SymmTensorI.H

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#include "Tensor.H"
 
// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt>::SymmTensor(Foam::zero)
:
    SymmTensor::vsType(Foam::zero{})
{}
 
 
template<class Cmpt>
template<class Cmpt2>
inline Foam::SymmTensor<Cmpt>::SymmTensor
(
    const VectorSpace<SymmTensor<Cmpt2>, Cmpt2, 6>& vs
)
:
    SymmTensor::vsType(vs)
{}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt>::SymmTensor(const SphericalTensor<Cmpt>& st)
{
    this->v_[XX] = st.ii(); this->v_[XY] = Zero; this->v_[XZ] = Zero;
    this->v_[YY] = st.ii(); this->v_[YZ] = Zero;
    this->v_[ZZ] = st.ii();
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt>::SymmTensor
(
    const Vector<Cmpt>& x,
    const Vector<Cmpt>& y,
    const Vector<Cmpt>& z,
    const bool transposed  /* ignored */
)
{
    this->rows(x, y, z);
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt>::SymmTensor
(
    const Cmpt txx, const Cmpt txy, const Cmpt txz,
                    const Cmpt tyy, const Cmpt tyz,
                                    const Cmpt tzz
)
{
    this->v_[XX] = txx; this->v_[XY] = txy; this->v_[XZ] = txz;
                        this->v_[YY] = tyy; this->v_[YZ] = tyz;
                                            this->v_[ZZ] = tzz;
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt>::SymmTensor(Istream& is)
:
    SymmTensor::vsType(is)
{}
 
 
// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::x() const
{
    return Vector<Cmpt>(this->v_[XX], this->v_[XY], this->v_[XZ]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::y() const
{
    return Vector<Cmpt>(this->v_[XY], this->v_[YY], this->v_[YZ]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::z() const
{
    return Vector<Cmpt>(this->v_[XZ], this->v_[YZ], this->v_[ZZ]);
}
 
 
template<class Cmpt>
template<Foam::direction Idx>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::row() const
{
    if (Idx == 0) return x();
    else if (Idx == 1) return y();
    else if (Idx == 2) return z();
 
    static_assert(Idx < 3, "Invalid row access");
    return Zero;
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::row(const direction r) const
{
    switch (r)
    {
        case 0: return x(); break;
        case 1: return y(); break;
        case 2: return z(); break;
        default:
            FatalErrorInFunction
                << "Invalid row access " << r << abort(FatalError);
    }
 
    return Zero;
}
 
 
template<class Cmpt>
template<Foam::direction Idx>
inline void Foam::SymmTensor<Cmpt>::row(const Vector<Cmpt>& v)
{
    if (Idx == 0)
    {
        this->v_[XX] = v.x(); this->v_[XY] = v.y(); this->v_[XZ] = v.z();
    }
    else if (Idx == 1)
    {
        this->v_[XY] = v.x(); this->v_[YY] = v.y(); this->v_[YZ] = v.z();
    }
    else if (Idx == 2)
    {
        this->v_[XZ] = v.x(); this->v_[YZ] = v.y(); this->v_[ZZ] = v.z();
    }
 
    static_assert(Idx < 3, "Invalid row access");
}
 
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::rows
(
    const Vector<Cmpt>& x,
    const Vector<Cmpt>& y,
    const Vector<Cmpt>& z
)
{
    this->v_[XX] = x.x(); this->v_[XY] = x.y(); this->v_[XZ] = x.z();
    this->v_[YY] = y.y(); this->v_[YZ] = y.z();
    this->v_[ZZ] = z.z();
}
 
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::row
(
    const direction r,
    const Vector<Cmpt>& v
)
{
    switch (r)
    {
        case 0: row<0>(v); break;
        case 1: row<1>(v); break;
        case 2: row<2>(v); break;
        default:
            FatalErrorInFunction
                << "Invalid row access " << r << abort(FatalError);
    }
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::SymmTensor<Cmpt>::diag() const
{
    return Vector<Cmpt>(this->v_[XX], this->v_[YY], this->v_[ZZ]);
}
 
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::diag(const Vector<Cmpt>& v)
{
    this->v_[XX] = v.x(); this->v_[YY] = v.y(); this->v_[ZZ] = v.z();
}
 
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::addDiag(const Vector<Cmpt>& v)
{
    this->v_[XX] += v.x(); this->v_[YY] += v.y(); this->v_[ZZ] += v.z();
}
 
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::subtractDiag(const Vector<Cmpt>& v)
{
    this->v_[XX] -= v.x(); this->v_[YY] -= v.y(); this->v_[ZZ] -= v.z();
}
 
 
template<class Cmpt>
inline Foam::scalar Foam::SymmTensor<Cmpt>::diagSqr() const
{
    return
    (
        Foam::magSqr(this->xx())
      + Foam::magSqr(this->yy())
      + Foam::magSqr(this->zz())
    );
}
 
 
template<class Cmpt>
inline bool Foam::SymmTensor<Cmpt>::is_identity(const scalar tol) const
{
    return
    (
        Foam::mag(xx() - pTraits<Cmpt>::one) < tol
     && Foam::mag(yy() - pTraits<Cmpt>::one) < tol
     && Foam::mag(zz() - pTraits<Cmpt>::one) < tol
     && Foam::mag(xy()) < tol
     && Foam::mag(xz()) < tol
     && Foam::mag(yz()) < tol
    );
}
 
 
template<class Cmpt>
inline Cmpt Foam::SymmTensor<Cmpt>::det() const
{
    return
    (
        xx()*yy()*zz() + xy()*yz()*xz()
      + xz()*xy()*yz() - xx()*yz()*yz()
      - xy()*xy()*zz() - xz()*yy()*xz()
    );
}
 
 
template<class Cmpt>
inline Cmpt Foam::SymmTensor<Cmpt>::det2D(const direction excludeCmpt) const
{
    switch (excludeCmpt)
    {
        case 0:  // Eliminate x
        {
            return (yy()*zz() - yz()*yz());
        }
 
        case 1:  // Eliminate y
        {
            return (xx()*zz() - xz()*xz());
        }
    }
 
    // Fall-through: Eliminate z
    return (xx()*yy() - xy()*xy());
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::adjunct() const
{
    // symmetric: cof() == adjunct()
    return SymmTensor<Cmpt>
    (
        yy()*zz() - yz()*yz(), xz()*yz() - xy()*zz(), xy()*yz() - xz()*yy(),
                               xx()*zz() - xz()*xz(), xy()*xz() - xx()*yz(),
                                                      xx()*yy() - xy()*xy()
    );
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::cof() const
{
    // symmetric: cof() == adjunct()
    return this->adjunct();
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::adjunct2D
(
    const direction excludeCmpt
) const
{
    switch (excludeCmpt)
    {
        case 0:  // Eliminate x
        {
            return SymmTensor<Cmpt>
            (
                Zero, Zero,  Zero,
                      zz(), -yz(),
                             yy()
            );
        }
 
        case 1:  // Eliminate y
        {
            return SymmTensor<Cmpt>
            (
                zz(), Zero, -xz(),
                      Zero,  Zero,
                             xx()
            );
        }
    }
 
    // Fall-through: Eliminate z
    return SymmTensor<Cmpt>
    (
        yy(), -xy(), Zero,
               xx(), Zero,
                     Zero
    );
}
 
 
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::inv2D
(
    const direction excludeCmpt
) const
{
    const Cmpt detval = this->det2D(excludeCmpt);
 
    return this->adjunct2D(excludeCmpt)/detval;
}
 
 
// Invert without much error handling
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::inv() const
{
    const Cmpt detval = this->det();
 
    #ifdef FULLDEBUG
    if (mag(detval) < VSMALL)
    {
        FatalErrorInFunction
            << "Tensor not properly invertible, determinant:"
            << detval << " tensor:" << *this << nl
            << abort(FatalError);
    }
    #endif
 
    return this->adjunct()/detval;
}
 
 
// Invert with some error handling
template<class Cmpt>
inline Foam::SymmTensor<Cmpt> Foam::SymmTensor<Cmpt>::safeInv() const
{
    {
        // Attempt to identify and handle 2-D cases.
        // - use diagSqr instead of magSqr for fewer operations
        const scalar magSqr_xx = Foam::magSqr(xx());
        const scalar magSqr_yy = Foam::magSqr(yy());
        const scalar magSqr_zz = Foam::magSqr(zz());
 
        // SMALL: 1e-15 (double), 1e-6 (float), but 1e-6 may be adequate
 
        const scalar threshold = SMALL * (magSqr_xx + magSqr_yy + magSqr_zz);
 
        const bool small_xx = (magSqr_xx < threshold);
        const bool small_yy = (magSqr_yy < threshold);
        const bool small_zz = (magSqr_zz < threshold);
 
        if (small_xx || small_yy || small_zz)
        {
            SymmTensor<Cmpt> work(*this);
 
            if (small_xx) { work.xx() += pTraits<Cmpt>::one; }
            if (small_yy) { work.yy() += pTraits<Cmpt>::one; }
            if (small_zz) { work.zz() += pTraits<Cmpt>::one; }
 
            const Cmpt detval = work.det();
 
            if (mag(detval) < ROOTVSMALL)
            {
                // Appears to be nearly zero - leave untouched?
                return SymmTensor<Cmpt>(Zero);
            }
 
            work = work.adjunct()/detval;
 
            if (small_xx) { work.xx() -= pTraits<Cmpt>::one; }
            if (small_yy) { work.yy() -= pTraits<Cmpt>::one; }
            if (small_zz) { work.zz() -= pTraits<Cmpt>::one; }
 
            return work;
        }
    }
 
    const Cmpt detval = this->det();
 
    if (mag(detval) < ROOTVSMALL)
    {
        // Appears to be nearly zero - leave untouched?
        return SymmTensor<Cmpt>(Zero);
    }
 
    return this->adjunct()/detval;
}
 
 
// * * * * * * * * * * * * * * * Member Operators  * * * * * * * * * * * * * //
 
template<class Cmpt>
inline void Foam::SymmTensor<Cmpt>::operator=(const SphericalTensor<Cmpt>& st)
{
    this->v_[XX] = st.ii(); this->v_[XY] = Zero;    this->v_[XZ] = Zero;
                            this->v_[YY] = st.ii(); this->v_[YZ] = Zero;
                                                    this->v_[ZZ] = st.ii();
}
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
namespace Foam
{
 
// * * * * * * * * * * * * * * * Global Functions  * * * * * * * * * * * * * //
 
//- Return the trace of a SymmTensor
template<class Cmpt>
inline Cmpt tr(const SymmTensor<Cmpt>& st)
{
    return st.xx() + st.yy() + st.zz();
}
 
 
//- Return the spherical part of a SymmTensor
template<class Cmpt>
inline SphericalTensor<Cmpt> sph(const SymmTensor<Cmpt>& st)
{
    return SphericalTensor<Cmpt>
    (
        (1.0/3.0)*tr(st)
    );
}
 
 
//- Return the symmetric part of a SymmTensor, i.e. itself
template<class Cmpt>
inline const SymmTensor<Cmpt>& symm(const SymmTensor<Cmpt>& st)
{
    return st;
}
 
 
//- Return twice the symmetric part of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> twoSymm(const SymmTensor<Cmpt>& st)
{
    return 2*st;
}
 
 
//- Return the deviatoric part of the symmetric part of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> devSymm(const SymmTensor<Cmpt>& st)
{
    return dev(st);
}
 
 
//- Return the deviatoric part of twice the symmetric part of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> devTwoSymm(const SymmTensor<Cmpt>& st)
{
    return dev2(st);
}
 
 
//- Return the deviatoric part of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> dev(const SymmTensor<Cmpt>& st)
{
    return st - sph(st);
}
 
 
//- Return the two-third deviatoric part of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> dev2(const SymmTensor<Cmpt>& st)
{
    return st - 2*sph(st);
}
 
 
//- Return the determinant of a SymmTensor
template<class Cmpt>
inline Cmpt det(const SymmTensor<Cmpt>& st)
{
    return st.det();
}
 
 
//- Return the cofactor of a SymmTensor as a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> cof(const SymmTensor<Cmpt>& st)
{
    return st.cof();
}
 
 
//- Return the inverse of a SymmTensor, using given determinant value
template<class Cmpt>
inline SymmTensor<Cmpt> inv(const SymmTensor<Cmpt>& st, const Cmpt detval)
{
    #ifdef FULLDEBUG
    if (mag(detval) < VSMALL)
    {
        FatalErrorInFunction
            << "SymmTensor not properly invertible, determinant:"
            << detval << " tensor:" << st << nl
            << abort(FatalError);
    }
    #endif
 
    return st.adjunct()/detval;
}
 
 
//- Return the inverse of a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> inv(const SymmTensor<Cmpt>& st)
{
    return st.inv();
}
 
 
//- Return the 1st invariant of a SymmTensor
template<class Cmpt>
inline Cmpt invariantI(const SymmTensor<Cmpt>& st)
{
    return tr(st);
}
 
 
//- Return the 2nd invariant of a SymmTensor
template<class Cmpt>
inline Cmpt invariantII(const SymmTensor<Cmpt>& st)
{
    return
    (
        st.xx()*st.yy() + st.yy()*st.zz() + st.xx()*st.zz()
      - st.xy()*st.xy() - st.yz()*st.yz() - st.xz()*st.xz()
    );
}
 
 
//- Return the 3rd invariant of a SymmTensor
template<class Cmpt>
inline Cmpt invariantIII(const SymmTensor<Cmpt>& st)
{
    return det(st);
}
 
 
//- Return the inner-product of a SymmTensor with itself
template<class Cmpt>
inline SymmTensor<Cmpt>
innerSqr(const SymmTensor<Cmpt>& st)
{
    return SymmTensor<Cmpt>
    (
        st.xx()*st.xx() + st.xy()*st.xy() + st.xz()*st.xz(),
        st.xx()*st.xy() + st.xy()*st.yy() + st.xz()*st.yz(),
        st.xx()*st.xz() + st.xy()*st.yz() + st.xz()*st.zz(),
 
        st.xy()*st.xy() + st.yy()*st.yy() + st.yz()*st.yz(),
        st.xy()*st.xz() + st.yy()*st.yz() + st.yz()*st.zz(),
 
        st.xz()*st.xz() + st.yz()*st.yz() + st.zz()*st.zz()
    );
}
 
 
//- Return the square of Frobenius norm of a SymmTensor
template<class Cmpt>
inline Foam::scalar magSqr(const SymmTensor<Cmpt>& st)
{
    return
    (
        magSqr(st.xx()) + 2*magSqr(st.xy()) + 2*magSqr(st.xz())
                        +   magSqr(st.yy()) + 2*magSqr(st.yz())
                                            +   magSqr(st.zz())
    );
}
 
 
//- Return the square of a Vector as a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt> sqr(const Vector<Cmpt>& v)
{
    return SymmTensor<Cmpt>
    (
        v.x()*v.x(), v.x()*v.y(), v.x()*v.z(),
                     v.y()*v.y(), v.y()*v.z(),
                                  v.z()*v.z()
    );
}
 
 
//- Linear interpolation of symmetric tensors a and b by factor t
template<class Cmpt>
inline SymmTensor<Cmpt> lerp
(
    const SymmTensor<Cmpt>& a,
    const SymmTensor<Cmpt>& b,
    const scalar t
)
{
    const scalar onet = (1-t);
 
    return SymmTensor<Cmpt>
    (
        onet*a.xx() + t*b.xx(),
        onet*a.xy() + t*b.xy(),
        onet*a.xz() + t*b.xz(),
        onet*a.yy() + t*b.yy(),
        onet*a.yz() + t*b.yz(),
        onet*a.zz() + t*b.zz()
    );
}
 
 
// * * * * * * * * * * * * * * * Global Operators  * * * * * * * * * * * * * //
 
//- Sum of a SphericalTensor and a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator+(const SphericalTensor<Cmpt>& spt1, const SymmTensor<Cmpt>& st2)
{
    return SymmTensor<Cmpt>
    (
        spt1.ii() + st2.xx(), st2.xy(),             st2.xz(),
                              spt1.ii() + st2.yy(), st2.yz(),
                                                    spt1.ii() + st2.zz()
    );
}
 
 
//- Sum of a SymmTensor and a SphericalTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator+(const SymmTensor<Cmpt>& st1, const SphericalTensor<Cmpt>& spt2)
{
    return SymmTensor<Cmpt>
    (
        st1.xx() + spt2.ii(), st1.xy(),             st1.xz(),
                              st1.yy() + spt2.ii(), st1.yz(),
                                                    st1.zz() + spt2.ii()
    );
}
 
 
//- Subtract a SymmTensor from a SphericalTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator-(const SphericalTensor<Cmpt>& spt1, const SymmTensor<Cmpt>& st2)
{
    return SymmTensor<Cmpt>
    (
        spt1.ii() - st2.xx(), -st2.xy(),             -st2.xz(),
                               spt1.ii() - st2.yy(), -st2.yz(),
                                                      spt1.ii() - st2.zz()
    );
}
 
 
//- Subtract a SphericalTensor from a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator-(const SymmTensor<Cmpt>& st1, const SphericalTensor<Cmpt>& spt2)
{
    return SymmTensor<Cmpt>
    (
        st1.xx() - spt2.ii(), st1.xy(),             st1.xz(),
                              st1.yy() - spt2.ii(), st1.yz(),
                                                    st1.zz() - spt2.ii()
    );
}
 
 
//- Return the Hodge dual of a SymmTensor as a Vector
template<class Cmpt>
inline Vector<Cmpt> operator*(const SymmTensor<Cmpt>& st)
{
    return Vector<Cmpt>(st.yz(), -st.xz(), st.xy());
}
 
 
//- Division of a SymmTensor by a Cmpt
template<class Cmpt>
inline SymmTensor<Cmpt>
operator/(const SymmTensor<Cmpt>& st, const Cmpt s)
{
    return SymmTensor<Cmpt>
    (
        st.xx()/s, st.xy()/s, st.xz()/s,
                   st.yy()/s, st.yz()/s,
                              st.zz()/s
    );
}
 
 
//- Inner-product of a SymmTensor and a SymmTensor
template<class Cmpt>
inline Tensor<Cmpt>
operator&(const SymmTensor<Cmpt>& st1, const SymmTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        st1.xx()*st2.xx() + st1.xy()*st2.xy() + st1.xz()*st2.xz(),
        st1.xx()*st2.xy() + st1.xy()*st2.yy() + st1.xz()*st2.yz(),
        st1.xx()*st2.xz() + st1.xy()*st2.yz() + st1.xz()*st2.zz(),
 
        st1.xy()*st2.xx() + st1.yy()*st2.xy() + st1.yz()*st2.xz(),
        st1.xy()*st2.xy() + st1.yy()*st2.yy() + st1.yz()*st2.yz(),
        st1.xy()*st2.xz() + st1.yy()*st2.yz() + st1.yz()*st2.zz(),
 
        st1.xz()*st2.xx() + st1.yz()*st2.xy() + st1.zz()*st2.xz(),
        st1.xz()*st2.xy() + st1.yz()*st2.yy() + st1.zz()*st2.yz(),
        st1.xz()*st2.xz() + st1.yz()*st2.yz() + st1.zz()*st2.zz()
    );
}
 
 
//- Inner-product of a SphericalTensor and a SymmTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator&(const SphericalTensor<Cmpt>& spt1, const SymmTensor<Cmpt>& st2)
{
    return SymmTensor<Cmpt>
    (
        spt1.ii()*st2.xx(), spt1.ii()*st2.xy(), spt1.ii()*st2.xz(),
                            spt1.ii()*st2.yy(), spt1.ii()*st2.yz(),
                                                spt1.ii()*st2.zz()
    );
}
 
 
//- Inner-product of a SymmTensor and a SphericalTensor
template<class Cmpt>
inline SymmTensor<Cmpt>
operator&(const SymmTensor<Cmpt>& st1, const SphericalTensor<Cmpt>& spt2)
{
    return SymmTensor<Cmpt>
    (
        st1.xx()*spt2.ii(), st1.xy()*spt2.ii(), st1.xz()*spt2.ii(),
                            st1.yy()*spt2.ii(), st1.yz()*spt2.ii(),
                                                st1.zz()*spt2.ii()
    );
}
 
 
//- Inner-product of a SymmTensor and a Vector
template<class Cmpt>
inline Vector<Cmpt>
operator&(const SymmTensor<Cmpt>& st, const Vector<Cmpt>& v)
{
    return Vector<Cmpt>
    (
        st.xx()*v.x() + st.xy()*v.y() + st.xz()*v.z(),
        st.xy()*v.x() + st.yy()*v.y() + st.yz()*v.z(),
        st.xz()*v.x() + st.yz()*v.y() + st.zz()*v.z()
    );
}
 
 
//- Inner-product of a Vector and a SymmTensor
template<class Cmpt>
inline Vector<Cmpt>
operator&(const Vector<Cmpt>& v, const SymmTensor<Cmpt>& st)
{
    return Vector<Cmpt>
    (
        v.x()*st.xx() + v.y()*st.xy() + v.z()*st.xz(),
        v.x()*st.xy() + v.y()*st.yy() + v.z()*st.yz(),
        v.x()*st.xz() + v.y()*st.yz() + v.z()*st.zz()
    );
}
 
 
//- Double-inner-product of a SymmTensor and a SymmTensor
template<class Cmpt>
inline Cmpt
operator&&(const SymmTensor<Cmpt>& st1, const SymmTensor<Cmpt>& st2)
{
    return
    (
        st1.xx()*st2.xx() + 2*st1.xy()*st2.xy() + 2*st1.xz()*st2.xz()
                          +   st1.yy()*st2.yy() + 2*st1.yz()*st2.yz()
                                                +   st1.zz()*st2.zz()
    );
}
 
 
//- Double-inner-product of a SphericalTensor and a SymmTensor
template<class Cmpt>
inline Cmpt
operator&&(const SphericalTensor<Cmpt>& spt1, const SymmTensor<Cmpt>& st2)
{
    return (spt1.ii()*st2.xx() + spt1.ii()*st2.yy() + spt1.ii()*st2.zz());
}
 
 
//- Double-inner-product of a SymmTensor and a SphericalTensor
template<class Cmpt>
inline Cmpt
operator&&(const SymmTensor<Cmpt>& st1, const SphericalTensor<Cmpt>& spt2)
{
    return (st1.xx()*spt2.ii() + st1.yy()*spt2.ii() + st1.zz()*spt2.ii());
}
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
template<class Cmpt>
class outerProduct<SymmTensor<Cmpt>, Cmpt>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
template<class Cmpt>
class outerProduct<Cmpt, SymmTensor<Cmpt>>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
template<class Cmpt>
class innerProduct<SymmTensor<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
template<class Cmpt>
class innerProduct<SymmTensor<Cmpt>, Vector<Cmpt>>
{
public:
 
    typedef Vector<Cmpt> type;
};
 
template<class Cmpt>
class innerProduct<Vector<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef Vector<Cmpt> type;
};
 
 
template<class Cmpt>
class typeOfSum<SphericalTensor<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
template<class Cmpt>
class typeOfSum<SymmTensor<Cmpt>, SphericalTensor<Cmpt>>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
template<class Cmpt>
class innerProduct<SphericalTensor<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
template<class Cmpt>
class innerProduct<SymmTensor<Cmpt>, SphericalTensor<Cmpt>>
{
public:
 
    typedef SymmTensor<Cmpt> type;
};
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
} // End namespace Foam
 
// ************************************************************************* //