lagrangepoints.hh 18.9 KB
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#pragma once

#include <cassert>
#include <array>

#include <dune/common/exceptions.hh>
#include <dune/geometry/type.hh>
#include <dune/localfunctions/lagrange/equidistantpoints.hh>

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namespace Dune
{

namespace Vtk
{
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namespace Impl {
  // forward declaration
  template <class K, unsigned int dim>
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  class LagrangePointSetBuilder;
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}


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/// \brief A set of lagrange points compatible with the numbering of VTK and Gmsh
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/**
 * \tparam K    Field-type for the coordinates
 * \tparam dim  Dimension of the coordinates
 **/
template <class K, unsigned int dim>
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class LagrangePointSet
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    : public EmptyPointSet<K, dim>
{
  using Super = EmptyPointSet<K, dim>;

public:
  static const unsigned int dimension = dim;

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  LagrangePointSet (std::size_t order)
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    : Super(order)
  {
    assert(order > 0);
  }

  /// Fill the lagrange points for the given geometry type
  void build (GeometryType gt)
  {
    assert(gt.dim() == dimension);
    builder_(gt, order(), points_);
  }

  /// Fill the lagrange points for the given topology type `Topology`
  template <class Topology>
  bool build ()
  {
    build(GeometryType(Topology{}));
    return true;
  }

  /// Returns whether the point set support the given topology type `Topology` and can
  /// generate point for the given order.
  template <class Topology>
  static bool supports (std::size_t order)
  {
    return true;
  }

  using Super::order;

private:
  using Super::points_;
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  Impl::LagrangePointSetBuilder<K,dim> builder_;
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};


namespace Impl {

// Build for lagrange point sets in different dimensions
// Specialized for dim=1,2,3
template <class K, unsigned int dim>
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class LagrangePointSetBuilder
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{
public:
  template <class Points>
  void operator()(GeometryType, unsigned int, Points& points) const
  {
    DUNE_THROW(Dune::NotImplemented,
      "Lagrange points not yet implemented for this GeometryType.");
  }
};

/**
 *  The implementation of the point set builder is directly derived from VTK.
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 *  Modification are a change in data-types and naming scheme. Additionally
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 *  a LocalKey is created to reflect the concept of a Dune PointSet.
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 *
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 *  Included is the license of the BSD 3-clause License included in the VTK
 *  source code from 2020/04/13 in commit b90dad558ce28f6d321420e4a6b17e23f5227a1c
 *  of git repository https://gitlab.kitware.com/vtk/vtk.
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 *
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    Program:   Visualization Toolkit
    Module:    Copyright.txt

    Copyright (c) 1993-2015 Ken Martin, Will Schroeder, Bill Lorensen
    All rights reserved.

    Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions are met:

    * Redistributions of source code must retain the above copyright notice,
      this list of conditions and the following disclaimer.

    * Redistributions in binary form must reproduce the above copyright notice,
      this list of conditions and the following disclaimer in the documentation
      and/or other materials provided with the distribution.

    * Neither name of Ken Martin, Will Schroeder, or Bill Lorensen nor the names
      of any contributors may be used to endorse or promote products derived
      from this software without specific prior written permission.

    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS''
    AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
    IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
    ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE FOR
    ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
    DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
    SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
    CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
    OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
    OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 **/

// Lagrange points on point geometries
template <class K>
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class LagrangePointSetBuilder<K,0>
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{
  static constexpr int dim = 0;
  using LP = LagrangePoint<K,dim>;
  using Vec = typename LP::Vector;
  using Key = LocalKey;

public:
  template <class Points>
  void operator()(GeometryType gt, int /*order*/, Points& points) const
  {
    assert(gt.isVertex());
    points.push_back(LP{Vec{},Key{0,0,0}});
  }
};


// Lagrange points on line geometries
template <class K>
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class LagrangePointSetBuilder<K,1>
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{
  static constexpr int dim = 1;
  using LP = LagrangePoint<K,dim>;
  using Vec = typename LP::Vector;
  using Key = LocalKey;

public:
  template <class Points>
  void operator()(GeometryType gt, int order, Points& points) const
  {
    assert(gt.isLine());

    // Vertex nodes
    points.push_back(LP{Vec{0.0}, Key{0,dim,0}});
    points.push_back(LP{Vec{1.0}, Key{1,dim,0}});

    if (order > 1) {
      // Inner nodes
      Vec p{0.0};
      for (unsigned int i = 0; i < order-1; i++)
      {
        p[0] += 1.0 / order;
        points.push_back(LP{p,Key{0,dim-1,i}});
      }
    }
  }
};


// Lagrange points on 2d geometries
template <class K>
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class LagrangePointSetBuilder<K,2>
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{
  static constexpr int dim = 2;
  using LP = LagrangePoint<K,dim>;
  using Vec = typename LP::Vector;
  using Key = LocalKey;

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  friend class LagrangePointSetBuilder<K,3>;
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public:
  template <class Points>
  void operator()(GeometryType gt, int order, Points& points) const
  {
    std::size_t nPoints = numLagrangePoints(gt.id(), dim, order);

    if (gt.isTriangle())
      buildTriangle(nPoints, order, points);
    else if (gt.isQuadrilateral())
      buildQuad(nPoints, order, points);
    else {
      DUNE_THROW(Dune::NotImplemented,
        "Lagrange points not yet implemented for this GeometryType.");
    }

    assert(points.size() == nPoints);
  }

private:
  // Construct the point set in a triangle element.
  // Loop from the outside to the inside
  template <class Points>
  void buildTriangle (std::size_t nPoints, int order, Points& points) const
  {
    points.reserve(nPoints);

    const int nVertexDOFs = 3;
    const int nEdgeDOFs = 3 * (order-1);

    static const unsigned int vertexPerm[3] = {0,1,2};
    static const unsigned int edgePerm[3]   = {0,2,1};

    auto calcKey = [&](int p) -> Key
    {
      if (p < nVertexDOFs) {
        return Key{vertexPerm[p], dim, 0};
      }
      else if (p < nVertexDOFs+nEdgeDOFs) {
        unsigned int entityIndex = (p - nVertexDOFs) / (order-1);
        unsigned int index = (p - nVertexDOFs) % (order-1);
        return Key{edgePerm[entityIndex], dim-1, index};
      }
      else {
        unsigned int index = p - (nVertexDOFs + nEdgeDOFs);
        return Key{0, dim-2, index};
      }
    };

    std::array<int,3> bindex;
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    double order_d = double(order);
    for (std::size_t p = 0; p < nPoints; ++p) {
      barycentricIndex(p, bindex, order);
      Vec point{bindex[0] / order_d, bindex[1] / order_d};
      points.push_back(LP{point, calcKey(p)});
    }
  }

  // "Barycentric index" is a triplet of integers, each running from 0 to
  // <Order>. It is the index of a point on the triangle in barycentric
  // coordinates.
  static void barycentricIndex (int index, std::array<int,3>& bindex, int order)
  {
    int max = order;
    int min = 0;

    // scope into the correct triangle
    while (index != 0 && index >= 3 * order)
    {
      index -= 3 * order;
      max -= 2;
      min++;
      order -= 3;
    }

    // vertex DOFs
    if (index < 3)
    {
      bindex[index] = bindex[(index + 1) % 3] = min;
      bindex[(index + 2) % 3] = max;
    }
    // edge DOFs
    else
    {
      index -= 3;
      int dim = index / (order - 1);
      int offset = (index - dim * (order - 1));
      bindex[(dim + 1) % 3] = min;
      bindex[(dim + 2) % 3] = (max - 1) - offset;
      bindex[dim] = (min + 1) + offset;
    }
  }


  // Construct the point set in the quad element
  // 1. build equispaced points with index tuple (i,j)
  // 2. map index tuple to DOF index and LocalKey
  template <class Points>
  void buildQuad(std::size_t nPoints, int order, Points& points) const
  {
    points.resize(nPoints);

    std::array<int,2> orders{order, order};
    std::array<Vec,4> nodes{Vec{0., 0.}, Vec{1., 0.}, Vec{1., 1.}, Vec{0., 1.}};

    for (int n = 0; n <= orders[1]; ++n) {
      for (int m = 0; m <= orders[0]; ++m) {
        // int idx = pointIndexFromIJ(m,n,orders);

        const double r = double(m) / orders[0];
        const double s = double(n) / orders[1];
        Vec p = (1.0 - r) * (nodes[3] * s + nodes[0] * (1.0 - s)) +
                r *         (nodes[2] * s + nodes[1] * (1.0 - s));

        auto [idx,key] = calcQuadKey(m,n,orders);
        points[idx] = LP{p, key};
        // points[idx] = LP{p, calcQuadKey(n,m,orders)};
      }
    }
  }

  // Obtain the VTK DOF index of the node (i,j) in the quad element
  // and construct a LocalKey
  static std::pair<int,Key> calcQuadKey (int i, int j, std::array<int,2> order)
  {
    const bool ibdy = (i == 0 || i == order[0]);
    const bool jbdy = (j == 0 || j == order[1]);
    const int nbdy = (ibdy ? 1 : 0) + (jbdy ? 1 : 0); // How many boundaries do we lie on at once?
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    int dof = 0;
    unsigned int entityIndex = 0;
    unsigned int index = 0;

    if (nbdy == 2) // Vertex DOF
    {
      dof = (i ? (j ? 2 : 1) : (j ? 3 : 0));
      entityIndex = (j ? (i ? 3 : 2) : (i ? 1 : 0));
      return std::make_pair(dof,Key{entityIndex, dim, 0});
    }

    int offset = 4;
    if (nbdy == 1) // Edge DOF
    {
      if (!ibdy) {
        dof = (i - 1) + (j ? order[0]-1 + order[1]-1 : 0) + offset;
        entityIndex = j ? 3 : 2;
        index = i-1;
      }
      else if (!jbdy) {
        dof = (j - 1) + (i ? order[0]-1 : 2 * (order[0]-1) + order[1]-1) + offset;
        entityIndex = i ? 1 : 0;
        index = j-1;
      }
      return std::make_pair(dof, Key{entityIndex, dim-1, index});
    }

    offset += 2 * (order[0]-1 + order[1]-1);

    // nbdy == 0: Face DOF
    dof = offset + (i - 1) + (order[0]-1) * ((j - 1));
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    Key innerKey = LagrangePointSetBuilder<K,2>::calcQuadKey(i-1,j-1,{order[0]-2, order[1]-2}).second;
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    return std::make_pair(dof, Key{0, dim-2, innerKey.index()});
  }
};


// Lagrange points on 3d geometries
template <class K>
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class LagrangePointSetBuilder<K,3>
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{
  static constexpr int dim = 3;
  using LP = LagrangePoint<K,dim>;
  using Vec = typename LP::Vector;
  using Key = LocalKey;

public:
  template <class Points>
  void operator() (GeometryType gt, unsigned int order, Points& points) const
  {
    std::size_t nPoints = numLagrangePoints(gt.id(), dim, order);

    if (gt.isTetrahedron())
      buildTetra(nPoints, order, points);
    else if (gt.isHexahedron())
      buildHex(nPoints, order, points);
    else {
      DUNE_THROW(Dune::NotImplemented,
        "Lagrange points not yet implemented for this GeometryType.");
    }

    assert(points.size() == nPoints);
  }

private:
  // Construct the point set in the tetrahedron element
  // 1. construct barycentric (index) coordinates
  // 2. obtains the DOF index, LocalKey and actual coordinate from barycentric index
  template <class Points>
  void buildTetra (std::size_t nPoints, int order, Points& points) const
  {
    points.reserve(nPoints);

    const int nVertexDOFs = 4;
    const int nEdgeDOFs = 6 * (order-1);
    const int nFaceDOFs = 4 * (order-1)*(order-2)/2;

    static const unsigned int vertexPerm[4] = {0,1,2,3};
    static const unsigned int edgePerm[6]   = {0,2,1,3,4,5};
    static const unsigned int facePerm[4]   = {1,2,0,3};

    auto calcKey = [&](int p) -> Key
    {
      if (p < nVertexDOFs) {
        return Key{vertexPerm[p], dim, 0};
      }
      else if (p < nVertexDOFs+nEdgeDOFs) {
        unsigned int entityIndex = (p - nVertexDOFs) / (order-1);
        unsigned int index = (p - nVertexDOFs) % (order-1);
        return Key{edgePerm[entityIndex], dim-1, index};
      }
      else if (p < nVertexDOFs+nEdgeDOFs+nFaceDOFs) {
        unsigned int index = (p - (nVertexDOFs + nEdgeDOFs)) % ((order-1)*(order-2)/2);
        unsigned int entityIndex = (p - (nVertexDOFs + nEdgeDOFs)) / ((order-1)*(order-2)/2);
        return Key{facePerm[entityIndex], dim-2, index};
      }
      else {
        unsigned int index = p - (nVertexDOFs + nEdgeDOFs + nFaceDOFs);
        return Key{0, dim-3, index};
      }
    };

    std::array<int,4> bindex;
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    double order_d = double(order);
    for (std::size_t p = 0; p < nPoints; ++p) {
      barycentricIndex(p, bindex, order);
      Vec point{bindex[0] / order_d, bindex[1] / order_d, bindex[2] / order_d};
      points.push_back(LP{point, calcKey(p)});
    }
  }

  // "Barycentric index" is a set of 4 integers, each running from 0 to
  // <Order>. It is the index of a point in the tetrahedron in barycentric
  // coordinates.
  static void barycentricIndex (std::size_t p, std::array<int,4>& bindex, int order)
  {
    const int nVertexDOFs = 4;
    const int nEdgeDOFs = 6 * (order-1);

    static const int edgeVertices[6][2]   = {{0,1}, {1,2}, {2,0}, {0,3}, {1,3}, {2,3}};
    static const int linearVertices[4][4] = {{0,0,0,1}, {1,0,0,0}, {0,1,0,0}, {0,0,1,0}};
    static const int vertexMaxCoords[4]   = {3,0,1,2};
    static const int faceBCoords[4][3]    = {{0,2,3}, {2,0,1}, {2,1,3}, {1,0,3}};
    static const int faceMinCoord[4]      = {1,3,0,2};

    int max = order;
    int min = 0;

    // scope into the correct tetra
    while (p >= 2 * (order * order + 1) && p != 0 && order > 3)
    {
      p -= 2 * (order * order + 1);
      max -= 3;
      min++;
      order -= 4;
    }

    // vertex DOFs
    if (p < nVertexDOFs)
    {
      for (int coord = 0; coord < 4; ++coord)
        bindex[coord] = (coord == vertexMaxCoords[p] ? max : min);
    }
    // edge DOFs
    else if (p < nVertexDOFs+nEdgeDOFs)
    {
      int edgeId = (p - nVertexDOFs) / (order-1);
      int vertexId = (p - nVertexDOFs) % (order-1);
      for (int coord = 0; coord < 4; ++coord)
      {
        bindex[coord] = min +
          (linearVertices[edgeVertices[edgeId][0]][coord] * (max - min - 1 - vertexId) +
            linearVertices[edgeVertices[edgeId][1]][coord] * (1 + vertexId));
      }
    }
    // face DOFs
    else
    {
      int faceId = (p - (nVertexDOFs+nEdgeDOFs)) / ((order-2)*(order-1)/2);
      int vertexId = (p - (nVertexDOFs+nEdgeDOFs)) % ((order-2)*(order-1)/2);

      std::array<int,3> projectedBIndex;
      if (order == 3)
        projectedBIndex[0] = projectedBIndex[1] = projectedBIndex[2] = 0;
      else
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        LagrangePointSetBuilder<K,2>::barycentricIndex(vertexId, projectedBIndex, order-3);
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      for (int i = 0; i < 3; i++)
        bindex[faceBCoords[faceId][i]] = (min + 1 + projectedBIndex[i]);
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      bindex[faceMinCoord[faceId]] = min;
    }
  }

private:
  // Construct the point set in the heyhedral element
  // 1. build equispaced points with index tuple (i,j,k)
  // 2. map index tuple to DOF index and LocalKey
  template <class Points>
  void buildHex (std::size_t nPoints, int order, Points& points) const
  {
    points.resize(nPoints);

    std::array<int,3> orders{order, order, order};
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    std::array<Vec,8> nodes{Vec{0., 0., 0.}, Vec{1., 0., 0.}, Vec{1., 1., 0.}, Vec{0., 1., 0.},
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                            Vec{0., 0., 1.}, Vec{1., 0., 1.}, Vec{1., 1., 1.}, Vec{0., 1., 1.}};

    for (int p = 0; p <= orders[2]; ++p) {
      for (int n = 0; n <= orders[1]; ++n) {
        for (int m = 0; m <= orders[0]; ++m) {
          const double r = double(m) / orders[0];
          const double s = double(n) / orders[1];
          const double t = double(p) / orders[2];
          Vec point = (1.0-r) * ((nodes[3] * (1.0-t) + nodes[7] * t) * s + (nodes[0] * (1.0-t) + nodes[4] * t) * (1.0-s)) +
                      r *       ((nodes[2] * (1.0-t) + nodes[6] * t) * s + (nodes[1] * (1.0-t) + nodes[5] * t) * (1.0-s));

          auto [idx,key] = calcHexKey(m,n,p,orders);
          points[idx] = LP{point, key};
        }
      }
    }
  }

  // Obtain the VTK DOF index of the node (i,j,k) in the hexahedral element
  static std::pair<int,Key> calcHexKey (int i, int j, int k, std::array<int,3> order)
  {
    const bool ibdy = (i == 0 || i == order[0]);
    const bool jbdy = (j == 0 || j == order[1]);
    const bool kbdy = (k == 0 || k == order[2]);
    const int nbdy = (ibdy ? 1 : 0) + (jbdy ? 1 : 0) + (kbdy ? 1 : 0); // How many boundaries do we lie on at once?

    int dof = 0;
    unsigned int entityIndex = 0;
    unsigned int index = 0;

    if (nbdy == 3) // Vertex DOF
    {
      dof = (i ? (j ? 2 : 1) : (j ? 3 : 0)) + (k ? 4 : 0);
      entityIndex = (i ? 1 : 0) + (j ? 2 : 0) + (k ? 4 : 0);
      return std::make_pair(dof, Key{entityIndex, dim, 0});
    }

    int offset = 8;
    if (nbdy == 2) // Edge DOF
    {
      entityIndex = (k ? 8 : 4);
      if (!ibdy)
      { // On i axis
        dof = (i - 1) + (j ? order[0]-1 + order[1]-1 : 0) + (k ? 2 * (order[0]-1 + order[1]-1) : 0) + offset;
        index = i;
        entityIndex += (i ? 1 : 0);
      }
      else if (!jbdy)
      { // On j axis
        dof = (j - 1) + (i ? order[0]-1 : 2 * (order[0]-1) + order[1]-1) + (k ? 2 * (order[0]-1 + order[1]-1) : 0) + offset;
        index = j;
        entityIndex += (j ? 3 : 2);
      }
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      else
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      { // !kbdy, On k axis
        offset += 4 * (order[0]-1) + 4 * (order[1]-1);
        dof = (k - 1) + (order[2]-1) * (i ? (j ? 3 : 1) : (j ? 2 : 0)) + offset;
        index = k;
        entityIndex = (i ? 1 : 0) + (j ? 2 : 0);
      }
      return std::make_pair(dof, Key{entityIndex, dim-1, index});
    }

    offset += 4 * (order[0]-1 + order[1]-1 + order[2]-1);
    if (nbdy == 1) // Face DOF
    {
      Key faceKey;
      if (ibdy) // On i-normal face
      {
        dof = (j - 1) + ((order[1]-1) * (k - 1)) + (i ? (order[1]-1) * (order[2]-1) : 0) + offset;
        entityIndex = (i ? 1 : 0);
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        faceKey = LagrangePointSetBuilder<K,2>::calcQuadKey(j-1,k-1,{order[1]-2, order[2]-2}).second;
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      }
      else {
        offset += 2 * (order[1] - 1) * (order[2] - 1);
        if (jbdy) // On j-normal face
        {
          dof = (i - 1) + ((order[0]-1) * (k - 1)) + (j ? (order[2]-1) * (order[0]-1) : 0) + offset;
          entityIndex = (j ? 3 : 2);
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          faceKey = LagrangePointSetBuilder<K,2>::calcQuadKey(i-1,k-1,{order[0]-2, order[2]-2}).second;
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        }
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        else
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        { // kbdy, On k-normal face
          offset += 2 * (order[2]-1) * (order[0]-1);
          dof = (i - 1) + ((order[0]-1) * (j - 1)) + (k ? (order[0]-1) * (order[1]-1) : 0) + offset;
          entityIndex = (k ? 5 : 4);
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          faceKey = LagrangePointSetBuilder<K,2>::calcQuadKey(i-1,j-1,{order[0]-2, order[1]-2}).second;
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        }
      }
      return std::make_pair(dof, Key{entityIndex, dim-2, faceKey.index()});
    }

    // nbdy == 0: Body DOF
    offset += 2 * ((order[1]-1) * (order[2]-1) + (order[2]-1) * (order[0]-1) + (order[0]-1) * (order[1]-1));
    dof = offset + (i - 1) + (order[0]-1) * ((j - 1) + (order[1]-1) * ((k - 1)));
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    Key innerKey = LagrangePointSetBuilder<K,3>::calcHexKey(i-1,j-1,k-1,{order[0]-2, order[1]-2, order[2]-2}).second;
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    return std::make_pair(dof, Key{0, dim-3, innerKey.index()});
  }
};

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}}} // end namespace Dune::Vtk::Impl