boxgeometryhelper.hh

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// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
//
// SPDX-FileCopyrightText: Copyright © DuMux Project contributors, see AUTHORS.md in root folder
// SPDX-License-Identifier: GPL-3.0-or-later
//
#ifndef DUMUX_DISCRETIZATION_BOX_GEOMETRY_HELPER_HH
#define DUMUX_DISCRETIZATION_BOX_GEOMETRY_HELPER_HH
 
#include <array>
 
#include <dune/common/exceptions.hh>
 
#include <dune/geometry/type.hh>
#include <dune/geometry/typeindex.hh>
#include <dune/geometry/referenceelements.hh>
#include <dune/geometry/multilineargeometry.hh>
 
#include <dune/common/rangeutilities.hh>
 
#include <dumux/common/indextraits.hh>
#include <dumux/common/math.hh>
#include <dumux/geometry/center.hh>
#include <dumux/discretization/pq1/dofhelper.hh>
#include <dumux/discretization/scvfnormal.hh>
 
namespace Dumux {

template <class ct>
struct BoxMLGeometryTraits : public Dune::MultiLinearGeometryTraits<ct>
{
    // we use static vectors to store the corners as we know
    // the number of corners in advance (2^(mydim) corners (1<<(mydim))
    template< int mydim, int cdim >
    struct CornerStorage
    {
        using Type = std::array< Dune::FieldVector< ct, cdim >, (1<<(mydim)) >;
    };
 
    // we know all scvfs will have the same geometry type
    template< int mydim >
    struct hasSingleGeometryType
    {
        static const bool v = true;
        static const unsigned int topologyId = Dune::GeometryTypes::cube(mydim).id();
    };
};
 
namespace Detail::Box {
 
template<Dune::GeometryType::Id gt>
struct ScvCorners;
 
template<>
struct ScvCorners<Dune::GeometryTypes::line>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 2>, 2> keys = {{
        { Key{0, 1}, Key{0, 0} },
        { Key{1, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvCorners<Dune::GeometryTypes::triangle>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 4>, 3> keys = {{
        { Key{0, 2}, Key{0, 1}, Key{1, 1}, Key{0, 0} },
        { Key{1, 2}, Key{2, 1}, Key{0, 1}, Key{0, 0} },
        { Key{2, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvCorners<Dune::GeometryTypes::quadrilateral>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 4>, 4> keys = {{
        { Key{0, 2}, Key{2, 1}, Key{0, 1}, Key{0, 0} },
        { Key{1, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} },
        { Key{2, 2}, Key{0, 1}, Key{3, 1}, Key{0, 0} },
        { Key{3, 2}, Key{3, 1}, Key{1, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvCorners<Dune::GeometryTypes::tetrahedron>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 8>, 4> keys = {{
        { Key{0, 3}, Key{0, 2}, Key{1, 2}, Key{0, 1}, Key{3, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} },
        { Key{1, 3}, Key{2, 2}, Key{0, 2}, Key{0, 1}, Key{4, 2}, Key{3, 1}, Key{1, 1}, Key{0, 0} },
        { Key{2, 3}, Key{1, 2}, Key{2, 2}, Key{0, 1}, Key{5, 2}, Key{2, 1}, Key{3, 1}, Key{0, 0} },
        { Key{3, 3}, Key{3, 2}, Key{5, 2}, Key{2, 1}, Key{4, 2}, Key{1, 1}, Key{3, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvCorners<Dune::GeometryTypes::prism>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 8>, 6> keys = {{
        { Key{0, 3}, Key{3, 2}, Key{4, 2}, Key{3, 1}, Key{0, 2}, Key{0, 1}, Key{1, 1}, Key{0, 0} },
        { Key{1, 3}, Key{5, 2}, Key{3, 2}, Key{3, 1}, Key{1, 2}, Key{2, 1}, Key{0, 1}, Key{0, 0} },
        { Key{2, 3}, Key{4, 2}, Key{5, 2}, Key{3, 1}, Key{2, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} },
        { Key{3, 3}, Key{7, 2}, Key{6, 2}, Key{4, 1}, Key{0, 2}, Key{1, 1}, Key{0, 1}, Key{0, 0} },
        { Key{4, 3}, Key{6, 2}, Key{8, 2}, Key{4, 1}, Key{1, 2}, Key{0, 1}, Key{2, 1}, Key{0, 0} },
        { Key{5, 3}, Key{8, 2}, Key{7, 2}, Key{4, 1}, Key{2, 2}, Key{2, 1}, Key{1, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvCorners<Dune::GeometryTypes::hexahedron>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 8>, 8> keys = {{
        { Key{0, 3}, Key{6, 2}, Key{4, 2}, Key{4, 1}, Key{0, 2}, Key{2, 1}, Key{0, 1}, Key{0, 0} },
        { Key{1, 3}, Key{5, 2}, Key{6, 2}, Key{4, 1}, Key{1, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} },
        { Key{2, 3}, Key{4, 2}, Key{7, 2}, Key{4, 1}, Key{2, 2}, Key{0, 1}, Key{3, 1}, Key{0, 0} },
        { Key{3, 3}, Key{7, 2}, Key{5, 2}, Key{4, 1}, Key{3, 2}, Key{3, 1}, Key{1, 1}, Key{0, 0} },
        { Key{4, 3}, Key{8, 2}, Key{10, 2}, Key{5, 1}, Key{0, 2}, Key{0, 1}, Key{2, 1}, Key{0, 0} },
        { Key{5, 3}, Key{10, 2}, Key{9, 2}, Key{5, 1}, Key{1, 2}, Key{2, 1}, Key{1, 1}, Key{0, 0} },
        { Key{6, 3}, Key{11, 2}, Key{8, 2}, Key{5, 1}, Key{2, 2}, Key{3, 1}, Key{0, 1}, Key{0, 0} },
        { Key{7, 3}, Key{9, 2}, Key{11, 2}, Key{5, 1}, Key{3, 2}, Key{1, 1}, Key{3, 1}, Key{0, 0} }
    }};
};
 
template<Dune::GeometryType::Id gt>
struct ScvfCorners;
 
template<>
struct ScvfCorners<Dune::GeometryTypes::line>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 1>, 1> keys = {{
        { Key{0, 0} }
    }};
};
 
template<>
struct ScvfCorners<Dune::GeometryTypes::triangle>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 2>, 3> keys = {{
        { Key{0, 0}, Key{0, 1} },
        { Key{1, 1}, Key{0, 0} },
        { Key{0, 0}, Key{2, 1} }
    }};
};
 
template<>
struct ScvfCorners<Dune::GeometryTypes::quadrilateral>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 2>, 4> keys = {{
        { Key{0, 1}, Key{0, 0} },
        { Key{0, 0}, Key{1, 1} },
        { Key{0, 0}, Key{2, 1} },
        { Key{3, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvfCorners<Dune::GeometryTypes::tetrahedron>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 4>, 6> keys = {{
        { Key{0, 2}, Key{0, 1}, Key{1, 1}, Key{0, 0} },
        { Key{0, 1}, Key{1, 2}, Key{0, 0}, Key{2, 1} },
        { Key{2, 2}, Key{0, 1}, Key{3, 1}, Key{0, 0} },
        { Key{2, 1}, Key{3, 2}, Key{0, 0}, Key{1, 1} },
        { Key{3, 1}, Key{0, 0}, Key{4, 2}, Key{1, 1} },
        { Key{5, 2}, Key{2, 1}, Key{3, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvfCorners<Dune::GeometryTypes::prism>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 4>, 9> keys = {{
        { Key{0, 2}, Key{0, 1}, Key{1, 1}, Key{0, 0} },
        { Key{1, 2}, Key{2, 1}, Key{0, 1}, Key{0, 0} },
        { Key{2, 2}, Key{1, 1}, Key{2, 1}, Key{0, 0} },
        { Key{3, 2}, Key{0, 1}, Key{3, 1}, Key{0, 0} },
        { Key{4, 2}, Key{3, 1}, Key{1, 1}, Key{0, 0} },
        { Key{5, 2}, Key{2, 1}, Key{3, 1}, Key{0, 0} },
        { Key{6, 2}, Key{4, 1}, Key{0, 1}, Key{0, 0} },
        { Key{7, 2}, Key{1, 1}, Key{4, 1}, Key{0, 0} },
        { Key{8, 2}, Key{4, 1}, Key{2, 1}, Key{0, 0} }
    }};
};
 
template<>
struct ScvfCorners<Dune::GeometryTypes::hexahedron>
{
    using Key = std::pair<std::uint8_t, std::uint8_t>; // (i, codim)
    static constexpr std::array<std::array<Key, 4>, 12> keys = {{
        { Key{0, 1}, Key{0, 2}, Key{0, 0}, Key{2, 1} },
        { Key{1, 1}, Key{0, 0}, Key{1, 2}, Key{2, 1} },
        { Key{3, 1}, Key{2, 2}, Key{0, 0}, Key{0, 1} },
        { Key{3, 2}, Key{3, 1}, Key{1, 1}, Key{0, 0} },
        { Key{4, 1}, Key{4, 2}, Key{0, 0}, Key{0, 1} },
        { Key{5, 2}, Key{4, 1}, Key{1, 1}, Key{0, 0} },
        { Key{6, 2}, Key{4, 1}, Key{2, 1}, Key{0, 0} },
        { Key{4, 1}, Key{7, 2}, Key{0, 0}, Key{3, 1} },
        { Key{0, 0}, Key{0, 1}, Key{5, 1}, Key{8, 2} },
        { Key{9, 2}, Key{1, 1}, Key{5, 1}, Key{0, 0} },
        { Key{10, 2}, Key{2, 1}, Key{5, 1}, Key{0, 0} },
        { Key{11, 2}, Key{5, 1}, Key{3, 1}, Key{0, 0} }
    }};
};
 
 
// convert key array to corner storage
template<class S, class ReferenceElement, class Transformation, class KeyArray, std::size_t... I>
S keyToCornerStorageImpl(const ReferenceElement& ref, Transformation&& trans, const KeyArray& key, std::index_sequence<I...>)
{
    // key is a pair of a local sub-entity index (first) and the sub-entity's codim (second)
    return { trans(ref.position(key[I].first, key[I].second))... };
}
 
// convert key array to corner storage
template<class S, class ReferenceElement, class Transformation, class T, std::size_t N, class Indices = std::make_index_sequence<N>>
S keyToCornerStorage(const ReferenceElement& ref, Transformation&& trans, const std::array<T, N>& key)
{
    return keyToCornerStorageImpl<S>(ref, trans, key, Indices{});
}
 
// convert key array to corner storage
// for the i-th sub-entity of codim c (e.g. the i-th facet/codim-1-entity for boundaries)
template<class S, class ReferenceElement, class Transformation, class KeyArray, std::size_t... I>
S subEntityKeyToCornerStorageImpl(const ReferenceElement& ref, Transformation&& trans,
                                  unsigned int i, unsigned int c, const KeyArray& key, std::index_sequence<I...>)
{
    // subEntity gives the subEntity number with respect to the codim-0 reference element
    // key is a pair of a local sub-entity index (first) and the sub-entity's codim (second) but here w.r.t. the sub-entity i/c
    return { trans(ref.position(ref.subEntity(i, c, key[I].first, c+key[I].second), c+key[I].second))... };
}
 
// convert key array to corner storage
// for the i-th sub-entity of codim c (e.g. the i-th facet/codim-1-entity for boundaries)
template<class S, class ReferenceElement, class Transformation, class T, std::size_t N, class Indices = std::make_index_sequence<N>>
S subEntityKeyToCornerStorage(const ReferenceElement& ref, Transformation&& trans,
                              unsigned int i, unsigned int c, const std::array<T, N>& key)
{
    return subEntityKeyToCornerStorageImpl<S>(ref, trans, i, c, key, Indices{});
}
 
} // end namespace Detail::Box

template<class GridView, int dim, class ScvType, class ScvfType>
class BoxGeometryHelper;

template <class GridView, class ScvType, class ScvfType>
class BoxGeometryHelper<GridView, 1, ScvType, ScvfType>
{
private:
    using Scalar = typename GridView::ctype;
    using GlobalPosition = typename Dune::FieldVector<Scalar, GridView::dimensionworld>;
    using ScvCornerStorage = typename ScvType::Traits::CornerStorage;
    using ScvfCornerStorage = typename ScvfType::Traits::CornerStorage;
    using ScvGeometry = typename ScvType::Traits::Geometry;
    using ScvfGeometry = typename ScvfType::Traits::Geometry;
    using LocalIndexType = typename ScvType::Traits::LocalIndexType;
 
    using Element = typename GridView::template Codim<0>::Entity;
    using Intersection = typename GridView::Intersection;
 
    static constexpr int dim = 1;
public:
    using DofHelper = PQ1LagrangeDofHelper<GridView>;
 
    explicit BoxGeometryHelper(const typename Element::Geometry& geometry)
    : geo_(geometry)
    {}

    ScvCornerStorage getScvCorners(unsigned int localScvIdx) const
    {
        return getScvCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localScvIdx);
    }

    template<class Transformation>
    static ScvCornerStorage getScvCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvIdx)
    {
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::line>;
        return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
    }
 
    ScvGeometry scvGeometry(unsigned int localScvIdx) const
    {
        return { Dune::GeometryTypes::line, getScvCorners(localScvIdx) };
    }

    ScvfCornerStorage getScvfCorners(unsigned int localScvfIdx) const
    {
        return getScvfCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localScvfIdx);
    }

    template<class Transformation>
    static ScvfCornerStorage getScvfCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvfIdx)
    {
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::line>;
        return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
    }

    ScvfCornerStorage getBoundaryScvfCorners(unsigned int localFacetIndex,
                                             unsigned int) const
    {
        return ScvfCornerStorage{{ geo_.corner(localFacetIndex) }};
    }

    template<class Transformation>
    static ScvfCornerStorage getBoundaryScvfCorners(Dune::GeometryType type,
                                                    Transformation&& trans,
                                                    unsigned int localFacetIndex,
                                                    unsigned int indexInFacet)
    {
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        return trans(ref.position(localFacetIndex, dim));
    }

    GlobalPosition normal(const ScvfCornerStorage& scvfCorners,
                          const std::array<LocalIndexType, 2>&) const
    {
        auto normal = geo_.corner(1) - geo_.corner(0);
        normal /= normal.two_norm();
        return normal;
    }

    std::size_t numInteriorScvf() const
    {
        return referenceElement(geo_).size(dim-1);
    }

    static auto numInteriorScvf(Dune::GeometryType type)
    {
        return Dune::referenceElement<Scalar, dim>(type).size(dim-1);
    }

    std::size_t numScv() const
    {
        return referenceElement(geo_).size(dim);
    }

    const typename Element::Geometry& elementGeometry() const
    { return geo_; }

    static Element::Geometry::LocalCoordinate localScvfCenter(Dune::GeometryType type, unsigned int localScvfIdx)
    {
        return Dumux::center(getScvfCorners_(type, [&](const auto& local){ return local; }, localScvfIdx));
    }

    static Element::Geometry::LocalCoordinate localBoundaryScvfCenter(Dune::GeometryType type, unsigned int localFacetIndex, unsigned int)
    {
        return Dune::referenceElement<Scalar, dim>(type).position(localFacetIndex, dim);
    }
 
private:
    const typename Element::Geometry& geo_; 
};

template <class GridView, class ScvType, class ScvfType>
class BoxGeometryHelper<GridView, 2, ScvType, ScvfType>
{
    using Scalar = typename GridView::ctype;
    using GlobalPosition = typename Dune::FieldVector<Scalar, GridView::dimensionworld>;
    using ScvCornerStorage = typename ScvType::Traits::CornerStorage;
    using ScvfCornerStorage = typename ScvfType::Traits::CornerStorage;
    using LocalIndexType = typename ScvType::Traits::LocalIndexType;
 
    using Element = typename GridView::template Codim<0>::Entity;
    using Intersection = typename GridView::Intersection;
 
    static constexpr auto dim = GridView::dimension;
    static constexpr auto dimWorld = GridView::dimensionworld;
public:
    using DofHelper = PQ1LagrangeDofHelper<GridView>;
 
    explicit BoxGeometryHelper(const typename Element::Geometry& geometry)
    : geo_(geometry)
    {}

    ScvCornerStorage getScvCorners(unsigned int localScvIdx) const
    {
        return getScvCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localScvIdx);
    }

    template<class Transformation>
    static ScvCornerStorage getScvCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvIdx)
    {
        // proceed according to number of corners of the element
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        if (type == Dune::GeometryTypes::triangle)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::triangle>;
            return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
        }
        else if (type == Dune::GeometryTypes::quadrilateral)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::quadrilateral>;
            return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
        }
        else
            DUNE_THROW(Dune::NotImplemented, "Box scv geometries for dim=" << dim
                                                            << " dimWorld=" << dimWorld
                                                            << " type=" << type);
    }

    ScvfCornerStorage getScvfCorners(unsigned int localScvfIdx) const
    {
        return getScvfCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localScvfIdx);
    }

    template<class Transformation>
    static ScvfCornerStorage getScvfCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvfIdx)
    {
        // proceed according to number of corners
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        if (type == Dune::GeometryTypes::triangle)
        {
            using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::triangle>;
            return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
        }
        else if (type == Dune::GeometryTypes::quadrilateral)
        {
            using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::quadrilateral>;
            return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
        }
        else
            DUNE_THROW(Dune::NotImplemented, "Box scvf geometries for dim=" << dim
                                                            << " dimWorld=" << dimWorld
                                                            << " type=" << type);
    }

    ScvfCornerStorage getBoundaryScvfCorners(unsigned int localFacetIndex,
                                             unsigned int indexInFacet) const
    {
        return getBoundaryScvfCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localFacetIndex, indexInFacet);
    }

    template<class Transformation>
    static ScvfCornerStorage getBoundaryScvfCorners(Dune::GeometryType type,
                                                    Transformation&& trans,
                                                    unsigned int localFacetIndex,
                                                    unsigned int indexInFacet)
    {
        // we have to use the corresponding facet geometry as the intersection geometry
        // might be rotated or flipped. This makes sure that the corners (dof location)
        // and corresponding scvfs are sorted in the same way
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::line>;
        constexpr int facetCodim = 1;
        return Detail::Box::subEntityKeyToCornerStorage<ScvfCornerStorage>(ref, trans, localFacetIndex, facetCodim, Corners::keys[indexInFacet]);
    }

    GlobalPosition normal(const ScvfCornerStorage& scvfCorners,
                          const std::array<LocalIndexType, 2>& scvIndices) const
    {
        auto normal = Detail::scvfUnitNormal(geo_, scvfCorners);

        const auto v = geo_.corner(scvIndices[1]) - geo_.corner(scvIndices[0]);
        const auto s = v*normal;
        if (std::signbit(s))
            normal *= -1;
 
        return normal;
    }

    std::size_t numInteriorScvf() const
    {
        return referenceElement(geo_).size(dim-1);
    }

    static auto numInteriorScvf(Dune::GeometryType type)
    {
        return Dune::referenceElement<Scalar, dim>(type).size(dim-1);
    }

    std::size_t numScv() const
    {
        return referenceElement(geo_).size(dim);
    }

    const typename Element::Geometry& elementGeometry() const
    { return geo_; }

    static Element::Geometry::LocalCoordinate localScvfCenter(Dune::GeometryType type, unsigned int localScvfIdx)
    {
        return Dumux::center(getScvfCorners(type, [&](const auto& local){ return local; }, localScvfIdx));
    }

    static Element::Geometry::LocalCoordinate localBoundaryScvfCenter(Dune::GeometryType type, unsigned int localFacetIndex, unsigned int indexInFace)
    {
        return Dumux::center(getBoundaryScvfCorners(type, [&](const auto& local){ return local; }, localFacetIndex, indexInFace));
    }
 
private:
    const typename Element::Geometry& geo_; 
};

template <class GridView, class ScvType, class ScvfType>
class BoxGeometryHelper<GridView, 3, ScvType, ScvfType>
{
    using Scalar = typename GridView::ctype;
    using GlobalPosition = typename Dune::FieldVector<Scalar, GridView::dimensionworld>;
    using ScvCornerStorage = typename ScvType::Traits::CornerStorage;
    using ScvfCornerStorage = typename ScvfType::Traits::CornerStorage;
    using LocalIndexType = typename ScvType::Traits::LocalIndexType;
 
    using Element = typename GridView::template Codim<0>::Entity;
    using Intersection = typename GridView::Intersection;
 
    static constexpr auto dim = GridView::dimension;
    static constexpr auto dimWorld = GridView::dimensionworld;
 
public:
    using DofHelper = PQ1LagrangeDofHelper<GridView>;
 
    explicit BoxGeometryHelper(const typename Element::Geometry& geometry)
    : geo_(geometry)
    {}

    ScvCornerStorage getScvCorners(unsigned int localScvIdx) const
    {
        return getScvCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); }, localScvIdx);
    }

    template<class Transformation>
    static ScvCornerStorage getScvCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvIdx)
    {
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        if (type == Dune::GeometryTypes::tetrahedron)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::tetrahedron>;
            return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
        }
        else if (type == Dune::GeometryTypes::prism)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::prism>;
            return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
        }
        else if (type == Dune::GeometryTypes::hexahedron)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::hexahedron>;
            return Detail::Box::keyToCornerStorage<ScvCornerStorage>(ref, trans, Corners::keys[localScvIdx]);
        }
        else
            DUNE_THROW(Dune::NotImplemented, "Box scv geometries for dim=" << dim
                                                            << " dimWorld=" << dimWorld
                                                            << " type=" << type);
    }

    ScvfCornerStorage getScvfCorners(unsigned int localScvfIdx) const
    {
        return getScvfCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); },  localScvfIdx);
    }

    template<class Transformation>
    static ScvfCornerStorage getScvfCorners(Dune::GeometryType type, Transformation&& trans, unsigned int localScvfIdx)
    {
        // proceed according to number of corners
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        if (type == Dune::GeometryTypes::tetrahedron)
        {
            using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::tetrahedron>;
            return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
        }
        else if (type == Dune::GeometryTypes::prism)
        {
            using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::prism>;
            return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
        }
        else if (type == Dune::GeometryTypes::hexahedron)
        {
            using Corners = Detail::Box::ScvfCorners<Dune::GeometryTypes::hexahedron>;
            return Detail::Box::keyToCornerStorage<ScvfCornerStorage>(ref, trans, Corners::keys[localScvfIdx]);
        }
        else
            DUNE_THROW(Dune::NotImplemented, "Box scvf geometries for dim=" << dim
                                                            << " dimWorld=" << dimWorld
                                                            << " type=" << type);
    }

    ScvfCornerStorage getBoundaryScvfCorners(unsigned int localFacetIndex,
                                             unsigned int indexInFacet) const
    {
        return getBoundaryScvfCorners(geo_.type(), [&](const auto& local){ return geo_.global(local); },  localFacetIndex, indexInFacet);
    }

    template<class Transformation>
    static ScvfCornerStorage getBoundaryScvfCorners(Dune::GeometryType type,
                                                    Transformation&& trans,
                                                    unsigned localFacetIndex,
                                                    unsigned int indexInFacet)
    {
        constexpr int facetCodim = 1;
 
        // we have to use the corresponding facet geometry as the intersection geometry
        // might be rotated or flipped. This makes sure that the corners (dof location)
        // and corresponding scvfs are sorted in the same way
        const auto& ref = Dune::referenceElement<Scalar, dim>(type);
        const auto facetType = ref.type(localFacetIndex, facetCodim);
        if (facetType == Dune::GeometryTypes::triangle)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::triangle>;
            return Detail::Box::subEntityKeyToCornerStorage<ScvfCornerStorage>(ref, trans, localFacetIndex, facetCodim, Corners::keys[indexInFacet]);
        }
        else if (facetType == Dune::GeometryTypes::quadrilateral)
        {
            using Corners = Detail::Box::ScvCorners<Dune::GeometryTypes::quadrilateral>;
            return Detail::Box::subEntityKeyToCornerStorage<ScvfCornerStorage>(ref, trans, localFacetIndex, facetCodim, Corners::keys[indexInFacet]);
        }
        else
            DUNE_THROW(Dune::NotImplemented, "Box boundary scvf geometries for dim=" << dim
                                                            << " dimWorld=" << dimWorld
                                                            << " type=" << facetType);
    }

    GlobalPosition normal(const ScvfCornerStorage& p,
                          const std::array<LocalIndexType, 2>& scvIndices) const
    {
        auto normal = Detail::scvfUnitNormal(geo_, p);
 
        const auto v = geo_.corner(scvIndices[1]) - geo_.corner(scvIndices[0]);
        const auto s = v*normal;
        if (std::signbit(s))
            normal *= -1;
 
        return normal;
    }

    std::size_t numInteriorScvf() const
    {
        return referenceElement(geo_).size(dim-1);
    }

    static auto numInteriorScvf(Dune::GeometryType type)
    {
        return Dune::referenceElement<Scalar, dim>(type).size(dim-1);
    }

    std::size_t numScv() const
    {
        return referenceElement(geo_).size(dim);
    }

    const typename Element::Geometry& elementGeometry() const
    { return geo_; }

    static Element::Geometry::LocalCoordinate localScvfCenter(Dune::GeometryType type, unsigned int localScvfIdx)
    {
        return Dumux::center(getScvfCorners(type, [&](const auto& local){ return local; }, localScvfIdx));
    }

    static Element::Geometry::LocalCoordinate localBoundaryScvfCenter(Dune::GeometryType type, unsigned int localFacetIndex, unsigned int indexInFace)
    {
        return Dumux::center(getBoundaryScvfCorners(type, [&](const auto& local){ return local; }, localFacetIndex, indexInFace));
    }
 
private:
    const typename Element::Geometry& geo_; 
};
 
} // end namespace Dumux
 
#endif