couplingmanager1d3d_projection.hh

Go to the documentation of this file.

// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
//
// SPDX-FileCopyrightInfo: Copyright © DuMux Project contributors, see AUTHORS.md in root folder
// SPDX-License-Identifier: GPL-3.0-or-later
//
#ifndef DUMUX_MULTIDOMAIN_EMBEDDED_COUPLINGMANAGER_1D3D_PROJECTION_HH
#define DUMUX_MULTIDOMAIN_EMBEDDED_COUPLINGMANAGER_1D3D_PROJECTION_HH
 
#include <vector>
#include <algorithm>
#include <string>
 
#include <dune/common/timer.hh>
#include <dune/common/exceptions.hh>
#include <dune/common/fvector.hh>
#include <dune/geometry/quadraturerules.hh>
 
#include <dumux/common/tag.hh>
#include <dumux/common/properties.hh>
#include <dumux/common/math.hh>
#include <dumux/common/indextraits.hh>
 
#include <dumux/geometry/refinementquadraturerule.hh>
#include <dumux/geometry/triangulation.hh>
#include <dumux/geometry/grahamconvexhull.hh>
 
#include <dumux/multidomain/embedded/couplingmanagerbase.hh>
#include <dumux/multidomain/embedded/localrefinementquadrature.hh>
 
namespace Dumux {
 
// some implementation details of the projection coupling manager
namespace Detail {
 
template <class GridGeometry>
class SegmentNetwork
{
    using GridView = typename GridGeometry::GridView;
    using GlobalPosition = typename GridView::template Codim<0>::Geometry::GlobalCoordinate;
    using GridIndex = typename IndexTraits<GridView>::GridIndex;
 
    struct Segment
    {
        GlobalPosition a, b;
        double r;
        GridIndex eIdx;
    };
 
public:
    template <class RadiusFunction>
    SegmentNetwork(const GridGeometry& gg, const RadiusFunction& radiusFunction)
    {
        const auto& gridView = gg.gridView();
        segments_.resize(gridView.size(0));
        for (const auto &element : elements(gridView))
        {
            const auto geometry = element.geometry();
            const auto eIdx = gg.elementMapper().index(element);
            const auto radius = radiusFunction(eIdx);
 
            segments_[eIdx] = Segment{ geometry.corner(0), geometry.corner(1), radius, eIdx };
        }
 
        std::cout << "-- Extracted " << segments_.size() << " segments.\n";
    }
 
    auto radius(std::size_t eIdx) const
    { return segments_[eIdx].r; }
 
    const auto& segment(std::size_t eIdx) const
    { return segments_[eIdx]; }
 
    const auto& segments() const
    { return segments_; }
 
    auto findClosestSegmentSurface(const GlobalPosition& globalPos) const
    {
        GridIndex eIdx = 0;
        double minSignedDist = std::numeric_limits<double>::max();
 
        for (const auto &segment : segments_)
        {
            const auto signedDist = computeSignedDistance_(globalPos, segment);
            if (signedDist < minSignedDist)
            {
                minSignedDist = signedDist;
                eIdx = segment.eIdx;
            }
        }
 
        using std::abs;
        return std::make_tuple(eIdx, abs(minSignedDist));
    }
 
    template <class IndexRange>
    auto findClosestSegmentSurface(const GlobalPosition& globalPos,
                                   const IndexRange& segIndices) const
    {
        GridIndex eIdx = 0;
        double minSignedDist = std::numeric_limits<double>::max();
 
        for (const auto index : segIndices)
        {
            const auto &segment = segments_[index];
            const auto signedDist = computeSignedDistance_(globalPos, segment);
            if (signedDist < minSignedDist)
            {
                minSignedDist = signedDist;
                eIdx = segment.eIdx;
            }
        }
 
        using std::abs;
        return std::make_tuple(eIdx, abs(minSignedDist));
    }
 
    // Compute the projection point of p onto the segment connecting a->b
    GlobalPosition projectionPointOnSegment(const GlobalPosition& p, std::size_t idx) const
    {
        const auto& segment = segments_[idx];
        return projectionPointOnSegment_(p, segment.a, segment.b);
    }
 
private:
    // Compute the projection of p onto the segment
    // returns the closest end point if the orthogonal projection onto the line implied by the segment
    // is outside the segment
    GlobalPosition projectionPointOnSegment_(const GlobalPosition& p, const GlobalPosition& a, const GlobalPosition& b) const
    {
        const auto v = b - a;
        const auto w = p - a;
 
        const auto proj1 = v*w;
        if (proj1 <= 0.0)
            return a;
 
        const auto proj2 = v.two_norm2();
        if (proj2 <= proj1)
            return b;
 
        const auto t = proj1 / proj2;
        auto x = a;
        x.axpy(t, v);
        return x;
    }
 
    // Compute the signed (!) distance to a cylinder segment surface and the projection point onto the centerline
    auto computeSignedDistance_(const GlobalPosition& p, const Segment& segment) const
    {
        const auto projPoint = projectionPointOnSegment_(p, segment.a, segment.b);
        return (p-projPoint).two_norm()-segment.r;
    }
 
    std::vector<Segment> segments_;
};
 
template<class Network>
class NetworkIndicatorFunction
{
public:
    NetworkIndicatorFunction(const Network& n)
    : network_(n)
    {}
 
    template<class Pos>
    auto operator()(const Pos& pos) const
    {
        const auto& [closestSegmentIdx, _] = network_.findClosestSegmentSurface(pos);
        return closestSegmentIdx;
    }
 
    template <class Pos, class HintContainer>
    auto operator()(const Pos& pos, const HintContainer& hints) const
    {
        const auto& [closestSegmentIdx, _] = network_.findClosestSegmentSurface(pos, hints);
        return closestSegmentIdx;
    }
 
private:
    const Network& network_;
};
 
class DebugIntersectionVTKOutput
{
public:
    void push(const std::array<Dune::FieldVector<double, 3>, 3>& corners, double data)
    {
        vtkCells_.emplace_back(std::array<std::size_t, 3>{
            { vtkPoints_.size(), vtkPoints_.size()+1, vtkPoints_.size()+2 }
        });
        for (const auto& c : corners)
            vtkPoints_.push_back(c);
        vtkCellData_.push_back(data);
    }
 
    void write(const std::string& name)
    {
        std::ofstream intersectionFile(name);
        intersectionFile << "# vtk DataFile Version 2.0\n";
        intersectionFile << "Really cool intersection data\n";
        intersectionFile << "ASCII\n";
        intersectionFile << "DATASET UNSTRUCTURED_GRID\n";
        intersectionFile << "POINTS " << vtkPoints_.size() << " float\n";
        for (const auto& p : vtkPoints_)
            intersectionFile << p[0] << " " << p[1] << " " << p[2] << "\n";
        intersectionFile << "\nCELLS " << vtkCells_.size() << " " << vtkCells_.size()*4 << "\n";
        for (const auto& c : vtkCells_)
            intersectionFile << "3 " << c[0] << " " << c[1] << " " << c[2] << "\n";
        intersectionFile << "\nCELL_TYPES " << vtkCells_.size() << "\n";
        for (int i = 0; i < vtkCells_.size(); ++i)
            intersectionFile << "5\n";
        intersectionFile << "\nCELL_DATA " << vtkCells_.size() << "\nSCALARS index float\nLOOKUP_TABLE default\n";
        for (const auto& c : vtkCellData_)
            intersectionFile << c << "\n";
    }
 
private:
    std::vector<Dune::FieldVector<double, 3>> vtkPoints_;
    std::vector<std::array<std::size_t, 3>> vtkCells_;
    std::vector<double> vtkCellData_;
};
 
} // end namespace Detail
 
namespace Embedded1d3dCouplingMode {
struct Projection : public Utility::Tag<Projection> {
    static std::string name() { return "projection"; }
};
 
inline constexpr Projection projection{};
} // end namespace Embedded1d3dCouplingMode
 
// forward declaration
template<class MDTraits, class CouplingMode>
class Embedded1d3dCouplingManager;
 
template<class MDTraits>
class Embedded1d3dCouplingManager<MDTraits, Embedded1d3dCouplingMode::Projection>
: public EmbeddedCouplingManagerBase<MDTraits, Embedded1d3dCouplingManager<MDTraits, Embedded1d3dCouplingMode::Projection>>
{
    using ThisType = Embedded1d3dCouplingManager<MDTraits, Embedded1d3dCouplingMode::Projection>;
    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;
    template<std::size_t id> using GridIndex = typename IndexTraits<GridView<id>>::GridIndex;
 
    static constexpr int lowDimDim = GridView<lowDimIdx>::dimension;
    static constexpr int bulkDim = GridView<bulkIdx>::dimension;
    static constexpr int dimWorld = GridView<bulkIdx>::dimensionworld;
 
    template<std::size_t id>
    static constexpr bool isBox()
    { return GridGeometry<id>::discMethod == DiscretizationMethods::box; }
 
    using GlobalPosition = typename Element<bulkIdx>::Geometry::GlobalCoordinate;
 
    using SegmentNetwork = Detail::SegmentNetwork<GridGeometry<lowDimIdx>>;
 
public:
    static constexpr Embedded1d3dCouplingMode::Projection couplingMode{};
 
    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);
    }
 
    /* \brief Compute integration point point sources and associated data
     * This method computes a source term on the explicitly given surface of the network grid
     * by integrating over the elements of the surface, projecting the integration points onto
     * the centerlines to evaluate 1d quantities and set point sources (minimum distance projection)
     *
     * \param order Unused in this algorithm! (passed from default 1D3D coupling manager)
     * \param verbose If the source computation is verbose
     */
    void computePointSourceData(std::size_t order = 1, bool verbose = false)
    {
        // initialize the maps
        // do some logging and profiling
        Dune::Timer watch;
        std::cout << "Initializing the coupling manager (projection)" << std::endl;
 
        // debug VTK output
        const bool enableIntersectionOutput = getParam<bool>("MixedDimension.Projection.EnableIntersectionOutput", false);
        std::unique_ptr<Detail::DebugIntersectionVTKOutput> vtkCoupledFaces =
            enableIntersectionOutput ? std::make_unique<Detail::DebugIntersectionVTKOutput>() : nullptr;
        std::unique_ptr<Detail::DebugIntersectionVTKOutput> vtkIntersections =
            enableIntersectionOutput ? std::make_unique<Detail::DebugIntersectionVTKOutput>() : nullptr;
 
        // temporarily represent the network domain as a list of segments
        // a more efficient data structure
        const auto& lowDimProblem = this->problem(lowDimIdx);
        const auto& lowDimFvGridGeometry = lowDimProblem.gridGeometry();
        const auto& lowDimGridView = lowDimFvGridGeometry.gridView();
        localAreaFactor_.resize(lowDimGridView.size(0), 0.0);
        const auto radiusFunc = [&](const GridIndex<lowDimIdx> eIdx) {
            return lowDimProblem.spatialParams().radius(eIdx);
        };
        const auto network = SegmentNetwork{
            lowDimFvGridGeometry, radiusFunc
        };
 
        // precompute the vertex indices for efficiency (box method)
        this->precomputeVertexIndices(bulkIdx);
        this->precomputeVertexIndices(lowDimIdx);
 
        const auto& bulkProblem = this->problem(bulkIdx);
        const auto& bulkFvGridGeometry = bulkProblem.gridGeometry();
        const auto& bulkGridView = bulkFvGridGeometry.gridView();
        bulkElementMarker_.assign(bulkGridView.size(0), 0);
        bulkVertexMarker_.assign(bulkGridView.size(bulkDim), 0);
 
        // construct a bounding box that may be used to determine coupled faces
        // the shrinking factor shrinks this bounding box
        static const auto bBoxShrinking
            = getParam<Scalar>("MixedDimension.Projection.CoupledBoundingBoxShrinkingFactor", 1e-2);
        const GlobalPosition threshold(
            bBoxShrinking*( bulkFvGridGeometry.bBoxMax()-bulkFvGridGeometry.bBoxMin() ).two_norm()
        );
        const auto bBoxMinSmall = bulkFvGridGeometry.bBoxMin() + threshold;
        const auto bBoxMaxSmall = bulkFvGridGeometry.bBoxMax() - threshold;
        auto insideBBox = [bBoxMin=bBoxMinSmall, bBoxMax=bBoxMaxSmall](const GlobalPosition& point) -> bool
        {
            static constexpr Scalar eps_ = 1.0e-7;
            const Scalar eps0 = eps_*(bBoxMax[0] - bBoxMin[0]);
            const Scalar eps1 = eps_*(bBoxMax[1] - bBoxMin[1]);
            const Scalar eps2 = eps_*(bBoxMax[2] - bBoxMin[2]);
            return (bBoxMin[0] - eps0 <= point[0] && point[0] <= bBoxMax[0] + eps0 &&
                    bBoxMin[1] - eps1 <= point[1] && point[1] <= bBoxMax[1] + eps1 &&
                    bBoxMin[2] - eps2 <= point[2] && point[2] <= bBoxMax[2] + eps2);
        };
 
        // setup simplex "quadrature" rule
        static const auto quadSimplexRefineMaxLevel
            = getParam<std::size_t>("MixedDimension.Projection.SimplexIntegrationRefine", 4);
        const auto indicator = Detail::NetworkIndicatorFunction { network };
 
        // estimate number of point sources for memory allocation
        static const auto estimateNumberOfPointSources
            = getParam<std::size_t>("MixedDimension.Projection.EstimateNumberOfPointSources", bulkFvGridGeometry.gridView().size(0));
 
        this->pointSources(bulkIdx).reserve(estimateNumberOfPointSources);
        this->pointSources(lowDimIdx).reserve(estimateNumberOfPointSources);
        this->pointSourceData().reserve(estimateNumberOfPointSources);
 
        // we go over the bulk surface intersection, check if we are coupled,
        // and if this is the case, we add integration point sources using
        // the local simplex refinement quadrature procedure (see paper Koch 2021)
        for (const auto& element : elements(bulkGridView))
        {
            const auto bulkElementIdx = bulkFvGridGeometry.elementMapper().index(element);
            const auto bulkGeometry = element.geometry();
            const auto refElement = Dune::referenceElement(element);
            for (const auto& intersection : intersections(bulkGridView, element))
            {
                // skip inner intersection
                if (!intersection.boundary())
                    continue;
 
                // check if we are close enough to the coupling interface
                const auto [isCoupled, closestSegmentIdx, minDist] = isCoupled_(network, intersection, insideBBox);
                if (!isCoupled)
                    continue;
 
                // we found an intersection on the coupling interface: mark as coupled element
                bulkElementMarker_[bulkElementIdx] = 1;
 
                const auto interfaceGeometry = intersection.geometry();
                for (int i = 0; i < interfaceGeometry.corners(); ++i)
                {
                    const auto localVIdx = refElement.subEntity(intersection.indexInInside(), 1, i, bulkDim);
                    const auto vIdx = bulkFvGridGeometry.vertexMapper().subIndex(element, localVIdx, bulkDim);
                    bulkVertexMarker_[vIdx] = 1;
                }
 
                // debug graphical intersection output
                if (enableIntersectionOutput)
                    vtkCoupledFaces->push({
                        { interfaceGeometry.corner(0), interfaceGeometry.corner(1), interfaceGeometry.corner(2)}
                    }, closestSegmentIdx);
 
                const auto quadSimplex = LocalRefinementSimplexQuadrature{ interfaceGeometry, quadSimplexRefineMaxLevel, indicator };
 
                // iterate over all quadrature points (children simplices)
                for (const auto& qp : quadSimplex)
                {
                    const auto surfacePoint = interfaceGeometry.global(qp.position());
                    const auto closestSegmentIdx = qp.indicator();
                    const auto projPoint = network.projectionPointOnSegment(surfacePoint, closestSegmentIdx);
 
                    addPointSourceAtIP_(bulkGeometry, interfaceGeometry, qp, bulkElementIdx, closestSegmentIdx, surfacePoint, projPoint);
                    addStencilEntries_(bulkElementIdx, closestSegmentIdx);
                }
            }
        }
 
        // 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()
                );
            }
        });
 
        // debug VTK output
        if (enableIntersectionOutput)
            vtkCoupledFaces->write("coupledfaces.vtk");
 
        // compare theoretical cylinder surface to actual discrete surface
        // of the coupled bulk interface facets
        for (const auto& element : elements(lowDimGridView))
        {
            const auto length = element.geometry().volume();
            const auto eIdx = lowDimFvGridGeometry.elementMapper().index(element);
            const auto radius = this->problem(lowDimIdx).spatialParams().radius(eIdx);
            const auto cylinderSurface = 2.0*M_PI*radius*length;
            localAreaFactor_[eIdx] /= cylinderSurface;
            localAreaFactor_[eIdx] -= 1.0;
        }
 
        this->pointSources(bulkIdx).shrink_to_fit();
        this->pointSources(lowDimIdx).shrink_to_fit();
        this->pointSourceData().shrink_to_fit();
 
        std::cout << "-- Coupling at " << this->pointSourceData().size() << " integration points." << std::endl;
        std::cout << "-- Finished initialization after " << watch.elapsed() << " seconds." << std::endl;
    }
 
    // \{
 
    Scalar radius(std::size_t id) const
    {
        const auto& data = this->pointSourceData()[id];
        return this->problem(lowDimIdx).spatialParams().radius(data.lowDimElementIdx());
    }
 
    // \}
 
    const std::vector<int>& bulkElementMarker() const
    { return bulkElementMarker_; }
 
    const std::vector<int>& bulkVertexMarker() const
    { return bulkVertexMarker_; }
 
    const std::vector<Scalar>& localAreaFactor() const
    { return localAreaFactor_; }
 
private:
    std::vector<int> bulkElementMarker_, bulkVertexMarker_;
    std::vector<double> localAreaFactor_;
 
    // add a point source containing all data needed to evaluate
    // the coupling source term from both domains
    template<class BulkGeometry, class InterfaceGeometry, class QP>
    void addPointSourceAtIP_(const BulkGeometry& bulkGeometry,
                             const InterfaceGeometry& ifGeometry,
                             const QP& qp,
                             const GridIndex<bulkIdx> bulkElementIdx,
                             const GridIndex<lowDimIdx> lowDimElementIdx,
                             const GlobalPosition& surfacePoint,
                             const GlobalPosition& projPoint)
    {
        // add a new point source id (for the associated data)
        const auto id = this->idCounter_++;
 
        // add a bulk point source
        const auto qpweight = qp.weight();
        const auto integrationElement = ifGeometry.integrationElement(qp.position());
        this->pointSources(bulkIdx).emplace_back(surfacePoint, id, qpweight, integrationElement, bulkElementIdx);
 
        // find the corresponding projection point and 1d element
        this->pointSources(lowDimIdx).emplace_back(projPoint, id, qpweight, integrationElement, lowDimElementIdx);
        localAreaFactor_[lowDimElementIdx] += qpweight*integrationElement;
 
        const auto& bulkFvGridGeometry = this->problem(bulkIdx).gridGeometry();
        const auto& lowDimFvGridGeometry = this->problem(lowDimIdx).gridGeometry();
 
        // pre compute additional data used for the evaluation of
        // the actual solution dependent source term
        PointSourceData psData;
 
        if constexpr (isBox<lowDimIdx>())
        {
            std::vector<Dune::FieldVector<Scalar, 1> > shapeValues;
            const auto lowDimGeometry = this->problem(lowDimIdx).gridGeometry().element(lowDimElementIdx).geometry();
            this->getShapeValues(lowDimIdx, lowDimFvGridGeometry, lowDimGeometry, projPoint, shapeValues);
            psData.addLowDimInterpolation(shapeValues, this->vertexIndices(lowDimIdx, lowDimElementIdx), lowDimElementIdx);
        }
        else
        {
            psData.addLowDimInterpolation(lowDimElementIdx);
        }
 
        if constexpr (isBox<bulkIdx>())
        {
            std::vector<Dune::FieldVector<Scalar, 1> > shapeValues;
            this->getShapeValues(bulkIdx, bulkFvGridGeometry, bulkGeometry, surfacePoint, 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));
    }
 
    void addStencilEntries_(const GridIndex<bulkIdx> bulkElementIdx, const GridIndex<lowDimIdx> lowDimElementIdx)
    {
        // export the bulk coupling stencil
        if constexpr (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);
        }
 
        // export the lowdim coupling stencil
        if constexpr (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);
        }
    }
 
    template<class BoundaryIntersection, class BBoxCheck>
    auto isCoupled_(const SegmentNetwork& network, const BoundaryIntersection& intersection, const BBoxCheck& insideBBox) const
    {
        const auto center = intersection.geometry().center();
        const auto [eIdx, minDist] = network.findClosestSegmentSurface(center);
 
        // all elements in the bounding box are coupled if this is enabled
        // this is a simplified check that can be enabled for box-shaped domains
        static const bool considerBBox = getParam<bool>("MixedDimension.Projection.ConsiderFacesWithinBoundingBoxCoupled", false);
        if (considerBBox && insideBBox(center))
            return std::make_tuple(true, eIdx, minDist);
 
        // general coupling face detection algorithm:
        // if the distance is under a certain threshold we are coupled (and the angle is not very big)
        static const auto rFactor = getParam<Scalar>("MixedDimension.Projection.CoupledRadiusFactor", 0.1);
        static const auto spFactor = getParam<Scalar>("MixedDimension.Projection.CoupledAngleFactor", 0.3);
        const auto projPoint = network.projectionPointOnSegment(center, eIdx);
        const auto distVec = projPoint - center;
        const bool isCoupled = minDist < rFactor * network.radius(eIdx) && distVec * intersection.centerUnitOuterNormal() > distVec.two_norm() * spFactor;
        return std::make_tuple(isCoupled, eIdx, minDist);
    }
};
 
template<class MDTraits>
struct CouplingManagerSupportsMultithreadedAssembly<Embedded1d3dCouplingManager<MDTraits, Embedded1d3dCouplingMode::Projection>>
: public std::true_type {};
 
} // end namespace Dumux
 
#endif