velocity.hh

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#ifndef DUMUX_POROUSMEDIUMFLOW_VELOCITY_HH
#define DUMUX_POROUSMEDIUMFLOW_VELOCITY_HH
 
#include <vector>
#include <type_traits>
 
#include <dune/common/fvector.hh>
#include <dune/common/float_cmp.hh>
#include <dune/geometry/type.hh>
#include <dune/geometry/referenceelements.hh>
 
#include <dumux/common/parameters.hh>
#include <dumux/discretization/extrusion.hh>
#include <dumux/discretization/method.hh>
#include <dumux/discretization/elementsolution.hh>
#include <dumux/flux/traits.hh>
 
namespace Dumux {
 
#ifndef DOXYGEN
namespace Detail {
// helper structs and functions detecting if the model is compositional
 
template <typename T, typename ...Ts>
using MoleFractionDetector = decltype(std::declval<T>().moleFraction(std::declval<Ts>()...));
 
template<class T, typename ...Args>
static constexpr bool hasMoleFraction()
{ return Dune::Std::is_detected<MoleFractionDetector, T, Args...>::value; }
 
template <typename T, typename ...Ts>
using MassFractionDetector = decltype(std::declval<T>().massFraction(std::declval<Ts>()...));
 
template<class T, typename ...Args>
static constexpr bool hasMassFraction()
{ return Dune::Std::is_detected<MassFractionDetector, T, Args...>::value; }
 
} // end namespace Detail
#endif // DOXYGEN

template<class GridVariables, class FluxVariables>
class PorousMediumFlowVelocity
{
    using GridGeometry = typename GridVariables::GridGeometry;
    using Extrusion = Extrusion_t<GridGeometry>;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using SubControlVolume = typename GridGeometry::SubControlVolume;
    using SubControlVolumeFace = typename GridGeometry::SubControlVolumeFace;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
    using GridVolumeVariables = typename GridVariables::GridVolumeVariables;
    using ElementFluxVarsCache = typename GridVariables::GridFluxVariablesCache::LocalView;
    using VolumeVariables = typename GridVariables::VolumeVariables;
    using ElementVolumeVariables = typename GridVolumeVariables::LocalView;
    using FluidSystem = typename VolumeVariables::FluidSystem;
    using Scalar = typename GridVariables::Scalar;
    using FluxTraits = typename Dumux::FluxTraits<FluxVariables>;
    using AdvectionType = typename FluxVariables::AdvectionType;
 
    static constexpr int dim = GridView::dimension;
    static constexpr int dimWorld = GridView::dimensionworld;
    static constexpr bool isBox = GridGeometry::discMethod == DiscretizationMethods::box;
    static constexpr int dofCodim = isBox ? dim : 0;
    static constexpr bool stationaryVelocityField = FluxTraits::hasStationaryVelocityField();
 
    using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
 
    using Problem = typename GridVolumeVariables::Problem;
 
    static constexpr bool modelIsCompositional = Detail::hasMoleFraction<typename GridVolumeVariables::VolumeVariables, int, int>() ||
                                                 Detail::hasMassFraction<typename GridVolumeVariables::VolumeVariables, int, int>();
 
    static_assert(VolumeVariables::numFluidPhases() >= 1, "Velocity output only makes sense for models with fluid phases.");
 
public:
    static constexpr int numFluidPhases = VolumeVariables::numFluidPhases();
    using VelocityVector = std::vector<Dune::FieldVector<Scalar, dimWorld>>;

    PorousMediumFlowVelocity(const GridVariables& gridVariables)
    : problem_(gridVariables.curGridVolVars().problem())
    , gridGeometry_(gridVariables.gridGeometry())
    , gridVariables_(gridVariables)
    {
        // set the number of scvs the vertices are connected to
        if constexpr (isBox && dim > 1)
        {
            // resize to the number of vertices of the grid
            cellNum_.assign(gridGeometry_.gridView().size(dim), 0);
 
            for (const auto& element : elements(gridGeometry_.gridView()))
                for (unsigned int vIdx = 0; vIdx < element.subEntities(dim); ++vIdx)
                    ++cellNum_[gridGeometry_.vertexMapper().subIndex(element, vIdx, dim)];
        }
    }

    void calculateVelocity(VelocityVector& velocity,
                           const Element& element,
                           const FVElementGeometry& fvGeometry,
                           const ElementVolumeVariables& elemVolVars,
                           const ElementFluxVarsCache& elemFluxVarsCache,
                           int phaseIdx) const
    {
        using Velocity = typename VelocityVector::value_type;
 
        const auto geometry = element.geometry();
        const Dune::GeometryType geomType = geometry.type();
 
        // the upwind term to be used for the volume flux evaluation
        auto upwindTerm = [phaseIdx](const auto& volVars) { return volVars.mobility(phaseIdx); };
 
        if constexpr (isBox && dim == 1)
        {
            Velocity tmpVelocity(0.0);
            tmpVelocity = (geometry.corner(1) - geometry.corner(0));
            tmpVelocity /= tmpVelocity.two_norm();
 
            for (auto&& scvf : scvfs(fvGeometry))
            {
                if (scvf.boundary())
                    continue;
 
                // insantiate the flux variables
                FluxVariables fluxVars;
                fluxVars.init(problem_, element, fvGeometry, elemVolVars, scvf, elemFluxVarsCache);
 
                // get the volume flux divided by the area of the
                // subcontrolvolume face in the reference element
                Scalar localArea = scvfReferenceArea_(geomType, scvf.index());
                Scalar flux = fluxVars.advectiveFlux(phaseIdx, upwindTerm) / localArea;
                const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
                flux /= insideVolVars.extrusionFactor() * Extrusion::area(scvf) / scvf.area();
                tmpVelocity *= flux;
 
                const int eIdxGlobal = gridGeometry_.elementMapper().index(element);
                velocity[eIdxGlobal] = tmpVelocity;
            }
            return;
        }
 
        // get the transposed Jacobian of the element mapping
        using Dune::referenceElement;
        const auto refElement = referenceElement(geometry);
        const auto& localPos = refElement.position(0, 0);
        const auto jacobianT2 = geometry.jacobianTransposed(localPos);
 
        if constexpr (isBox)
        {
            using ScvVelocities = Dune::BlockVector<Velocity>;
            ScvVelocities scvVelocities(fvGeometry.numScv());
            scvVelocities = 0;
 
            for (auto&& scvf : scvfs(fvGeometry))
            {
                if (scvf.boundary())
                    continue;
 
                // local position of integration point
                const auto localPosIP = geometry.local(scvf.ipGlobal());
 
                // Transformation of the global normal vector to normal vector in the reference element
                const auto jacobianT1 = geometry.jacobianTransposed(localPosIP);
                const auto globalNormal = scvf.unitOuterNormal();
                GlobalPosition localNormal(0);
                jacobianT1.mv(globalNormal, localNormal);
                localNormal /= localNormal.two_norm();
 
                // instantiate the flux variables
                FluxVariables fluxVars;
                fluxVars.init(problem_, element, fvGeometry, elemVolVars, scvf, elemFluxVarsCache);
 
                // get the volume flux divided by the area of the
                // subcontrolvolume face in the reference element
                Scalar localArea = scvfReferenceArea_(geomType, scvf.index());
                Scalar flux = fluxVars.advectiveFlux(phaseIdx, upwindTerm) / localArea;
                const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
                flux /= insideVolVars.extrusionFactor() * Extrusion::area(scvf) / scvf.area();
 
                // transform the volume flux into a velocity vector
                Velocity tmpVelocity = localNormal;
                tmpVelocity *= flux;
 
                scvVelocities[scvf.insideScvIdx()] += tmpVelocity;
                scvVelocities[scvf.outsideScvIdx()] += tmpVelocity;
            }
 
            // transform vertex velocities from local to global coordinates
            for (auto&& scv : scvs(fvGeometry))
            {
                int vIdxGlobal = scv.dofIndex();
 
                // calculate the subcontrolvolume velocity by the Piola transformation
                Velocity scvVelocity(0);
 
                jacobianT2.mtv(scvVelocities[scv.indexInElement()], scvVelocity);
                scvVelocity /= geometry.integrationElement(localPos)*cellNum_[vIdxGlobal];
                // add up the wetting phase subcontrolvolume velocities for each vertex
                velocity[vIdxGlobal] += scvVelocity;
            }
        }
        else
        {
            // For the number of scvfs per facet (mpfa) we simply obtain the number of
            // corners of the first facet as prisms/pyramids are not supported here anyway
            // -> for prisms/pyramids the number of scvfs would differ from facet to facet
            static constexpr bool isMpfa = GridGeometry::discMethod == DiscretizationMethods::ccmpfa;
            const int numScvfsPerFace = isMpfa ? element.template subEntity<1>(0).geometry().corners() : 1;
 
            if (fvGeometry.numScvf() != element.subEntities(1)*numScvfsPerFace)
                DUNE_THROW(Dune::NotImplemented, "Velocity output for non-conforming grids");
 
            if (!geomType.isCube() && !geomType.isSimplex())
                DUNE_THROW(Dune::NotImplemented, "Velocity output for other geometry types than cube and simplex");
 
            // first we extract the corner indices for each scv for the CIV method
            // for network grids there might be multiple intersection with the same geometryInInside
            // we identify those by the indexInInside for now (assumes conforming grids at branching facets)
            // here we keep track of them
            std::vector<bool> handledScvf;
            if (dim < dimWorld)
                handledScvf.resize(element.subEntities(1), false);
 
            // find the local face indices of the scvfs (for conforming meshes)
            std::vector<unsigned int> scvfIndexInInside(fvGeometry.numScvf());
            int localScvfIdx = 0;
            for (const auto& intersection : intersections(gridGeometry_.gridView(), element))
            {
                if (dim < dimWorld)
                    if (handledScvf[intersection.indexInInside()])
                        continue;
 
                if (intersection.neighbor() || intersection.boundary())
                {
                    for (int i = 0; i < numScvfsPerFace; ++i)
                        scvfIndexInInside[localScvfIdx++] = intersection.indexInInside();
 
                    // for surface and network grids mark that we handled this face
                    if (dim < dimWorld)
                        handledScvf[intersection.indexInInside()] = true;
                }
            }
 
            std::vector<Scalar> scvfFluxes(element.subEntities(1), 0.0);
            localScvfIdx = 0;
            for (auto&& scvf : scvfs(fvGeometry))
            {
                const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
                if (!scvf.boundary())
                {
                    FluxVariables fluxVars;
                    fluxVars.init(problem_, element, fvGeometry, elemVolVars, scvf, elemFluxVarsCache);
                    scvfFluxes[scvfIndexInInside[localScvfIdx]] += fluxVars.advectiveFlux(phaseIdx, upwindTerm)/insideVolVars.extrusionFactor();
                }
                else
                {
                    auto bcTypes = problem_.boundaryTypes(element, scvf);
                    if (bcTypes.hasOnlyDirichlet())
                    {
                        FluxVariables fluxVars;
                        fluxVars.init(problem_, element, fvGeometry, elemVolVars, scvf, elemFluxVarsCache);
                        scvfFluxes[scvfIndexInInside[localScvfIdx]] += fluxVars.advectiveFlux(phaseIdx, upwindTerm)/insideVolVars.extrusionFactor();
                    }
                }
 
                // increment scvf counter
                localScvfIdx++;
            }
 
            // Correct boundary fluxes in case of Neumann conditions.
            // In this general setting, it would be very difficult to
            // calculate correct phase, i.e., volume, fluxes from arbitrary
            // Neumann conditions. We approximate the Neumann flux by the
            // flux on the opposite face. For extremely distorted grids this can
            // lead to unexpected results (but then TPFA also leads to unexpected results).
            localScvfIdx = 0;
            for (auto&& scvf : scvfs(fvGeometry))
            {
                if (scvf.boundary())
                {
                    auto bcTypes = problem_.boundaryTypes(element, scvf);
                    if (bcTypes.hasNeumann())
                    {
                        // for stationary velocity fields we can easily compute the correct velocity
                        // this special treatment makes sure that the velocity field is also correct on Neumann boundaries
                        // of tracer problems where the velocity field is given.
                        // (For Dirichlet boundaries no special treatment is necessary.)
                        if (stationaryVelocityField)
                        {
                            const auto flux = AdvectionType::flux(problem_, element, fvGeometry, elemVolVars,
                                                                  scvf, phaseIdx, elemFluxVarsCache);
 
                            const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
                            scvfFluxes[scvfIndexInInside[localScvfIdx]] += flux / insideVolVars.extrusionFactor();
                        }
                        else
                        {
                            // check if we have Neumann no flow, we can just use 0
                            const auto neumannFlux = problem_.neumann(element, fvGeometry, elemVolVars, elemFluxVarsCache, scvf);
                            using NumEqVector = std::decay_t<decltype(neumannFlux)>;
                            if (Dune::FloatCmp::eq<NumEqVector, Dune::FloatCmp::CmpStyle::absolute>(neumannFlux, NumEqVector(0.0), 1e-30))
                                scvfFluxes[scvfIndexInInside[localScvfIdx]] = 0;
 
                            // otherwise, we try some reconstruction
                            // for cubes
                            else if (dim == 1 || geomType.isCube())
                            {
                                const auto fIdx = scvfIndexInInside[localScvfIdx];
 
                                if constexpr (!modelIsCompositional)
                                {
                                    // We assume that the density at the face equals the one at the cell center and reconstruct a volume flux from the Neumann mass flux.
                                    const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
                                    const auto eqIdx = VolumeVariables::Indices::conti0EqIdx + phaseIdx;
                                    scvfFluxes[fIdx] += neumannFlux[eqIdx] / insideVolVars.density(phaseIdx) * scvf.area() * insideVolVars.extrusionFactor();
                                }
                                else
                                {
                                    // For compositional models, we generally can't reconstruct the volume flow from the Neumann flux (which is a species flux rather
                                    // than a phase flux here). Instead, we use the velocity of the opposing face.
                                    const auto fIdxOpposite = fIdx%2 ? fIdx-1 : fIdx+1;
                                    scvfFluxes[fIdx] = -scvfFluxes[fIdxOpposite];
                                }
                            }
 
                            // for simplices
                            else if (geomType.isSimplex())
                                scvfFluxes[scvfIndexInInside[localScvfIdx]] = 0;
 
                            else
                                DUNE_THROW(Dune::NotImplemented, "Velocity computation at Neumann boundaries for cell-centered and prism/pyramid");
                        }
                    }
                }
 
                // increment scvf counter
                localScvfIdx++;
            }
 
 
            Velocity refVelocity;
            // cubes: On the reference element simply average over opposite fluxes
            // note that this is equal to a corner velocity interpolation method
            if (dim == 1 || geomType.isCube())
            {
                for (int i = 0; i < dim; i++)
                    refVelocity[i] = 0.5 * (scvfFluxes[2*i + 1] - scvfFluxes[2*i]);
            }
            // simplices: Raviart-Thomas-0 interpolation evaluated at the cell center
            else if (geomType.isSimplex())
            {
                for (int dimIdx = 0; dimIdx < dim; dimIdx++)
                {
                    refVelocity[dimIdx] = -scvfFluxes[dim - 1 - dimIdx];
                    for (int fIdx = 0; fIdx < dim + 1; fIdx++)
                        refVelocity[dimIdx] += scvfFluxes[fIdx]/(dim + 1);
                }
            }
            // 3D prism and pyramids
            else
                DUNE_THROW(Dune::NotImplemented, "Velocity computation for cell-centered and prism/pyramid");
 
            Velocity scvVelocity(0);
            jacobianT2.mtv(refVelocity, scvVelocity);
 
            scvVelocity /= geometry.integrationElement(localPos);
 
            int eIdxGlobal = gridGeometry_.elementMapper().index(element);
 
            velocity[eIdxGlobal] = scvVelocity;
 
        } // cell-centered
    }
 
private:
    // The area of a subcontrolvolume face in a reference element.
    // The 3d non-cube values have been calculated with quadrilateralArea3D
    // of boxfvelementgeometry.hh.
    static Scalar scvfReferenceArea_(Dune::GeometryType geomType, int fIdx)
    {
        if (dim == 1 || geomType.isCube())
        {
            return 1.0/(1 << (dim-1));
        }
        else if (geomType.isTriangle())
        {
            static const Scalar faceToArea[] = {0.372677996249965,
                                                0.372677996249965,
                                                0.235702260395516};
            return faceToArea[fIdx];
        }
        else if (geomType.isTetrahedron())
        {
            static const Scalar faceToArea[] = {0.102062072615966,
                                                0.102062072615966,
                                                0.058925565098879,
                                                0.102062072615966,
                                                0.058925565098879,
                                                0.058925565098879};
            return faceToArea[fIdx];
        }
        else if (geomType.isPyramid())
        {
            static const Scalar faceToArea[] = {0.130437298687488,
                                                0.130437298687488,
                                                0.130437298687488,
                                                0.130437298687488,
                                                0.150923085635624,
                                                0.1092906420717,
                                                0.1092906420717,
                                                0.0781735959970571};
            return faceToArea[fIdx];
        }
        else if (geomType.isPrism())
        {
            static const Scalar faceToArea[] = {0.166666666666667,
                                                0.166666666666667,
                                                0.166666666666667,
                                                0.186338998124982,
                                                0.186338998124982,
                                                0.117851130197758,
                                                0.186338998124982,
                                                0.186338998124982,
                                                0.117851130197758};
            return faceToArea[fIdx];
        }
        else {
            DUNE_THROW(Dune::NotImplemented, "scvf area for unknown GeometryType");
        }
    }
 
private:
    const Problem& problem_;
    const GridGeometry& gridGeometry_;
    const GridVariables& gridVariables_;
 
    std::vector<int> cellNum_;
};
 
} // end namespace Dumux
 
#endif