primitiveMeshTools.C

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    Copyright (C) 2012-2016 OpenFOAM Foundation
    Copyright (C) 2017-2022 OpenCFD Ltd.
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    the Free Software Foundation, either version 3 of the License, or
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    ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
    for more details.
 
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#include "primitiveMeshTools.H"
#include "primitiveMesh.H"
#include "syncTools.H"
#include "pyramidPointFaceRef.H"
#include "PrecisionAdaptor.H"
 
// * * * * * * * * * * * * * Static Member Functions * * * * * * * * * * * * //
 
void Foam::primitiveMeshTools::updateFaceCentresAndAreas
(
    const primitiveMesh& mesh,
    const UList<label>& faceIDs,
    const pointField& p,
    vectorField& fCtrs,
    vectorField& fAreas
)
{
    const faceList& fs = mesh.faces();
 
    for (const label facei : faceIDs)
    {
        const labelList& f = fs[facei];
        const label nPoints = f.size();
 
        // If the face is a triangle, do a direct calculation for efficiency
        // and to avoid round-off error-related problems
        if (nPoints == 3)
        {
            fCtrs[facei] = (1.0/3.0)*(p[f[0]] + p[f[1]] + p[f[2]]);
            fAreas[facei] = 0.5*((p[f[1]] - p[f[0]])^(p[f[2]] - p[f[0]]));
        }
        else
        {
            typedef Vector<solveScalar> solveVector;
 
            solveVector sumN = Zero;
            solveScalar sumA = 0.0;
            solveVector sumAc = Zero;
 
            solveVector fCentre = p[f[0]];
            for (label pi = 1; pi < nPoints; ++pi)
            {
                fCentre += solveVector(p[f[pi]]);
            }
 
            fCentre /= nPoints;
 
            for (label pi = 0; pi < nPoints; ++pi)
            {
                const label nextPi(pi == nPoints-1 ? 0 : pi+1);
                const solveVector nextPoint(p[f[nextPi]]);
                const solveVector thisPoint(p[f[pi]]);
 
                solveVector c = thisPoint + nextPoint + fCentre;
                solveVector n = (nextPoint - thisPoint)^(fCentre - thisPoint);
                solveScalar a = mag(n);
                sumN += n;
                sumA += a;
                sumAc += a*c;
            }
 
            // This is to deal with zero-area faces. Mark very small faces
            // to be detected in e.g., processorPolyPatch.
            if (sumA < ROOTVSMALL)
            {
                fCtrs[facei] = fCentre;
                fAreas[facei] = Zero;
            }
            else
            {
                fCtrs[facei] = (1.0/3.0)*sumAc/sumA;
                fAreas[facei] = 0.5*sumN;
            }
        }
    }
}
 
 
void Foam::primitiveMeshTools::updateCellCentresAndVols
(
    const primitiveMesh& mesh,
    const vectorField& fCtrs,
    const vectorField& fAreas,
    const List<label>& cellIDs,
    const List<cell>& cells,
    vectorField& cellCtrs_s,
    scalarField& cellVols_s
)
{
    typedef Vector<solveScalar> solveVector;
 
    PrecisionAdaptor<solveVector, vector> tcellCtrs(cellCtrs_s, false);
    PrecisionAdaptor<solveScalar, scalar> tcellVols(cellVols_s, false);
    Field<solveVector>& cellCtrs = tcellCtrs.ref();
    Field<solveScalar>& cellVols = tcellVols.ref();
 
 
    // Use current cell centres as estimates for the new cell centres
    // Note - not currently used
    // - Introduces a small difference (seen when write precision extended to
    //   16) to the cell centre and volume values, typically last 4 digits
    //const vectorField Cc0(cellCtrs, cellIDs);
 
    // Estimate cell centres using current face centres
    // - Same method as used by makeCellCentresAndVols()
    vectorField Cc0(cellIDs.size(), Zero);
    {
        labelField nCellFaces(cellIDs.size(), Zero);
        const auto& cells = mesh.cells();
 
        forAll(cellIDs, i)
        {
            const label celli = cellIDs[i];
            const cell& c = cells[celli];
            for (const auto facei : c)
            {
                Cc0[i] += fCtrs[facei];
              ++nCellFaces[i];
            }
        }
 
        forAll(Cc0, i)
        {
            Cc0[i] /= nCellFaces[i];
        }
    }
 
    // Clear the fields for accumulation
    for (const label celli : cellIDs)
    {
        cellCtrs[celli] = Zero;
        cellVols[celli] = Zero;
    }
 
    const auto& own = mesh.faceOwner();
 
    forAll(cellIDs, i)
    {
        const label celli = cellIDs[i];
        const auto& c = cells[celli];
        const solveVector cc(Cc0[i]);
 
        for (const label facei : c)
        {
            const solveVector fc(fCtrs[facei]);
            const solveVector fA(fAreas[facei]);
 
            const solveScalar pyr3Vol = own[facei] == celli
              ? fA & (fc - cc)
              : fA & (cc - fc);
 
            // Calculate face-pyramid centre
            const solveVector pc = (3.0/4.0)*fc + (1.0/4.0)*cc;
 
            // Accumulate volume-weighted face-pyramid centre
            cellCtrs[celli] += pyr3Vol*pc;
 
            // Accumulate face-pyramid volume
            cellVols[celli] += pyr3Vol;
        }
 
        if (mag(cellVols[celli]) > VSMALL)
        {
            cellCtrs[celli] /= cellVols[celli];
            cellVols[celli] *= (1.0/3.0);
        }
        else
        {
            cellCtrs[celli] = Cc0[i];
        }
    }
}
 
 
void Foam::primitiveMeshTools::makeFaceCentresAndAreas
(
    const UList<face>& fcs,
    const pointField& p,
    vectorField& fCtrs,
    vectorField& fAreas
)
{
    // Safety first - ensure properly sized
    fCtrs.resize_nocopy(fcs.size());
    fAreas.resize_nocopy(fcs.size());
 
    forAll(fcs, facei)
    {
        const face& f = fcs[facei];
        const label nPoints = f.size();
 
        // If the face is a triangle, do a direct calculation for efficiency
        // and to avoid round-off error-related problems
        if (nPoints == 3)
        {
            fCtrs[facei] = (1.0/3.0)*(p[f[0]] + p[f[1]] + p[f[2]]);
            fAreas[facei] = 0.5*((p[f[1]] - p[f[0]])^(p[f[2]] - p[f[0]]));
        }
        else
        {
            solveVector sumN = Zero;
            solveScalar sumA = Zero;
            solveVector sumAc = Zero;
 
            solveVector fCentre = p[f[0]];
            for (label pi = 1; pi < nPoints; ++pi)
            {
                fCentre += solveVector(p[f[pi]]);
            }
            fCentre /= nPoints;
 
            for (label pi = 0; pi < nPoints; ++pi)
            {
                const solveVector thisPoint(p[f.thisLabel(pi)]);
                const solveVector nextPoint(p[f.nextLabel(pi)]);
 
                solveVector c = thisPoint + nextPoint + fCentre;
                solveVector n = (nextPoint - thisPoint)^(fCentre - thisPoint);
                solveScalar a = mag(n);
 
                sumN += n;
                sumA += a;
                sumAc += a*c;
            }
 
            // This is to deal with zero-area faces. Mark very small faces
            // to be detected in e.g., processorPolyPatch.
            if (sumA < ROOTVSMALL)
            {
                fCtrs[facei] = fCentre;
                fAreas[facei] = Zero;
            }
            else
            {
                fCtrs[facei] = (1.0/3.0)*sumAc/sumA;
                fAreas[facei] = 0.5*sumN;
            }
        }
    }
}
 
 
void Foam::primitiveMeshTools::makeFaceCentresAndAreas
(
    const primitiveMesh& mesh,
    const pointField& p,
    vectorField& fCtrs,
    vectorField& fAreas
)
{
    makeFaceCentresAndAreas(mesh.faces(), p, fCtrs, fAreas);
}
 
 
void Foam::primitiveMeshTools::makeCellCentresAndVols
(
    const primitiveMesh& mesh,
    const vectorField& fCtrs,
    const vectorField& fAreas,
    vectorField& cellCtrs_s,
    scalarField& cellVols_s
)
{
    PrecisionAdaptor<solveVector, vector> tcellCtrs(cellCtrs_s, false);
    PrecisionAdaptor<solveScalar, scalar> tcellVols(cellVols_s, false);
    Field<solveVector>& cellCtrs = tcellCtrs.ref();
    Field<solveScalar>& cellVols = tcellVols.ref();
 
    // Clear the fields for accumulation
    cellCtrs = Zero;
    cellVols = Zero;
 
    const labelList& own = mesh.faceOwner();
    const labelList& nei = mesh.faceNeighbour();
 
    // first estimate the approximate cell centre as the average of
    // face centres
 
    Field<solveVector> cEst(mesh.nCells(), Zero);
    labelField nCellFaces(mesh.nCells(), Zero);
 
    forAll(own, facei)
    {
        cEst[own[facei]] += solveVector(fCtrs[facei]);
        ++nCellFaces[own[facei]];
    }
 
    forAll(nei, facei)
    {
        cEst[nei[facei]] += solveVector(fCtrs[facei]);
        ++nCellFaces[nei[facei]];
    }
 
    forAll(cEst, celli)
    {
        cEst[celli] /= nCellFaces[celli];
    }
 
    forAll(own, facei)
    {
        const solveVector fc(fCtrs[facei]);
        const solveVector fA(fAreas[facei]);
 
        // Calculate 3*face-pyramid volume
        solveScalar pyr3Vol = fA & (fc - cEst[own[facei]]);
 
        // Calculate face-pyramid centre
        solveVector pc = (3.0/4.0)*fc + (1.0/4.0)*cEst[own[facei]];
 
        // Accumulate volume-weighted face-pyramid centre
        cellCtrs[own[facei]] += pyr3Vol*pc;
 
        // Accumulate face-pyramid volume
        cellVols[own[facei]] += pyr3Vol;
    }
 
    forAll(nei, facei)
    {
        const solveVector fc(fCtrs[facei]);
        const solveVector fA(fAreas[facei]);
 
        // Calculate 3*face-pyramid volume
        solveScalar pyr3Vol = fA & (cEst[nei[facei]] - fc);
 
        // Calculate face-pyramid centre
        solveVector pc = (3.0/4.0)*fc + (1.0/4.0)*cEst[nei[facei]];
 
        // Accumulate volume-weighted face-pyramid centre
        cellCtrs[nei[facei]] += pyr3Vol*pc;
 
        // Accumulate face-pyramid volume
        cellVols[nei[facei]] += pyr3Vol;
    }
 
    forAll(cellCtrs, celli)
    {
        if (mag(cellVols[celli]) > VSMALL)
        {
            cellCtrs[celli] /= cellVols[celli];
        }
        else
        {
            cellCtrs[celli] = cEst[celli];
        }
    }
 
    cellVols *= (1.0/3.0);
}
 
 
Foam::scalar Foam::primitiveMeshTools::faceSkewness
(
    const UList<face>& fcs,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& fAreas,
 
    const label facei,
    const point& ownCc,
    const point& neiCc
)
{
    const face& f = fcs[facei];
 
    vector Cpf = fCtrs[facei] - ownCc;
    vector d = neiCc - ownCc;
 
    // Skewness vector
    vector sv =
        Cpf
      - ((fAreas[facei] & Cpf)/((fAreas[facei] & d) + ROOTVSMALL))*d;
    vector svHat = sv/(mag(sv) + ROOTVSMALL);
 
    // Normalisation distance calculated as the approximate distance
    // from the face centre to the edge of the face in the direction
    // of the skewness
    scalar fd = 0.2*mag(d) + ROOTVSMALL;
    forAll(f, pi)
    {
        fd = max(fd, mag(svHat & (p[f[pi]] - fCtrs[facei])));
    }
 
    // Normalised skewness
    return mag(sv)/fd;
}
 
 
Foam::scalar Foam::primitiveMeshTools::boundaryFaceSkewness
(
    const UList<face>& fcs,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& fAreas,
 
    const label facei,
    const point& ownCc
)
{
    const face& f = fcs[facei];
 
    vector Cpf = fCtrs[facei] - ownCc;
 
    vector normal = normalised(fAreas[facei]);
    vector d = normal*(normal & Cpf);
 
    // Skewness vector
    vector sv =
        Cpf
      - ((fAreas[facei] & Cpf)/((fAreas[facei] & d) + ROOTVSMALL))*d;
    vector svHat = sv/(mag(sv) + ROOTVSMALL);
 
    // Normalisation distance calculated as the approximate distance
    // from the face centre to the edge of the face in the direction
    // of the skewness
    scalar fd = 0.4*mag(d) + ROOTVSMALL;
    forAll(f, pi)
    {
        fd = max(fd, mag(svHat & (p[f[pi]] - fCtrs[facei])));
    }
 
    // Normalised skewness
    return mag(sv)/fd;
}
 
 
Foam::scalar Foam::primitiveMeshTools::faceSkewness
(
    const primitiveMesh& mesh,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& fAreas,
 
    const label facei,
    const point& ownCc,
    const point& neiCc
)
{
    return faceSkewness(mesh.faces(), p, fCtrs, fAreas, facei, ownCc, neiCc);
}
 
 
Foam::scalar Foam::primitiveMeshTools::boundaryFaceSkewness
(
    const primitiveMesh& mesh,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& fAreas,
 
    const label facei,
    const point& ownCc
)
{
    return boundaryFaceSkewness(mesh.faces(), p, fCtrs, fAreas, facei, ownCc);
}
 
 
Foam::scalar Foam::primitiveMeshTools::faceOrthogonality
(
    const point& ownCc,
    const point& neiCc,
    const vector& s
)
{
    vector d = neiCc - ownCc;
 
    return (d & s)/(mag(d)*mag(s) + ROOTVSMALL);
}
 
 
// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //
 
Foam::tmp<Foam::scalarField> Foam::primitiveMeshTools::faceOrthogonality
(
    const primitiveMesh& mesh,
    const vectorField& areas,
    const vectorField& cc
)
{
    const labelList& own = mesh.faceOwner();
    const labelList& nei = mesh.faceNeighbour();
 
    auto tortho = tmp<scalarField>::New(mesh.nInternalFaces());
    auto& ortho = tortho.ref();
 
    // Internal faces
    forAll(nei, facei)
    {
        ortho[facei] = faceOrthogonality
        (
            cc[own[facei]],
            cc[nei[facei]],
            areas[facei]
        );
    }
 
    return tortho;
}
 
 
Foam::tmp<Foam::scalarField> Foam::primitiveMeshTools::faceSkewness
(
    const primitiveMesh& mesh,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& fAreas,
    const vectorField& cellCtrs
)
{
    const labelList& own = mesh.faceOwner();
    const labelList& nei = mesh.faceNeighbour();
    const faceList& faces = mesh.faces();
 
    auto tskew = tmp<scalarField>::New(mesh.nFaces());
    auto& skew = tskew.ref();
 
    // Internal faces
    for (label facei = 0; facei < mesh.nInternalFaces(); ++facei)
    {
        skew[facei] = faceSkewness
        (
            faces,
            p,
            fCtrs,
            fAreas,
 
            facei,
            cellCtrs[own[facei]],
            cellCtrs[nei[facei]]
        );
    }
 
 
    // Boundary faces: consider them to have only skewness error.
    // (i.e. treat as if mirror cell on other side)
 
    for (label facei = mesh.nInternalFaces(); facei < mesh.nFaces(); ++facei)
    {
        skew[facei] = boundaryFaceSkewness
        (
            faces,
            p,
            fCtrs,
            fAreas,
            facei,
            cellCtrs[own[facei]]
        );
    }
 
    return tskew;
}
 
 
void Foam::primitiveMeshTools::facePyramidVolume
(
    const primitiveMesh& mesh,
    const pointField& points,
    const vectorField& ctrs,
 
    scalarField& ownPyrVol,
    scalarField& neiPyrVol
)
{
    const labelList& own = mesh.faceOwner();
    const labelList& nei = mesh.faceNeighbour();
    const faceList& f = mesh.faces();
 
    ownPyrVol.setSize(mesh.nFaces());
    neiPyrVol.setSize(mesh.nInternalFaces());
 
    forAll(f, facei)
    {
        // Create the owner pyramid
        ownPyrVol[facei] = -pyramidPointFaceRef
        (
            f[facei],
            ctrs[own[facei]]
        ).mag(points);
 
        if (mesh.isInternalFace(facei))
        {
            // Create the neighbour pyramid - it will have positive volume
            neiPyrVol[facei] = pyramidPointFaceRef
            (
                f[facei],
                ctrs[nei[facei]]
            ).mag(points);
        }
    }
}
 
 
void Foam::primitiveMeshTools::cellClosedness
(
    const primitiveMesh& mesh,
    const Vector<label>& meshD,
    const vectorField& areas,
    const scalarField& vols,
 
    scalarField& openness,
    scalarField& aratio
)
{
    const labelList& own = mesh.faceOwner();
    const labelList& nei = mesh.faceNeighbour();
 
    // Loop through cell faces and sum up the face area vectors for each cell.
    // This should be zero in all vector components
 
    vectorField sumClosed(mesh.nCells(), Zero);
    vectorField sumMagClosed(mesh.nCells(), Zero);
 
    forAll(own, facei)
    {
        // Add to owner
        sumClosed[own[facei]] += areas[facei];
        sumMagClosed[own[facei]] += cmptMag(areas[facei]);
    }
 
    forAll(nei, facei)
    {
        // Subtract from neighbour
        sumClosed[nei[facei]] -= areas[facei];
        sumMagClosed[nei[facei]] += cmptMag(areas[facei]);
    }
 
 
    label nDims = 0;
    for (direction dir = 0; dir < vector::nComponents; dir++)
    {
        if (meshD[dir] == 1)
        {
            nDims++;
        }
    }
 
 
    // Check the sums
    openness.setSize(mesh.nCells());
    aratio.setSize(mesh.nCells());
 
    forAll(sumClosed, celli)
    {
        scalar maxOpenness = 0;
 
        for (direction cmpt=0; cmpt<vector::nComponents; cmpt++)
        {
            maxOpenness = max
            (
                maxOpenness,
                mag(sumClosed[celli][cmpt])
               /(sumMagClosed[celli][cmpt] + ROOTVSMALL)
            );
        }
        openness[celli] = maxOpenness;
 
        // Calculate the aspect ration as the maximum of Cartesian component
        // aspect ratio to the total area hydraulic area aspect ratio
        scalar minCmpt = VGREAT;
        scalar maxCmpt = -VGREAT;
        for (direction dir = 0; dir < vector::nComponents; dir++)
        {
            if (meshD[dir] == 1)
            {
                minCmpt = min(minCmpt, sumMagClosed[celli][dir]);
                maxCmpt = max(maxCmpt, sumMagClosed[celli][dir]);
            }
        }
 
        scalar aspectRatio = maxCmpt/(minCmpt + ROOTVSMALL);
        if (nDims == 3)
        {
            scalar v = max(ROOTVSMALL, vols[celli]);
 
            aspectRatio = max
            (
                aspectRatio,
                1.0/6.0*cmptSum(sumMagClosed[celli])/pow(v, 2.0/3.0)
            );
        }
 
        aratio[celli] = aspectRatio;
    }
}
 
 
Foam::tmp<Foam::scalarField> Foam::primitiveMeshTools::faceConcavity
(
    const scalar maxSin,
    const primitiveMesh& mesh,
    const pointField& p,
    const vectorField& faceAreas
)
{
    const faceList& fcs = mesh.faces();
 
    vectorField faceNormals(faceAreas);
    faceNormals /= mag(faceNormals) + ROOTVSMALL;
 
    auto tfaceAngles = tmp<scalarField>::New(mesh.nFaces());
    auto&& faceAngles = tfaceAngles.ref();
 
    forAll(fcs, facei)
    {
        const face& f = fcs[facei];
 
        // Normalized vector from f[size-1] to f[0];
        vector ePrev(p[f.first()] - p[f.last()]);
        scalar magEPrev = mag(ePrev);
        ePrev /= magEPrev + ROOTVSMALL;
 
        scalar maxEdgeSin = 0.0;
 
        forAll(f, fp0)
        {
            // Normalized vector between two consecutive points
            vector e10(p[f.nextLabel(fp0)] - p[f.thisLabel(fp0)]);
            scalar magE10 = mag(e10);
            e10 /= magE10 + ROOTVSMALL;
 
            if (magEPrev > SMALL && magE10 > SMALL)
            {
                vector edgeNormal = ePrev ^ e10;
                scalar magEdgeNormal = mag(edgeNormal);
 
                if (magEdgeNormal < maxSin)
                {
                    // Edges (almost) aligned -> face is ok.
                }
                else
                {
                    // Check normal
                    edgeNormal /= magEdgeNormal;
 
                    if ((edgeNormal & faceNormals[facei]) < SMALL)
                    {
                        maxEdgeSin = max(maxEdgeSin, magEdgeNormal);
                    }
                }
            }
 
            ePrev = e10;
            magEPrev = magE10;
        }
 
        faceAngles[facei] = maxEdgeSin;
    }
 
    return tfaceAngles;
}
 
 
Foam::tmp<Foam::scalarField> Foam::primitiveMeshTools::faceFlatness
(
    const primitiveMesh& mesh,
    const pointField& p,
    const vectorField& fCtrs,
    const vectorField& faceAreas
)
{
    const faceList& fcs = mesh.faces();
 
    // Areas are calculated as the sum of areas. (see
    // primitiveMeshFaceCentresAndAreas.C)
    scalarField magAreas(mag(faceAreas));
 
    auto tfaceFlatness = tmp<scalarField>::New(mesh.nFaces(), scalar(1));
    auto& faceFlatness = tfaceFlatness.ref();
 
    forAll(fcs, facei)
    {
        const face& f = fcs[facei];
 
        if (f.size() > 3 && magAreas[facei] > ROOTVSMALL)
        {
            const solveVector fc = fCtrs[facei];
 
            // Calculate the sum of magnitude of areas and compare to magnitude
            // of sum of areas.
 
            solveScalar sumA = 0.0;
 
            forAll(f, fp)
            {
                const solveVector thisPoint = p[f[fp]];
                const solveVector nextPoint = p[f.nextLabel(fp)];
 
                // Triangle around fc.
                solveVector n = 0.5*((nextPoint - thisPoint)^(fc - thisPoint));
                sumA += mag(n);
            }
 
            faceFlatness[facei] = magAreas[facei]/(sumA + ROOTVSMALL);
        }
    }
 
    return tfaceFlatness;
}
 
 
Foam::tmp<Foam::scalarField> Foam::primitiveMeshTools::cellDeterminant
(
    const primitiveMesh& mesh,
    const Vector<label>& meshD,
    const vectorField& faceAreas,
    const bitSet& internalOrCoupledFace
)
{
    // Determine number of dimensions and (for 2D) missing dimension
    label nDims = 0;
    label twoD = -1;
    for (direction dir = 0; dir < vector::nComponents; dir++)
    {
        if (meshD[dir] == 1)
        {
            nDims++;
        }
        else
        {
            twoD = dir;
        }
    }
 
    auto tcellDeterminant = tmp<scalarField>::New(mesh.nCells());
    auto& cellDeterminant = tcellDeterminant.ref();
 
    const cellList& c = mesh.cells();
 
    if (nDims == 1)
    {
        cellDeterminant = 1.0;
    }
    else
    {
        forAll(c, celli)
        {
            const labelList& curFaces = c[celli];
 
            // Calculate local normalization factor
            scalar avgArea = 0;
 
            label nInternalFaces = 0;
 
            forAll(curFaces, i)
            {
                if (internalOrCoupledFace.test(curFaces[i]))
                {
                    avgArea += mag(faceAreas[curFaces[i]]);
 
                    nInternalFaces++;
                }
            }
 
            if (nInternalFaces == 0 || avgArea < ROOTVSMALL)
            {
                cellDeterminant[celli] = 0;
            }
            else
            {
                avgArea /= nInternalFaces;
 
                symmTensor areaTensor(Zero);
 
                forAll(curFaces, i)
                {
                    if (internalOrCoupledFace.test(curFaces[i]))
                    {
                        areaTensor += sqr(faceAreas[curFaces[i]]/avgArea);
                    }
                }
 
                if (nDims == 2)
                {
                    // Add the missing eigenvector (such that it does not
                    // affect the determinant)
                    if (twoD == 0)
                    {
                        areaTensor.xx() = 1;
                    }
                    else if (twoD == 1)
                    {
                        areaTensor.yy() = 1;
                    }
                    else
                    {
                        areaTensor.zz() = 1;
                    }
                }
 
                // Note:
                // - normalise to be 0..1 (since cube has eigenvalues 2 2 2)
                // - we use the determinant (i.e. 3rd invariant) and not e.g.
                //   condition number (= max ev / min ev) since we are
                //   interested in the minimum connectivity and not the
                //   uniformity. Using the condition number on corner cells
                //   leads to uniformity 1 i.e. equally bad in all three
                //   directions which is not what we want.
                cellDeterminant[celli] = mag(det(areaTensor))/8.0;
            }
        }
    }
 
    return tcellDeterminant;
}
 
 
// ************************************************************************* //