekDouble> tmp;
        // calculate mean value of vel mag.
        for (size_t i = 0; i < nel; ++i)
        {
            size_t nq    = m_fields[0]->GetExp(i)->GetTotPoints();
            tmp          = Velmag + cnt;
            diffcoeff[i] = m_fields[0]->GetExp(i)->Integral(tmp);
            Vmath::Fill(nq, 1.0, tmp, 1);
            NekDouble area = m_fields[0]->GetExp(i)->Integral(tmp);
            diffcoeff[i]   = diffcoeff[i] / area;
            cnt += nq;
        }
    }
    else
    {
        nvel = m_expdim;
    }
 
    for (size_t e = 0; e < nel; e++)
    {
        LocalRegions::ExpansionSharedPtr exp = m_fields[0]->GetExp(e);
        NekDouble h                          = 0;
 
        // Find maximum length of edge.
        size_t nEdge = exp->GetGeom()->GetNumEdges();
        for (size_t i = 0; i < nEdge; ++i)
        {
            h = max(h, exp->GetGeom()->GetEdge(i)->GetVertex(0)->dist(
                           *(exp->GetGeom()->GetEdge(i)->GetVertex(1))));
        }
 
        int p = 0;
        for (int i = 0; i < m_expdim; ++i)
        {
            p = max(p, exp->GetBasisNumModes(i) - 1);
        }
 
        diffcoeff[e] *= h / p;
    }
}
 
void VelocityCorrectionScheme::AppendSVVFactors(
    StdRegions::ConstFactorMap &factors,
    MultiRegions::VarFactorsMap &varFactorsMap)
{
 
    if (m_useSpecVanVisc)
    {
        factors[StdRegions::eFactorSVVCutoffRatio] = m_sVVCutoffRatio;
        factors[StdRegions::eFactorSVVDiffCoeff]   = m_sVVDiffCoeff / m_kinvis;
        if (m_svvVarDiffCoeff != NullNekDouble1DArray)
        {
            if (m_IsSVVPowerKernel)
            {
                varFactorsMap[StdRegions::eFactorSVVPowerKerDiffCoeff] =
                    m_svvVarDiffCoeff;
            }
            else
            {
                varFactorsMap[StdRegions::eFactorSVVDGKerDiffCoeff] =
                    m_svvVarDiffCoeff;
            }
        }
    }
}
 
/*
 * Calculate Normal velocity on Trace (boundary of elements)
 * for GJP stabilisation.
 * We only use this for explicit stabilisation. The Semi-Implicit
 * operator uses a constant u_{norm} = 1.0
 */
void VelocityCorrectionScheme::ComputeGJPNormalVelocity(
    const Array<OneD, const Array<OneD, NekDouble>> &inarray,
    StdRegions::VarCoeffMap &varcoeffs)
{
    if (m_useGJPNormalVel)
    {
        MultiRegions::ContFieldSharedPtr cfield =
            std::dynamic_pointer_cast<MultiRegions::ContField>(m_fields[0]);
 
        MultiRegions::GJPStabilisationSharedPtr GJPData =
            cfield->GetGJPForcing();
 
        int nTracePts = GJPData->GetNumTracePts();
        Array<OneD, NekDouble> unorm(nTracePts, 1.0);
        Array<OneD, NekDouble> Fwd(nTracePts), Bwd(nTracePts);
        Array<OneD, Array<OneD, NekDouble>> traceNormals =
            GJPData->GetTraceNormals();
 
        m_fields[0]->GetFwdBwdTracePhys(inarray[0], Fwd, Bwd, true, true);
        Vmath::Vmul(nTracePts, Fwd, 1, traceNormals[0], 1, unorm, 1);
 
        // Evaluate u.n on trace
        for (int f = 1; f < m_fields[0]->GetCoordim(0); ++f)
        {
            m_fields[0]->GetFwdBwdTracePhys(inarray[f], Fwd, Bwd, true, true);
            Vmath::Vvtvp(nTracePts, Fwd, 1, traceNormals[f], 1, unorm, 1, unorm,
                         1);
        }
        Vmath::Vabs(nTracePts, unorm, 1, unorm, 1);
        varcoeffs[StdRegions::eVarCoeffGJPNormVel] = unorm;
    }
}
} // namespace Nektar