#include <IncBaseCondition.h>

Inheritance diagram for Nektar::IncBaseCondition:

Public Member Functions
virtual	~IncBaseCondition ()=default

void	Initialise (const LibUtilities::SessionReaderSharedPtr &pSession)

void	Update (const Array< OneD, const Array< OneD, NekDouble > > &fields, const Array< OneD, const Array< OneD, NekDouble > > &Adv, std::map< std::string, NekDouble > &params)

Protected Member Functions
	IncBaseCondition (const LibUtilities::SessionReaderSharedPtr pSession, Array< OneD, MultiRegions::ExpListSharedPtr > pFields, Array< OneD, SpatialDomains::BoundaryConditionShPtr > cond, Array< OneD, MultiRegions::ExpListSharedPtr > exp, int nbnd, int spacedim, int bnddim)

virtual void	v_Initialise (const LibUtilities::SessionReaderSharedPtr &pSession)

virtual void	v_Update (const Array< OneD, const Array< OneD, NekDouble > > &fields, const Array< OneD, const Array< OneD, NekDouble > > &Adv, std::map< std::string, NekDouble > &params)

void	ExtrapolateArray (const int numCalls, Array< OneD, Array< OneD, Array< OneD, NekDouble > > > &array)

void	RollOver (Array< OneD, Array< OneD, Array< OneD, NekDouble > > > &input)

void	AddVisPressureBCs (const Array< OneD, const Array< OneD, NekDouble > > &fields, Array< OneD, Array< OneD, NekDouble > > &N, std::map< std::string, NekDouble > &params)

void	AddRigidBodyAcc (Array< OneD, Array< OneD, NekDouble > > &N, std::map< std::string, NekDouble > &params, int npts0)

void	RigidBodyVelocity (Array< OneD, Array< OneD, NekDouble > > &velocities, std::map< std::string, NekDouble > &params, int npts0)

void	InitialiseCoords (std::map< std::string, NekDouble > &params)

void	SetNumPointsOnPlane0 (int &npointsPlane0)

Protected Attributes
int	m_spacedim

int	m_bnddim
	bounday dimensionality

int	m_nbnd

int	m_numCalls

int	m_intSteps

std::map< int, SpatialDomains::BoundaryConditionShPtr >	m_BndConds

std::map< int, MultiRegions::ExpListSharedPtr >	m_BndExp

int	m_npoints

Array< OneD, Array< OneD, NekDouble > >	m_coords

MultiRegions::ExpListSharedPtr	m_bndElmtExps

MultiRegions::ExpListSharedPtr	m_field

Array< OneD, Array< OneD, Array< OneD, NekDouble > > >	m_viscous

int	m_pressure

std::string	classname

Static Protected Attributes
static NekDouble	StifflyStable_Betaq_Coeffs [3][3]

static NekDouble	StifflyStable_Alpha_Coeffs [3][3]

static NekDouble	StifflyStable_Gamma0_Coeffs [3]

Detailed Description

Definition at line 65 of file IncBaseCondition.h.

Constructor & Destructor Documentation

◆ ~IncBaseCondition()

virtual Nektar::IncBaseCondition::~IncBaseCondition ( )

virtualdefault

◆ IncBaseCondition()

Nektar::IncBaseCondition::IncBaseCondition	(	const LibUtilities::SessionReaderSharedPtr	pSession,
		Array< OneD, MultiRegions::ExpListSharedPtr >	pFields,
		Array< OneD, SpatialDomains::BoundaryConditionShPtr >	cond,
		Array< OneD, MultiRegions::ExpListSharedPtr >	exp,
		int	nbnd,
		int	spacedim,
		int	bnddim
	)

protected

Definition at line 54 of file IncBaseCondition.cpp.

    : m_spacedim(spacedim), m_bnddim(bnddim), m_nbnd(nbnd), m_field(pFields[0])
{
}

Member Function Documentation

◆ AddRigidBodyAcc()

void Nektar::IncBaseCondition::AddRigidBodyAcc	(	Array< OneD, Array< OneD, NekDouble > > &	N,
		std::map< std::string, NekDouble > &	params,
		int	npts0
	)

protected

Definition at line 170 of file IncBaseCondition.cpp.

{
    if (npts0 == 0)
    {
        return;
    }
    Array<OneD, Array<OneD, NekDouble>> acceleration(m_spacedim);
    for (size_t k = 0; k < m_spacedim; ++k)
    {
        acceleration[k] = Array<OneD, NekDouble>(npts0, 0.0);
    }
 
    // set up pressure condition
    if (params.find("Omega_z") != params.end())
    {
        NekDouble Wz2 = params["Omega_z"] * params["Omega_z"];
        NekDouble dWz = 0.;
        if (params.find("DOmega_z") != params.end())
        {
            dWz = params["DOmega_z"];
        }
        Vmath::Svtsvtp(npts0, Wz2, m_coords[0], 1, dWz, m_coords[1], 1, N[0],
                       1);
        Vmath::Svtsvtp(npts0, Wz2, m_coords[1], 1, -dWz, m_coords[0], 1, N[1],
                       1);
    }
    std::vector<std::string> vars = {"A_x", "A_y", "A_z"};
    for (int k = 0; k < m_bnddim; ++k)
    {
        if (params.find(vars[k]) != params.end())
        {
            Vmath::Sadd(npts0, -params[vars[k]], N[k], 1, N[k], 1);
        }
    }
}

References m_bnddim, m_coords, m_spacedim, Vmath::Sadd(), and Vmath::Svtsvtp().

Referenced by Nektar::MovingFrameWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ AddVisPressureBCs()

void Nektar::IncBaseCondition::AddVisPressureBCs	(	const Array< OneD, const Array< OneD, NekDouble > > &	fields,
		Array< OneD, Array< OneD, NekDouble > > &	N,
		std::map< std::string, NekDouble > &	params
	)

protected

Definition at line 208 of file IncBaseCondition.cpp.

{
    if (m_intSteps == 0 || params.find("Kinvis") == params.end() ||
        params["Kinvis"] <= 0. || fields.size() == 0)
    {
        return;
    }
    NekDouble kinvis = params["Kinvis"];
    m_bndElmtExps->SetWaveSpace(m_field->GetWaveSpace());
    Array<OneD, Array<OneD, NekDouble>> Velocity(m_spacedim);
    Array<OneD, Array<OneD, NekDouble>> Q(m_spacedim);
    // Loop all boundary conditions
    int nq = m_bndElmtExps->GetTotPoints();
    for (int i = 0; i < m_spacedim; i++)
    {
        Q[i] = Array<OneD, NekDouble>(nq, 0.0);
    }
 
    for (int i = 0; i < m_spacedim; i++)
    {
        m_field->ExtractPhysToBndElmt(m_nbnd, fields[i], Velocity[i]);
    }
 
    // CurlCurl
    m_bndElmtExps->CurlCurl(Velocity, Q);
 
    Array<OneD, NekDouble> temp(m_npoints);
    for (int i = 0; i < m_bnddim; i++)
    {
        m_field->ExtractElmtToBndPhys(m_nbnd, Q[i],
                                      m_viscous[m_intSteps - 1][i]);
    }
    ExtrapolateArray(m_numCalls, m_viscous);
    for (int i = 0; i < m_bnddim; i++)
    {
        Vmath::Svtvp(m_npoints, -kinvis, m_viscous[m_intSteps - 1][i], 1, N[i],
                     1, N[i], 1);
    }
}

References ExtrapolateArray(), m_bnddim, m_bndElmtExps, m_field, m_intSteps, m_nbnd, m_npoints, m_numCalls, m_spacedim, m_viscous, and Vmath::Svtvp().

Referenced by Nektar::MovingFrameWall::v_Update(), Nektar::StaticWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ ExtrapolateArray()

void Nektar::IncBaseCondition::ExtrapolateArray	(	const int	numCalls,
		Array< OneD, Array< OneD, Array< OneD, NekDouble > > > &	array
	)

protected

Definition at line 136 of file IncBaseCondition.cpp.

{
    if (m_intSteps == 1)
    {
        return;
    }
    int nint    = std::min(numCalls, m_intSteps);
    int nlevels = array.size();
    int dim     = array[0].size();
    int nPts    = array[0][0].size();
    // Check integer for time levels
    // Note that ExtrapolateArray assumes m_pressureCalls is >= 1
    // meaning v_EvaluatePressureBCs has been called previously
    ASSERTL0(nint > 0, "nint must be > 0 when calling ExtrapolateArray.");
    // Update array
    RollOver(array);
    // Extrapolate to outarray
    for (int i = 0; i < dim; ++i)
    {
        Vmath::Smul(nPts, StifflyStable_Betaq_Coeffs[nint - 1][nint - 1],
                    array[nint - 1][i], 1, array[nlevels - 1][i], 1);
    }
    for (int n = 0; n < nint - 1; ++n)
    {
        for (int i = 0; i < dim; ++i)
        {
            Vmath::Svtvp(nPts, StifflyStable_Betaq_Coeffs[nint - 1][n],
                         array[n][i], 1, array[nlevels - 1][i], 1,
                         array[nlevels - 1][i], 1);
        }
    }
}

References ASSERTL0, m_intSteps, RollOver(), Vmath::Smul(), StifflyStable_Betaq_Coeffs, and Vmath::Svtvp().

Referenced by AddVisPressureBCs().

◆ Initialise()

void Nektar::IncBaseCondition::Initialise ( const LibUtilities::SessionReaderSharedPtr & pSession )

inline

Definition at line 70 of file IncBaseCondition.h.

    {
        v_Initialise(pSession);
    }

References v_Initialise().

◆ InitialiseCoords()

void Nektar::IncBaseCondition::InitialiseCoords ( std::map< std::string, NekDouble > & params )

protected

Definition at line 104 of file IncBaseCondition.cpp.

{
    MultiRegions::ExpListSharedPtr bndexp = m_BndExp.begin()->second;
    if (m_coords.size() == 0)
    {
        m_coords = Array<OneD, Array<OneD, NekDouble>>(m_spacedim);
        for (size_t k = 0; k < m_spacedim; ++k)
        {
            m_coords[k] = Array<OneD, NekDouble>(m_npoints, 0.0);
        }
        if (m_spacedim == 2)
        {
            bndexp->GetCoords(m_coords[0], m_coords[1]);
        }
        else
        {
            bndexp->GetCoords(m_coords[0], m_coords[1], m_coords[2]);
        }
        // move the centre to the location of pivot
        std::vector<std::string> xyz = {"X0", "Y0", "Z0"};
        for (int i = 0; i < m_spacedim; ++i)
        {
            if (params.find(xyz[i]) != params.end())
            {
                Vmath::Sadd(m_npoints, -params[xyz[i]], m_coords[i], 1,
                            m_coords[i], 1);
            }
        }
    }
}

References m_BndExp, m_coords, m_npoints, m_spacedim, and Vmath::Sadd().

Referenced by Nektar::MovingFrameWall::v_Update().

◆ RigidBodyVelocity()

void Nektar::IncBaseCondition::RigidBodyVelocity	(	Array< OneD, Array< OneD, NekDouble > > &	velocities,
		std::map< std::string, NekDouble > &	params,
		int	npts0
	)

protected

Definition at line 264 of file IncBaseCondition.cpp.

{
    if (npts0 == 0)
    {
        return;
    }
    // for the wall we need to calculate:
    // [V_wall]_xyz = [V_frame]_xyz + [Omega X r]_xyz
    // Note all vectors must be in moving frame coordinates xyz
    // not in inertial frame XYZ
 
    // vx = OmegaY*z-OmegaZ*y
    // vy = OmegaZ*x-OmegaX*z
    // vz = OmegaX*y-OmegaY*x
    if (params.find("Omega_z") != params.end())
    {
        NekDouble Wz = params["Omega_z"];
        if (m_BndExp.find(0) != m_BndExp.end())
        {
            Vmath::Smul(npts0, -Wz, m_coords[1], 1, velocities[0], 1);
        }
        if (m_BndExp.find(1) != m_BndExp.end())
        {
            Vmath::Smul(npts0, Wz, m_coords[0], 1, velocities[1], 1);
        }
    }
    if (m_bnddim == 3)
    {
        if (params.find("Omega_x") != params.end())
        {
            NekDouble Wx = params["Omega_x"];
            if (m_BndExp.find(2) != m_BndExp.end())
            {
                Vmath::Smul(npts0, Wx, m_coords[1], 1, velocities[2], 1);
            }
            if (m_BndExp.find(1) != m_BndExp.end())
            {
                Vmath::Svtvp(npts0, -Wx, m_coords[2], 1, velocities[1], 1,
                             velocities[1], 1);
            }
        }
        if (params.find("Omega_y") != params.end())
        {
            NekDouble Wy = params["Omega_x"];
            if (m_BndExp.find(0) != m_BndExp.end())
            {
                Vmath::Svtvp(npts0, Wy, m_coords[2], 1, velocities[0], 1,
                             velocities[0], 1);
            }
            if (m_BndExp.find(2) != m_BndExp.end())
            {
                Vmath::Svtvp(npts0, -Wy, m_coords[0], 1, velocities[2], 1,
                             velocities[2], 1);
            }
        }
    }
 
    // add the translation velocity
    std::vector<std::string> vars = {"U", "V", "W"};
    for (int k = 0; k < m_bnddim; ++k)
    {
        if (params.find(vars[k]) != params.end() &&
            m_BndExp.find(k) != m_BndExp.end())
        {
            Vmath::Sadd(npts0, params[vars[k]], velocities[k], 1, velocities[k],
                        1);
        }
    }
}

References m_bnddim, m_BndExp, m_coords, Vmath::Sadd(), Vmath::Smul(), and Vmath::Svtvp().

Referenced by Nektar::MovingFrameWall::v_Update().

◆ RollOver()

void Nektar::IncBaseCondition::RollOver ( Array< OneD, Array< OneD, Array< OneD, NekDouble > > > & input )

protected

Definition at line 251 of file IncBaseCondition.cpp.

{
    int nlevels = input.size();
    Array<OneD, Array<OneD, NekDouble>> tmp;
    tmp = input[nlevels - 1];
    for (int n = nlevels - 1; n > 0; --n)
    {
        input[n] = input[n - 1];
    }
    input[0] = tmp;
}

Referenced by ExtrapolateArray().

◆ SetNumPointsOnPlane0()

void Nektar::IncBaseCondition::SetNumPointsOnPlane0 ( int & npointsPlane0 )

protected

Definition at line 85 of file IncBaseCondition.cpp.

{
    if (m_BndExp.begin()->second->GetExpType() == MultiRegions::e2DH1D)
    {
        if (m_field->GetZIDs()[0] == 0)
        {
            npointsPlane0 = m_BndExp.begin()->second->GetPlane(0)->GetNpoints();
        }
        else
        {
            npointsPlane0 = 0;
        }
    }
    else
    {
        npointsPlane0 = m_BndExp.begin()->second->GetNpoints();
    }
}

References Nektar::MultiRegions::e2DH1D, m_BndExp, and m_field.

Referenced by Nektar::MovingFrameFar::v_Update(), Nektar::MovingFrameWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ Update()

void Nektar::IncBaseCondition::Update	(	const Array< OneD, const Array< OneD, NekDouble > > &	fields,
		const Array< OneD, const Array< OneD, NekDouble > > &	Adv,
		std::map< std::string, NekDouble > &	params
	)

inline

Definition at line 75 of file IncBaseCondition.h.

    {
        v_Update(fields, Adv, params);
    };

References v_Update().

◆ v_Initialise()

void Nektar::IncBaseCondition::v_Initialise ( const LibUtilities::SessionReaderSharedPtr & pSession )

protectedvirtual

Reimplemented in Nektar::MovingFrameFar, Nektar::MovingFrameWall, Nektar::StaticWall, and Nektar::TransMovingWall.

Definition at line 65 of file IncBaseCondition.cpp.

{
    m_npoints  = m_BndExp.begin()->second->GetNpoints();
    m_numCalls = 0;
    if (pSession->DefinesParameter("ExtrapolateOrder"))
    {
        m_intSteps = std::round(pSession->GetParameter("ExtrapolateOrder"));
        m_intSteps = std::min(3, std::max(0, m_intSteps));
    }
    else if (pSession->DefinesTimeIntScheme())
    {
        m_intSteps = pSession->GetTimeIntScheme().order;
    }
    else
    {
        m_intSteps = 0;
    }
}

References m_BndExp, m_intSteps, m_npoints, and m_numCalls.

Referenced by Initialise(), Nektar::MovingFrameFar::v_Initialise(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ v_Update()

virtual void Nektar::IncBaseCondition::v_Update	(	const Array< OneD, const Array< OneD, NekDouble > > &	fields,
		const Array< OneD, const Array< OneD, NekDouble > > &	Adv,
		std::map< std::string, NekDouble > &	params
	)

inlineprotectedvirtual

Reimplemented in Nektar::MovingFrameFar, Nektar::MovingFrameWall, Nektar::StaticWall, and Nektar::TransMovingWall.

Definition at line 92 of file IncBaseCondition.h.

97 {

98 }

Referenced by Update().

Member Data Documentation

◆ classname

std::string Nektar::IncBaseCondition::classname

protected

Definition at line 138 of file IncBaseCondition.h.

Referenced by Nektar::MovingFrameFar::MovingFrameFar(), Nektar::MovingFrameWall::MovingFrameWall(), Nektar::StaticWall::StaticWall(), and Nektar::TransMovingWall::TransMovingWall().

◆ m_BndConds

std::map<int, SpatialDomains::BoundaryConditionShPtr> Nektar::IncBaseCondition::m_BndConds

protected

Definition at line 126 of file IncBaseCondition.h.

Referenced by Nektar::MovingFrameFar::MovingFrameFar(), Nektar::TransMovingWall::TransMovingWall(), Nektar::MovingFrameFar::v_Initialise(), and Nektar::TransMovingWall::v_Update().

◆ m_bnddim

int Nektar::IncBaseCondition::m_bnddim

protected

bounday dimensionality

Definition at line 121 of file IncBaseCondition.h.

Referenced by AddRigidBodyAcc(), AddVisPressureBCs(), Nektar::MovingFrameWall::MovingFrameWall(), RigidBodyVelocity(), Nektar::TransMovingWall::TransMovingWall(), Nektar::MovingFrameWall::v_Initialise(), Nektar::StaticWall::v_Initialise(), Nektar::MovingFrameWall::v_Update(), Nektar::StaticWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ m_bndElmtExps

MultiRegions::ExpListSharedPtr Nektar::IncBaseCondition::m_bndElmtExps

protected

Definition at line 130 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ m_BndExp

std::map<int, MultiRegions::ExpListSharedPtr> Nektar::IncBaseCondition::m_BndExp

protected

Definition at line 127 of file IncBaseCondition.h.

Referenced by InitialiseCoords(), Nektar::MovingFrameFar::MovingFrameFar(), Nektar::MovingFrameWall::MovingFrameWall(), RigidBodyVelocity(), SetNumPointsOnPlane0(), Nektar::StaticWall::StaticWall(), Nektar::TransMovingWall::TransMovingWall(), v_Initialise(), Nektar::StaticWall::v_Initialise(), Nektar::TransMovingWall::v_Initialise(), Nektar::MovingFrameFar::v_Update(), Nektar::MovingFrameWall::v_Update(), Nektar::StaticWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ m_coords

Array<OneD, Array<OneD, NekDouble> > Nektar::IncBaseCondition::m_coords

protected

Definition at line 129 of file IncBaseCondition.h.

Referenced by AddRigidBodyAcc(), InitialiseCoords(), and RigidBodyVelocity().

◆ m_field

MultiRegions::ExpListSharedPtr Nektar::IncBaseCondition::m_field

protected

Definition at line 131 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), SetNumPointsOnPlane0(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ m_intSteps

int Nektar::IncBaseCondition::m_intSteps

protected

Definition at line 125 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), ExtrapolateArray(), v_Initialise(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ m_nbnd

int Nektar::IncBaseCondition::m_nbnd

protected

Definition at line 122 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ m_npoints

int Nektar::IncBaseCondition::m_npoints

protected

Definition at line 128 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), InitialiseCoords(), v_Initialise(), Nektar::MovingFrameWall::v_Initialise(), Nektar::StaticWall::v_Initialise(), Nektar::MovingFrameWall::v_Update(), Nektar::StaticWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ m_numCalls

int Nektar::IncBaseCondition::m_numCalls

protected

Definition at line 124 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), v_Initialise(), Nektar::MovingFrameWall::v_Update(), Nektar::StaticWall::v_Update(), and Nektar::TransMovingWall::v_Update().

◆ m_pressure

int Nektar::IncBaseCondition::m_pressure

protected

Definition at line 133 of file IncBaseCondition.h.

Referenced by Nektar::StaticWall::StaticWall(), and Nektar::StaticWall::v_Update().

◆ m_spacedim

int Nektar::IncBaseCondition::m_spacedim

protected

Definition at line 119 of file IncBaseCondition.h.

Referenced by AddRigidBodyAcc(), AddVisPressureBCs(), InitialiseCoords(), Nektar::MovingFrameFar::MovingFrameFar(), Nektar::MovingFrameFar::v_Initialise(), and Nektar::MovingFrameFar::v_Update().

◆ m_viscous

Array<OneD, Array<OneD, Array<OneD, NekDouble> > > Nektar::IncBaseCondition::m_viscous

protected

Definition at line 132 of file IncBaseCondition.h.

Referenced by AddVisPressureBCs(), Nektar::MovingFrameWall::v_Initialise(), and Nektar::StaticWall::v_Initialise().

◆ StifflyStable_Alpha_Coeffs

NekDouble Nektar::IncBaseCondition::StifflyStable_Alpha_Coeffs

staticprotected

Initial value:

= {

{1.0, 0.0, 0.0}, {2.0, -0.5, 0.0}, {3.0, -1.5, 1.0 / 3.0}}

Definition at line 136 of file IncBaseCondition.h.

◆ StifflyStable_Betaq_Coeffs

NekDouble Nektar::IncBaseCondition::StifflyStable_Betaq_Coeffs

staticprotected

Initial value:

= {

{1.0, 0.0, 0.0}, {2.0, -1.0, 0.0}, {3.0, -3.0, 1.0}}

Definition at line 135 of file IncBaseCondition.h.

Referenced by ExtrapolateArray().

◆ StifflyStable_Gamma0_Coeffs

NekDouble Nektar::IncBaseCondition::StifflyStable_Gamma0_Coeffs

staticprotected

Initial value:

= {1.0, 1.5,

11.0 / 6.0}

Definition at line 137 of file IncBaseCondition.h.

Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ ~IncBaseCondition()

◆ IncBaseCondition()

Member Function Documentation

◆ AddRigidBodyAcc()

◆ AddVisPressureBCs()

◆ ExtrapolateArray()

◆ Initialise()

◆ InitialiseCoords()

◆ RigidBodyVelocity()

◆ RollOver()

◆ SetNumPointsOnPlane0()

◆ Update()

◆ v_Initialise()

◆ v_Update()

Member Data Documentation

◆ classname

◆ m_BndConds

◆ m_bnddim

◆ m_bndElmtExps

◆ m_BndExp

◆ m_coords

◆ m_field

◆ m_intSteps

◆ m_nbnd

◆ m_npoints

◆ m_numCalls

◆ m_pressure

◆ m_spacedim

◆ m_viscous

◆ StifflyStable_Alpha_Coeffs

◆ StifflyStable_Betaq_Coeffs

◆ StifflyStable_Gamma0_Coeffs