LinearisedAdvection.cpp

Go to the documentation of this file.

///////////////////////////////////////////////////////////////////////////////
//
// File: LinearisedAdvection.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: Evaluation of the linearised advective term
//
///////////////////////////////////////////////////////////////////////////////
 
#include <IncNavierStokesSolver/AdvectionTerms/LinearisedAdvection.h>
#include <StdRegions/StdSegExp.h>
#include <boost/format.hpp>
 
using namespace std;
 
namespace Nektar
{
string LinearisedAdvection::className =
    SolverUtils::GetAdvectionFactory().RegisterCreatorFunction(
        "Linearised", LinearisedAdvection::create,
        "Linearised Non-Conservative");
 
/**
 * Constructor. Creates ...
 *
 * \param
 * \param
 */
 
LinearisedAdvection::LinearisedAdvection() : Advection()
{
}
 
void LinearisedAdvection::v_InitObject(
    LibUtilities::SessionReaderSharedPtr pSession,
    Array<OneD, MultiRegions::ExpListSharedPtr> pFields)
{
    Advection::v_InitObject(pSession, pFields);
 
    m_session = pSession;
    m_boundaryConditions =
        MemoryManager<SpatialDomains::BoundaryConditions>::AllocateSharedPtr(
            m_session, pFields[0]->GetGraph());
    m_spacedim = pFields[0]->GetGraph()->GetSpaceDimension();
    m_expdim   = pFields[0]->GetGraph()->GetMeshDimension();
 
    // Setting parameters for homogeneous problems
    m_HomoDirec       = 0;
    m_useFFT          = false;
    m_HomogeneousType = eNotHomogeneous;
    m_singleMode      = false;
    m_halfMode        = false;
    m_multipleModes   = false;
 
    if (m_session->DefinesSolverInfo("HOMOGENEOUS"))
    {
        std::string HomoStr = m_session->GetSolverInfo("HOMOGENEOUS");
        m_spacedim          = 3;
 
        if ((HomoStr == "HOMOGENEOUS1D") || (HomoStr == "Homogeneous1D") ||
            (HomoStr == "1D") || (HomoStr == "Homo1D"))
        {
            m_HomogeneousType = eHomogeneous1D;
            m_LhomZ           = m_session->GetParameter("LZ");
            m_HomoDirec       = 1;
 
            ASSERTL0(m_session->DefinesSolverInfo("ModeType"),
                     "Need to specify ModeType as HalfMode,SingleMode or "
                     "MultipleModes");
 
            m_session->MatchSolverInfo("ModeType", "SingleMode", m_singleMode,
                                       false);
            m_session->MatchSolverInfo("ModeType", "HalfMode", m_halfMode,
                                       false);
            m_session->MatchSolverInfo("ModeType", "MultipleModes",
                                       m_multipleModes, false);
 
            if (m_singleMode)
            {
                m_npointsZ = 2;
            }
            else if (m_halfMode)
            {
                m_npointsZ = 1;
            }
            else if (m_multipleModes)
            {
                m_npointsZ = m_session->GetParameter("HomModesZ");
            }
        }
 
        if ((HomoStr == "HOMOGENEOUS2D") || (HomoStr == "Homogeneous2D") ||
            (HomoStr == "2D") || (HomoStr == "Homo2D"))
        {
            m_HomogeneousType = eHomogeneous2D;
            m_session->LoadParameter("HomModesY", m_npointsY);
            m_session->LoadParameter("LY", m_LhomY);
            m_session->LoadParameter("HomModesZ", m_npointsZ);
            m_session->LoadParameter("LZ", m_LhomZ);
            m_HomoDirec = 2;
        }
 
        if ((HomoStr == "HOMOGENEOUS3D") || (HomoStr == "Homogeneous3D") ||
            (HomoStr == "3D") || (HomoStr == "Homo3D"))
        {
            m_HomogeneousType = eHomogeneous3D;
            m_session->LoadParameter("HomModesX", m_npointsX);
            m_session->LoadParameter("LX", m_LhomX);
            m_session->LoadParameter("HomModesY", m_npointsY);
            m_session->LoadParameter("LY", m_LhomY);
            m_session->LoadParameter("HomModesZ", m_npointsZ);
            m_session->LoadParameter("LZ", m_LhomZ);
            m_HomoDirec = 3;
        }
 
        if (m_session->DefinesSolverInfo("USEFFT"))
        {
            m_useFFT = true;
        }
    }
    else
    {
        m_npointsZ = 1; // set to default value so can use to identify 2d or 3D
                        // (homogeneous) expansions
    }
 
    size_t nvar = m_session->GetVariables().size();
    m_baseflow  = Array<OneD, Array<OneD, NekDouble>>(nvar);
    for (size_t i = 0; i < nvar; ++i)
    {
        m_baseflow[i] = Array<OneD, NekDouble>(pFields[i]->GetTotPoints(), 0.0);
    }
 
    size_t nBaseDerivs = (m_halfMode || m_singleMode) ? 2 : m_spacedim;
    m_gradBase = Array<OneD, Array<OneD, NekDouble>>(nvar * nBaseDerivs);
    for (size_t i = 0; i < nvar; ++i)
    {
        for (size_t j = 0; j < nBaseDerivs; ++j)
        {
            m_gradBase[i * nBaseDerivs + j] =
                Array<OneD, NekDouble>(pFields[i]->GetTotPoints(), 0.0);
        }
    }
 
    ASSERTL0(m_session->DefinesFunction("BaseFlow"),
             "Base flow must be defined for linearised forms.");
    string file = m_session->GetFunctionFilename("BaseFlow", 0);
 
    // Periodic base flows
    if (m_session->DefinesParameter("N_slices"))
    {
        m_session->LoadParameter("N_slices", m_slices);
        if (m_slices > 1)
        {
            ASSERTL0(m_session->GetFunctionType("BaseFlow", 0) ==
                         LibUtilities::eFunctionTypeFile,
                     "Base flow should be a sequence of files.");
            m_session->LoadParameter("BaseFlow_interporder", m_interporder, 0);
            ASSERTL0(m_interporder <= m_slices,
                     "BaseFlow_interporder should be smaller than or equal to "
                     "N_slices.");
            m_isperiodic = m_interporder < 2;
            m_session->LoadParameter("N_start", m_start, 0);
            m_session->LoadParameter("N_skip", m_skip, 1);
            DFT(file, pFields);
        }
        else
        {
            ASSERTL0(false, "Number of slices must be a positive number "
                            "greater than 1");
        }
    }
    // Steady base-flow
    else
    {
        m_slices = 1;
 
        // BaseFlow from file
        if (m_session->GetFunctionType("BaseFlow", m_session->GetVariable(0)) ==
            LibUtilities::eFunctionTypeFile)
        {
            ImportFldBase(file, pFields, 1);
        }
        // analytic base flow
        else
        {
            size_t nq = pFields[0]->GetNpoints();
            Array<OneD, NekDouble> x0(nq);
            Array<OneD, NekDouble> x1(nq);
            Array<OneD, NekDouble> x2(nq);
 
            // get the coordinates (assuming all fields have the same
            // discretisation)
            pFields[0]->GetCoords(x0, x1, x2);
 
            for (size_t i = 0; i < pFields.size(); i++)
            {
                LibUtilities::EquationSharedPtr ifunc =
                    m_session->GetFunction("BaseFlow", i);
 
                ifunc->Evaluate(x0, x1, x2, m_baseflow[i]);
            }
        }
    }
 
    for (size_t i = 0; i < nvar; ++i)
    {
        UpdateGradBase(i, pFields[i]);
    }
 
    if (m_session->DefinesParameter("period"))
    {
        m_period = m_session->GetParameter("period");
    }
    else
    {
        m_period =
            (m_session->GetParameter("TimeStep") * m_slices) / (m_slices - 1.);
    }
    if (m_session->GetComm()->GetRank() == 0)
    {
        cout << "baseflow info : interpolation order " << m_interporder
             << ", period " << m_period << ", periodicity ";
        if (m_isperiodic)
        {
            cout << "yes\n";
        }
        else
        {
            cout << "no\n";
        }
        cout << "baseflow info : files from " << m_start << " to "
             << (m_start + (m_slices - 1) * m_skip) << " (skip " << m_skip
             << ") with " << (m_slices - (m_interporder > 1))
             << " time intervals" << endl;
    }
}
 
LinearisedAdvection::~LinearisedAdvection()
{
}
 
// Advection function
 
void LinearisedAdvection::v_Advect(
    const int nConvectiveFields,
    const Array<OneD, MultiRegions::ExpListSharedPtr> &fields,
    const Array<OneD, Array<OneD, NekDouble>> &advVel,
    const Array<OneD, Array<OneD, NekDouble>> &inarray,
    Array<OneD, Array<OneD, NekDouble>> &outarray, const NekDouble &time,
    [[maybe_unused]] const Array<OneD, Array<OneD, NekDouble>> &pFwd,
    [[maybe_unused]] const Array<OneD, Array<OneD, NekDouble>> &pBwd)
{
    ASSERTL1(nConvectiveFields == inarray.size(),
             "Number of convective fields and Inarray are not compatible");
 
    size_t nPointsTot  = fields[0]->GetNpoints();
    size_t ndim        = advVel.size();
    size_t nBaseDerivs = (m_halfMode || m_singleMode) ? 2 : m_spacedim;
    size_t nDerivs     = (m_halfMode) ? 2 : m_spacedim;
 
    Array<OneD, Array<OneD, NekDouble>> velocity(ndim);
    for (size_t i = 0; i < ndim; ++i)
    {
        if (fields[i]->GetWaveSpace() && !m_singleMode && !m_halfMode)
        {
            velocity[i] = Array<OneD, NekDouble>(nPointsTot, 0.0);
            fields[i]->HomogeneousBwdTrans(nPointsTot, advVel[i], velocity[i]);
        }
        else
        {
            velocity[i] = advVel[i];
        }
    }
 
    Array<OneD, Array<OneD, NekDouble>> grad(nDerivs);
    for (size_t i = 0; i < nDerivs; ++i)
    {
        grad[i] = Array<OneD, NekDouble>(nPointsTot);
    }
 
    // Evaluation of the base flow for periodic cases
    if (m_slices > 1)
    {
        for (size_t i = 0; i < ndim; ++i)
        {
            UpdateBase(m_interp[i], m_baseflow[i], time);
            UpdateGradBase(i, fields[i]);
        }
    }
 
    // Evaluate the linearised advection term
    for (size_t i = 0; i < (size_t)nConvectiveFields; ++i)
    {
        // Calculate gradient
        switch (nDerivs)
        {
            case 1:
            {
                fields[i]->PhysDeriv(inarray[i], grad[0]);
            }
            break;
            case 2:
            {
                fields[i]->PhysDeriv(inarray[i], grad[0], grad[1]);
            }
            break;
            case 3:
            {
                fields[i]->PhysDeriv(inarray[i], grad[0], grad[1], grad[2]);
                if (m_multipleModes)
                {
                    // transform gradients into physical Fourier space
                    fields[i]->HomogeneousBwdTrans(nPointsTot, grad[0],
                                                   grad[0]);
                    fields[i]->HomogeneousBwdTrans(nPointsTot, grad[1],
                                                   grad[1]);
                    fields[i]->HomogeneousBwdTrans(nPointsTot, grad[2],
                                                   grad[2]);
                }
            }
            break;
        }
 
        // Calculate U_j du'_i/dx_j
        Vmath::Vmul(nPointsTot, grad[0], 1, m_baseflow[0], 1, outarray[i], 1);
        for (size_t j = 1; j < nDerivs; ++j)
        {
            Vmath::Vvtvp(nPointsTot, grad[j], 1, m_baseflow[j], 1, outarray[i],
                         1, outarray[i], 1);
        }
 
        // Add u'_j dU_i/dx_j
        size_t lim = (m_halfMode || m_singleMode) ? 2 : ndim;
        if (m_halfMode && i == 2)
        {
            lim = 0;
        }
        for (size_t j = 0; j < lim; ++j)
        {
            Vmath::Vvtvp(nPointsTot, m_gradBase[i * nBaseDerivs + j], 1,
                         velocity[j], 1, outarray[i], 1, outarray[i], 1);
        }
 
        if (m_multipleModes)
        {
            fields[i]->HomogeneousFwdTrans(nPointsTot, outarray[i],
                                           outarray[i]);
        }
        Vmath::Neg(nPointsTot, outarray[i], 1);
    }
}
 
void LinearisedAdvection::v_SetBaseFlow(
    const Array<OneD, Array<OneD, NekDouble>> &inarray,
    const Array<OneD, MultiRegions::ExpListSharedPtr> &fields)
{
    if (m_session->GetSolverInfo("EqType") == "UnsteadyNavierStokes")
    {
        // The SFD method is only applied to the velocity variables in
        // incompressible
        ASSERTL1(inarray.size() == (m_baseflow.size() - 1),
                 "Number of base flow variables does not match what is "
                 "expected.");
    }
    else
    {
        ASSERTL1(
            inarray.size() == (m_baseflow.size()),
            "Number of base flow variables does not match what is expected.");
    }
 
    size_t npts = inarray[0].size();
 
    for (size_t i = 0; i < inarray.size(); ++i)
    {
        ASSERTL1(npts == m_baseflow[i].size(),
                 "Size of base flow array does not match expected.");
        Vmath::Vcopy(npts, inarray[i], 1, m_baseflow[i], 1);
        UpdateGradBase(i, fields[i]);
    }
}
 
/**
 * Import field from infile and load into \a m_fields. This routine will
 * also perform a \a BwdTrans to ensure data is in both the physical and
 * coefficient storage.
 * @param   pInFile          Filename to read.
 * @param   pFields          Array of expansion lists
 */
void LinearisedAdvection::ImportFldBase(
    std::string pInfile, Array<OneD, MultiRegions::ExpListSharedPtr> &pFields,
    int pSlice)
{
    std::vector<LibUtilities::FieldDefinitionsSharedPtr> FieldDef;
    std::vector<std::vector<NekDouble>> FieldData;
 
    size_t nqtot = m_baseflow[0].size();
    Array<OneD, NekDouble> tmp_coeff(pFields[0]->GetNcoeffs(), 0.0);
 
    size_t numexp = pFields[0]->GetExpSize();
    Array<OneD, int> ElementGIDs(numexp);
 
    // Define list of global element ids
    for (size_t i = 0; i < numexp; ++i)
    {
        ElementGIDs[i] = pFields[0]->GetExp(i)->GetGeom()->GetGlobalID();
    }
 
    // Get Homogeneous
    LibUtilities::FieldIOSharedPtr fld =
        LibUtilities::FieldIO::CreateForFile(m_session, pInfile);
    fld->Import(pInfile, FieldDef, FieldData,
                LibUtilities::NullFieldMetaDataMap, ElementGIDs);
 
    size_t nSessionVar     = m_session->GetVariables().size();
    size_t nSessionConvVar = nSessionVar - 1;
    size_t nFileVar        = FieldDef[0]->m_fields.size();
 
    std::unordered_map<int, int> zIdToPlane;
    if (m_singleMode || m_halfMode)
    {
        zIdToPlane[0] = 0;
    }
 
    for (size_t j = 0; j < nFileVar; ++j)
    {
        size_t k = 0;
        for (; k < nSessionConvVar; ++k)
        {
            if (m_session->GetVariable(k) == FieldDef[0]->m_fields[j])
            {
                break;
            }
        }
        if (k == nSessionConvVar)
        {
            continue;
        }
        for (size_t i = 0; i < FieldDef.size(); ++i)
        {
            pFields[j]->ExtractDataToCoeffs(FieldDef[i], FieldData[i],
                                            FieldDef[i]->m_fields[j], tmp_coeff,
                                            zIdToPlane);
        }
 
        if (m_singleMode || m_halfMode)
        {
            pFields[j]->GetPlane(0)->BwdTrans(tmp_coeff, m_baseflow[k]);
 
            if (m_singleMode)
            {
                // copy the bwd trans into the second plane for single
                // Mode Analysis
                int ncplane = (pFields[0]->GetNpoints()) / m_npointsZ;
                Vmath::Vcopy(ncplane, &m_baseflow[k][0], 1,
                             &m_baseflow[k][ncplane], 1);
            }
        }
        else // fully 3D base flow - put in physical space.
        {
            bool oldwavespace = pFields[j]->GetWaveSpace();
            pFields[j]->SetWaveSpace(false);
            pFields[j]->BwdTrans(tmp_coeff, m_baseflow[k]);
            pFields[j]->SetWaveSpace(oldwavespace);
        }
    }
 
    // If time-periodic, put loaded data into the slice storage.
    if (m_slices > 1)
    {
        for (size_t i = 0; i < nSessionConvVar; ++i)
        {
            Vmath::Vcopy(nqtot, &m_baseflow[i][0], 1,
                         &m_interp[i][pSlice * nqtot], 1);
        }
    }
}
 
void LinearisedAdvection::UpdateBase(
    const Array<OneD, const NekDouble> &inarray,
    Array<OneD, NekDouble> &outarray, const NekDouble time)
{
    int npoints = m_baseflow[0].size();
    if (m_isperiodic)
    {
        NekDouble BetaT = 2 * M_PI * fmod(time, m_period) / m_period;
        NekDouble phase;
        Array<OneD, NekDouble> auxiliary(npoints);
 
        Vmath::Vcopy(npoints, &inarray[0], 1, &outarray[0], 1);
        Vmath::Svtvp(npoints, cos(0.5 * m_slices * BetaT), &inarray[npoints], 1,
                     &outarray[0], 1, &outarray[0], 1);
 
        for (int i = 2; i < m_slices; i += 2)
        {
            phase = (i >> 1) * BetaT;
 
            Vmath::Svtvp(npoints, cos(phase), &inarray[i * npoints], 1,
                         &outarray[0], 1, &outarray[0], 1);
            Vmath::Svtvp(npoints, -sin(phase), &inarray[(i + 1) * npoints], 1,
                         &outarray[0], 1, &outarray[0], 1);
        }
    }
    else
    {
        NekDouble x = time;
        x           = x / m_period * (m_slices - 1);
        int ix      = x;
        if (ix < 0)
        {
            ix = 0;
        }
        if (ix > m_slices - 2)
        {
            ix = m_slices - 2;
        }
        int padleft = m_interporder / 2 - 1;
        if (padleft > ix)
        {
            padleft = ix;
        }
        int padright = m_interporder - 1 - padleft;
        if (padright > m_slices - 1 - ix)
        {
            padright = m_slices - 1 - ix;
        }
        padleft = m_interporder - 1 - padright;
        Array<OneD, NekDouble> coeff(m_interporder, 1.);
        for (int i = 0; i < m_interporder; ++i)
        {
            for (int j = 0; j < m_interporder; ++j)
            {
                if (i != j)
                {
                    coeff[i] *= (x - ix + padleft - (NekDouble)j) /
                                ((NekDouble)i - (NekDouble)j);
                }
            }
        }
        Vmath::Zero(npoints, &outarray[0], 1);
        for (int i = ix - padleft; i < ix + padright + 1; ++i)
        {
            Vmath::Svtvp(npoints, coeff[i - ix + padleft],
                         &inarray[i * npoints], 1, &outarray[0], 1,
                         &outarray[0], 1);
        }
    }
}
 
void LinearisedAdvection::UpdateGradBase(
    const int var, const MultiRegions::ExpListSharedPtr &field)
{
    int nBaseDerivs = (m_halfMode || m_singleMode) ? 2 : m_spacedim;
    switch (m_spacedim)
    {
        case 1: // 1D
        {
            field->PhysDeriv(m_baseflow[var],
                             m_gradBase[var * nBaseDerivs + 0]);
        }
        break;
        case 2: // 2D
        {
            field->PhysDeriv(m_baseflow[var], m_gradBase[var * nBaseDerivs + 0],
                             m_gradBase[var * nBaseDerivs + 1]);
        }
        break;
        case 3:
        {
            if (m_halfMode) // can assume W = 0 and  d/dz == 0
            {
                if (var < 2)
                {
                    field->PhysDeriv(m_baseflow[var],
                                     m_gradBase[var * nBaseDerivs + 0],
                                     m_gradBase[var * nBaseDerivs + 1]);
                }
            }
            else if (m_singleMode) // single mode where d/dz = 0
            {
                field->PhysDeriv(m_baseflow[var],
                                 m_gradBase[var * nBaseDerivs + 0],
                                 m_gradBase[var * nBaseDerivs + 1]);
            }
            else
            {
                // Differentiate base flow in physical space
                bool oldwavespace = field->GetWaveSpace();
                field->SetWaveSpace(false);
                field->PhysDeriv(m_baseflow[var],
                                 m_gradBase[var * nBaseDerivs + 0],
                                 m_gradBase[var * nBaseDerivs + 1],
                                 m_gradBase[var * nBaseDerivs + 2]);
                field->SetWaveSpace(oldwavespace);
            }
        }
        break;
    }
}
 
DNekBlkMatSharedPtr LinearisedAdvection::GetFloquetBlockMatrix(
    [[maybe_unused]] FloquetMatType mattype,
    [[maybe_unused]] bool UseContCoeffs) const
{
    DNekMatSharedPtr loc_mat;
    DNekBlkMatSharedPtr BlkMatrix;
    size_t n_exp = 0;
 
    n_exp = m_baseflow[0].size(); // will operatore on m_phys
 
    Array<OneD, unsigned int> nrows(n_exp);
    Array<OneD, unsigned int> ncols(n_exp);
 
    nrows = Array<OneD, unsigned int>(n_exp, m_slices);
    ncols = Array<OneD, unsigned int>(n_exp, m_slices);
 
    MatrixStorage blkmatStorage = eDIAGONAL;
    BlkMatrix = MemoryManager<DNekBlkMat>::AllocateSharedPtr(nrows, ncols,
                                                             blkmatStorage);
 
    const LibUtilities::PointsKey Pkey(m_slices,
                                       LibUtilities::eFourierEvenlySpaced);
    const LibUtilities::BasisKey BK(LibUtilities::eFourier, m_slices, Pkey);
    StdRegions::StdSegExp StdSeg(BK);
 
    StdRegions::StdMatrixKey matkey(StdRegions::eFwdTrans,
                                    StdSeg.DetShapeType(), StdSeg);
 
    loc_mat = StdSeg.GetStdMatrix(matkey);
 
    // set up array of block matrices.
    for (size_t i = 0; i < n_exp; ++i)
    {
        BlkMatrix->SetBlock(i, i, loc_mat);
    }
 
    return BlkMatrix;
}
 
// Discrete Fourier Transform for Floquet analysis
void LinearisedAdvection::DFT(
    const string file, Array<OneD, MultiRegions::ExpListSharedPtr> &pFields)
{
    size_t ConvectedFields = m_baseflow.size() - 1;
    size_t npoints         = m_baseflow[0].size();
    m_interp = Array<OneD, Array<OneD, NekDouble>>(ConvectedFields);
 
    for (size_t i = 0; i < ConvectedFields; ++i)
    {
        m_interp[i] = Array<OneD, NekDouble>(npoints * m_slices, 0.0);
    }
 
    // Import the slides into auxiliary vector
    // The base flow should be stored in the form "filename_%d.ext"
    // A subdirectory can also be included, such as "dir/filename_%d.ext"
    size_t found = file.find("%d");
    ASSERTL0(found != string::npos &&
                 file.find("%d", found + 1) == string::npos,
             "Since N_slices is specified, the filename provided for function "
             "'BaseFlow' must include exactly one instance of the format "
             "specifier '%d', to index the time-slices.");
    size_t nstart = m_start;
    for (size_t i = nstart; i < nstart + m_slices * m_skip; i += m_skip)
    {
        boost::format filename(file);
        filename % i;
        ImportFldBase(filename.str(), pFields, (i - nstart) / m_skip);
        if (m_session->GetComm()->GetRank() == 0)
        {
            cout << "read base flow file " << filename.str() << endl;
        }
    }
    if (!m_isperiodic)
    {
        return;
    }
 
    // Discrete Fourier Transform of the fields
    for (size_t k = 0; k < ConvectedFields; ++k)
    {
#ifdef NEKTAR_USING_FFTW
 
        // Discrete Fourier Transform using FFTW
        Array<OneD, NekDouble> fft_in(npoints * m_slices);
        Array<OneD, NekDouble> fft_out(npoints * m_slices);
 
        Array<OneD, NekDouble> m_tmpIN(m_slices);
        Array<OneD, NekDouble> m_tmpOUT(m_slices);
 
        // Shuffle the data
        for (int j = 0; j < m_slices; ++j)
        {
            Vmath::Vcopy(npoints, &m_interp[k][j * npoints], 1, &(fft_in[j]),
                         m_slices);
        }
 
        m_FFT = LibUtilities::GetNektarFFTFactory().CreateInstance("NekFFTW",
                                                                   m_slices);
 
        // FFT Transform
        for (size_t i = 0; i < npoints; i++)
        {
            m_FFT->FFTFwdTrans(m_tmpIN  = fft_in + i * m_slices,
                               m_tmpOUT = fft_out + i * m_slices);
        }
 
        // Reshuffle data
        for (int s = 0; s < m_slices; ++s)
        {
            Vmath::Vcopy(npoints, &fft_out[s], m_slices,
                         &m_interp[k][s * npoints], 1);
        }
 
        Vmath::Zero(fft_in.size(), &fft_in[0], 1);
        Vmath::Zero(fft_out.size(), &fft_out[0], 1);
#else
        // Discrete Fourier Transform using MVM
        DNekBlkMatSharedPtr blkmat;
        blkmat = GetFloquetBlockMatrix(eForwardsPhys);
 
        int nrows = blkmat->GetRows();
        int ncols = blkmat->GetColumns();
 
        Array<OneD, NekDouble> sortedinarray(ncols);
        Array<OneD, NekDouble> sortedoutarray(nrows);
 
        // Shuffle the data
        for (int j = 0; j < m_slices; ++j)
        {
            Vmath::Vcopy(npoints, &m_interp[k][j * npoints], 1,
                         &(sortedinarray[j]), m_slices);
        }
 
        // Create NekVectors from the given data arrays
        NekVector<NekDouble> in(ncols, sortedinarray, eWrapper);
        NekVector<NekDouble> out(nrows, sortedoutarray, eWrapper);
 
        // Perform matrix-vector multiply.
        out = (*blkmat) * in;
 
        // Reshuffle data
        for (int s = 0; s < m_slices; ++s)
        {
            Vmath::Vcopy(npoints, &sortedoutarray[s], m_slices,
                         &m_interp[k][s * npoints], 1);
        }
 
        for (size_t r = 0; r < sortedinarray.size(); ++r)
        {
            sortedinarray[0]  = 0;
            sortedoutarray[0] = 0;
        }
 
#endif
    }
}
 
} // namespace Nektar