TensorI.H

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-------------------------------------------------------------------------------
    Copyright (C) 2011-2016 OpenFOAM Foundation
    Copyright (C) 2016-2023 OpenCFD Ltd.
-------------------------------------------------------------------------------
License
    This file is part of OpenFOAM.
 
    OpenFOAM is free software: you can redistribute it and/or modify it
    under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.
 
    OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
    ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
    for more details.
 
    You should have received a copy of the GNU General Public License
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#include <type_traits>
 
#include "SymmTensor.H"
 
// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor(Foam::zero)
:
    Tensor::msType(Foam::zero{})
{}
 
 
template<class Cmpt>
template<class Cmpt2>
inline Foam::Tensor<Cmpt>::Tensor
(
    const MatrixSpace<Tensor<Cmpt2>, Cmpt2, 3, 3>& vs
)
:
    Tensor::msType(vs)
{}
 
 
template<class Cmpt>
template<class Cmpt2>
inline Foam::Tensor<Cmpt>::Tensor
(
    const VectorSpace<Tensor<Cmpt2>, Cmpt2, 9>& vs
)
:
    Tensor::msType(vs)
{}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor(const SphericalTensor<Cmpt>& st)
{
    this->v_[XX] = st.ii();    this->v_[XY] = Zero;      this->v_[XZ] = Zero;
    this->v_[YX] = Zero;       this->v_[YY] = st.ii();   this->v_[YZ] = Zero;
    this->v_[ZX] = Zero;       this->v_[ZY] = Zero;      this->v_[ZZ] = st.ii();
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor(const SymmTensor<Cmpt>& st)
{
    this->v_[XX] = st.xx(); this->v_[XY] = st.xy(); this->v_[XZ] = st.xz();
    this->v_[YX] = st.xy(); this->v_[YY] = st.yy(); this->v_[YZ] = st.yz();
    this->v_[ZX] = st.xz(); this->v_[ZY] = st.yz(); this->v_[ZZ] = st.zz();
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor
(
    const Vector<Vector<Cmpt>>& vecs,
    const bool transposed
)
:
    Tensor<Cmpt>(vecs.x(), vecs.y(), vecs.z(), transposed)
{}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor
(
    const Vector<Cmpt>& x,
    const Vector<Cmpt>& y,
    const Vector<Cmpt>& z,
    const bool transposed
)
{
    if (transposed)
    {
        this->cols(x, y, z);
    }
    else
    {
        this->rows(x, y, z);
    }
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor
(
    const Cmpt txx, const Cmpt txy, const Cmpt txz,
    const Cmpt tyx, const Cmpt tyy, const Cmpt tyz,
    const Cmpt tzx, const Cmpt tzy, const Cmpt tzz
)
{
    this->v_[XX] = txx; this->v_[XY] = txy; this->v_[XZ] = txz;
    this->v_[YX] = tyx; this->v_[YY] = tyy; this->v_[YZ] = tyz;
    this->v_[ZX] = tzx; this->v_[ZY] = tzy; this->v_[ZZ] = tzz;
}
 
 
template<class Cmpt>
template
<
    template<class, Foam::direction, Foam::direction> class Block2,
    Foam::direction BRowStart,
    Foam::direction BColStart
>
inline Foam::Tensor<Cmpt>::Tensor
(
    const Block2<Tensor<Cmpt>, BRowStart, BColStart>& block
)
:
    Tensor::msType(block)
{}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt>::Tensor(Istream& is)
:
    Tensor::msType(is)
{}
 
 
// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::x() const
{
    return Vector<Cmpt>(this->v_[XX], this->v_[XY], this->v_[XZ]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::y() const
{
    return Vector<Cmpt>(this->v_[YX], this->v_[YY], this->v_[YZ]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::z() const
{
    return Vector<Cmpt>(this->v_[ZX], this->v_[ZY], this->v_[ZZ]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::cx() const
{
    return Vector<Cmpt>(this->v_[XX], this->v_[YX], this->v_[ZX]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::cy() const
{
    return Vector<Cmpt>(this->v_[XY], this->v_[YY], this->v_[ZY]);
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::cz() 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::Tensor<Cmpt>::col() const
{
    if (Idx == 0) return cx();
    else if (Idx == 1) return cy();
    else if (Idx == 2) return cz();
 
    static_assert(Idx < 3, "Invalid column access");
    return Zero;
}
 
 
template<class Cmpt>
inline Foam::Vector<Cmpt> Foam::Tensor<Cmpt>::col(const direction c) const
{
    switch (c)
    {
        case 0: return cx(); break;
        case 1: return cy(); break;
        case 2: return cz(); break;
        default:
            FatalErrorInFunction
                << "Invalid column access " << c << abort(FatalError);
    }
 
    return Zero;
}
 
 
template<class Cmpt>
template<Foam::direction Idx>
inline Foam::Vector<Cmpt> Foam::Tensor<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::Tensor<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::Tensor<Cmpt>::col(const Vector<Cmpt>& v)
{
    if (Idx == 0)
    {
        this->v_[XX] = v.x();
        this->v_[YX] = v.y();
        this->v_[ZX] = v.z();
    }
    else if (Idx == 1)
    {
        this->v_[XY] = v.x();
        this->v_[YY] = v.y();
        this->v_[ZY] = 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 column access");
}
 
 
template<class Cmpt>
template<Foam::direction Idx>
inline void Foam::Tensor<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_[YX] = v.x(); this->v_[YY] = v.y(); this->v_[YZ] = v.z();
    }
    else if (Idx == 2)
    {
        this->v_[ZX] = v.x(); this->v_[ZY] = v.y(); this->v_[ZZ] = v.z();
    }
 
    static_assert(Idx < 3, "Invalid row access");
}
 
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::cols
(
    const Vector<Cmpt>& x,
    const Vector<Cmpt>& y,
    const Vector<Cmpt>& z
)
{
    this->v_[XX] = x.x(); this->v_[XY] = y.x(); this->v_[XZ] = z.x();
    this->v_[YX] = x.y(); this->v_[YY] = y.y(); this->v_[YZ] = z.y();
    this->v_[ZX] = x.z(); this->v_[ZY] = y.z(); this->v_[ZZ] = z.z();
}
 
 
template<class Cmpt>
inline void Foam::Tensor<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_[YX] = y.x(); this->v_[YY] = y.y(); this->v_[YZ] = y.z();
    this->v_[ZX] = z.x(); this->v_[ZY] = z.y(); this->v_[ZZ] = z.z();
}
 
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::col
(
    const direction c,
    const Vector<Cmpt>& v
)
{
    switch (c)
    {
        case 0: col<0>(v); break;
        case 1: col<1>(v); break;
        case 2: col<2>(v); break;
        default:
            FatalErrorInFunction
                << "Invalid column access " << c << abort(FatalError);
    }
}
 
 
template<class Cmpt>
inline void Foam::Tensor<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::Tensor<Cmpt>::diag() const
{
    return Vector<Cmpt>(this->v_[XX], this->v_[YY], this->v_[ZZ]);
}
 
 
template<class Cmpt>
inline void Foam::Tensor<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::Tensor<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::Tensor<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::Tensor<Cmpt>::diagSqr() const
{
    return
    (
        Foam::magSqr(this->xx())
      + Foam::magSqr(this->yy())
      + Foam::magSqr(this->zz())
    );
}
 
 
template<class Cmpt>
inline bool Foam::Tensor<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(yx()) < tol && Foam::mag(yz()) < tol
     && Foam::mag(zx()) < tol && Foam::mag(zy()) < tol
    );
}
 
 
// * * * * * * * * * * * * * * * Member Operations * * * * * * * * * * * * * //
 
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<Cmpt>::T() const
{
    return Tensor<Cmpt>
    (
        xx(), yx(), zx(),
        xy(), yy(), zy(),
        xz(), yz(), zz()
    );
}
 
 
template<class Cmpt>
inline Cmpt Foam::Tensor<Cmpt>::det() const
{
    return
    (
        xx()*yy()*zz() + xy()*yz()*zx()
      + xz()*yx()*zy() - xx()*yz()*zy()
      - xy()*yx()*zz() - xz()*yy()*zx()
    );
}
 
 
template<class Cmpt>
inline Cmpt Foam::Tensor<Cmpt>::det2D(const direction excludeCmpt) const
{
    switch (excludeCmpt)
    {
        case 0:  // Eliminate x
        {
            return (yy()*zz() - yz()*zy());
        }
 
        case 1:  // Eliminate y
        {
            return (xx()*zz() - xz()*zx());
        }
    }
 
    // Fall-through: Eliminate z
    return (xx()*yy() - xy()*yx());
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<Cmpt>::adjunct() const
{
    return Tensor<Cmpt>
    (
        yy()*zz() - zy()*yz(), xz()*zy() - xy()*zz(), xy()*yz() - xz()*yy(),
        zx()*yz() - yx()*zz(), xx()*zz() - xz()*zx(), yx()*xz() - xx()*yz(),
        yx()*zy() - yy()*zx(), xy()*zx() - xx()*zy(), xx()*yy() - yx()*xy()
    );
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<Cmpt>::cof() const
{
    return this->adjunct().T();
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<Cmpt>::adjunct2D
(
    const direction excludeCmpt
) const
{
    switch (excludeCmpt)
    {
        case 0:  // Eliminate x
        {
            return Tensor<Cmpt>
            (
                Zero,  Zero,  Zero,
                Zero,  zz(), -yz(),
                Zero, -zy(),  yy()
            );
        }
 
        case 1:  // Eliminate y
        {
            return Tensor<Cmpt>
            (
                zz(), Zero, -xz(),
                Zero, Zero,  Zero,
               -zx(), Zero,  xx()
            );
        }
    }
 
    // Fall-through: Eliminate z
    return Tensor<Cmpt>
    (
        yy(), -xy(), Zero,
       -yx(),  xx(), Zero,
        Zero,  Zero, Zero
    );
}
 
 
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<Cmpt>::inv2D
(
    const direction excludeCmpt
) const
{
    const Cmpt detval = this->det2D(excludeCmpt);
 
    return this->adjunct2D(excludeCmpt)/detval;
}
 
 
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Foam::Tensor<Cmpt>
Foam::Tensor<Cmpt>::inner(const Tensor<Cmpt>& t2) const
{
    const Tensor<Cmpt>& t1 = *this;
 
    return Tensor<Cmpt>
    (
        t1.xx()*t2.xx() + t1.xy()*t2.yx() + t1.xz()*t2.zx(),
        t1.xx()*t2.xy() + t1.xy()*t2.yy() + t1.xz()*t2.zy(),
        t1.xx()*t2.xz() + t1.xy()*t2.yz() + t1.xz()*t2.zz(),
 
        t1.yx()*t2.xx() + t1.yy()*t2.yx() + t1.yz()*t2.zx(),
        t1.yx()*t2.xy() + t1.yy()*t2.yy() + t1.yz()*t2.zy(),
        t1.yx()*t2.xz() + t1.yy()*t2.yz() + t1.yz()*t2.zz(),
 
        t1.zx()*t2.xx() + t1.zy()*t2.yx() + t1.zz()*t2.zx(),
        t1.zx()*t2.xy() + t1.zy()*t2.yy() + t1.zz()*t2.zy(),
        t1.zx()*t2.xz() + t1.zy()*t2.yz() + t1.zz()*t2.zz()
    );
}
 
 
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Foam::Tensor<Cmpt>
Foam::Tensor<Cmpt>::schur(const Tensor<Cmpt>& t2) const
{
    const Tensor<Cmpt>& t1 = *this;
 
    return Tensor<Cmpt>
    (
        t1.xx()*t2.xx(), t1.xy()*t2.xy(), t1.xz()*t2.xz(),
        t1.yx()*t2.yx(), t1.yy()*t2.yy(), t1.yz()*t2.yz(),
        t1.zx()*t2.zx(), t1.zy()*t2.zy(), t1.zz()*t2.zz()
    );
}
 
 
// Invert without much error handling
template<class Cmpt>
inline Foam::Tensor<Cmpt> Foam::Tensor<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::Tensor<Cmpt> Foam::Tensor<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)
        {
            Tensor<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 Tensor<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 Tensor<Cmpt>(Zero);
    }
 
    return this->adjunct()/detval;
}
 
 
// * * * * * * * * * * * * * * * Member Operators  * * * * * * * * * * * * * //
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::operator&=(const Tensor<Cmpt>& t)
{
    *this = this->inner(t);
}
 
 
template<class Cmpt>
template<class Cmpt2>
inline void Foam::Tensor<Cmpt>::operator=
(
    const VectorSpace<Tensor<Cmpt2>, Cmpt2, 9>& vs
)
{
    VectorSpace<Tensor<Cmpt>, Cmpt, 9>::operator=(vs);
}
 
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::operator=(const SphericalTensor<Cmpt>& st)
{
    this->v_[XX] = st.ii();  this->v_[XY] = Zero;     this->v_[XZ] = Zero;
    this->v_[YX] = Zero;     this->v_[YY] = st.ii();  this->v_[YZ] = Zero;
    this->v_[ZX] = Zero;     this->v_[ZY] = Zero;     this->v_[ZZ] = st.ii();
}
 
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::operator=(const SymmTensor<Cmpt>& st)
{
    this->v_[XX] = st.xx(); this->v_[XY] = st.xy(); this->v_[XZ] = st.xz();
    this->v_[YX] = st.xy(); this->v_[YY] = st.yy(); this->v_[YZ] = st.yz();
    this->v_[ZX] = st.xz(); this->v_[ZY] = st.yz(); this->v_[ZZ] = st.zz();
}
 
 
template<class Cmpt>
inline void Foam::Tensor<Cmpt>::operator=(const Vector<Vector<Cmpt>>& tr)
{
    this->v_[XX] = tr.x().x();
    this->v_[XY] = tr.x().y();
    this->v_[XZ] = tr.x().z();
 
    this->v_[YX] = tr.y().x();
    this->v_[YY] = tr.y().y();
    this->v_[YZ] = tr.y().z();
 
    this->v_[ZX] = tr.z().x();
    this->v_[ZY] = tr.z().y();
    this->v_[ZZ] = tr.z().z();
}
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
namespace Foam
{
 
// * * * * * * * * * * * * * * * Global Functions  * * * * * * * * * * * * * //
 
//- Return the trace of a Tensor
template<class Cmpt>
inline Cmpt tr(const Tensor<Cmpt>& t)
{
    return t.xx() + t.yy() + t.zz();
}
 
 
//- Return the spherical part of a Tensor
template<class Cmpt>
inline SphericalTensor<Cmpt> sph(const Tensor<Cmpt>& t)
{
    return SphericalTensor<Cmpt>
    (
        (1.0/3.0)*tr(t)
    );
}
 
 
//- Return the symmetric part of a Tensor
template<class Cmpt>
inline SymmTensor<Cmpt> symm(const Tensor<Cmpt>& t)
{
    return SymmTensor<Cmpt>
    (
        t.xx(), 0.5*(t.xy() + t.yx()), 0.5*(t.xz() + t.zx()),
                t.yy(),                0.5*(t.yz() + t.zy()),
                                       t.zz()
    );
}
 
 
//- Return twice the symmetric part of a Tensor
template<class Cmpt>
inline SymmTensor<Cmpt> twoSymm(const Tensor<Cmpt>& t)
{
    return SymmTensor<Cmpt>
    (
        2*t.xx(), (t.xy() + t.yx()), (t.xz() + t.zx()),
                  2*t.yy(),          (t.yz() + t.zy()),
                                     2*t.zz()
    );
}
 
 
//- Return the deviatoric part of the symmetric part of a Tensor
template<class Cmpt>
inline SymmTensor<Cmpt> devSymm(const Tensor<Cmpt>& t)
{
    const Cmpt sph(tr(t)/3.0);
 
    return SymmTensor<Cmpt>
    (
        t.xx() - sph, 0.5*(t.xy() + t.yx()), 0.5*(t.xz() + t.zx()),
                      t.yy() - sph,          0.5*(t.yz() + t.zy()),
                                             t.zz() - sph
    );
}
 
 
//- Return the deviatoric part of twice the symmetric part of a Tensor
template<class Cmpt>
inline SymmTensor<Cmpt> devTwoSymm(const Tensor<Cmpt>& t)
{
    const Cmpt sph((2.0/3.0)*tr(t));
 
    return SymmTensor<Cmpt>
    (
        2*t.xx() - sph, (t.xy() + t.yx()), (t.xz() + t.zx()),
                        2*t.yy() - sph,    (t.yz() + t.zy()),
                                           2*t.zz() - sph
    );
}
 
 
//- Return the skew-symmetric part of a Tensor
template<class Cmpt>
inline Tensor<Cmpt> skew(const Tensor<Cmpt>& t)
{
    return Tensor<Cmpt>
    (
        Zero,                  0.5*(t.xy() - t.yx()), 0.5*(t.xz() - t.zx()),
        0.5*(t.yx() - t.xy()), Zero,                  0.5*(t.yz() - t.zy()),
        0.5*(t.zx() - t.xz()), 0.5*(t.zy() - t.yz()), Zero
    );
}
 
 
//- Return the skew-symmetric part of a SymmTensor as a Tensor
template<class Cmpt>
inline const Tensor<Cmpt>& skew(const SymmTensor<Cmpt>& st)
{
    return Tensor<Cmpt>::zero;
}
 
 
//- Return the deviatoric part of a Tensor
template<class Cmpt>
inline Tensor<Cmpt> dev(const Tensor<Cmpt>& t)
{
    return t - sph(t);
}
 
 
//- Return the two-third deviatoric part of a Tensor
template<class Cmpt>
inline Tensor<Cmpt> dev2(const Tensor<Cmpt>& t)
{
    return t - 2*sph(t);
}
 
 
//- Return the determinant of a Tensor
template<class Cmpt>
inline Cmpt det(const Tensor<Cmpt>& t)
{
    return t.det();
}
 
 
//- Return the cofactor Tensor of a Tensor
template<class Cmpt>
inline Tensor<Cmpt> cof(const Tensor<Cmpt>& t)
{
    return t.cof();
}
 
 
//- Return the inverse of a Tensor, using the given determinant value
template<class Cmpt>
inline Tensor<Cmpt> inv(const Tensor<Cmpt>& t, const Cmpt detval)
{
    #ifdef FULLDEBUG
    if (mag(detval) < VSMALL)
    {
        FatalErrorInFunction
            << "Tensor not properly invertible, determinant:"
            << detval << " tensor:" << t << nl
            << abort(FatalError);
    }
    #endif
 
    return t.adjunct()/detval;
}
 
 
//- Return the inverse of a Tensor
template<class Cmpt>
inline Tensor<Cmpt> inv(const Tensor<Cmpt>& t)
{
    return t.inv();
}
 
 
//- Return the 1st invariant of a Tensor
template<class Cmpt>
inline Cmpt invariantI(const Tensor<Cmpt>& t)
{
    return tr(t);
}
 
 
//- Return the 2nd invariant of a Tensor
template<class Cmpt>
inline Cmpt invariantII(const Tensor<Cmpt>& t)
{
    return
    (
        t.xx()*t.yy() + t.yy()*t.zz() + t.xx()*t.zz()
      - t.xy()*t.yx() - t.yz()*t.zy() - t.xz()*t.zx()
    );
}
 
 
//- Return the 3rd invariant of a Tensor
template<class Cmpt>
inline Cmpt invariantIII(const Tensor<Cmpt>& t)
{
    return det(t);
}
 
 
//- Linear interpolation of tensors a and b by factor t
template<class Cmpt>
inline Tensor<Cmpt> lerp
(
    const Tensor<Cmpt>& a,
    const Tensor<Cmpt>& b,
    const scalar t
)
{
    const scalar onet = (1-t);
 
    return Tensor<Cmpt>
    (
        onet*a.xx() + t*b.xx(),
        onet*a.xy() + t*b.xy(),
        onet*a.xz() + t*b.xz(),
        onet*a.yx() + t*b.yx(),
        onet*a.yy() + t*b.yy(),
        onet*a.yz() + t*b.yz(),
        onet*a.zx() + t*b.zx(),
        onet*a.zy() + t*b.zy(),
        onet*a.zz() + t*b.zz()
    );
}
 
 
// * * * * * * * * * * * * * * * Global Operators  * * * * * * * * * * * * * //
 
//- Sum of a SphericalTensor and a Tensor
template<class Cmpt>
inline Tensor<Cmpt>
operator+(const SphericalTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.ii() + t2.xx(), t2.xy(),            t2.xz(),
        t2.yx(),            st1.ii() + t2.yy(), t2.yz(),
        t2.zx(),            t2.zy(),            st1.ii() + t2.zz()
    );
}
 
 
//- Sum of a Tensor and a SphericalTensor
template<class Cmpt>
inline Tensor<Cmpt>
operator+(const Tensor<Cmpt>& t1, const SphericalTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx() + st2.ii(), t1.xy(),            t1.xz(),
        t1.yx(),            t1.yy() + st2.ii(), t1.yz(),
        t1.zx(),            t1.zy(),            t1.zz() + st2.ii()
    );
}
 
 
//- Sum of a SymmTensor and a Tensor
template<class Cmpt>
inline Tensor<Cmpt>
operator+(const SymmTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.xx() + t2.xx(), st1.xy() + t2.xy(), st1.xz() + t2.xz(),
        st1.xy() + t2.yx(), st1.yy() + t2.yy(), st1.yz() + t2.yz(),
        st1.xz() + t2.zx(), st1.yz() + t2.zy(), st1.zz() + t2.zz()
    );
}
 
 
//- Sum of a Tensor and a SymmTensor
template<class Cmpt>
inline Tensor<Cmpt>
operator+(const Tensor<Cmpt>& t1, const SymmTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx() + st2.xx(), t1.xy() + st2.xy(), t1.xz() + st2.xz(),
        t1.yx() + st2.xy(), t1.yy() + st2.yy(), t1.yz() + st2.yz(),
        t1.zx() + st2.xz(), t1.zy() + st2.yz(), t1.zz() + st2.zz()
    );
}
 
 
//- Subtract a Tensor from a SphericalTensor
template<class Cmpt>
inline Tensor<Cmpt>
operator-(const SphericalTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.ii() - t2.xx(), -t2.xy(),            -t2.xz(),
       -t2.yx(),             st1.ii() - t2.yy(), -t2.yz(),
       -t2.zx(),            -t2.zy(),             st1.ii() - t2.zz()
    );
}
 
 
//- Subtract a SphericalTensor from a Tensor
template<class Cmpt>
inline Tensor<Cmpt>
operator-(const Tensor<Cmpt>& t1, const SphericalTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx() - st2.ii(), t1.xy(),            t1.xz(),
        t1.yx(),            t1.yy() - st2.ii(), t1.yz(),
        t1.zx(),            t1.zy(),            t1.zz() - st2.ii()
    );
}
 
 
//- Subtract a Tensor from a SymmTensor
template<class Cmpt>
inline Tensor<Cmpt>
operator-(const SymmTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.xx() - t2.xx(), st1.xy() - t2.xy(), st1.xz() - t2.xz(),
        st1.xy() - t2.yx(), st1.yy() - t2.yy(), st1.yz() - t2.yz(),
        st1.xz() - t2.zx(), st1.yz() - t2.zy(), st1.zz() - t2.zz()
    );
}
 
 
//- Subtract a SymmTensor from a Tensor
template<class Cmpt>
inline Tensor<Cmpt>
operator-(const Tensor<Cmpt>& t1, const SymmTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx() - st2.xx(), t1.xy() - st2.xy(), t1.xz() - st2.xz(),
        t1.yx() - st2.xy(), t1.yy() - st2.yy(), t1.yz() - st2.yz(),
        t1.zx() - st2.xz(), t1.zy() - st2.yz(), t1.zz() - st2.zz()
    );
}
 
 
//- Return the Hodge dual of a Tensor as a Vector
template<class Cmpt>
inline Vector<Cmpt> operator*(const Tensor<Cmpt>& t)
{
    return Vector<Cmpt>(t.yz(), -t.xz(), t.xy());
}
 
 
//- Return the Hodge dual of a Vector as a Tensor
template<class Cmpt>
inline Tensor<Cmpt> operator*(const Vector<Cmpt>& v)
{
    return Tensor<Cmpt>
    (
         Zero,  -v.z(),  v.y(),
         v.z(),  Zero,  -v.x(),
        -v.y(),  v.x(),  Zero
    );
}
 
 
//- Division of a Vector by a Tensor
template<class Cmpt>
inline typename innerProduct<Vector<Cmpt>, Tensor<Cmpt>>::type
operator/(const Vector<Cmpt>& v, const Tensor<Cmpt>& t)
{
    return inv(t) & v;
}
 
 
//- Division of a Tensor by a Cmpt
template<class Cmpt>
inline Tensor<Cmpt>
operator/(const Tensor<Cmpt>& t, const Cmpt s)
{
    #ifdef FULLDEBUG
    if (mag(s) < VSMALL)
    {
        FatalErrorInFunction
            << "Tensor = " << t
            << " is not divisible due to a zero value in Cmpt:"
            << "Cmpt = " << s
            << abort(FatalError);
    }
    #endif
 
    return Tensor<Cmpt>
    (
        t.xx()/s, t.xy()/s, t.xz()/s,
        t.yx()/s, t.yy()/s, t.yz()/s,
        t.zx()/s, t.zy()/s, t.zz()/s
    );
}
 
 
//- Inner-product of a Tensor and a Tensor
template<class Cmpt>
inline typename innerProduct<Tensor<Cmpt>, Tensor<Cmpt>>::type
operator&(const Tensor<Cmpt>& t1, const Tensor<Cmpt>& t2)
{
    return t1.inner(t2);
}
 
 
//- Inner-product of a SphericalTensor and a Tensor
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Tensor<Cmpt>
operator&(const SphericalTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.ii()*t2.xx(), st1.ii()*t2.xy(), st1.ii()*t2.xz(),
        st1.ii()*t2.yx(), st1.ii()*t2.yy(), st1.ii()*t2.yz(),
        st1.ii()*t2.zx(), st1.ii()*t2.zy(), st1.ii()*t2.zz()
    );
}
 
 
//- Inner-product of a Tensor and a SphericalTensor
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Tensor<Cmpt>
operator&(const Tensor<Cmpt>& t1, const SphericalTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx()*st2.ii(), t1.xy()*st2.ii(), t1.xz()*st2.ii(),
        t1.yx()*st2.ii(), t1.yy()*st2.ii(), t1.yz()*st2.ii(),
        t1.zx()*st2.ii(), t1.zy()*st2.ii(), t1.zz()*st2.ii()
    );
}
 
 
//- Inner-product of a SymmTensor and a Tensor
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Tensor<Cmpt>
operator&(const SymmTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return Tensor<Cmpt>
    (
        st1.xx()*t2.xx() + st1.xy()*t2.yx() + st1.xz()*t2.zx(),
        st1.xx()*t2.xy() + st1.xy()*t2.yy() + st1.xz()*t2.zy(),
        st1.xx()*t2.xz() + st1.xy()*t2.yz() + st1.xz()*t2.zz(),
 
        st1.xy()*t2.xx() + st1.yy()*t2.yx() + st1.yz()*t2.zx(),
        st1.xy()*t2.xy() + st1.yy()*t2.yy() + st1.yz()*t2.zy(),
        st1.xy()*t2.xz() + st1.yy()*t2.yz() + st1.yz()*t2.zz(),
 
        st1.xz()*t2.xx() + st1.yz()*t2.yx() + st1.zz()*t2.zx(),
        st1.xz()*t2.xy() + st1.yz()*t2.yy() + st1.zz()*t2.zy(),
        st1.xz()*t2.xz() + st1.yz()*t2.yz() + st1.zz()*t2.zz()
    );
}
 
 
//- Inner-product of a Tensor and a SymmTensor
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Tensor<Cmpt>
operator&(const Tensor<Cmpt>& t1, const SymmTensor<Cmpt>& st2)
{
    return Tensor<Cmpt>
    (
        t1.xx()*st2.xx() + t1.xy()*st2.xy() + t1.xz()*st2.xz(),
        t1.xx()*st2.xy() + t1.xy()*st2.yy() + t1.xz()*st2.yz(),
        t1.xx()*st2.xz() + t1.xy()*st2.yz() + t1.xz()*st2.zz(),
 
        t1.yx()*st2.xx() + t1.yy()*st2.xy() + t1.yz()*st2.xz(),
        t1.yx()*st2.xy() + t1.yy()*st2.yy() + t1.yz()*st2.yz(),
        t1.yx()*st2.xz() + t1.yy()*st2.yz() + t1.yz()*st2.zz(),
 
        t1.zx()*st2.xx() + t1.zy()*st2.xy() + t1.zz()*st2.xz(),
        t1.zx()*st2.xy() + t1.zy()*st2.yy() + t1.zz()*st2.yz(),
        t1.zx()*st2.xz() + t1.zy()*st2.yz() + t1.zz()*st2.zz()
    );
}
 
 
//- Inner-product of a Tensor and a Vector
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Vector<Cmpt>
operator&(const Tensor<Cmpt>& t, const Vector<Cmpt>& v)
{
    return Vector<Cmpt>
    (
        t.xx()*v.x() + t.xy()*v.y() + t.xz()*v.z(),
        t.yx()*v.x() + t.yy()*v.y() + t.yz()*v.z(),
        t.zx()*v.x() + t.zy()*v.y() + t.zz()*v.z()
    );
}
 
 
//- Inner-product of a Vector and a Tensor
template<class Cmpt>
#if defined(__GNUC__) && !defined(__clang__)
// Workaround for gcc (11+) that fails to handle tensor dot vector
__attribute__((optimize("no-tree-vectorize")))
#endif
inline Vector<Cmpt>
operator&(const Vector<Cmpt>& v, const Tensor<Cmpt>& t)
{
    return Vector<Cmpt>
    (
        v.x()*t.xx() + v.y()*t.yx() + v.z()*t.zx(),
        v.x()*t.xy() + v.y()*t.yy() + v.z()*t.zy(),
        v.x()*t.xz() + v.y()*t.yz() + v.z()*t.zz()
    );
}
 
 
//- Double-inner-product of a SphericalTensor and a Tensor
template<class Cmpt>
inline Cmpt
operator&&(const SphericalTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return (st1.ii()*t2.xx() + st1.ii()*t2.yy() + st1.ii()*t2.zz());
}
 
 
//- Double-inner-product of a Tensor and a SphericalTensor
template<class Cmpt>
inline Cmpt
operator&&(const Tensor<Cmpt>& t1, const SphericalTensor<Cmpt>& st2)
{
    return (t1.xx()*st2.ii() + t1.yy()*st2.ii() + t1.zz()*st2.ii());
}
 
 
//- Double-inner-product of a SymmTensor and a Tensor
template<class Cmpt>
inline Cmpt
operator&&(const SymmTensor<Cmpt>& st1, const Tensor<Cmpt>& t2)
{
    return
    (
        st1.xx()*t2.xx() + st1.xy()*t2.xy() + st1.xz()*t2.xz() +
        st1.xy()*t2.yx() + st1.yy()*t2.yy() + st1.yz()*t2.yz() +
        st1.xz()*t2.zx() + st1.yz()*t2.zy() + st1.zz()*t2.zz()
    );
}
 
 
//- Double-inner-product of a Tensor and a SymmTensor
template<class Cmpt>
inline Cmpt
operator&&(const Tensor<Cmpt>& t1, const SymmTensor<Cmpt>& st2)
{
    return
    (
        t1.xx()*st2.xx() + t1.xy()*st2.xy() + t1.xz()*st2.xz() +
        t1.yx()*st2.xy() + t1.yy()*st2.yy() + t1.yz()*st2.yz() +
        t1.zx()*st2.xz() + t1.zy()*st2.yz() + t1.zz()*st2.zz()
    );
}
 
 
//- Outer-product of a Vector and a Vector
template<class Cmpt>
inline typename outerProduct<Vector<Cmpt>, Vector<Cmpt>>::type
operator*(const Vector<Cmpt>& v1, const Vector<Cmpt>& v2)
{
    return Tensor<Cmpt>
    (
        v1.x()*v2.x(), v1.x()*v2.y(), v1.x()*v2.z(),
        v1.y()*v2.x(), v1.y()*v2.y(), v1.y()*v2.z(),
        v1.z()*v2.x(), v1.z()*v2.y(), v1.z()*v2.z()
    );
}
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
template<class Cmpt>
class typeOfSum<SphericalTensor<Cmpt>, Tensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class typeOfSum<Tensor<Cmpt>, SphericalTensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class innerProduct<SphericalTensor<Cmpt>, Tensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class innerProduct<Tensor<Cmpt>, SphericalTensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class typeOfSum<SymmTensor<Cmpt>, Tensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class typeOfSum<Tensor<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class innerProduct<SymmTensor<Cmpt>, Tensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
template<class Cmpt>
class innerProduct<Tensor<Cmpt>, SymmTensor<Cmpt>>
{
public:
 
    typedef Tensor<Cmpt> type;
};
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
} // End namespace Foam
 
// ************************************************************************* //