CompressibleFlowSystem.cpp

Go to the documentation of this file.

///////////////////////////////////////////////////////////////////////////////
//
// File CompressibleFlowSystem.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: Compressible flow system base class with auxiliary functions
//
///////////////////////////////////////////////////////////////////////////////
 
#include <boost/core/ignore_unused.hpp>
 
#include <CompressibleFlowSolver/EquationSystems/CompressibleFlowSystem.h>
 
using namespace std;
 
namespace Nektar
{
    CompressibleFlowSystem::CompressibleFlowSystem(
        const LibUtilities::SessionReaderSharedPtr& pSession,
        const SpatialDomains::MeshGraphSharedPtr& pGraph)
        : UnsteadySystem(pSession, pGraph),
          AdvectionSystem(pSession, pGraph)
    {
    }
 
    /**
     * @brief Initialization object for CompressibleFlowSystem class.
     */
    void CompressibleFlowSystem::v_InitObject()
    {
        AdvectionSystem::v_InitObject();
 
        m_varConv = MemoryManager<VariableConverter>::AllocateSharedPtr(
                    m_session, m_spacedim);
 
        ASSERTL0(m_session->DefinesSolverInfo("UPWINDTYPE"),
                 "No UPWINDTYPE defined in session.");
 
        // Do not forwards transform initial condition
        m_homoInitialFwd = false;
 
        // Set up locations of velocity vector.
        m_vecLocs = Array<OneD, Array<OneD, NekDouble> >(1);
        m_vecLocs[0] = Array<OneD, NekDouble>(m_spacedim);
        for (int i = 0; i < m_spacedim; ++i)
        {
            m_vecLocs[0][i] = 1 + i;
        }
 
        // Loading parameters from session file
        InitialiseParameters();
 
        // Setting up advection and diffusion operators
        InitAdvection();
 
        // Create artificial diffusion
        if (m_shockCaptureType != "Off")
        {
            m_artificialDiffusion = GetArtificialDiffusionFactory()
                                    .CreateInstance(m_shockCaptureType,
                                                    m_session,
                                                    m_fields,
                                                    m_spacedim);
        }
 
        // Forcing terms for the sponge region
        m_forcing = SolverUtils::Forcing::Load(m_session, shared_from_this(),
                                        m_fields, m_fields.num_elements());
 
        // User-defined boundary conditions
        int cnt = 0;
        for (int n = 0; n < m_fields[0]->GetBndConditions().num_elements(); ++n)
        {
            std::string type =
                m_fields[0]->GetBndConditions()[n]->GetUserDefined();
 
            if (m_fields[0]->GetBndConditions()[n]->GetBoundaryConditionType()
                == SpatialDomains::ePeriodic)
            {
                continue;
            }
 
            if(!type.empty())
            {
                m_bndConds.push_back(GetCFSBndCondFactory().CreateInstance(
                        type,
                        m_session,
                        m_fields,
                        m_traceNormals,
                        m_spacedim,
                        n,
                        cnt));
            }
            cnt += m_fields[0]->GetBndCondExpansions()[n]->GetExpSize();
        }
 
        if (m_explicitAdvection)
        {
            m_ode.DefineOdeRhs    (&CompressibleFlowSystem::DoOdeRhs, this);
            m_ode.DefineProjection(&CompressibleFlowSystem::DoOdeProjection, this);
        }
        else
        {
            ASSERTL0(false, "Implicit CFS not set up.");
        }
    }
 
    /**
     * @brief Destructor for CompressibleFlowSystem class.
     */
    CompressibleFlowSystem::~CompressibleFlowSystem()
    {
 
    }
 
    /**
     * @brief Load CFS parameters from the session file
     */
    void CompressibleFlowSystem::InitialiseParameters()
    {
        // Get gamma parameter from session file.
        m_session->LoadParameter("Gamma", m_gamma, 1.4);
 
        // Shock capture
        m_session->LoadSolverInfo("ShockCaptureType",
                                  m_shockCaptureType, "Off");
 
        // Load parameters for exponential filtering
        m_session->MatchSolverInfo("ExponentialFiltering","True",
                                   m_useFiltering, false);
        if(m_useFiltering)
        {
            m_session->LoadParameter ("FilterAlpha", m_filterAlpha, 36);
            m_session->LoadParameter ("FilterExponent", m_filterExponent, 16);
            m_session->LoadParameter ("FilterCutoff", m_filterCutoff, 0);
        }
 
        // Load CFL for local time-stepping (for steady state)
        m_session->MatchSolverInfo("LocalTimeStep","True",
                                   m_useLocalTimeStep, false);
        if(m_useLocalTimeStep)
        {
            ASSERTL0(m_cflSafetyFactor != 0,
                    "Local time stepping requires CFL parameter.");
        }
    }
 
    /**
     * @brief Create advection and diffusion objects for CFS
     */
    void CompressibleFlowSystem::InitAdvection()
    {
        // Check if projection type is correct
        ASSERTL0(m_projectionType == MultiRegions::eDiscontinuous,
                "Unsupported projection type.");
 
        string advName, riemName;
        m_session->LoadSolverInfo("AdvectionType", advName, "WeakDG");
 
        m_advObject = SolverUtils::GetAdvectionFactory()
                                    .CreateInstance(advName, advName);
 
        if (m_specHP_dealiasing)
        {
            m_advObject->SetFluxVector(&CompressibleFlowSystem::
                                       GetFluxVectorDeAlias, this);
        }
        else
        {
            m_advObject->SetFluxVector  (&CompressibleFlowSystem::
                                          GetFluxVector, this);
        }
 
        // Setting up Riemann solver for advection operator
        m_session->LoadSolverInfo("UpwindType", riemName, "Average");
 
        SolverUtils::RiemannSolverSharedPtr riemannSolver;
        riemannSolver = SolverUtils::GetRiemannSolverFactory()
                                    .CreateInstance(riemName, m_session);
 
        // Setting up parameters for advection operator Riemann solver
        riemannSolver->SetParam (
            "gamma",   &CompressibleFlowSystem::GetGamma,   this);
        riemannSolver->SetAuxVec(
            "vecLocs", &CompressibleFlowSystem::GetVecLocs, this);
        riemannSolver->SetVector(
            "N",       &CompressibleFlowSystem::GetNormals, this);
 
        // Concluding initialisation of advection / diffusion operators
        m_advObject->SetRiemannSolver   (riemannSolver);
        m_advObject->InitObject         (m_session, m_fields);
    }
 
    /**
     * @brief Compute the right-hand side.
     */
    void CompressibleFlowSystem::DoOdeRhs(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray,
              Array<OneD,       Array<OneD, NekDouble> > &outarray,
        const NekDouble                                   time)
    {
        int i;
        int nvariables = inarray.num_elements();
        int npoints    = GetNpoints();
        int nTracePts  = GetTraceTotPoints();
 
        // Store forwards/backwards space along trace space
        Array<OneD, Array<OneD, NekDouble> > Fwd    (nvariables);
        Array<OneD, Array<OneD, NekDouble> > Bwd    (nvariables);
 
        if (m_HomogeneousType == eHomogeneous1D)
        {
            Fwd = NullNekDoubleArrayofArray;
            Bwd = NullNekDoubleArrayofArray;
        }
        else
        {
            for(i = 0; i < nvariables; ++i)
            {
                Fwd[i]     = Array<OneD, NekDouble>(nTracePts, 0.0);
                Bwd[i]     = Array<OneD, NekDouble>(nTracePts, 0.0);
                m_fields[i]->GetFwdBwdTracePhys(inarray[i], Fwd[i], Bwd[i]);
            }
        }
 
        // Calculate advection
        DoAdvection(inarray, outarray, time, Fwd, Bwd);
 
        // Negate results
        for (i = 0; i < nvariables; ++i)
        {
            Vmath::Neg(npoints, outarray[i], 1);
        }
 
        // Add diffusion terms
        DoDiffusion(inarray, outarray, Fwd, Bwd);
 
        // Add forcing terms
        for (auto &x : m_forcing)
        {
            x->Apply(m_fields, inarray, outarray, time);
        }
 
        if (m_useLocalTimeStep)
        {
            int nElements = m_fields[0]->GetExpSize();
            int nq, offset;
            NekDouble fac;
            Array<OneD, NekDouble> tmp;
 
            Array<OneD, NekDouble> tstep (nElements, 0.0);
            GetElmtTimeStep(inarray, tstep);
 
            // Loop over elements
            for(int n = 0; n < nElements; ++n)
            {
                nq     = m_fields[0]->GetExp(n)->GetTotPoints();
                offset = m_fields[0]->GetPhys_Offset(n);
                fac    = tstep[n] / m_timestep;
                for(i = 0; i < nvariables; ++i)
                {
                    Vmath::Smul(nq, fac, outarray[i] + offset, 1,
                                         tmp = outarray[i] + offset, 1);
                }
            }
        }
    }
 
    /**
     * @brief Compute the projection and call the method for imposing the
     * boundary conditions in case of discontinuous projection.
     */
    void CompressibleFlowSystem::DoOdeProjection(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray,
              Array<OneD,       Array<OneD, NekDouble> > &outarray,
        const NekDouble                                   time)
    {
        int i;
        int nvariables = inarray.num_elements();
 
        switch(m_projectionType)
        {
            case MultiRegions::eDiscontinuous:
            {
                // Just copy over array
                int npoints = GetNpoints();
 
                for(i = 0; i < nvariables; ++i)
                {
                    Vmath::Vcopy(npoints, inarray[i], 1, outarray[i], 1);
                    if(m_useFiltering)
                    {
                        m_fields[i]->ExponentialFilter(outarray[i],
                            m_filterAlpha, m_filterExponent, m_filterCutoff);
                    }
                }
                SetBoundaryConditions(outarray, time);
                break;
            }
            case MultiRegions::eGalerkin:
            case MultiRegions::eMixed_CG_Discontinuous:
            {
                ASSERTL0(false, "No Continuous Galerkin for full compressible "
                                "Navier-Stokes equations");
                break;
            }
            default:
                ASSERTL0(false, "Unknown projection scheme");
                break;
        }
    }
 
    /**
     * @brief Compute the advection terms for the right-hand side
     */
    void CompressibleFlowSystem::DoAdvection(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray,
              Array<OneD,       Array<OneD, NekDouble> > &outarray,
        const NekDouble                                   time,
        const Array<OneD, Array<OneD, NekDouble> >       &pFwd,
        const Array<OneD, Array<OneD, NekDouble> >       &pBwd)
    {
        int nvariables = inarray.num_elements();
        Array<OneD, Array<OneD, NekDouble> > advVel(m_spacedim);
 
        m_advObject->Advect(nvariables, m_fields, advVel, inarray,
                            outarray, time, pFwd, pBwd);
    }
 
    /**
     * @brief Add the diffusions terms to the right-hand side
     */
    void CompressibleFlowSystem::DoDiffusion(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray,
              Array<OneD,       Array<OneD, NekDouble> > &outarray,
            const Array<OneD, Array<OneD, NekDouble> >   &pFwd,
            const Array<OneD, Array<OneD, NekDouble> >   &pBwd)
    {
        v_DoDiffusion(inarray, outarray, pFwd, pBwd);
 
        if (m_shockCaptureType != "Off")
        {
            // Get min h/p
            m_artificialDiffusion->SetElmtHP(GetElmtMinHP());
            m_artificialDiffusion->DoArtificialDiffusion(inarray, outarray);
        }
    }
 
    void CompressibleFlowSystem::SetBoundaryConditions(
            Array<OneD, Array<OneD, NekDouble> >             &physarray,
            NekDouble                                         time)
    {
        int nTracePts  = GetTraceTotPoints();
        int nvariables = physarray.num_elements();
 
        Array<OneD, Array<OneD, NekDouble> > Fwd(nvariables);
        for (int i = 0; i < nvariables; ++i)
        {
            Fwd[i] = Array<OneD, NekDouble>(nTracePts);
            m_fields[i]->ExtractTracePhys(physarray[i], Fwd[i]);
        }
 
        if (m_bndConds.size())
        {
            // Loop over user-defined boundary conditions
            for (auto &x : m_bndConds)
            {
                x->Apply(Fwd, physarray, time);
            }
        }
    }
 
    /**
     * @brief Return the flux vector for the compressible Euler equations.
     *
     * @param physfield   Fields.
     * @param flux        Resulting flux.
     */
    void CompressibleFlowSystem::GetFluxVector(
        const Array<OneD, Array<OneD, NekDouble> >               &physfield,
              Array<OneD, Array<OneD, Array<OneD, NekDouble> > > &flux)
    {
        int i, j;
        int nq = physfield[0].num_elements();
        int nVariables = m_fields.num_elements();
 
        Array<OneD, NekDouble> pressure(nq);
        Array<OneD, Array<OneD, NekDouble> > velocity(m_spacedim);
 
        // Flux vector for the rho equation
        for (i = 0; i < m_spacedim; ++i)
        {
            velocity[i] = Array<OneD, NekDouble>(nq);
            Vmath::Vcopy(nq, physfield[i+1], 1, flux[0][i], 1);
        }
 
        m_varConv->GetVelocityVector(physfield, velocity);
        m_varConv->GetPressure(physfield, pressure);
 
        // Flux vector for the velocity fields
        for (i = 0; i < m_spacedim; ++i)
        {
            for (j = 0; j < m_spacedim; ++j)
            {
                Vmath::Vmul(nq, velocity[j], 1, physfield[i+1], 1,
                            flux[i+1][j], 1);
            }
 
            // Add pressure to appropriate field
            Vmath::Vadd(nq, flux[i+1][i], 1, pressure, 1, flux[i+1][i], 1);
        }
 
        // Flux vector for energy.
        Vmath::Vadd(nq, physfield[m_spacedim+1], 1, pressure, 1,
                    pressure, 1);
 
        for (j = 0; j < m_spacedim; ++j)
        {
            Vmath::Vmul(nq, velocity[j], 1, pressure, 1,
                        flux[m_spacedim+1][j], 1);
        }
 
        // For the smooth viscosity model
        if (nVariables == m_spacedim+3)
        {
            // Add a zero row for the advective fluxes
            for (j = 0; j < m_spacedim; ++j)
            {
                Vmath::Zero(nq, flux[m_spacedim+2][j], 1);
            }
        }
    }
 
    /**
     * @brief Return the flux vector for the compressible Euler equations
     * by using the de-aliasing technique.
     *
     * @param physfield   Fields.
     * @param flux        Resulting flux.
     */
    void CompressibleFlowSystem::GetFluxVectorDeAlias(
        const Array<OneD, Array<OneD, NekDouble> >               &physfield,
              Array<OneD, Array<OneD, Array<OneD, NekDouble> > > &flux)
    {
        int i, j;
        int nq = physfield[0].num_elements();
        int nVariables = m_fields.num_elements();
 
        // Factor to rescale 1d points in dealiasing
        NekDouble OneDptscale = 2;
        nq = m_fields[0]->Get1DScaledTotPoints(OneDptscale);
 
        Array<OneD, NekDouble> pressure(nq);
        Array<OneD, Array<OneD, NekDouble> > velocity(m_spacedim);
 
        Array<OneD, Array<OneD, NekDouble> > physfield_interp(nVariables);
        Array<OneD, Array<OneD, Array<OneD, NekDouble> > > flux_interp(
                                                            nVariables);
 
        for (i = 0; i < nVariables; ++ i)
        {
            physfield_interp[i] = Array<OneD, NekDouble>(nq);
            flux_interp[i] = Array<OneD, Array<OneD, NekDouble> >(m_spacedim);
            m_fields[0]->PhysInterp1DScaled(
                OneDptscale, physfield[i], physfield_interp[i]);
 
            for (j = 0; j < m_spacedim; ++j)
            {
                flux_interp[i][j] = Array<OneD, NekDouble>(nq);
            }
        }
 
        // Flux vector for the rho equation
        for (i = 0; i < m_spacedim; ++i)
        {
            velocity[i] = Array<OneD, NekDouble>(nq);
 
            // Galerkin project solution back to original space
            m_fields[0]->PhysGalerkinProjection1DScaled(
                OneDptscale, physfield_interp[i+1], flux[0][i]);
        }
 
        m_varConv->GetVelocityVector(physfield_interp, velocity);
        m_varConv->GetPressure      (physfield_interp, pressure);
 
        // Evaluation of flux vector for the velocity fields
        for (i = 0; i < m_spacedim; ++i)
        {
            for (j = 0; j < m_spacedim; ++j)
            {
                Vmath::Vmul(nq, velocity[j], 1, physfield_interp[i+1], 1,
                            flux_interp[i+1][j], 1);
            }
 
            // Add pressure to appropriate field
            Vmath::Vadd(nq, flux_interp[i+1][i], 1, pressure,1,
                        flux_interp[i+1][i], 1);
        }
 
        // Galerkin project solution back to original space
        for (i = 0; i < m_spacedim; ++i)
        {
            for (j = 0; j < m_spacedim; ++j)
            {
                m_fields[0]->PhysGalerkinProjection1DScaled(
                    OneDptscale, flux_interp[i+1][j], flux[i+1][j]);
            }
        }
 
        // Evaluation of flux vector for energy
        Vmath::Vadd(nq, physfield_interp[m_spacedim+1], 1, pressure, 1,
                    pressure, 1);
 
        for (j = 0; j < m_spacedim; ++j)
        {
            Vmath::Vmul(nq, velocity[j], 1, pressure, 1,
                        flux_interp[m_spacedim+1][j], 1);
 
            // Galerkin project solution back to original space
            m_fields[0]->PhysGalerkinProjection1DScaled(
                OneDptscale,
                flux_interp[m_spacedim+1][j],
                flux[m_spacedim+1][j]);
        }
    }
 
    /**
     * @brief Calculate the maximum timestep on each element
     *        subject to CFL restrictions.
     */
    void CompressibleFlowSystem::GetElmtTimeStep(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray,
              Array<OneD, NekDouble> &tstep)
    {
        boost::ignore_unused(inarray);
 
        int n;
        int nElements = m_fields[0]->GetExpSize();
 
        // Change value of m_timestep (in case it is set to zero)
        NekDouble tmp = m_timestep;
        m_timestep    = 1.0;
 
        Array<OneD, NekDouble> cfl(nElements);
        cfl = GetElmtCFLVals();
 
        // Factors to compute the time-step limit
        NekDouble alpha     = MaxTimeStepEstimator();
 
        // Loop over elements to compute the time-step limit for each element
        for(n = 0; n < nElements; ++n)
        {
            tstep[n] = m_cflSafetyFactor * alpha / cfl[n];
        }
 
        // Restore value of m_timestep
        m_timestep = tmp;
    }
 
    /**
     * @brief Calculate the maximum timestep subject to CFL restrictions.
     */
    NekDouble CompressibleFlowSystem::v_GetTimeStep(
        const Array<OneD, const Array<OneD, NekDouble> > &inarray)
    {
        int nElements = m_fields[0]->GetExpSize();
        Array<OneD, NekDouble> tstep (nElements, 0.0);
 
        GetElmtTimeStep(inarray, tstep);
 
        // Get the minimum time-step limit and return the time-step
        NekDouble TimeStep = Vmath::Vmin(nElements, tstep, 1);
        m_comm->AllReduce(TimeStep, LibUtilities::ReduceMin);
        return TimeStep;
    }
 
    /**
     * @brief Set up logic for residual calculation.
     */
    void CompressibleFlowSystem::v_SetInitialConditions(
        NekDouble initialtime,
        bool      dumpInitialConditions,
        const int domain)
    {
        boost::ignore_unused(domain);
 
        EquationSystem::v_SetInitialConditions(initialtime, false);
 
        // insert white noise in initial condition
        NekDouble Noise;
        int phystot = m_fields[0]->GetTotPoints();
        Array<OneD, NekDouble> noise(phystot);
 
        m_session->LoadParameter("Noise", Noise,0.0);
        int m_nConvectiveFields =  m_fields.num_elements();
 
        if (Noise > 0.0)
        {
            int seed = - m_comm->GetRank()*m_nConvectiveFields;
            for (int i = 0; i < m_nConvectiveFields; i++)
            {
                Vmath::FillWhiteNoise(phystot, Noise, noise, 1,
                                      seed);
                --seed;
                Vmath::Vadd(phystot, m_fields[i]->GetPhys(), 1,
                            noise, 1, m_fields[i]->UpdatePhys(), 1);
                m_fields[i]->FwdTrans_IterPerExp(m_fields[i]->GetPhys(),
                                                 m_fields[i]->UpdateCoeffs());
            }
        }
 
        if (dumpInitialConditions && m_checksteps)
        {
            Checkpoint_Output(m_nchk);
            m_nchk++;
        }
    }
 
    /**
     * @brief Compute the advection velocity in the standard space
     * for each element of the expansion.
     */
    Array<OneD, NekDouble> CompressibleFlowSystem::v_GetMaxStdVelocity()
    {
        int nTotQuadPoints = GetTotPoints();
        int n_element      = m_fields[0]->GetExpSize();
        int expdim         = m_fields[0]->GetGraph()->GetMeshDimension();
        int nfields        = m_fields.num_elements();
        int offset;
        Array<OneD, NekDouble> tmp;
 
        Array<OneD, Array<OneD, NekDouble> > physfields(nfields);
        for (int i = 0; i < nfields; ++i)
        {
            physfields[i] = m_fields[i]->GetPhys();
        }
 
        Array<OneD, NekDouble> stdV(n_element, 0.0);
 
        // Getting the velocity vector on the 2D normal space
        Array<OneD, Array<OneD, NekDouble> > velocity   (m_spacedim);
        Array<OneD, Array<OneD, NekDouble> > stdVelocity(m_spacedim);
        Array<OneD, Array<OneD, NekDouble> > stdSoundSpeed(m_spacedim);
        Array<OneD, NekDouble>               soundspeed (nTotQuadPoints);
        LibUtilities::PointsKeyVector        ptsKeys;
 
        for (int i = 0; i < m_spacedim; ++i)
        {
            velocity   [i]   = Array<OneD, NekDouble>(nTotQuadPoints);
            stdVelocity[i]   = Array<OneD, NekDouble>(nTotQuadPoints, 0.0);
            stdSoundSpeed[i] = Array<OneD, NekDouble>(nTotQuadPoints, 0.0);
        }
 
        m_varConv->GetVelocityVector(physfields, velocity);
        m_varConv->GetSoundSpeed    (physfields, soundspeed);
 
        for(int el = 0; el < n_element; ++el)
        {
            ptsKeys = m_fields[0]->GetExp(el)->GetPointsKeys();
            offset  = m_fields[0]->GetPhys_Offset(el);
            int nq = m_fields[0]->GetExp(el)->GetTotPoints();
 
            const SpatialDomains::GeomFactorsSharedPtr metricInfo =
                m_fields[0]->GetExp(el)->GetGeom()->GetMetricInfo();
            const Array<TwoD, const NekDouble> &gmat =
                m_fields[0]->GetExp(el)->GetGeom()->GetMetricInfo()
                                                  ->GetDerivFactors(ptsKeys);
 
            // Convert to standard element
            //    consider soundspeed in all directions
            //    (this might overestimate the cfl)
            if(metricInfo->GetGtype() == SpatialDomains::eDeformed)
            {
                // d xi/ dx = gmat = 1/J * d x/d xi
                for (int i = 0; i < expdim; ++i)
                {
                    Vmath::Vmul(nq, gmat[i], 1,
                                    velocity[0] + offset, 1,
                                    tmp = stdVelocity[i] + offset, 1);
                    Vmath::Vmul(nq, gmat[i], 1,
                                    soundspeed + offset, 1,
                                    tmp = stdSoundSpeed[i] + offset, 1);
                    for (int j = 1; j < expdim; ++j)
                    {
                        Vmath::Vvtvp(nq, gmat[expdim*j+i], 1,
                                         velocity[j] + offset, 1,
                                         stdVelocity[i] + offset, 1,
                                         tmp = stdVelocity[i] + offset, 1);
                        Vmath::Vvtvp(nq, gmat[expdim*j+i], 1,
                                         soundspeed + offset, 1,
                                         stdSoundSpeed[i] + offset, 1,
                                         tmp = stdSoundSpeed[i] + offset, 1);
                    }
                }
            }
            else
            {
                for (int i = 0; i < expdim; ++i)
                {
                    Vmath::Smul(nq, gmat[i][0],
                                    velocity[0] + offset, 1,
                                    tmp = stdVelocity[i] + offset, 1);
                    Vmath::Smul(nq, gmat[i][0],
                                    soundspeed + offset, 1,
                                    tmp = stdSoundSpeed[i] + offset, 1);
                    for (int j = 1; j < expdim; ++j)
                    {
                        Vmath::Svtvp(nq, gmat[expdim*j+i][0],
                                         velocity[j] + offset, 1,
                                         stdVelocity[i] + offset, 1,
                                         tmp = stdVelocity[i] + offset, 1);
                        Vmath::Svtvp(nq, gmat[expdim*j+i][0],
                                         soundspeed + offset, 1,
                                         stdSoundSpeed[i] + offset, 1,
                                         tmp = stdSoundSpeed[i] + offset, 1);
                    }
                }
            }
 
            NekDouble vel;
            for (int i = 0; i < nq; ++i)
            {
                NekDouble pntVelocity = 0.0;
                for (int j = 0; j < expdim; ++j)
                {
                    // Add sound speed
                    vel = std::abs(stdVelocity[j][offset + i]) +
                          std::abs(stdSoundSpeed[j][offset + i]);
                    pntVelocity += vel * vel;
                }
                pntVelocity = sqrt(pntVelocity);
                if (pntVelocity > stdV[el])
                {
                    stdV[el] = pntVelocity;
                }
            }
        }
 
        return stdV;
    }
 
    /**
     * @brief Set the denominator to compute the time step when a cfl
     * control is employed. This function is no longer used but is still
     * here for being utilised in the future.
     *
     * @param n   Order of expansion element by element.
     */
    NekDouble CompressibleFlowSystem::GetStabilityLimit(int n)
    {
        ASSERTL0(n <= 20, "Illegal modes dimension for CFL calculation "
                          "(P has to be less then 20)");
 
        NekDouble CFLDG[21] = {  2.0000,   6.0000,  11.8424,  19.1569,
                                27.8419,  37.8247,  49.0518,  61.4815,
                                75.0797,  89.8181, 105.6700, 122.6200,
                               140.6400, 159.7300, 179.8500, 201.0100,
                               223.1800, 246.3600, 270.5300, 295.6900,
                               321.8300}; //CFLDG 1D [0-20]
        NekDouble CFL = 0.0;
 
        if (m_projectionType == MultiRegions::eDiscontinuous)
        {
            CFL = CFLDG[n];
        }
        else
        {
            ASSERTL0(false, "Continuous Galerkin stability coefficients "
                            "not introduced yet.");
        }
 
        return CFL;
    }
 
    /**
     * @brief Compute the vector of denominators to compute the time step
     * when a cfl control is employed. This function is no longer used but
     * is still here for being utilised in the future.
     *
     * @param ExpOrder   Order of expansion element by element.
     */
    Array<OneD, NekDouble> CompressibleFlowSystem::GetStabilityLimitVector(
        const Array<OneD,int> &ExpOrder)
    {
        int i;
        Array<OneD,NekDouble> returnval(m_fields[0]->GetExpSize(), 0.0);
        for (i =0; i<m_fields[0]->GetExpSize(); i++)
        {
            returnval[i] = GetStabilityLimit(ExpOrder[i]);
        }
        return returnval;
    }
 
    void CompressibleFlowSystem::v_ExtraFldOutput(
        std::vector<Array<OneD, NekDouble> > &fieldcoeffs,
        std::vector<std::string>             &variables)
    {
        bool extraFields;
        m_session->MatchSolverInfo("OutputExtraFields","True",
                                   extraFields, true);
        if (extraFields)
        {
            const int nPhys   = m_fields[0]->GetNpoints();
            const int nCoeffs = m_fields[0]->GetNcoeffs();
            Array<OneD, Array<OneD, NekDouble> > tmp(m_fields.num_elements());
 
            for (int i = 0; i < m_fields.num_elements(); ++i)
            {
                tmp[i] = m_fields[i]->GetPhys();
            }
 
            Array<OneD, Array<OneD, NekDouble> > velocity(m_spacedim);
            Array<OneD, Array<OneD, NekDouble> > velFwd  (m_spacedim);
            for (int i = 0; i < m_spacedim; ++i)
            {
                velocity[i] = Array<OneD, NekDouble> (nPhys);
                velFwd[i]   = Array<OneD, NekDouble> (nCoeffs);
            }
 
            Array<OneD, NekDouble> pressure(nPhys), temperature(nPhys);
            Array<OneD, NekDouble> entropy(nPhys);
            Array<OneD, NekDouble> soundspeed(nPhys), mach(nPhys);
            Array<OneD, NekDouble> sensor(nPhys), SensorKappa(nPhys);
 
            m_varConv->GetVelocityVector(tmp, velocity);
            m_varConv->GetPressure  (tmp, pressure);
            m_varConv->GetTemperature(tmp, temperature);
            m_varConv->GetEntropy   (tmp, entropy);
            m_varConv->GetSoundSpeed(tmp, soundspeed);
            m_varConv->GetMach      (tmp, soundspeed, mach);
 
            int sensorOffset;
            m_session->LoadParameter ("SensorOffset", sensorOffset, 1);
            m_varConv->GetSensor (m_fields[0], tmp, sensor, SensorKappa,
                                    sensorOffset);
 
            Array<OneD, NekDouble> pFwd(nCoeffs), TFwd(nCoeffs);
            Array<OneD, NekDouble> sFwd(nCoeffs);
            Array<OneD, NekDouble> aFwd(nCoeffs), mFwd(nCoeffs);
            Array<OneD, NekDouble> sensFwd(nCoeffs);
 
            string velNames[3] = {"u", "v", "w"};
            for (int i = 0; i < m_spacedim; ++i)
            {
                m_fields[0]->FwdTrans_IterPerExp(velocity[i], velFwd[i]);
                variables.push_back(velNames[i]);
                fieldcoeffs.push_back(velFwd[i]);
            }
 
            m_fields[0]->FwdTrans_IterPerExp(pressure,   pFwd);
            m_fields[0]->FwdTrans_IterPerExp(temperature,TFwd);
            m_fields[0]->FwdTrans_IterPerExp(entropy,    sFwd);
            m_fields[0]->FwdTrans_IterPerExp(soundspeed, aFwd);
            m_fields[0]->FwdTrans_IterPerExp(mach,       mFwd);
            m_fields[0]->FwdTrans_IterPerExp(sensor,     sensFwd);
 
            variables.push_back  ("p");
            variables.push_back  ("T");
            variables.push_back  ("s");
            variables.push_back  ("a");
            variables.push_back  ("Mach");
            variables.push_back  ("Sensor");
            fieldcoeffs.push_back(pFwd);
            fieldcoeffs.push_back(TFwd);
            fieldcoeffs.push_back(sFwd);
            fieldcoeffs.push_back(aFwd);
            fieldcoeffs.push_back(mFwd);
            fieldcoeffs.push_back(sensFwd);
 
            if (m_artificialDiffusion)
            {
                // Get min h/p
                m_artificialDiffusion->SetElmtHP(GetElmtMinHP());
                // reuse pressure
                Array<OneD, NekDouble> sensorFwd(nCoeffs);
                m_artificialDiffusion->GetArtificialViscosity(tmp, pressure);
                m_fields[0]->FwdTrans_IterPerExp(pressure,   sensorFwd);
 
                variables.push_back  ("ArtificialVisc");
                fieldcoeffs.push_back(sensorFwd);
            }
        }
    }
 
    /**
     *
     */
    void CompressibleFlowSystem::GetPressure(
        const Array<OneD, const Array<OneD, NekDouble> > &physfield,
              Array<OneD, NekDouble>                     &pressure)
    {
        m_varConv->GetPressure(physfield, pressure);
    }
 
    /**
     *
     */
    void CompressibleFlowSystem::GetDensity(
        const Array<OneD, const Array<OneD, NekDouble> > &physfield,
              Array<OneD, NekDouble>                     &density)
    {
        density = physfield[0];
    }
 
    /**
     *
     */
    void CompressibleFlowSystem::GetVelocity(
        const Array<OneD, const Array<OneD, NekDouble> > &physfield,
              Array<OneD, Array<OneD, NekDouble> >       &velocity)
    {
        m_varConv->GetVelocityVector(physfield, velocity);
    }
 
/**
 * @brief Compute an estimate of minimum h/p
 * for each element of the expansion.
 */
Array<OneD, NekDouble>  CompressibleFlowSystem::GetElmtMinHP(void)
{
    int nElements               = m_fields[0]->GetExpSize();
    Array<OneD, NekDouble> hOverP(nElements, 1.0);
 
    // Determine h/p scaling
    Array<OneD, int> pOrderElmt = m_fields[0]->EvalBasisNumModesMaxPerExp();
    for (int e = 0; e < nElements; e++)
    {
        NekDouble h = 1.0e+10;
        switch(m_expdim)
        {
            case 3:
            {
                LocalRegions::Expansion3DSharedPtr exp3D;
                exp3D = m_fields[0]->GetExp(e)->as<LocalRegions::Expansion3D>();
                for(int i = 0; i < exp3D->GetNedges(); ++i)
                {
                    h = min(h, exp3D->GetGeom3D()->GetEdge(i)->GetVertex(0)->
                        dist(*(exp3D->GetGeom3D()->GetEdge(i)->GetVertex(1))));
                }
            break;
            }
 
            case 2:
            {
                LocalRegions::Expansion2DSharedPtr exp2D;
                exp2D = m_fields[0]->GetExp(e)->as<LocalRegions::Expansion2D>();
                for(int i = 0; i < exp2D->GetNedges(); ++i)
                {
                    h = min(h, exp2D->GetGeom2D()->GetEdge(i)->GetVertex(0)->
                        dist(*(exp2D->GetGeom2D()->GetEdge(i)->GetVertex(1))));
                }
            break;
            }
            case 1:
            {
                LocalRegions::Expansion1DSharedPtr exp1D;
                exp1D = m_fields[0]->GetExp(e)->as<LocalRegions::Expansion1D>();
 
                h = min(h, exp1D->GetGeom1D()->GetVertex(0)->
                    dist(*(exp1D->GetGeom1D()->GetVertex(1))));
 
            break;
            }
            default:
            {
                ASSERTL0(false,"Dimension out of bound.")
            }
        }
 
        // Determine h/p scaling
        hOverP[e] = h/max(pOrderElmt[e]-1,1);
 
    }
    return hOverP;
}
}