couplingmanager1d3d.hh

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#ifndef DUMUX_MULTIDOMAIN_EMBEDDED_COUPLINGMANAGER_1D3D_HH
#define DUMUX_MULTIDOMAIN_EMBEDDED_COUPLINGMANAGER_1D3D_HH
 
#include <vector>
 
#include <dune/common/timer.hh>
#include <dune/geometry/quadraturerules.hh>
 
#include <dumux/common/properties.hh>
#include <dumux/multidomain/embedded/pointsourcedata.hh>
#include <dumux/multidomain/embedded/integrationpointsource.hh>
#include <dumux/multidomain/embedded/couplingmanagerbase.hh>
#include <dumux/multidomain/embedded/circlepoints.hh>
#include <dumux/multidomain/embedded/extendedsourcestencil.hh>
 
namespace Dumux {
 
enum class EmbeddedCouplingMode
{ line, average, cylindersources, kernel };
 
template<class MDTraits>
struct CircleAveragePointSourceTraits
{
private:
    template<std::size_t i> using SubDomainTypeTag = typename MDTraits::template SubDomain<i>::TypeTag;
    template<std::size_t i> using GridGeometry = GetPropType<SubDomainTypeTag<i>, Properties::GridGeometry>;
    template<std::size_t i> using NumEqVector = GetPropType<SubDomainTypeTag<i>, Properties::NumEqVector>;
public:
    template<std::size_t i>
    using PointSource = IntegrationPointSource<typename GridGeometry<i>::GlobalCoordinate, NumEqVector<i>>;
 
    template<std::size_t i>
    using PointSourceHelper = IntegrationPointSourceHelper;
 
    using PointSourceData = PointSourceDataCircleAverage<MDTraits>;
};
 
template<class MDTraits, EmbeddedCouplingMode mode>
class EmbeddedCouplingManager1d3d;
 
template<class MDTraits>
class EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::line>
: public EmbeddedCouplingManagerBase<MDTraits, EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::line>>
{
    using ThisType = EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::line>;
    using ParentType = EmbeddedCouplingManagerBase<MDTraits, ThisType>;
 
    using Scalar = typename MDTraits::Scalar;
    using SolutionVector = typename MDTraits::SolutionVector;
 
    static constexpr auto bulkIdx = typename MDTraits::template SubDomain<0>::Index();
    static constexpr auto lowDimIdx = typename MDTraits::template SubDomain<1>::Index();
 
    // the sub domain type aliases
    template<std::size_t id> using SubDomainTypeTag = typename MDTraits::template SubDomain<id>::TypeTag;
    template<std::size_t id> using Problem = GetPropType<SubDomainTypeTag<id>, Properties::Problem>;
    template<std::size_t id> using GridGeometry = GetPropType<SubDomainTypeTag<id>, Properties::GridGeometry>;
    template<std::size_t id> using GridView = typename GridGeometry<id>::GridView;
    template<std::size_t id> using Element = typename GridView<id>::template Codim<0>::Entity;
 
public:
    static constexpr EmbeddedCouplingMode couplingMode = EmbeddedCouplingMode::line;
 
    using ParentType::ParentType;
 
    void init(std::shared_ptr<Problem<bulkIdx>> bulkProblem,
              std::shared_ptr<Problem<lowDimIdx>> lowDimProblem,
              const SolutionVector& curSol)
    {
        ParentType::init(bulkProblem, lowDimProblem, curSol);
        computeLowDimVolumeFractions();
    }
 
    void computeLowDimVolumeFractions()
    {
        // resize the storage vector
        lowDimVolumeInBulkElement_.resize(this->gridView(bulkIdx).size(0));
 
        // compute the low dim volume fractions
        for (const auto& is : intersections(this->glue()))
        {
            // all inside elements are identical...
            const auto& inside = is.targetEntity(0);
            const auto intersectionGeometry = is.geometry();
            const std::size_t lowDimElementIdx = this->problem(lowDimIdx).gridGeometry().elementMapper().index(inside);
 
            // compute the volume the low-dim domain occupies in the bulk domain if it were full-dimensional
            const auto radius = this->problem(lowDimIdx).spatialParams().radius(lowDimElementIdx);
            for (std::size_t outsideIdx = 0; outsideIdx < is.numDomainNeighbors(); ++outsideIdx)
            {
                const auto& outside = is.domainEntity(outsideIdx);
                const std::size_t bulkElementIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(outside);
                lowDimVolumeInBulkElement_[bulkElementIdx] += intersectionGeometry.volume()*M_PI*radius*radius;
            }
        }
    }
 
    // \{
 
    Scalar radius(std::size_t id) const
    {
        const auto& data = this->pointSourceData()[id];
        return this->problem(lowDimIdx).spatialParams().radius(data.lowDimElementIdx());
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolume(const Element<bulkIdx>& element) const
    {
        const auto eIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(element);
        return lowDimVolumeInBulkElement_[eIdx];
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolumeFraction(const Element<bulkIdx>& element) const
    {
        const auto totalVolume = element.geometry().volume();
        return lowDimVolume(element) / totalVolume;
    }
 
    // \}
 
private:
    std::vector<Scalar> lowDimVolumeInBulkElement_;
};
 
template<class MDTraits>
class EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::average>
: public EmbeddedCouplingManagerBase<MDTraits, EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::average>,
                                     CircleAveragePointSourceTraits<MDTraits>>
{
    using ThisType = EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::average>;
    using ParentType = EmbeddedCouplingManagerBase<MDTraits, ThisType, CircleAveragePointSourceTraits<MDTraits>>;
    using Scalar = typename MDTraits::Scalar;
    using SolutionVector = typename MDTraits::SolutionVector;
    using PointSourceData = typename ParentType::PointSourceTraits::PointSourceData;
 
    static constexpr auto bulkIdx = typename MDTraits::template SubDomain<0>::Index();
    static constexpr auto lowDimIdx = typename MDTraits::template SubDomain<1>::Index();
 
    // the sub domain type aliases
    template<std::size_t id> using SubDomainTypeTag = typename MDTraits::template SubDomain<id>::TypeTag;
    template<std::size_t id> using Problem = GetPropType<SubDomainTypeTag<id>, Properties::Problem>;
    template<std::size_t id> using GridGeometry = GetPropType<SubDomainTypeTag<id>, Properties::GridGeometry>;
    template<std::size_t id> using GridView = typename GridGeometry<id>::GridView;
    template<std::size_t id> using Element = typename GridView<id>::template Codim<0>::Entity;
 
    using GlobalPosition = typename Element<bulkIdx>::Geometry::GlobalCoordinate;
 
    template<std::size_t id>
    static constexpr bool isBox()
    { return GridGeometry<id>::discMethod == DiscretizationMethod::box; }
 
 
public:
    enum {
        bulkDim = GridView<bulkIdx>::dimension,
        lowDimDim = GridView<lowDimIdx>::dimension,
        dimWorld = GridView<bulkIdx>::dimensionworld
    };
 
    static constexpr EmbeddedCouplingMode couplingMode = EmbeddedCouplingMode::average;
 
    using ParentType::ParentType;
 
    void init(std::shared_ptr<Problem<bulkIdx>> bulkProblem,
              std::shared_ptr<Problem<lowDimIdx>> lowDimProblem,
              const SolutionVector& curSol)
    {
        ParentType::init(bulkProblem, lowDimProblem, curSol);
        computeLowDimVolumeFractions();
    }
 
    template<std::size_t id, class JacobianPattern>
    void extendJacobianPattern(Dune::index_constant<id> domainI, JacobianPattern& pattern) const
    {
        extendedSourceStencil_.extendJacobianPattern(*this, domainI, pattern);
    }
 
    template<std::size_t i, class LocalAssemblerI, class JacobianMatrixDiagBlock, class GridVariables>
    void evalAdditionalDomainDerivatives(Dune::index_constant<i> domainI,
                                         const LocalAssemblerI& localAssemblerI,
                                         const typename LocalAssemblerI::LocalResidual::ElementResidualVector&,
                                         JacobianMatrixDiagBlock& A,
                                         GridVariables& gridVariables)
    {
        extendedSourceStencil_.evalAdditionalDomainDerivatives(*this, domainI, localAssemblerI, this->curSol(), A, gridVariables);
    }
 
    /* \brief Compute integration point point sources and associated data
     *
     * This method uses grid glue to intersect the given grids. Over each intersection
     * we later need to integrate a source term. This method places point sources
     * at each quadrature point and provides the point source with the necessary
     * information to compute integrals (quadrature weight and integration element)
     * \param order The order of the quadrature rule for integration of sources over an intersection
     * \param verbose If the point source computation is verbose
     */
    void computePointSourceData(std::size_t order = 1, bool verbose = false)
    {
        // Initialize the bulk bounding box tree
        const auto& bulkTree = this->problem(bulkIdx).gridGeometry().boundingBoxTree();
 
        // initilize the maps
        // do some logging and profiling
        Dune::Timer watch;
        std::cout << "Initializing the point sources..." << std::endl;
 
        // clear all internal members like pointsource vectors and stencils
        // initializes the point source id counter
        this->clear();
        extendedSourceStencil_.stencil().clear();
 
        // precompute the vertex indices for efficiency
        this->preComputeVertexIndices(bulkIdx);
        this->preComputeVertexIndices(lowDimIdx);
 
        // iterate over all lowdim elements
        const auto& lowDimProblem = this->problem(lowDimIdx);
        for (const auto& lowDimElement : elements(this->gridView(lowDimIdx)))
        {
            // get the Gaussian quadrature rule for the low dim element
            const auto lowDimGeometry = lowDimElement.geometry();
            const auto& quad = Dune::QuadratureRules<Scalar, lowDimDim>::rule(lowDimGeometry.type(), order);
 
            const auto lowDimElementIdx = lowDimProblem.gridGeometry().elementMapper().index(lowDimElement);
 
            // apply the Gaussian quadrature rule and define point sources at each quadrature point
            // note that the approximation is not optimal if
            // (a) the one-dimensional elements are too large,
            // (b) whenever a one-dimensional element is split between two or more elements,
            // (c) when gradients of important quantities in the three-dimensional domain are large.
 
            // iterate over all quadrature points
            for (auto&& qp : quad)
            {
                // global position of the quadrature point
                const auto globalPos = lowDimGeometry.global(qp.position());
 
                const auto bulkElementIndices = intersectingEntities(globalPos, bulkTree);
 
                // do not add a point source if the qp is outside of the 3d grid
                // this is equivalent to having a source of zero for that qp
                if (bulkElementIndices.empty())
                    continue;
 
                // get circle average connectivity and interpolation data
 
                static const auto numIp = getParam<int>("MixedDimension.NumCircleSegments");
                const auto radius = lowDimProblem.spatialParams().radius(lowDimElementIdx);
                const auto normal = lowDimGeometry.corner(1)-lowDimGeometry.corner(0);
                const auto weight = 2*M_PI*radius/numIp;
 
                const auto circlePoints = EmbeddedCoupling::circlePoints(globalPos, normal, radius, numIp);
                std::vector<Scalar> circleIpWeight(circlePoints.size());
                std::vector<std::size_t> circleStencil(circlePoints.size());
                // for box
                std::unordered_map<std::size_t, std::vector<std::size_t> > circleCornerIndices;
                using ShapeValues = std::vector<Dune::FieldVector<Scalar, 1> >;
                std::unordered_map<std::size_t, ShapeValues> circleShapeValues;
 
                for (int k = 0; k < circlePoints.size(); ++k)
                {
                    const auto circleBulkElementIndices = intersectingEntities(circlePoints[k], bulkTree);
                    if (circleBulkElementIndices.empty())
                        continue;
 
                    const auto bulkElementIdx = circleBulkElementIndices[0];
                    circleStencil[k] = bulkElementIdx;
                    circleIpWeight[k] = weight;
 
                    if (isBox<bulkIdx>())
                    {
                        if (!static_cast<bool>(circleCornerIndices.count(bulkElementIdx)))
                        {
                            const auto bulkElement = this->problem(bulkIdx).gridGeometry().element(bulkElementIdx);
                            circleCornerIndices[bulkElementIdx] = this->vertexIndices(bulkIdx, bulkElementIdx);
 
                            // evaluate shape functions at the integration point
                            const auto bulkGeometry = bulkElement.geometry();
                            this->getShapeValues(bulkIdx, this->problem(bulkIdx).gridGeometry(), bulkGeometry, circlePoints[k], circleShapeValues[bulkElementIdx]);
                        }
                    }
                }
 
                // export low dim circle stencil
                if (isBox<bulkIdx>())
                {
                    // we insert all vertices and make it unique later
                    for (const auto& vertices : circleCornerIndices)
                    {
                        this->couplingStencils(lowDimIdx)[lowDimElementIdx].insert(this->couplingStencils(lowDimIdx)[lowDimElementIdx].end(),
                                                                                   vertices.second.begin(), vertices.second.end());
 
                    }
                }
                else
                {
                    this->couplingStencils(lowDimIdx)[lowDimElementIdx].insert(this->couplingStencils(lowDimIdx)[lowDimElementIdx].end(),
                                                                               circleStencil.begin(), circleStencil.end());
                }
 
                // loop over the bulk elements at the integration points (usually one except when it is on a face or edge or vertex)
                for (auto bulkElementIdx : bulkElementIndices)
                {
                    const auto id = this->idCounter_++;
                    const auto ie = lowDimGeometry.integrationElement(qp.position());
                    const auto qpweight = qp.weight();
 
                    this->pointSources(bulkIdx).emplace_back(globalPos, id, qpweight, ie, std::vector<std::size_t>({bulkElementIdx}));
                    this->pointSources(bulkIdx).back().setEmbeddings(bulkElementIndices.size());
                    this->pointSources(lowDimIdx).emplace_back(globalPos, id, qpweight, ie, std::vector<std::size_t>({lowDimElementIdx}));
                    this->pointSources(lowDimIdx).back().setEmbeddings(bulkElementIndices.size());
 
                    // pre compute additional data used for the evaluation of
                    // the actual solution dependent source term
                    PointSourceData psData;
 
                    if (isBox<lowDimIdx>())
                    {
                        ShapeValues shapeValues;
                        this->getShapeValues(lowDimIdx, this->problem(lowDimIdx).gridGeometry(), lowDimGeometry, globalPos, shapeValues);
                        psData.addLowDimInterpolation(shapeValues, this->vertexIndices(lowDimIdx, lowDimElementIdx), lowDimElementIdx);
                    }
                    else
                    {
                        psData.addLowDimInterpolation(lowDimElementIdx);
                    }
 
                    // add data needed to compute integral over the circle
                    if (isBox<bulkIdx>())
                    {
                        psData.addCircleInterpolation(circleCornerIndices, circleShapeValues, circleIpWeight, circleStencil);
 
                        const auto bulkGeometry = this->problem(bulkIdx).gridGeometry().element(bulkElementIdx).geometry();
                        ShapeValues shapeValues;
                        this->getShapeValues(bulkIdx, this->problem(bulkIdx).gridGeometry(), bulkGeometry, globalPos, shapeValues);
                        psData.addBulkInterpolation(shapeValues, this->vertexIndices(bulkIdx, bulkElementIdx), bulkElementIdx);
                    }
                    else
                    {
                        psData.addCircleInterpolation(circleIpWeight, circleStencil);
                        psData.addBulkInterpolation(bulkElementIdx);
                    }
 
                    // publish point source data in the global vector
                    this->pointSourceData().emplace_back(std::move(psData));
 
                    // export the bulk coupling stencil
                    if (isBox<lowDimIdx>())
                    {
                        this->couplingStencils(bulkIdx)[bulkElementIdx].insert(this->couplingStencils(bulkIdx)[bulkElementIdx].end(),
                                                                               this->vertexIndices(lowDimIdx, lowDimElementIdx).begin(),
                                                                               this->vertexIndices(lowDimIdx, lowDimElementIdx).end());
 
                    }
                    else
                    {
                        this->couplingStencils(bulkIdx)[bulkElementIdx].push_back(lowDimElementIdx);
                    }
 
                    // export bulk circle stencil
                    if (isBox<bulkIdx>())
                    {
                        // we insert all vertices and make it unique later
                        for (const auto& vertices : circleCornerIndices)
                        {
                            extendedSourceStencil_.stencil()[bulkElementIdx].insert(extendedSourceStencil_.stencil()[bulkElementIdx].end(),
                                                                       vertices.second.begin(), vertices.second.end());
 
                        }
                    }
                    else
                    {
                        extendedSourceStencil_.stencil()[bulkElementIdx].insert(extendedSourceStencil_.stencil()[bulkElementIdx].end(),
                                                                   circleStencil.begin(), circleStencil.end());
                    }
                }
            }
        }
 
        // make the circle stencil unique (for source derivatives)
        for (auto&& stencil : extendedSourceStencil_.stencil())
        {
            std::sort(stencil.second.begin(), stencil.second.end());
            stencil.second.erase(std::unique(stencil.second.begin(), stencil.second.end()), stencil.second.end());
 
            // remove the vertices element (box)
            if (isBox<bulkIdx>())
            {
                const auto& indices = this->vertexIndices(bulkIdx, stencil.first);
                stencil.second.erase(std::remove_if(stencil.second.begin(), stencil.second.end(),
                                                   [&](auto i){ return std::find(indices.begin(), indices.end(), i) != indices.end(); }),
                                     stencil.second.end());
            }
            // remove the own element (cell-centered)
            else
            {
                stencil.second.erase(std::remove_if(stencil.second.begin(), stencil.second.end(),
                                                   [&](auto i){ return i == stencil.first; }),
                                     stencil.second.end());
            }
        }
 
        // make stencils unique
        using namespace Dune::Hybrid;
        forEach(integralRange(Dune::index_constant<2>{}), [&](const auto domainIdx)
        {
            for (auto&& stencil : this->couplingStencils(domainIdx))
            {
                std::sort(stencil.second.begin(), stencil.second.end());
                stencil.second.erase(std::unique(stencil.second.begin(), stencil.second.end()), stencil.second.end());
            }
        });
 
        std::cout << "took " << watch.elapsed() << " seconds." << std::endl;
    }
 
    void computeLowDimVolumeFractions()
    {
        // resize the storage vector
        lowDimVolumeInBulkElement_.resize(this->gridView(bulkIdx).size(0));
 
        // compute the low dim volume fractions
        for (const auto& is : intersections(this->glue()))
        {
            // all inside elements are identical...
            const auto& inside = is.targetEntity(0);
            const auto intersectionGeometry = is.geometry();
            const std::size_t lowDimElementIdx = this->problem(lowDimIdx).gridGeometry().elementMapper().index(inside);
 
            // compute the volume the low-dim domain occupies in the bulk domain if it were full-dimensional
            const auto radius = this->problem(lowDimIdx).spatialParams().radius(lowDimElementIdx);
            for (std::size_t outsideIdx = 0; outsideIdx < is.numDomainNeighbors(); ++outsideIdx)
            {
                const auto& outside = is.domainEntity(outsideIdx);
                const std::size_t bulkElementIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(outside);
                lowDimVolumeInBulkElement_[bulkElementIdx] += intersectionGeometry.volume()*M_PI*radius*radius;
            }
        }
    }
 
    // \{
 
    Scalar radius(std::size_t id) const
    {
        const auto& data = this->pointSourceData()[id];
        return this->problem(lowDimIdx).spatialParams().radius(data.lowDimElementIdx());
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolume(const Element<bulkIdx>& element) const
    {
        const auto eIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(element);
        return lowDimVolumeInBulkElement_[eIdx];
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolumeFraction(const Element<bulkIdx>& element) const
    {
        const auto totalVolume = element.geometry().volume();
        return lowDimVolume(element) / totalVolume;
    }
 
    // \}
 
private:
    EmbeddedCoupling::ExtendedSourceStencil<ThisType> extendedSourceStencil_;
 
    std::vector<Scalar> lowDimVolumeInBulkElement_;
};
 
 
template<class MDTraits>
class EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::cylindersources>
: public EmbeddedCouplingManagerBase<MDTraits, EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::cylindersources>>
{
    using ThisType = EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::cylindersources>;
    using ParentType = EmbeddedCouplingManagerBase<MDTraits, ThisType>;
    using Scalar = typename MDTraits::Scalar;
    using SolutionVector = typename MDTraits::SolutionVector;
    using PointSourceData = typename ParentType::PointSourceTraits::PointSourceData;
 
    static constexpr auto bulkIdx = typename MDTraits::template SubDomain<0>::Index();
    static constexpr auto lowDimIdx = typename MDTraits::template SubDomain<1>::Index();
 
    // the sub domain type aliases
    template<std::size_t id> using SubDomainTypeTag = typename MDTraits::template SubDomain<id>::TypeTag;
    template<std::size_t id> using Problem = GetPropType<SubDomainTypeTag<id>, Properties::Problem>;
    template<std::size_t id> using GridGeometry = GetPropType<SubDomainTypeTag<id>, Properties::GridGeometry>;
    template<std::size_t id> using GridView = typename GridGeometry<id>::GridView;
    template<std::size_t id> using Element = typename GridView<id>::template Codim<0>::Entity;
 
    using GlobalPosition = typename Element<bulkIdx>::Geometry::GlobalCoordinate;
 
    template<std::size_t id>
    static constexpr bool isBox()
    { return GridGeometry<id>::discMethod == DiscretizationMethod::box; }
 
    enum {
        bulkDim = GridView<bulkIdx>::dimension,
        lowDimDim = GridView<lowDimIdx>::dimension,
        dimWorld = GridView<bulkIdx>::dimensionworld
    };
public:
    static constexpr EmbeddedCouplingMode couplingMode = EmbeddedCouplingMode::cylindersources;
 
    using ParentType::ParentType;
 
    void init(std::shared_ptr<Problem<bulkIdx>> bulkProblem,
              std::shared_ptr<Problem<lowDimIdx>> lowDimProblem,
              const SolutionVector& curSol)
    {
        ParentType::init(bulkProblem, lowDimProblem, curSol);
        computeLowDimVolumeFractions();
    }
 
    /* \brief Compute integration point point sources and associated data
     *
     * This method uses grid glue to intersect the given grids. Over each intersection
     * we later need to integrate a source term. This method places point sources
     * at each quadrature point and provides the point source with the necessary
     * information to compute integrals (quadrature weight and integration element)
     * \param order The order of the quadrature rule for integration of sources over an intersection
     * \param verbose If the point source computation is verbose
     */
    void computePointSourceData(std::size_t order = 1, bool verbose = false)
    {
        // Initialize the bulk bounding box tree
        const auto& bulkTree = this->problem(bulkIdx).gridGeometry().boundingBoxTree();
 
        // initilize the maps
        // do some logging and profiling
        Dune::Timer watch;
        std::cout << "Initializing the point sources..." << std::endl;
 
        // clear all internal members like pointsource vectors and stencils
        // initializes the point source id counter
        this->clear();
 
        // precompute the vertex indices for efficiency
        this->preComputeVertexIndices(bulkIdx);
        this->preComputeVertexIndices(lowDimIdx);
 
        // iterate over all lowdim elements
        const auto& lowDimProblem = this->problem(lowDimIdx);
        for (const auto& lowDimElement : elements(this->gridView(lowDimIdx)))
        {
            // get the Gaussian quadrature rule for the low dim element
            const auto lowDimGeometry = lowDimElement.geometry();
            const auto& quad = Dune::QuadratureRules<Scalar, lowDimDim>::rule(lowDimGeometry.type(), order);
 
            const auto lowDimElementIdx = lowDimProblem.gridGeometry().elementMapper().index(lowDimElement);
 
            // apply the Gaussian quadrature rule and define point sources at each quadrature point
            // note that the approximation is not optimal if
            // (a) the one-dimensional elements are too large,
            // (b) whenever a one-dimensional element is split between two or more elements,
            // (c) when gradients of important quantities in the three-dimensional domain are large.
 
            // iterate over all quadrature points
            for (auto&& qp : quad)
            {
                // global position of the quadrature point
                const auto globalPos = lowDimGeometry.global(qp.position());
 
                const auto bulkElementIndices = intersectingEntities(globalPos, bulkTree);
 
                // do not add a point source if the qp is outside of the 3d grid
                // this is equivalent to having a source of zero for that qp
                if (bulkElementIndices.empty())
                    continue;
 
                // get points on the cylinder surface at the integration point
 
                static const auto numIp = getParam<int>("MixedDimension.NumCircleSegments", 25);
                const auto radius = lowDimProblem.spatialParams().radius(lowDimElementIdx);
                const auto normal = lowDimGeometry.corner(1)-lowDimGeometry.corner(0);
                const auto integrationElement = lowDimGeometry.integrationElement(qp.position())*2*M_PI*radius/Scalar(numIp);
                const auto weight = qp.weight()/(2*M_PI*radius);
 
                const auto circlePoints = EmbeddedCoupling::circlePoints(globalPos, normal, radius, numIp);
                for (int k = 0; k < circlePoints.size(); ++k)
                {
                    const auto& circlePos = circlePoints[k];
                    const auto circleBulkElementIndices = intersectingEntities(circlePos, bulkTree);
                    if (circleBulkElementIndices.empty())
                        continue;
 
                    // loop over the bulk elements at the integration points (usually one except when it is on a face or edge or vertex)
                    // and add a point source at every point on the circle
                    for (const auto bulkElementIdx : circleBulkElementIndices)
                    {
                        const auto id = this->idCounter_++;
                        this->pointSources(bulkIdx).emplace_back(circlePos, id, weight, integrationElement, std::vector<std::size_t>({bulkElementIdx}));
                        this->pointSources(bulkIdx).back().setEmbeddings(circleBulkElementIndices.size());
                        this->pointSources(lowDimIdx).emplace_back(globalPos, id, weight, integrationElement, std::vector<std::size_t>({lowDimElementIdx}));
                        this->pointSources(lowDimIdx).back().setEmbeddings(circleBulkElementIndices.size());
 
                        // pre compute additional data used for the evaluation of
                        // the actual solution dependent source term
                        PointSourceData psData;
 
                        if (isBox<lowDimIdx>())
                        {
                            using ShapeValues = std::vector<Dune::FieldVector<Scalar, 1> >;
                            ShapeValues shapeValues;
                            this->getShapeValues(lowDimIdx, this->problem(lowDimIdx).gridGeometry(), lowDimGeometry, globalPos, shapeValues);
                            psData.addLowDimInterpolation(shapeValues, this->vertexIndices(lowDimIdx, lowDimElementIdx), lowDimElementIdx);
                        }
                        else
                        {
                            psData.addLowDimInterpolation(lowDimElementIdx);
                        }
 
                        // add data needed to compute integral over the circle
                        if (isBox<bulkIdx>())
                        {
                            using ShapeValues = std::vector<Dune::FieldVector<Scalar, 1> >;
                            const auto bulkGeometry = this->problem(bulkIdx).gridGeometry().element(bulkElementIdx).geometry();
                            ShapeValues shapeValues;
                            this->getShapeValues(bulkIdx, this->problem(bulkIdx).gridGeometry(), bulkGeometry, circlePos, shapeValues);
                            psData.addBulkInterpolation(shapeValues, this->vertexIndices(bulkIdx, bulkElementIdx), bulkElementIdx);
                        }
                        else
                        {
                            psData.addBulkInterpolation(bulkElementIdx);
                        }
 
                        // publish point source data in the global vector
                        this->pointSourceData().emplace_back(std::move(psData));
 
                        // export the lowdim coupling stencil
                        // we insert all vertices / elements and make it unique later
                        if (isBox<bulkIdx>())
                        {
                            const auto& vertices = this->vertexIndices(bulkIdx, bulkElementIdx);
                            this->couplingStencils(lowDimIdx)[lowDimElementIdx].insert(this->couplingStencils(lowDimIdx)[lowDimElementIdx].end(),
                                                                                       vertices.begin(), vertices.end());
                        }
                        else
                        {
                            this->couplingStencils(lowDimIdx)[lowDimElementIdx].push_back(bulkElementIdx);
                        }
 
                        // export the bulk coupling stencil
                        // we insert all vertices / elements and make it unique later
                        if (isBox<lowDimIdx>())
                        {
                            const auto& vertices = this->vertexIndices(lowDimIdx, lowDimElementIdx);
                            this->couplingStencils(bulkIdx)[bulkElementIdx].insert(this->couplingStencils(bulkIdx)[bulkElementIdx].end(),
                                                                                   vertices.begin(), vertices.end());
 
                        }
                        else
                        {
                            this->couplingStencils(bulkIdx)[bulkElementIdx].push_back(lowDimElementIdx);
                        }
 
                    }
                }
            }
        }
 
        // make stencils unique
        using namespace Dune::Hybrid;
        forEach(integralRange(Dune::index_constant<2>{}), [&](const auto domainIdx)
        {
            for (auto&& stencil : this->couplingStencils(domainIdx))
            {
                std::sort(stencil.second.begin(), stencil.second.end());
                stencil.second.erase(std::unique(stencil.second.begin(), stencil.second.end()), stencil.second.end());
            }
        });
 
        std::cout << "took " << watch.elapsed() << " seconds." << std::endl;
    }
 
    void computeLowDimVolumeFractions()
    {
        // resize the storage vector
        lowDimVolumeInBulkElement_.resize(this->gridView(bulkIdx).size(0));
 
        // compute the low dim volume fractions
        for (const auto& is : intersections(this->glue()))
        {
            // all inside elements are identical...
            const auto& inside = is.targetEntity(0);
            const auto intersectionGeometry = is.geometry();
            const std::size_t lowDimElementIdx = this->problem(lowDimIdx).gridGeometry().elementMapper().index(inside);
 
            // compute the volume the low-dim domain occupies in the bulk domain if it were full-dimensional
            const auto radius = this->problem(lowDimIdx).spatialParams().radius(lowDimElementIdx);
            for (std::size_t outsideIdx = 0; outsideIdx < is.numDomainNeighbors(); ++outsideIdx)
            {
                const auto& outside = is.domainEntity(outsideIdx);
                const std::size_t bulkElementIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(outside);
                lowDimVolumeInBulkElement_[bulkElementIdx] += intersectionGeometry.volume()*M_PI*radius*radius;
            }
        }
    }
 
    // \{
 
    Scalar radius(std::size_t id) const
    {
        const auto& data = this->pointSourceData()[id];
        return this->problem(lowDimIdx).spatialParams().radius(data.lowDimElementIdx());
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolume(const Element<bulkIdx>& element) const
    {
        const auto eIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(element);
        return lowDimVolumeInBulkElement_[eIdx];
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolumeFraction(const Element<bulkIdx>& element) const
    {
        const auto totalVolume = element.geometry().volume();
        return lowDimVolume(element) / totalVolume;
    }
 
    // \}
 
private:
 
    std::vector<Scalar> lowDimVolumeInBulkElement_;
};
 
 
template<class MDTraits>
class EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::kernel>
: public EmbeddedCouplingManagerBase<MDTraits, EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::kernel>>
{
    using ThisType = EmbeddedCouplingManager1d3d<MDTraits, EmbeddedCouplingMode::kernel>;
    using ParentType = EmbeddedCouplingManagerBase<MDTraits, ThisType>;
    using Scalar = typename MDTraits::Scalar;
    using SolutionVector = typename MDTraits::SolutionVector;
    using PointSourceData = typename ParentType::PointSourceTraits::PointSourceData;
 
    static constexpr auto bulkIdx = typename MDTraits::template SubDomain<0>::Index();
    static constexpr auto lowDimIdx = typename MDTraits::template SubDomain<1>::Index();
 
    // the sub domain type aliases
    template<std::size_t id> using SubDomainTypeTag = typename MDTraits::template SubDomain<id>::TypeTag;
    template<std::size_t id> using Problem = GetPropType<SubDomainTypeTag<id>, Properties::Problem>;
    template<std::size_t id> using GridGeometry = GetPropType<SubDomainTypeTag<id>, Properties::GridGeometry>;
    template<std::size_t id> using GridView = typename GridGeometry<id>::GridView;
    template<std::size_t id> using Element = typename GridView<id>::template Codim<0>::Entity;
 
    using GlobalPosition = typename Element<bulkIdx>::Geometry::GlobalCoordinate;
 
    template<std::size_t id>
    static constexpr bool isBox()
    { return GridGeometry<id>::discMethod == DiscretizationMethod::box; }
 
    enum {
        bulkDim = GridView<bulkIdx>::dimension,
        lowDimDim = GridView<lowDimIdx>::dimension,
        dimWorld = GridView<bulkIdx>::dimensionworld
    };
public:
    static constexpr EmbeddedCouplingMode couplingMode = EmbeddedCouplingMode::kernel;
 
    using ParentType::ParentType;
 
    void init(std::shared_ptr<Problem<bulkIdx>> bulkProblem,
              std::shared_ptr<Problem<lowDimIdx>> lowDimProblem,
              const SolutionVector& curSol)
    {
        ParentType::init(bulkProblem, lowDimProblem, curSol);
        computeLowDimVolumeFractions();
    }
 
    template<std::size_t id, class JacobianPattern>
    void extendJacobianPattern(Dune::index_constant<id> domainI, JacobianPattern& pattern) const
    {
        extendedSourceStencil_.extendJacobianPattern(*this, domainI, pattern);
    }
 
    template<std::size_t i, class LocalAssemblerI, class JacobianMatrixDiagBlock, class GridVariables>
    void evalAdditionalDomainDerivatives(Dune::index_constant<i> domainI,
                                         const LocalAssemblerI& localAssemblerI,
                                         const typename LocalAssemblerI::LocalResidual::ElementResidualVector&,
                                         JacobianMatrixDiagBlock& A,
                                         GridVariables& gridVariables)
    {
        extendedSourceStencil_.evalAdditionalDomainDerivatives(*this, domainI, localAssemblerI, this->curSol(), A, gridVariables);
    }
 
    /* \brief Compute integration point point sources and associated data
     *
     * This method uses grid glue to intersect the given grids. Over each intersection
     * we later need to integrate a source term. This method places point sources
     * at each quadrature point and provides the point source with the necessary
     * information to compute integrals (quadrature weight and integration element)
     * \param order The order of the quadrature rule for integration of sources over an intersection
     * \param verbose If the point source computation is verbose
     */
    void computePointSourceData(std::size_t order = 1, bool verbose = false)
    {
        // initilize the maps
        // do some logging and profiling
        Dune::Timer watch;
        std::cout << "Initializing the point sources..." << std::endl;
 
        // clear all internal members like pointsource vectors and stencils
        // initializes the point source id counter
        this->clear();
        bulkSourceIds_.clear();
        bulkSourceWeights_.clear();
        extendedSourceStencil_.stencil().clear();
 
        // precompute the vertex indices for efficiency for the box method
        this->preComputeVertexIndices(bulkIdx);
        this->preComputeVertexIndices(lowDimIdx);
 
        const auto& bulkFvGridGeometry = this->problem(bulkIdx).gridGeometry();
        const auto& lowDimFvGridGeometry = this->problem(lowDimIdx).gridGeometry();
 
        bulkSourceIds_.resize(this->gridView(bulkIdx).size(0));
        bulkSourceWeights_.resize(this->gridView(bulkIdx).size(0));
 
        // intersect the bounding box trees
        this->glueGrids();
 
        this->pointSourceData().reserve(this->glue().size());
        this->averageDistanceToBulkCell().reserve(this->glue().size());
        const Scalar kernelWidth = getParam<Scalar>("MixedDimension.KernelWidth");
        for (const auto& is : intersections(this->glue()))
        {
            // all inside elements are identical...
            const auto& inside = is.targetEntity(0);
            // get the intersection geometry for integrating over it
            const auto intersectionGeometry = is.geometry();
 
            // get the Gaussian quadrature rule for the local intersection
            const auto& quad = Dune::QuadratureRules<Scalar, lowDimDim>::rule(intersectionGeometry.type(), order);
            const std::size_t lowDimElementIdx = lowDimFvGridGeometry.elementMapper().index(inside);
 
            // iterate over all quadrature points and place a source
            // for 1d: make a new point source
            // for 3d: make a new kernel volume source
            for (auto&& qp : quad)
            {
                // compute the coupling stencils
                for (std::size_t outsideIdx = 0; outsideIdx < is.numDomainNeighbors(); ++outsideIdx)
                {
                    const auto& outside = is.domainEntity(outsideIdx);
                    const std::size_t bulkElementIdx = bulkFvGridGeometry.elementMapper().index(outside);
 
                    // each quadrature point will be a point source for the sub problem
                    const auto globalPos = intersectionGeometry.global(qp.position());
                    const auto id = this->idCounter_++;
                    const auto qpweight = qp.weight();
                    const auto ie = intersectionGeometry.integrationElement(qp.position());
                    this->pointSources(lowDimIdx).emplace_back(globalPos, id, qpweight, ie, std::vector<std::size_t>({lowDimElementIdx}));
                    this->pointSources(lowDimIdx).back().setEmbeddings(is.numDomainNeighbors());
                    computeBulkSource(globalPos, kernelWidth, id, lowDimElementIdx, bulkElementIdx, qpweight*ie/is.numDomainNeighbors());
 
                    // pre compute additional data used for the evaluation of
                    // the actual solution dependent source term
                    PointSourceData psData;
 
                    if (isBox<lowDimIdx>())
                    {
                        using ShapeValues = std::vector<Dune::FieldVector<Scalar, 1> >;
                        const auto lowDimGeometry = this->problem(lowDimIdx).gridGeometry().element(lowDimElementIdx).geometry();
                        ShapeValues shapeValues;
                        this->getShapeValues(lowDimIdx, this->problem(lowDimIdx).gridGeometry(), lowDimGeometry, globalPos, shapeValues);
                        psData.addLowDimInterpolation(shapeValues, this->vertexIndices(lowDimIdx, lowDimElementIdx), lowDimElementIdx);
                    }
                    else
                    {
                        psData.addLowDimInterpolation(lowDimElementIdx);
                    }
 
                    // add data needed to compute integral over the circle
                    if (isBox<bulkIdx>())
                    {
                        using ShapeValues = std::vector<Dune::FieldVector<Scalar, 1> >;
                        const auto bulkGeometry = this->problem(bulkIdx).gridGeometry().element(bulkElementIdx).geometry();
                        ShapeValues shapeValues;
                        this->getShapeValues(bulkIdx, this->problem(bulkIdx).gridGeometry(), bulkGeometry, globalPos, shapeValues);
                        psData.addBulkInterpolation(shapeValues, this->vertexIndices(bulkIdx, bulkElementIdx), bulkElementIdx);
                    }
                    else
                    {
                        psData.addBulkInterpolation(bulkElementIdx);
                    }
 
                    // publish point source data in the global vector
                    this->pointSourceData().emplace_back(std::move(psData));
 
                    // compute average distance to bulk cell
                    this->averageDistanceToBulkCell().push_back(this->computeDistance(outside.geometry(), globalPos));
 
                    // export the lowdim coupling stencil
                    // we insert all vertices / elements and make it unique later
                    if (isBox<bulkIdx>())
                    {
                        const auto& vertices = this->vertexIndices(bulkIdx, bulkElementIdx);
                        this->couplingStencils(lowDimIdx)[lowDimElementIdx].insert(this->couplingStencils(lowDimIdx)[lowDimElementIdx].end(),
                                                                                   vertices.begin(), vertices.end());
                    }
                    else
                    {
                        this->couplingStencils(lowDimIdx)[lowDimElementIdx].push_back(bulkElementIdx);
                    }
                }
            }
        }
 
        // make extra stencils unique
        for (auto&& stencil : extendedSourceStencil_.stencil())
        {
            std::sort(stencil.second.begin(), stencil.second.end());
            stencil.second.erase(std::unique(stencil.second.begin(), stencil.second.end()), stencil.second.end());
 
            // remove the vertices element (box)
            if (isBox<bulkIdx>())
            {
                const auto& indices = this->vertexIndices(bulkIdx, stencil.first);
                stencil.second.erase(std::remove_if(stencil.second.begin(), stencil.second.end(),
                                                   [&](auto i){ return std::find(indices.begin(), indices.end(), i) != indices.end(); }),
                                     stencil.second.end());
            }
            // remove the own element (cell-centered)
            else
            {
                stencil.second.erase(std::remove_if(stencil.second.begin(), stencil.second.end(),
                                                   [&](auto i){ return i == stencil.first; }),
                                     stencil.second.end());
            }
        }
 
        // make stencils unique
        using namespace Dune::Hybrid;
        forEach(integralRange(Dune::index_constant<2>{}), [&](const auto domainIdx)
        {
            for (auto&& stencil : this->couplingStencils(domainIdx))
            {
                std::sort(stencil.second.begin(), stencil.second.end());
                stencil.second.erase(std::unique(stencil.second.begin(), stencil.second.end()), stencil.second.end());
            }
        });
 
        if (!this->pointSources(bulkIdx).empty())
            DUNE_THROW(Dune::InvalidStateException, "Kernel method shouldn't have point sources in the bulk domain but only volume sources!");
 
        std::cout << "took " << watch.elapsed() << " seconds." << std::endl;
    }
 
    void computeLowDimVolumeFractions()
    {
        // resize the storage vector
        lowDimVolumeInBulkElement_.resize(this->gridView(bulkIdx).size(0));
 
        // compute the low dim volume fractions
        for (const auto& is : intersections(this->glue()))
        {
            // all inside elements are identical...
            const auto& inside = is.targetEntity(0);
            const auto intersectionGeometry = is.geometry();
            const std::size_t lowDimElementIdx = this->problem(lowDimIdx).gridGeometry().elementMapper().index(inside);
 
            // compute the volume the low-dim domain occupies in the bulk domain if it were full-dimensional
            const auto radius = this->problem(lowDimIdx).spatialParams().radius(lowDimElementIdx);
            for (std::size_t outsideIdx = 0; outsideIdx < is.numDomainNeighbors(); ++outsideIdx)
            {
                const auto& outside = is.domainEntity(outsideIdx);
                const std::size_t bulkElementIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(outside);
                lowDimVolumeInBulkElement_[bulkElementIdx] += intersectionGeometry.volume()*M_PI*radius*radius;
            }
        }
    }
 
    // \{
 
    Scalar radius(std::size_t id) const
    {
        const auto& data = this->pointSourceData()[id];
        return this->problem(lowDimIdx).spatialParams().radius(data.lowDimElementIdx());
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolume(const Element<bulkIdx>& element) const
    {
        const auto eIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(element);
        return lowDimVolumeInBulkElement_[eIdx];
    }
 
    // For one-dimensional low dim domain we assume radial tubes
    Scalar lowDimVolumeFraction(const Element<bulkIdx>& element) const
    {
        const auto totalVolume = element.geometry().volume();
        return lowDimVolume(element) / totalVolume;
    }
 
    const std::vector<std::size_t> bulkSourceIds(std::size_t eIdx) const
    { return bulkSourceIds_[eIdx]; }
 
    const std::vector<Scalar> bulkSourceWeights(std::size_t eIdx) const
    { return bulkSourceWeights_[eIdx]; }
 
    // \}
 
private:
    void computeBulkSource(const GlobalPosition& globalPos, const Scalar kernelWidth,
                           std::size_t id, std::size_t lowDimElementIdx, std::size_t coupledBulkElementIdx,
                           Scalar pointSourceWeight)
    {
        // make sure it is mass conservative
        // i.e. the point source in the 1d domain needs to have the exact same integral as the distributed
        // kernel source integral in the 3d domain. Correct the integration formula by balancing the error
        // by scaling the kernel with the checksum inverse
        Scalar checkSum = 0.0;
        std::vector<bool> mask(this->gridView(bulkIdx).size(0), false);
        for (const auto& element : elements(this->gridView(bulkIdx)))
        {
            Scalar weight = 0.0;
            const auto geometry = element.geometry();
            const auto& quad = Dune::QuadratureRules<Scalar, bulkDim>::rule(geometry.type(), 3);
            for (auto&& qp : quad)
            {
                const auto qpweight = qp.weight();
                const auto ie = geometry.integrationElement(qp.position());
                weight += evalKernel(globalPos, geometry.global(qp.position()), kernelWidth)*qpweight*ie;
            }
 
            if (weight > 1e-13)
            {
                const auto bulkElementIdx = this->problem(bulkIdx).gridGeometry().elementMapper().index(element);
                bulkSourceIds_[bulkElementIdx].push_back(id);
                bulkSourceWeights_[bulkElementIdx].push_back(weight*pointSourceWeight);
                mask[bulkElementIdx] = true;
 
                // add lowDim dofs that the source is related to to the bulk stencil
                if (isBox<lowDimIdx>())
                {
                    const auto& vertices = this->vertexIndices(lowDimIdx, lowDimElementIdx);
                    this->couplingStencils(bulkIdx)[bulkElementIdx].insert(this->couplingStencils(bulkIdx)[bulkElementIdx].end(),
                                                                           vertices.begin(), vertices.end());
 
                }
                else
                {
                    this->couplingStencils(bulkIdx)[bulkElementIdx].push_back(lowDimElementIdx);
                }
 
                // tpfa
                extendedSourceStencil_.stencil()[bulkElementIdx].push_back(coupledBulkElementIdx);
 
                // compute check sum -> should sum up to 1.0 to be mass conservative
                checkSum += weight;
            }
        }
 
        for (std::size_t eIdx = 0; eIdx < bulkSourceWeights_.size(); ++eIdx)
            if (mask[eIdx]) bulkSourceWeights_[eIdx].back() /= checkSum;
 
        // balance error of the quadrature rule -> TODO: what to do at boundaries
        // const auto diff = 1.0 - checkSum;
        // std::cout << "Integrated kernel with integration error of " << diff << std::endl;
    }
 
    inline Scalar evalKernel(const GlobalPosition& origin,
                             const GlobalPosition& pos,
                             const Scalar width) const noexcept
    {
        const auto r = (pos-origin).two_norm();
        const auto r2 = r*r;
        const auto r3 = r2*r;
 
        if (r > width)
            return 0.0;
 
        const Scalar w2 = width*width;
        const Scalar w3 = w2*width;
        const Scalar k = 15.0/(4*M_PI*w3);
        const Scalar a = 2.0/w3;
        const Scalar b = 3.0/w2;
 
        return k*(a*r3 - b*r2 + 1.0);
    }
 
    EmbeddedCoupling::ExtendedSourceStencil<ThisType> extendedSourceStencil_;
    std::vector<Scalar> lowDimVolumeInBulkElement_;
    std::vector<std::vector<std::size_t>> bulkSourceIds_;
    std::vector<std::vector<Scalar>> bulkSourceWeights_;
};
 
} // end namespace Dumux
 
#endif