NCMesh3D.cpp

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#include <polyfem/mesh/mesh3D/NCMesh3D.hpp>
#include <polyfem/mesh/MeshUtils.hpp>
#include <polyfem/utils/StringUtils.hpp>
 
#include <polyfem/utils/Logger.hpp>
 
#include <igl/writeMESH.h>
 
#include <geogram/mesh/mesh_io.h>
#include <fstream>
 
using namespace polyfem::utils;
 
namespace polyfem
{
    namespace mesh
    {
        int NCMesh3D::face_edge(const int f_id, const int le_id) const
        {
            const int v0 = faces[valid_to_all_face(f_id)].vertices(le_id);
            const int v1 = faces[valid_to_all_face(f_id)].vertices((le_id + 1) % 3);
            const int e_id = find_edge(v0, v1);
            return all_to_valid_edge(e_id);
        }
 
        void NCMesh3D::refine(const int n_refinement, const double t)
        {
            if (n_refinement <= 0)
                return;
            std::vector<bool> refine_mask(elements.size(), false);
            for (int i = 0; i < elements.size(); i++)
                if (elements[i].is_valid())
                    refine_mask[i] = true;
 
            for (int i = 0; i < refine_mask.size(); i++)
                if (refine_mask[i])
                    refine_element(i);
 
            refine(n_refinement - 1, t);
        }
 
        bool NCMesh3D::is_boundary_element(const int element_global_id) const
        {
            assert(index_prepared);
            for (int lv = 0; lv < n_cell_edges(element_global_id); lv++)
                if (is_boundary_vertex(cell_vertex(element_global_id, lv)))
                    return true;
 
            return false;
        }
 
        bool NCMesh3D::build_from_matrices(const Eigen::MatrixXd &V, const Eigen::MatrixXi &F)
        {
            n_elements = 0;
            elements.clear();
            vertices.clear();
            edges.clear();
            midpointMap.clear();
            edgeMap.clear();
            faceMap.clear();
            refineHistory.clear();
            index_prepared = false;
            adj_prepared = false;
 
            vertices.reserve(V.rows());
            for (int i = 0; i < V.rows(); i++)
            {
                vertices.emplace_back(V.row(i));
            }
            for (int i = 0; i < F.rows(); i++)
            {
                add_element(F.row(i), -1);
            }
 
            prepare_mesh();
 
            return true;
        }
 
        void NCMesh3D::attach_higher_order_nodes(const Eigen::MatrixXd &V, const std::vector<std::vector<int>> &nodes)
        {
            for (int f = 0; f < n_cells(); ++f)
                if (nodes[f].size() != 4)
                    throw std::runtime_error("NCMesh doesn't support high order mesh!");
        }
 
        void NCMesh3D::normalize()
        {
            polyfem::RowVectorNd min, max;
            bounding_box(min, max);
 
            auto extent = max - min;
            double scale = extent.maxCoeff();
 
            for (auto &v : vertices)
                v.pos = (v.pos - min.transpose()) / scale;
        }
 
        void NCMesh3D::compute_elements_tag()
        {
            elements_tag_.assign(n_cells(), ElementType::SIMPLEX);
        }
        void NCMesh3D::update_elements_tag()
        {
            elements_tag_.assign(n_cells(), ElementType::SIMPLEX);
        }
        double NCMesh3D::edge_length(const int gid) const
        {
            const int v1 = edge_vertex(gid, 0);
            const int v2 = edge_vertex(gid, 1);
 
            return (point(v1) - point(v2)).norm();
        }
 
        RowVectorNd NCMesh3D::point(const int global_index) const
        {
            return vertices[valid_to_all_vertex(global_index)].pos.transpose();
        }
 
        void NCMesh3D::set_point(const int global_index, const RowVectorNd &p)
        {
            vertices[valid_to_all_vertex(global_index)].pos = p;
        }
 
        RowVectorNd NCMesh3D::edge_barycenter(const int e) const
        {
            const int v1 = edge_vertex(e, 0);
            const int v2 = edge_vertex(e, 1);
 
            return 0.5 * (point(v1) + point(v2));
        }
        RowVectorNd NCMesh3D::face_barycenter(const int f) const
        {
            const int v1 = face_vertex(f, 0);
            const int v2 = face_vertex(f, 1);
            const int v3 = face_vertex(f, 2);
 
            return (point(v1) + point(v2) + point(v3)) / 3.;
        }
        RowVectorNd NCMesh3D::cell_barycenter(const int c) const
        {
            const int v1 = face_vertex(c, 0);
            const int v2 = face_vertex(c, 1);
            const int v3 = face_vertex(c, 2);
 
            return (point(v1) + point(v2) + point(v3)) / 3.;
        }
 
        void NCMesh3D::bounding_box(RowVectorNd &min, RowVectorNd &max) const
        {
            min = vertices[0].pos;
            max = vertices[0].pos;
 
            for (const auto &v : vertices)
            {
                for (int d = 0; d < 3; d++)
                {
                    if (v.pos[d] > max[d])
                        max[d] = v.pos[d];
                    if (v.pos[d] < min[d])
                        min[d] = v.pos[d];
                }
            }
        }
 
        void NCMesh3D::compute_boundary_ids(const std::function<int(const size_t, const std::vector<int> &, const RowVectorNd &, bool)> &marker)
        {
            boundary_ids_.resize(n_faces());
            std::fill(boundary_ids_.begin(), boundary_ids_.end(), -1);
 
            for (int f = 0; f < n_faces(); ++f)
            {
                const bool is_boundary = is_boundary_face(f);
                std::vector<int> vs(n_face_vertices(f));
                const auto p = face_barycenter(f);
 
                for (int vid = 0; vid < vs.size(); ++vid)
                    vs[vid] = face_vertex(f, vid);
 
                std::sort(vs.begin(), vs.end());
                boundary_ids_[f] = marker(f, vs, p, is_boundary);
 
                faces[valid_to_all_face(f)].boundary_id = boundary_ids_[f];
            }
        }
 
        void NCMesh3D::compute_body_ids(const std::function<int(const size_t, const std::vector<int> &, const RowVectorNd &)> &marker)
        {
            body_ids_.resize(n_cells());
            std::fill(body_ids_.begin(), body_ids_.end(), -1);
 
            for (int e = 0; e < n_cells(); ++e)
            {
                const auto bary = cell_barycenter(e);
                body_ids_[e] = marker(e, element_vertices(e), bary);
                elements[valid_to_all_elem(e)].body_id = body_ids_[e];
            }
        }
        void NCMesh3D::set_boundary_ids(const std::vector<int> &boundary_ids)
        {
            assert(boundary_ids.size() == n_faces());
            for (int i = 0; i < boundary_ids.size(); i++)
            {
                faces[valid_to_all_face(i)].boundary_id = boundary_ids[i];
            }
        }
        void NCMesh3D::set_body_ids(const std::vector<int> &body_ids)
        {
            assert(body_ids.size() == n_cells());
            for (int i = 0; i < body_ids.size(); i++)
            {
                elements[valid_to_all_elem(i)].body_id = body_ids[i];
            }
        }
 
        // void NCMesh3D::triangulate_faces(Eigen::MatrixXi &tris, Eigen::MatrixXd &pts, std::vector<int> &ranges) const
        // {
        //  ranges.clear();
 
        //  std::vector<Eigen::MatrixXi> local_tris(n_cells());
        //  std::vector<Eigen::MatrixXd> local_pts(n_cells());
        //  Eigen::MatrixXi tets;
 
        //  int total_tris = 0;
        //  int total_pts = 0;
 
        //  ranges.push_back(0);
 
        //  Eigen::MatrixXd face_barys;
        //  face_barycenters(face_barys);
 
        //  Eigen::MatrixXd cell_barys;
        //  cell_barycenters(cell_barys);
 
        //  for (std::size_t e = 0; e < n_cells(); ++e)
        //  {
        //      const int n_vertices = n_cell_vertices(e);
        //      const int n_faces = n_cell_faces(e);
 
        //      Eigen::MatrixXd local_pt(n_vertices + n_faces, 3);
 
        //      std::map<int, int> global_to_local;
 
        //      for (int i = 0; i < n_vertices; ++i)
        //      {
        //          const int global_index = cell_vertex(e, i);
        //          local_pt.row(i) = point(global_index);
        //          global_to_local[global_index] = i;
        //      }
 
        //      int n_local_faces = 0;
        //      for (int i = 0; i < n_faces; ++i)
        //      {
        //          const int f_id = cell_face(e, i);
        //          n_local_faces += n_face_vertices(f_id);
 
        //          local_pt.row(n_vertices + i) = face_barys.row(f_id);
        //      }
 
        //      Eigen::MatrixXi local_faces(n_local_faces, 3);
 
        //      int face_index = 0;
        //      for (int i = 0; i < n_faces; ++i)
        //      {
        //          const int f_id = cell_face(e, i);
 
        //          const Eigen::RowVector3d e0 = (point(face_vertex(f_id, 0)) - local_pt.row(n_vertices + i));
        //          const Eigen::RowVector3d e1 = (point(face_vertex(f_id, 1)) - local_pt.row(n_vertices + i));
        //          const Eigen::RowVector3d normal = e0.cross(e1);
        //          // const Eigen::RowVector3d check_dir = (node_from_element(e)-p);
        //          const Eigen::RowVector3d check_dir = (cell_barys.row(e) - point(face_vertex(f_id, 1)));
 
        //          const bool reverse_order = normal.dot(check_dir) > 0;
 
        //          for (int j = 0; j < n_face_vertices(f_id); ++j)
        //          {
        //              const int jp = (j + 1) % n_face_vertices(f_id);
        //              if (reverse_order)
        //              {
        //                  local_faces(face_index, 0) = global_to_local[face_vertex(f_id, jp)];
        //                  local_faces(face_index, 1) = global_to_local[face_vertex(f_id, j)];
        //              }
        //              else
        //              {
        //                  local_faces(face_index, 0) = global_to_local[face_vertex(f_id, j)];
        //                  local_faces(face_index, 1) = global_to_local[face_vertex(f_id, jp)];
        //              }
        //              local_faces(face_index, 2) = n_vertices + i;
 
        //              ++face_index;
        //          }
        //      }
 
        //      local_pts[e] = local_pt;
        //      local_tris[e] = local_faces;
 
        //      total_tris += local_tris[e].rows();
        //      total_pts += local_pts[e].rows();
 
        //      ranges.push_back(total_tris);
 
        //      assert(local_pts[e].rows() == local_pt.rows());
        //  }
 
        //  tris.resize(total_tris, 3);
        //  pts.resize(total_pts, 3);
 
        //  int tri_index = 0;
        //  int pts_index = 0;
        //  for (std::size_t i = 0; i < local_tris.size(); ++i)
        //  {
        //      tris.block(tri_index, 0, local_tris[i].rows(), local_tris[i].cols()) = local_tris[i].array() + pts_index;
        //      tri_index += local_tris[i].rows();
 
        //      pts.block(pts_index, 0, local_pts[i].rows(), local_pts[i].cols()) = local_pts[i];
        //      pts_index += local_pts[i].rows();
        //  }
        // }
 
        RowVectorNd NCMesh3D::kernel(const int cell_id) const
        {
            assert(false);
            return RowVectorNd();
        }
        Navigation3D::Index NCMesh3D::get_index_from_element(int hi, int lf, int lv) const
        {
            Navigation3D::Index idx;
 
            idx.element = hi;
 
            idx.element_patch = lf;
            idx.face = cell_face(hi, lf);
 
            if (lv >= 4)
                lv = lv % 4;
            idx.face_corner = lv;
            idx.vertex = face_vertex(idx.face, idx.face_corner);
 
            idx.edge = face_edge(idx.face, idx.face_corner);
 
            return idx;
        }
        Navigation3D::Index NCMesh3D::get_index_from_element(int hi) const
        {
            Navigation3D::Index idx;
 
            idx.element = hi;
            idx.element_patch = 0;
            idx.face = cell_face(hi, 0);
            idx.face_corner = 0;
            idx.vertex = face_vertex(idx.face, idx.face_corner);
            idx.edge = face_edge(idx.face, 0);
 
            return idx;
        }
 
        Navigation3D::Index NCMesh3D::get_index_from_element_edge(int hi, int v0, int v1) const
        {
            Navigation3D::Index idx;
 
            idx.element = hi;
            idx.vertex = v0;
            idx.edge = all_to_valid_edge(find_edge(valid_to_all_vertex(v0), valid_to_all_vertex(v1)));
 
            for (int i = 0; i < 4; i++)
            {
                const int f_id = cell_face(idx.element, i);
                for (int j = 0; j < 3; j++)
                {
                    const int e_id = face_edge(f_id, j);
                    if (idx.edge == e_id)
                    {
                        idx.element_patch = i;
                        idx.face = f_id;
 
                        for (int d = 0; d < n_face_vertices(f_id); d++)
                            if (face_vertex(f_id, d) == idx.vertex)
                                idx.face_corner = d;
 
                        assert(switch_vertex(idx).vertex == v1);
 
                        return idx;
                    }
                }
            }
 
            assert(false);
            return idx;
        }
        Navigation3D::Index NCMesh3D::get_index_from_element_face(int hi, int v0, int v1, int v2) const
        {
            Navigation3D::Index idx;
 
            idx.element = hi;
            idx.vertex = v0;
            idx.face = all_to_valid_face(find_face(valid_to_all_vertex(v0), valid_to_all_vertex(v1), valid_to_all_vertex(v2)));
 
            for (int d = 0; d < n_face_vertices(idx.face); d++)
                if (face_vertex(idx.face, d) == idx.vertex)
                    idx.face_corner = d;
 
            int v0_ = v0, v1_ = v1;
 
            // let v0_ < v1_
            if (v0_ > v1_)
                std::swap(v0_, v1_);
 
            for (int i = 0; i < 4; i++)
            {
                const int fid = cell_face(idx.element, i);
                if (fid != idx.face)
                    continue;
 
                idx.element_patch = i;
 
                for (int j = 0; j < 3; j++)
                {
                    const int eid = face_edge(fid, j);
                    const int ev0 = edge_vertex(eid, 0);
                    const int ev1 = edge_vertex(eid, 1);
 
                    if ((ev0 == v0_ && ev1 == v1_) || (ev0 == v1_ && ev1 == v0_))
                    {
                        idx.edge = eid;
 
                        assert(switch_vertex(idx).vertex == v1);
                        assert(switch_vertex(switch_edge(idx)).vertex == v2);
 
                        return idx;
                    }
                }
            }
 
            assert(false);
            return idx;
        }
 
        std::vector<uint32_t> NCMesh3D::vertex_neighs(const int v_gid) const
        {
            std::vector<uint32_t> hs;
            for (auto h : vertices[valid_to_all_vertex(v_gid)].elem_list)
                hs.push_back(all_to_valid_elem(h));
 
            return hs;
        }
        std::vector<uint32_t> NCMesh3D::edge_neighs(const int e_gid) const
        {
            std::vector<uint32_t> hs;
            for (auto h : edges[valid_to_all_edge(e_gid)].elem_list)
                hs.push_back(all_to_valid_elem(h));
 
            return hs;
        }
 
        Navigation3D::Index NCMesh3D::switch_vertex(Navigation3D::Index idx) const
        {
            if (idx.vertex == edge_vertex(idx.edge, 0))
                idx.vertex = edge_vertex(idx.edge, 1);
            else
                idx.vertex = edge_vertex(idx.edge, 0);
 
            for (int d = 0; d < n_face_vertices(idx.face); d++)
                if (face_vertex(idx.face, d) == idx.vertex)
                    idx.face_corner = d;
 
            return idx;
        }
        Navigation3D::Index NCMesh3D::switch_edge(Navigation3D::Index idx) const
        {
            if (idx.edge == face_edge(idx.face, idx.face_corner))
                idx.edge = face_edge(idx.face, (idx.face_corner + 2) % 3);
            else
                idx.edge = face_edge(idx.face, idx.face_corner);
            return idx;
        }
        Navigation3D::Index NCMesh3D::switch_face(Navigation3D::Index idx) const
        {
            for (int i = 0; i < 4; i++)
            {
                const int fid = cell_face(idx.element, i);
                if (fid == idx.face)
                    continue;
                if (idx.edge == face_edge(fid, 0) || idx.edge == face_edge(fid, 1) || idx.edge == face_edge(fid, 2))
                {
                    idx.face = fid;
                    idx.element_patch = i;
 
                    for (int d = 0; d < n_face_vertices(fid); d++)
                        if (face_vertex(fid, d) == idx.vertex)
                            idx.face_corner = d;
 
                    break;
                }
            }
            return idx;
        }
        Navigation3D::Index NCMesh3D::switch_element(Navigation3D::Index idx) const
        {
            const auto &face = faces[valid_to_all_face(idx.face)];
            if (face.n_elem() != 2)
            {
                idx.element = -1;
                return idx;
            }
            if (all_to_valid_elem(face.get_element()) == idx.element)
                idx.element = all_to_valid_elem(face.find_opposite_element(face.get_element()));
            else
                idx.element = all_to_valid_elem(face.get_element());
 
            for (int f = 0; f < n_cell_faces(idx.element); f++)
                if (cell_face(idx.element, f) == idx.face)
                    idx.element_patch = f;
            return idx;
        }
 
        Navigation3D::Index NCMesh3D::next_around_edge(Navigation3D::Index idx) const
        {
            return switch_element(switch_face(idx));
        }
        Navigation3D::Index NCMesh3D::next_around_face(Navigation3D::Index idx) const
        {
            return switch_edge(switch_vertex(idx));
        }
 
        void NCMesh3D::get_vertex_elements_neighs(const int v_id, std::vector<int> &ids) const
        {
            ids.clear();
            for (auto h : vertices[valid_to_all_vertex(v_id)].elem_list)
                ids.push_back(all_to_valid_elem(h));
        }
        void NCMesh3D::get_edge_elements_neighs(const int e_id, std::vector<int> &ids) const
        {
            ids.clear();
            for (auto h : edges[valid_to_all_edge(e_id)].elem_list)
                ids.push_back(all_to_valid_elem(h));
        }
        void NCMesh3D::get_face_elements_neighs(const int f_id, std::vector<int> &ids) const
        {
            ids.clear();
            for (auto h : faces[valid_to_all_face(f_id)].elem_list)
                ids.push_back(all_to_valid_elem(h));
        }
 
        void NCMesh3D::refine_element(int id_full)
        {
            assert(elements[id_full].is_valid());
            if (elements[id_full].is_not_valid())
                throw std::runtime_error("Cannot refine an invalid element!");
 
            const auto v = elements[id_full].vertices;
            elements[id_full].is_refined = true;
            n_elements--;
 
            for (int f = 0; f < elements[id_full].faces.size(); f++)
                faces[elements[id_full].faces(f)].remove_element(id_full);
 
            for (int e = 0; e < elements[id_full].edges.size(); e++)
                edges[elements[id_full].edges(e)].remove_element(id_full);
 
            for (int i = 0; i < v.size(); i++)
                vertices[v(i)].remove_element(id_full);
 
            if (elements[id_full].children(0) >= 0)
            {
                for (int c = 0; c < elements[id_full].children.size(); c++)
                {
                    const int child_id = elements[id_full].children(c);
                    auto &elem = elements[child_id];
                    elem.is_ghost = false;
                    n_elements++;
 
                    for (int f = 0; f < elem.faces.size(); f++)
                        faces[elem.faces(f)].add_element(child_id);
 
                    for (int e = 0; e < elem.edges.size(); e++)
                        edges[elem.edges(e)].add_element(child_id);
 
                    for (int v = 0; v < elem.vertices.size(); v++)
                        vertices[elem.vertices(v)].add_element(child_id);
                }
            }
            else
            {
                // create mid-points if not exist
                const int v1 = v[0];
                const int v2 = v[1];
                const int v3 = v[2];
                const int v4 = v[3];
                const int v5 = get_vertex(Eigen::Vector2i(v1, v2));
                const int v8 = get_vertex(Eigen::Vector2i(v3, v2));
                const int v6 = get_vertex(Eigen::Vector2i(v1, v3));
                const int v7 = get_vertex(Eigen::Vector2i(v1, v4));
                const int v9 = get_vertex(Eigen::Vector2i(v4, v2));
                const int v10 = get_vertex(Eigen::Vector2i(v3, v4));
 
                // inherite line singularity flag from parent edge
                for (int i = 0; i < v.size(); i++)
                    for (int j = 0; j < i; j++)
                    {
                        int mid_id = find_vertex(v[i], v[j]);
                        int edge_id = find_edge(v[i], v[j]);
                        int edge1 = get_edge(v[i], mid_id);
                        int edge2 = get_edge(v[j], mid_id);
                        edges[edge1].boundary_id = edges[edge_id].boundary_id;
                        edges[edge2].boundary_id = edges[edge_id].boundary_id;
                    }
 
                for (int i = 0; i < v.size(); i++)
                    for (int j = 0; j < i; j++)
                        for (int k = 0; k < j; k++)
                        {
                            int fid = find_face(v[i], v[j], v[k]);
                            int vij = find_vertex(v[i], v[j]);
                            int vjk = find_vertex(v[j], v[k]);
                            int vik = find_vertex(v[i], v[k]);
                            int facei = get_face(v[i], vij, vik);
                            int facej = get_face(v[j], vjk, vij);
                            int facek = get_face(v[k], vjk, vik);
                            int facem = get_face(vij, vjk, vik);
                            faces[facei].boundary_id = faces[fid].boundary_id;
                            faces[facej].boundary_id = faces[fid].boundary_id;
                            faces[facek].boundary_id = faces[fid].boundary_id;
                            faces[facem].boundary_id = faces[fid].boundary_id;
                        }
 
                // create children
                elements[id_full].children(0) = elements.size();
                add_element(Eigen::Vector4i(v1, v5, v6, v7), id_full);
                elements[id_full].children(1) = elements.size();
                add_element(Eigen::Vector4i(v5, v2, v8, v9), id_full);
                elements[id_full].children(2) = elements.size();
                add_element(Eigen::Vector4i(v6, v8, v3, v10), id_full);
                elements[id_full].children(3) = elements.size();
                add_element(Eigen::Vector4i(v7, v9, v10, v4), id_full);
                elements[id_full].children(4) = elements.size();
                add_element(Eigen::Vector4i(v5, v6, v7, v9), id_full);
                elements[id_full].children(5) = elements.size();
                add_element(Eigen::Vector4i(v5, v9, v8, v6), id_full);
                elements[id_full].children(6) = elements.size();
                add_element(Eigen::Vector4i(v6, v7, v9, v10), id_full);
                elements[id_full].children(7) = elements.size();
                add_element(Eigen::Vector4i(v6, v10, v9, v8), id_full);
            }
 
            refineHistory.push_back(id_full);
        }
        void NCMesh3D::refine_elements(const std::vector<int> &ids)
        {
            std::vector<int> full_ids(ids.size());
            for (int i = 0; i < ids.size(); i++)
                full_ids[i] = valid_to_all_elem(ids[i]);
 
            for (int i : full_ids)
                refine_element(i);
        }
 
        void NCMesh3D::coarsen_element(int id_full)
        {
            const int parent_id = elements[id_full].parent;
            auto &parent = elements[parent_id];
 
            for (int i = 0; i < parent.children.size(); i++)
                if (elements[parent.children(i)].is_not_valid())
                    throw std::runtime_error("Invalid siblings in coarsening!");
 
            // remove elements
            for (int c = 0; c < parent.children.size(); c++)
            {
                auto &elem = elements[parent.children(c)];
                elem.is_ghost = true;
                n_elements--;
 
                for (int f = 0; f < elem.faces.size(); f++)
                    faces[elem.faces(f)].remove_element(parent.children(c));
 
                for (int e = 0; e < elem.edges.size(); e++)
                    edges[elem.edges(e)].remove_element(parent.children(c));
 
                for (int v = 0; v < elem.vertices.size(); v++)
                    vertices[elem.vertices(v)].remove_element(parent.children(c));
            }
 
            // add element
            parent.is_refined = false;
            n_elements++;
 
            for (int f = 0; f < parent.faces.size(); f++)
                faces[parent.faces(f)].add_element(parent_id);
 
            for (int e = 0; e < parent.edges.size(); e++)
                edges[parent.edges(e)].add_element(parent_id);
 
            for (int v = 0; v < parent.vertices.size(); v++)
                vertices[parent.vertices(v)].add_element(parent_id);
 
            refineHistory.push_back(parent_id);
        }
 
        void NCMesh3D::mark_boundary()
        {
            for (auto &face : faces)
            {
                if (face.n_elem() == 1)
                    face.isboundary = true;
                else
                    face.isboundary = false;
            }
 
            for (auto &face : faces)
            {
                if (face.leader >= 0)
                {
                    face.isboundary = false;
                    faces[face.leader].isboundary = false;
                }
            }
 
            for (auto &vert : vertices)
                vert.isboundary = false;
 
            for (auto &edge : edges)
                edge.isboundary = false;
 
            for (auto &face : faces)
            {
                if (face.isboundary && face.n_elem())
                {
                    for (int j = 0; j < 3; j++)
                        vertices[face.vertices(j)].isboundary = true;
 
                    for (int j = 0; j < 3; j++)
                        for (int i = 0; i < j; i++)
                            edges[find_edge(face.vertices(i), face.vertices(j))].isboundary = true;
                }
            }
 
            // for (int v = 0; v < n_vertices(); v++)
            //     std::cout << v << ": " << point(v) << "\n";
 
            // for (int f = 0; f < n_faces(); f++)
            // {
            //     for (int v = 0; v < n_face_vertices(f); v++)
            //         std::cout << face_vertex(f, v) << ", ";
            //     std::cout << faces[valid_to_all_face(f)].isboundary << std::endl;
            // }
        }
 
        void NCMesh3D::build_index_mapping()
        {
            all_to_valid_elemMap.assign(elements.size(), -1);
            valid_to_all_elemMap.resize(n_elements);
 
            for (int i = 0, e = 0; i < elements.size(); i++)
            {
                if (elements[i].is_not_valid())
                    continue;
                all_to_valid_elemMap[i] = e;
                valid_to_all_elemMap[e] = i;
                e++;
            }
 
            const int n_verts = n_vertices();
 
            all_to_valid_vertexMap.assign(vertices.size(), -1);
            valid_to_all_vertexMap.resize(n_verts);
 
            for (int i = 0, j = 0; i < vertices.size(); i++)
            {
                if (vertices[i].n_elem() == 0)
                    continue;
                all_to_valid_vertexMap[i] = j;
                valid_to_all_vertexMap[j] = i;
                j++;
            }
 
            all_to_valid_edgeMap.assign(edges.size(), -1);
            valid_to_all_edgeMap.resize(n_edges());
 
            for (int i = 0, j = 0; i < edges.size(); i++)
            {
                if (edges[i].n_elem() == 0)
                    continue;
                all_to_valid_edgeMap[i] = j;
                valid_to_all_edgeMap[j] = i;
                j++;
            }
 
            all_to_valid_faceMap.assign(faces.size(), -1);
            valid_to_all_faceMap.resize(n_faces());
 
            for (int i = 0, j = 0; i < faces.size(); i++)
            {
                if (faces[i].n_elem() == 0)
                    continue;
                all_to_valid_faceMap[i] = j;
                valid_to_all_faceMap[j] = i;
                j++;
            }
            index_prepared = true;
        }
 
        void NCMesh3D::append(const Mesh &mesh)
        {
            assert(typeid(mesh) == typeid(NCMesh3D));
            Mesh::append(mesh);
 
            const NCMesh3D &mesh3d = dynamic_cast<const NCMesh3D &>(mesh);
 
            const int n_v = n_vertices();
            const int n_f = n_cells();
 
            vertices.reserve(n_v + mesh3d.n_vertices());
            for (int i = 0; i < mesh3d.n_vertices(); i++)
            {
                vertices.emplace_back(mesh3d.vertices[i].pos);
            }
            for (int i = 0; i < mesh3d.n_cells(); i++)
            {
                Eigen::Vector4i cell = mesh3d.elements[i].vertices;
                cell = cell.array() + n_v;
                add_element(cell, -1);
            }
 
            prepare_mesh();
        }
 
        std::unique_ptr<Mesh> NCMesh3D::copy() const
        {
            return std::make_unique<NCMesh3D>(*this);
        }
 
        bool NCMesh3D::load(const std::string &path)
        {
            if (!StringUtils::endswith(path, ".HYBRID"))
            {
                GEO::Mesh M;
                GEO::mesh_load(path, M);
                return load(M);
            }
            return false;
        }
        bool NCMesh3D::load(const GEO::Mesh &M)
        {
            assert(M.vertices.dimension() == 3);
 
            Eigen::MatrixXd V(M.vertices.nb(), 2);
            Eigen::MatrixXi C(M.cells.nb(), 4);
 
            for (int v = 0; v < V.rows(); v++)
                V.row(v) << M.vertices.point(v)[0], M.vertices.point(v)[1], M.vertices.point(v)[2];
 
            for (int c = 0; c < C.rows(); c++)
            {
                if (M.cells.type(c) != GEO::MESH_TET)
                    throw std::runtime_error("NCMesh3D only supports tet mesh!");
                for (int i = 0; i < C.cols(); i++)
                    C(c, i) = M.cells.vertex(c, i);
            }
 
            n_elements = 0;
            vertices.reserve(V.rows());
            for (int i = 0; i < V.rows(); i++)
            {
                vertices.emplace_back(V.row(i));
            }
            for (int i = 0; i < C.rows(); i++)
            {
                add_element(C.row(i), -1);
            }
 
            prepare_mesh();
 
            return true;
        }
 
        int NCMesh3D::find_vertex(Eigen::Vector2i v) const
        {
            std::sort(v.data(), v.data() + v.size());
            auto search = midpointMap.find(v);
            if (search != midpointMap.end())
                return search->second;
            else
                return -1;
        }
        int NCMesh3D::get_vertex(Eigen::Vector2i v)
        {
            std::sort(v.data(), v.data() + v.size());
            int id = find_vertex(v);
            if (id < 0)
            {
                Eigen::VectorXd v_mid = (vertices[v[0]].pos + vertices[v[1]].pos) / 2.;
                id = vertices.size();
                vertices.emplace_back(v_mid);
                midpointMap.emplace(v, id);
            }
            return id;
        }
        int NCMesh3D::find_edge(Eigen::Vector2i v) const
        {
            std::sort(v.data(), v.data() + v.size());
            auto search = edgeMap.find(v);
            if (search != edgeMap.end())
                return search->second;
            else
                return -1;
        }
        int NCMesh3D::get_edge(Eigen::Vector2i v)
        {
            std::sort(v.data(), v.data() + v.size());
            int id = find_edge(v);
            if (id < 0)
            {
                edges.emplace_back(v);
                id = edges.size() - 1;
                edgeMap.emplace(v, id);
            }
            return id;
        }
        int NCMesh3D::find_face(Eigen::Vector3i v) const
        {
            std::sort(v.data(), v.data() + v.size());
            auto search = faceMap.find(v);
            if (search != faceMap.end())
                return search->second;
            else
                return -1;
        }
        int NCMesh3D::get_face(Eigen::Vector3i v)
        {
            std::sort(v.data(), v.data() + v.size());
            int id = find_face(v);
            if (id < 0)
            {
                faces.emplace_back(v);
                id = faces.size() - 1;
                faceMap.emplace(v, id);
            }
            return id;
        }
        void NCMesh3D::traverse_edge(Eigen::Vector2i v, double p1, double p2, int depth, std::vector<follower_edge> &list) const
        {
            int v_mid = find_vertex(v);
            std::vector<follower_edge> list1, list2;
            if (v_mid >= 0)
            {
                double p_mid = (p1 + p2) / 2;
                traverse_edge(Eigen::Vector2i(v[0], v_mid), p1, p_mid, depth + 1, list1);
                list.insert(
                    list.end(),
                    std::make_move_iterator(list1.begin()),
                    std::make_move_iterator(list1.end()));
                traverse_edge(Eigen::Vector2i(v_mid, v[1]), p_mid, p2, depth + 1, list2);
                list.insert(
                    list.end(),
                    std::make_move_iterator(list2.begin()),
                    std::make_move_iterator(list2.end()));
            }
            if (depth > 0)
            {
                int follower_id = find_edge(v);
                if (follower_id >= 0 && edges[follower_id].n_elem() > 0)
                    list.emplace_back(follower_id, p1, p2);
            }
        }
        void NCMesh3D::build_edge_follower_chain()
        {
            for (auto &edge : edges)
            {
                edge.leader = -1;
                edge.followers.clear();
                edge.weights.setConstant(-1);
            }
 
            for (int e_id = 0; e_id < edges.size(); e_id++)
            {
                auto &edge = edges[e_id];
                if (edge.n_elem() == 0)
                    continue;
                std::vector<follower_edge> followers;
                traverse_edge(edge.vertices, 0, 1, 0, followers);
                for (auto &s : followers)
                {
                    if (edges[s.id].leader >= 0 && std::abs(edges[s.id].weights(1) - edges[s.id].weights(0)) < std::abs(s.p2 - s.p1))
                        continue;
                    edge.followers.push_back(s.id);
                    edges[s.id].leader = e_id;
                    edges[s.id].weights << s.p1, s.p2;
                }
            }
 
            // In 3d, it's possible for one edge to have both leader and follower edges, but we don't care this case.
            for (auto &edge : edges)
                if (edge.leader >= 0 && edge.followers.size())
                    edge.followers.clear();
        }
        void NCMesh3D::traverse_face(int v1, int v2, int v3, Eigen::Vector2d p1, Eigen::Vector2d p2, Eigen::Vector2d p3, int depth, std::vector<follower_face> &face_list, std::vector<int> &edge_list) const
        {
            int v12 = find_vertex(Eigen::Vector2i(v1, v2));
            int v23 = find_vertex(Eigen::Vector2i(v3, v2));
            int v31 = find_vertex(Eigen::Vector2i(v1, v3));
            std::vector<follower_face> list1, list2, list3, list4;
            std::vector<int> list1_, list2_, list3_, list4_;
            if (depth > 0)
            {
                edge_list.push_back(find_edge(v1, v2));
                edge_list.push_back(find_edge(v3, v2));
                edge_list.push_back(find_edge(v1, v3));
            }
            if (v12 >= 0 && v23 >= 0 && v31 >= 0)
            {
                auto p12 = (p1 + p2) / 2, p23 = (p2 + p3) / 2, p31 = (p1 + p3) / 2;
                traverse_face(v1, v12, v31, p1, p12, p31, depth + 1, list1, list1_);
                traverse_face(v12, v2, v23, p12, p2, p23, depth + 1, list2, list2_);
                traverse_face(v31, v23, v3, p31, p23, p3, depth + 1, list3, list3_);
                traverse_face(v12, v23, v31, p12, p23, p31, depth + 1, list4, list4_);
 
                face_list.insert(face_list.end(), std::make_move_iterator(list1.begin()), std::make_move_iterator(list1.end()));
                face_list.insert(face_list.end(), std::make_move_iterator(list2.begin()), std::make_move_iterator(list2.end()));
                face_list.insert(face_list.end(), std::make_move_iterator(list3.begin()), std::make_move_iterator(list3.end()));
                face_list.insert(face_list.end(), std::make_move_iterator(list4.begin()), std::make_move_iterator(list4.end()));
 
                edge_list.insert(edge_list.end(), std::make_move_iterator(list1_.begin()), std::make_move_iterator(list1_.end()));
                edge_list.insert(edge_list.end(), std::make_move_iterator(list2_.begin()), std::make_move_iterator(list2_.end()));
                edge_list.insert(edge_list.end(), std::make_move_iterator(list3_.begin()), std::make_move_iterator(list3_.end()));
                edge_list.insert(edge_list.end(), std::make_move_iterator(list4_.begin()), std::make_move_iterator(list4_.end()));
            }
            if (depth > 0)
            {
                int follower_id = find_face(Eigen::Vector3i(v1, v2, v3));
                if (follower_id >= 0 && faces[follower_id].n_elem() > 0)
                    face_list.emplace_back(follower_id, p1, p2, p3);
            }
        }
        void NCMesh3D::build_face_follower_chain()
        {
            Eigen::Matrix<int, 4, 3> fv;
            fv.row(0) << 0, 1, 2;
            fv.row(1) << 0, 1, 3;
            fv.row(2) << 1, 2, 3;
            fv.row(3) << 2, 0, 3;
 
            for (auto &face : faces)
            {
                face.leader = -1;
                face.followers.clear();
            }
 
            for (auto &edge : edges)
            {
                edge.leader_face = -1;
            }
 
            for (int f_id = 0; f_id < faces.size(); f_id++)
            {
                auto &face = faces[f_id];
                if (face.n_elem() == 0)
                    continue;
                std::vector<follower_face> followers;
                std::vector<int> interior_edges;
                traverse_face(face.vertices(0), face.vertices(1), face.vertices(2), Eigen::Vector2d(0, 0), Eigen::Vector2d(1, 0), Eigen::Vector2d(0, 1), 0, followers, interior_edges); // order is important
                for (auto &s : followers)
                {
                    faces[s.id].leader = f_id;
                    face.followers.push_back(s.id);
                }
                for (int s : interior_edges)
                    if (s >= 0 && edges[s].leader < 0 && edges[s].n_elem() > 0)
                        edges[s].leader_face = f_id;
            }
        }
        void NCMesh3D::build_element_vertex_adjacency()
        {
            for (auto &vert : vertices)
            {
                vert.edge = -1;
                vert.face = -1;
            }
 
            for (auto &small_edge : edges)
            {
                // invalid edges
                if (small_edge.n_elem() == 0)
                    continue;
 
                // not follower edges
                int large_edge = small_edge.leader;
                if (large_edge < 0)
                    continue;
                assert(edges[large_edge].leader < 0);
 
                // follower edges
                for (int j = 0; j < 2; j++)
                {
                    const int v_id = small_edge.vertices(j);
                    // hanging nodes
                    if (v_id != edges[large_edge].vertices(0) && v_id != edges[large_edge].vertices(1))
                        vertices[v_id].edge = large_edge;
                }
            }
 
            for (auto &small_face : faces)
            {
                // invalid faces
                if (small_face.n_elem() == 0)
                    continue;
 
                // not follower faces
                int large_face = small_face.leader;
                if (large_face < 0)
                    continue;
 
                // follower faces
                for (int j = 0; j < 3; j++)
                {
                    const int v_id = small_face.vertices(j);
                    // hanging nodes
                    if (v_id != faces[large_face].vertices(0) && v_id != faces[large_face].vertices(1) && v_id != faces[large_face].vertices(2))
                        vertices[v_id].face = large_face;
                }
            }
        }
        std::array<int, 4> NCMesh3D::get_ordered_vertices_from_tet(const int element_index) const
        {
            return std::array<int, 4>({{cell_vertex(element_index, 0), cell_vertex(element_index, 1), cell_vertex(element_index, 2), cell_vertex(element_index, 3)}});
        }
        int NCMesh3D::add_element(Eigen::Vector4i v, int parent)
        {
            const int id = elements.size();
            const int level = (parent < 0) ? 0 : elements[parent].level + 1;
 
            Eigen::Vector3d e1 = vertices[v[1]].pos - vertices[v[0]].pos;
            Eigen::Vector3d e2 = vertices[v[2]].pos - vertices[v[0]].pos;
            Eigen::Vector3d e3 = vertices[v[3]].pos - vertices[v[0]].pos;
            if ((e1.cross(e2)).dot(e3) < 0)
                std::swap(v[2], v[3]);
 
            e2 = vertices[v[2]].pos - vertices[v[0]].pos;
            e3 = vertices[v[3]].pos - vertices[v[0]].pos;
            assert((e1.cross(e2)).dot(e3) > 0);
 
            elements.emplace_back(3, v, level, parent);
 
            if (parent >= 0)
                elements[id].body_id = elements[parent].body_id;
 
            // add faces if not exist
            const int face012 = get_face(v[0], v[1], v[2]);
            const int face013 = get_face(v[0], v[1], v[3]);
            const int face123 = get_face(v[1], v[2], v[3]);
            const int face203 = get_face(v[2], v[0], v[3]);
 
            faces[face012].add_element(id);
            faces[face013].add_element(id);
            faces[face123].add_element(id);
            faces[face203].add_element(id);
 
            elements[id].faces << face012, face013, face123, face203;
 
            // add edges if not exist
            const int edge01 = get_edge(v[0], v[1]);
            const int edge12 = get_edge(v[1], v[2]);
            const int edge20 = get_edge(v[2], v[0]);
            const int edge03 = get_edge(v[0], v[3]);
            const int edge13 = get_edge(v[1], v[3]);
            const int edge23 = get_edge(v[2], v[3]);
 
            edges[edge01].add_element(id);
            edges[edge12].add_element(id);
            edges[edge20].add_element(id);
            edges[edge03].add_element(id);
            edges[edge13].add_element(id);
            edges[edge23].add_element(id);
 
            elements[id].edges << edge01, edge12, edge20, edge03, edge13, edge23;
 
            n_elements++;
 
            for (int i = 0; i < v.size(); i++)
                vertices[v[i]].add_element(id);
 
            return id;
        }
    } // namespace mesh
} // namespace polyfem