getfem_mesh_level_set.cc

/*===========================================================================

 Copyright (C) 2005-2017 Julien Pommier

 This file is a part of GetFEM++

 GetFEM++  is  free software;  you  can  redistribute  it  and/or modify it
 under  the  terms  of the  GNU  Lesser General Public License as published
 by  the  Free Software Foundation;  either version 3 of the License,  or
 (at your option) any later version along with the GCC Runtime Library
 Exception either version 3.1 or (at your option) any later version.
 This program  is  distributed  in  the  hope  that it will be useful,  but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or  FITNESS  FOR  A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License and GCC Runtime Library Exception for more details.
 You  should  have received a copy of the GNU Lesser General Public License
 along  with  this program;  if not, write to the Free Software Foundation,
 Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA.

===========================================================================*/

#include "getfem/getfem_mesh_level_set.h"


namespace getfem {

  std::ostream &operator<<(std::ostream &os, const mesh_level_set::zone &z) {
    os << "zone[";
    for (mesh_level_set::zone::const_iterator it = z.begin();
         it != z.end(); ++it) {
      if (it != z.begin()) os << ", ";
      os << **it; 
    }
    os << "]";
    return os;
  }

  std::ostream &operator<<(std::ostream &os,const mesh_level_set::zoneset &zs) {
    os << "zoneset[";
    for (mesh_level_set::zoneset::const_iterator it = zs.begin(); 
         it != zs.end(); ++it) {
      if (it != zs.begin()) os << "; ";
      os << **it;
    }
    os << "]";
    return os;
  }
  

#ifdef DEBUG_LS
#include <asm/msr.h>
struct Chrono {
  unsigned long long tprev;
  unsigned long long acc;
  unsigned long long tmax;
  bool running; unsigned cnt;
  Chrono() : acc(0), tmax(0), running(false), cnt(0) {}
  void tic() { rdtscll(tprev); running = true; ++cnt; }
  double toc() {
    if (!running) return 0.; running = false;
    unsigned long long t; rdtscll(t);
    t -= tprev;
    acc += t; tmax = std::max(tmax, t);
    return double(t)/2.8e9;
  }
  double total() { return double(acc) / 2.8e9; }
  double max() { return double(tmax) / 2.8e9; }
  double mean() { return cnt ? total() / cnt : 0.; }
  unsigned count() { return cnt; }
};
  
  Chrono interpolate_face_chrono;
#endif

  static bool noisy = false;
  void getfem_mesh_level_set_noisy(void) { noisy = true; }

  void mesh_level_set::clear(void) {
    cut_cv.clear();
    is_adapted_ = false; touch();
  }

  const dal::bit_vector &mesh_level_set::crack_tip_convexes() const {
    return crack_tip_convexes_;
  }

  void mesh_level_set::init_with_mesh(mesh &me) {
    GMM_ASSERT1(linked_mesh_ == 0, "mesh_level_set already initialized");
    linked_mesh_ = &me;
    this->add_dependency(me);
    is_adapted_ = false;
  }

  mesh_level_set::mesh_level_set(mesh &me)
  { linked_mesh_ = 0; init_with_mesh(me); }

  mesh_level_set::mesh_level_set(void)
  { linked_mesh_ = 0; is_adapted_ = false; }


  mesh_level_set::~mesh_level_set() {}

  static void interpolate_face(const bgeot::pgeometric_trans &pgt,
                               mesh &m, dal::bit_vector& ptdone, 
                               const std::vector<size_type>& ipts,
                               const bgeot::pconvex_structure &cvs, 
                               size_type nb_vertices,
                               const std::vector<dal::bit_vector> &constraints,
                               const std::vector<const
                               mesher_signed_distance *> &list_constraints) {
    if (cvs->dim() == 0) return;
    else if (cvs->dim() > 1) {
      std::vector<size_type> fpts;
      for (short_type f=0; f < cvs->nb_faces(); ++f) {
        fpts.resize(cvs->nb_points_of_face(f));
        for (size_type k=0; k < fpts.size(); ++k)
          fpts[k] = ipts[cvs->ind_points_of_face(f)[k]];
        interpolate_face(pgt, m,ptdone,fpts,cvs->faces_structure()[f],
                         nb_vertices, constraints, list_constraints);
      }
    }

    if (noisy) {
      cout << "ipts.size() = " << ipts.size() << endl;
      cout << " nb_vertices = " <<  nb_vertices << endl;
    }

    dal::bit_vector cts; size_type cnt = 0;
    for (size_type i=0; i < ipts.size(); ++i) {
      // cout << "ipts[i] = " << ipts[i] << endl;
      if (ipts[i] < nb_vertices) {
        if (noisy)
          cout << "point " << i << " coordinates "
               << m.points()[ipts[i]] << " constraints[ipts[i]] = "
               << constraints[ipts[i]] << endl;
        if (cnt == 0) cts = constraints[ipts[i]];
        else cts &= constraints[ipts[i]];
        ++cnt;
      }
    }

    if (noisy) cout << "cts = " << cts << endl;
      
    if (cts.card()) {
      // dal::bit_vector new_cts;
      for (size_type i=0; i < ipts.size(); ++i) {
        if (ipts[i] >= nb_vertices && !ptdone[ipts[i]]) {
          base_node P = m.points()[ipts[i]];
          // if (cts.card() > 1)
          //   cout << "WARNING, projection sur " << cts << endl;
          if (!pure_multi_constraint_projection(list_constraints, P, cts)) {
            GMM_WARNING1("Pure multi has failed in interpolate_face !! "
                         "Original point " << m.points()[ipts[i]]
                         << " projection " << P);
          } else {
            if (pgt->convex_ref()->is_in(P) > 1E-8) {
              GMM_WARNING1("Projected point outside the reference convex ! "
                           "Projection canceled. P = " << P);
            } else m.points()[ipts[i]] = P;
          }
          ptdone[ipts[i]] = true;
          // dist(P, new_cts);
        }
      }
    }
  }


  struct point_stock {
    
    bgeot::node_tab points;
    std::vector<dal::bit_vector> constraints_;
    std::vector<scalar_type> radius_;
    const std::vector<const mesher_signed_distance*> &list_constraints;
    scalar_type radius_cv;

    void clear(void) { points.clear(); constraints_.clear();radius_.clear(); }
    scalar_type radius(size_type i) const { return radius_[i]; }
    const dal::bit_vector &constraints(size_type i) const
    { return constraints_[i]; }
    size_type size(void) const { return points.size(); }
    const base_node &operator[](size_type i) const { return points[i]; }

    size_type add(const base_node &pt, const dal::bit_vector &bv,
                    scalar_type r) {
      size_type j = points.search_node(pt);
      if (j == size_type(-1)) {
        j = points.add_node(pt);
        constraints_.push_back(bv);
        radius_.push_back(r);
      }
      return j;
    }
    size_type add(const base_node &pt, scalar_type r) {
      size_type j = points.search_node(pt);
      if (j == size_type(-1)) {
        dal::bit_vector bv;
        for (size_type i = 0; i < list_constraints.size(); ++i)
          if (gmm::abs((*(list_constraints[i]))(pt)) < 1E-8*radius_cv)
            bv.add(i);
        j = points.add_node(pt);
        constraints_.push_back(bv);
        radius_.push_back(r);
      }
      return j;
    }
    size_type add(const base_node &pt) {
      size_type j = points.search_node(pt);
      if (j == size_type(-1)) {
        dal::bit_vector bv;
        for (size_type i = 0; i < list_constraints.size(); ++i)
          if (gmm::abs((*(list_constraints[i]))(pt)) < 1E-8*radius_cv)
            bv.add(i);
        j = points.add_node(pt);
        constraints_.push_back(bv);
        scalar_type r = min_curvature_radius_estimate(list_constraints,pt,bv);
        radius_.push_back(r);
      }
      return j;
    }

    dal::bit_vector decimate(const mesher_signed_distance &ref_element,
                             scalar_type dmin) const {
      dal::bit_vector retained_points;
      if (size() != 0) {
        size_type n = points[0].size();

        bgeot::kdtree points_tree;
        points_tree.reserve(size());
        for (size_type i = 0; i < size(); ++i)
          points_tree.add_point_with_id(points[i], i);

        for (size_type nb_co = n; nb_co != size_type(-1); nb_co --) {
          for (size_type i = 0; i < size(); ++i) {
            if (!(retained_points.is_in(i)) &&
                (constraints(i).card() == nb_co ||
                 (nb_co == n && constraints(i).card() > n)) &&
                ref_element(points[i]) < 1E-8) {
              bool kept = true;
              scalar_type d = (nb_co == 0) ? (dmin * 1.5)
                : std::min(radius(i)*0.25, dmin);
              base_node min = points[i], max = min;
              for (size_type m = 0; m < n; ++m) { min[m]-=d; max[m]+=d; }
              bgeot::kdtree_tab_type inpts;
              points_tree.points_in_box(inpts, min, max);
              for (size_type j = 0; j < inpts.size(); ++j)
                if (retained_points.is_in(inpts[j].i) &&
                    constraints(inpts[j].i).contains(constraints(i))
                    && gmm::vect_dist2(points[i], points[inpts[j].i]) < d)
                  { kept = false; break; }
              if (kept) {
                if (noisy) cout << "kept point : " << points[i] << " co = "
                                << constraints(i) << endl;
                retained_points.add(i);
              }
            }
          }
        }
      }
      return retained_points;
    }

    point_stock(const std::vector<const mesher_signed_distance*> &ls,
                scalar_type rcv)
      : points(scalar_type(10000000)), list_constraints(ls),
        radius_cv(rcv) {}
  };


  static pmesher_signed_distance new_ref_element(bgeot::pgeometric_trans pgt) {
    dim_type n = pgt->structure()->dim();
    size_type nbp = pgt->basic_structure()->nb_points();
    /* Identifying simplexes.                                          */
    if (nbp == size_type(n+1) &&
        pgt->basic_structure() == bgeot::simplex_structure(n)) {
        return new_mesher_simplex_ref(n);
    }
    
    /* Identifying parallelepiped.                                     */
    if (nbp == (size_type(1) << n) &&
        pgt->basic_structure() == bgeot::parallelepiped_structure(n)) {
      base_node rmin(n), rmax(n);
      std::fill(rmax.begin(), rmax.end(), scalar_type(1));
      return new_mesher_rectangle(rmin, rmax);
    }
    
    /* Identifying prisms.                                             */
    if (nbp == size_type(2 * n) &&
        pgt->basic_structure() == bgeot::prism_P1_structure(n)) {
      return new_mesher_prism_ref(n);
    }
    
    GMM_ASSERT1(false, "This element is not taken into account. Contact us");
  }


  struct global_mesh_for_mesh_level_set : public mesh {};
  static mesh& global_mesh() {
    return dal::singleton<global_mesh_for_mesh_level_set>::instance();
  }

  void mesh_level_set::run_delaunay(std::vector<base_node> &fixed_points,
                                    gmm::dense_matrix<size_type> &simplexes,
                                    std::vector<dal::bit_vector> &
                                    /* fixed_points_constraints */) {
    double t0=gmm::uclock_sec();
    if (noisy) cout << "running delaunay with " << fixed_points.size()
                    << " points.." << std::flush;
    bgeot::qhull_delaunay(fixed_points, simplexes);
    if (noisy) cout << " -> " << gmm::mat_ncols(simplexes)
                    << " simplexes [" << gmm::uclock_sec()-t0 << "sec]\n";
  }

  static bool intersects(const mesh_level_set::zone &z1, 
                         const mesh_level_set::zone &z2) {
    for (std::set<const std::string *>::const_iterator it = z1.begin(); it != z1.end();
         ++it)
      if (z2.find(*it) != z2.end()) return true;
    return false;
  }

  void mesh_level_set::merge_zoneset(zoneset &zones1,
                                     const zoneset &zones2) const {
    for (std::set<const zone *>::const_iterator it2 = zones2.begin(); 
         it2 != zones2.end(); ++it2) {
      zone z = *(*it2);
      for (std::set<const zone *>::iterator it1 = zones1.begin(); 
           it1 != zones1.end(); ) {
        if (intersects(z, *(*it1))) {
          z.insert((*it1)->begin(), (*it1)->end());
          zones1.erase(it1++);
        }
        else ++it1;
      }
      zones1.insert(&(*(allzones.insert(z).first)));
    }
  }

  /* recursively replace '0' by '+' and '-', and the add the new zones */
  static void add_sub_zones_no_zero(std::string &s, 
                                    mesh_level_set::zone &z, 
                                    std::set<std::string> &allsubzones) {
    size_t i = s.find('0');
    if (i != size_t(-1)) {
      s[i] = '+'; add_sub_zones_no_zero(s, z, allsubzones);
      s[i] = '-'; add_sub_zones_no_zero(s, z, allsubzones);
    } else {
      z.insert(&(*(allsubzones.insert(s).first)));
    }
  }

  void mesh_level_set::merge_zoneset(zoneset &zones1,
                                     const std::string &subz) const {
    // very sub-optimal
    zone z; std::string s(subz);
    add_sub_zones_no_zero(s, z, allsubzones);
    zoneset zs;
    zs.insert(&(*(allzones.insert(z).first)));
    merge_zoneset(zones1, zs);
  }

  /* prezone was filled for the whole convex by find_crossing_level_set. 
     This information is now refined for each sub-convex.
  */
  void mesh_level_set::find_zones_of_element(size_type cv,
                                             std::string &prezone,
                                             scalar_type radius) {
    convex_info &cvi = cut_cv[cv];
    cvi.zones.clear();
    for (dal::bv_visitor i(cvi.pmsh->convex_index()); !i.finished();++i) {
      // If the sub element is too small, the zone is not taken into account
      if (cvi.pmsh->convex_area_estimate(i) > 1e-8) {
        std::string subz = prezone;
        //cout << "prezone for convex " << cv << " : " << subz << endl;
        for (size_type j = 0; j < level_sets.size(); ++j) {
          if (subz[j] == '*' || subz[j] == '0') {
            int s = sub_simplex_is_not_crossed_by(cv, level_sets[j], i,radius);
            // cout << "sub_simplex_is_not_crossed_by = " << s << endl;
            subz[j] = (s < 0) ? '-' : ((s > 0) ? '+' : '0');
          }
        }
        merge_zoneset(cvi.zones, subz);
      }
    }
    if (noisy) cout << "Number of zones for convex " << cv << " : "
                    << cvi.zones.size() << endl;
  }


  void mesh_level_set::cut_element(size_type cv,
                                   const dal::bit_vector &primary,
                                   const dal::bit_vector &secondary,
                                   scalar_type radius_cv) {
    
    cut_cv[cv] = convex_info();
    cut_cv[cv].pmsh = std::make_shared<mesh>();
    if (noisy) cout << "cutting element " << cv << endl;
    bgeot::pgeometric_trans pgt = linked_mesh().trans_of_convex(cv);
    pmesher_signed_distance ref_element = new_ref_element(pgt);
    std::vector<pmesher_signed_distance> mesher_level_sets;
    
    size_type n = pgt->structure()->dim();
    size_type nbtotls = primary.card() + secondary.card();
    pintegration_method exactint = classical_exact_im(pgt);
    GMM_ASSERT1(nbtotls <= 16,
                "An element is cut by more than 16 level_set, aborting");

    /* 
     * Step 1 : build the signed distances, estimate the curvature radius
     *          and the degree K.
     */
    dim_type K = 0; // Max. degree of the level sets.
    scalar_type r0 = 1E+10; // min curvature radius
    std::vector<const mesher_signed_distance*> list_constraints;
    point_stock mesh_points(list_constraints, radius_cv);

    ref_element->register_constraints(list_constraints);
    size_type nbeltconstraints = list_constraints.size();
    mesher_level_sets.reserve(nbtotls);
    for (size_type ll = 0; ll < level_sets.size(); ++ll) {
      if (primary[ll]) {
        base_node X(n); gmm::fill_random(X);
        K = std::max(K, (level_sets[ll])->degree());
        mesher_level_sets.push_back(level_sets[ll]->mls_of_convex(cv, 0));
        pmesher_signed_distance mls(mesher_level_sets.back());
        list_constraints.push_back(mesher_level_sets.back().get());
        r0 = std::min(r0, curvature_radius_estimate(*mls, X, true));
        GMM_ASSERT1(gmm::abs(r0) >= 1e-13, "Something wrong in your level "
                    "set ... curvature radius = " << r0);
        if (secondary[ll]) {
          mesher_level_sets.push_back(level_sets[ll]->mls_of_convex(cv, 1));
          pmesher_signed_distance mls2(mesher_level_sets.back());
          list_constraints.push_back(mesher_level_sets.back().get());
          r0 = std::min(r0, curvature_radius_estimate(*mls2, X, true));
        }
      }
    }
    
    /* 
     * Step 2 : projection of points of a grid on each signed distance.
     */
    scalar_type h0 = std::min(1./(K+1), 0.2 * r0), dmin = 0.55;
    bool h0_is_ok = true;

    do {

      h0_is_ok = true;
      mesh_points.clear();
      scalar_type geps = .001*h0; 
      size_type nbpt = 1;
      std::vector<size_type> gridnx(n);
      for (size_type i=0; i < n; ++i)
        { gridnx[i]=1+(size_type)(1./h0); nbpt *= gridnx[i]; }
      base_node P(n), Q(n), V(n);
      dal::bit_vector co;
      std::vector<size_type> co_v;

      for (size_type i=0; i < nbpt; ++i) {
        for (size_type k=0, r = i; k < n; ++k) { // building grid point
          unsigned p =  unsigned(r % gridnx[k]);
          P[k] = p * scalar_type(1) / scalar_type(gridnx[k]-1);
          r /= gridnx[k];
        }
        co.clear(); co_v.resize(0);
        if ((*ref_element)(P) < geps) {
          bool kept = false;
          for (size_type k=list_constraints.size()-1; k!=size_type(-1); --k) {
            // for (size_type k=0; k < list_constraints.size(); ++k) {
            // Rerversed to project on level set before on convex boundaries
            gmm::copy(P, Q);
            scalar_type d = list_constraints[k]->grad(P, V);
            if (gmm::vect_norm2(V)*h0*7 > gmm::abs(d))
              if (try_projection(*(list_constraints[k]), Q, true)) {
                if (k >= nbeltconstraints
                    && gmm::vect_dist2(P, Q) < h0 * 3.5) kept = true;
                if (gmm::vect_dist2(P, Q) < h0 / 1.5) {
                  co.add(k); co_v.push_back(k); 
                  if ((*ref_element)(Q) < 1E-8) {
                    scalar_type r =
                      curvature_radius_estimate(*(list_constraints[k]), Q);
                    r0 = std::min(r0, r);
                    if (r0 < 4.*h0) { h0 = 0.2 * r0; h0_is_ok = false; break; }
                    if (k >= nbeltconstraints || kept) mesh_points.add(Q, r);
                  }
                }
              }
          }
          if (kept) mesh_points.add(P, 1.E10);
        }
        size_type nb_co = co.card();
        dal::bit_vector nn;
        if (h0_is_ok)
          for (size_type count = 0; count < (size_type(1) << nb_co); ++count) {
            nn.clear();
            for (size_type j = 0; j < nb_co; ++j)
              if (count & (size_type(1) << j)) nn.add(co_v[j]);
            if (nn.card() > 1 && nn.card() <= n) {
              if (noisy) cout << "trying set of constraints" << nn << endl;
              gmm::copy(P, Q);
              bool ok=pure_multi_constraint_projection(list_constraints,Q,nn);
              if (ok && (*ref_element)(Q) < 1E-9) {
                if (noisy) cout << "Intersection point found " << Q << " with "
                                << nn << endl;
                mesh_points.add(Q);
              }
            }
          }
      }
      
      if (!h0_is_ok) continue;
      
    /* 
     * Step 3 : Delaunay, test if a simplex cut a level set and adapt
     *          the mesh to the curved level sets.
     */

      dmin = 2.*h0;
      if (noisy) cout << "dmin = " << dmin << " h0 = " << h0 << endl;
      if (noisy) cout << "convex " << cv << endl;
      
      dal::bit_vector retained_points
        = mesh_points.decimate(*ref_element, dmin);
      
      bool delaunay_again = true;
      
      std::vector<base_node> fixed_points;
      std::vector<dal::bit_vector> fixed_points_constraints;
      mesh &msh(*(cut_cv[cv].pmsh));
        
      mesh_region &ls_border_faces(cut_cv[cv].ls_border_faces);
      std::vector<base_node> cvpts;

      size_type nb_delaunay = 0;

      while (delaunay_again) {
        if (++nb_delaunay > 15)
          { h0_is_ok = false; h0 /= 2.0; dmin = 2.*h0; break; }
        
        fixed_points.resize(0);
        fixed_points_constraints.resize(0);
        // std::vector<base_node> fixed_points;
        // std::vector<dal::bit_vector> fixed_points_constraints;
        fixed_points.reserve(retained_points.card());
        fixed_points_constraints.reserve(retained_points.card());
        for (dal::bv_visitor i(retained_points); !i.finished(); ++i) {
          fixed_points.push_back(mesh_points[i]);
          fixed_points_constraints.push_back(mesh_points.constraints(i));
        }

        gmm::dense_matrix<size_type> simplexes;
        run_delaunay(fixed_points, simplexes, fixed_points_constraints);
        delaunay_again = false;
        
        msh.clear(); 
        
        for (size_type i = 0; i <  fixed_points.size(); ++i) {
          size_type j = msh.add_point(fixed_points[i]);
          assert(j == i);
        }
        
        cvpts.resize(0);
        for (size_type i = 0; i < gmm::mat_ncols(simplexes); ++i) {
          size_type j = msh.add_convex(bgeot::simplex_geotrans(n,1),
                                       gmm::vect_const_begin(gmm::mat_col(simplexes, i)));
          /* remove flat convexes => beware the final mesh may not be conformal ! */
          if (msh.convex_quality_estimate(j) < 1E-10) msh.sup_convex(j);
          else {
            std::vector<scalar_type> signs(list_constraints.size());
            std::vector<size_type> prev_point(list_constraints.size());
            for (size_type ii = 0; ii <= n; ++ii) {
              for (size_type jj = 0; jj < list_constraints.size(); ++jj) {
                scalar_type dd =
                  (*(list_constraints[jj]))(msh.points_of_convex(j)[ii]);
                if (gmm::abs(dd) > radius_cv * 1E-7) {
                  if (dd * signs[jj] < 0.0) {
                    if (noisy) cout << "Intersection found ... " << jj
                                    << " dd = " << dd << " convex quality = "
                                    << msh.convex_quality_estimate(j) << "\n";
                    // calcul d'intersection
                    base_node X = msh.points_of_convex(j)[ii], G;
                    base_node VV = msh.points_of_convex(j)[prev_point[jj]] - X;
                    if (dd > 0.) gmm::scale(VV, -1.);
                    dd = (*(list_constraints[jj])).grad(X, G);
                    size_type nbit = 0;
                    while (gmm::abs(dd) > 1e-16*radius_cv && (++nbit < 20)) { // Newton
                      scalar_type nG = gmm::vect_sp(G, VV);
                      if (gmm::abs(nG) < 1E-8) nG = 1E-8;
                      if (nG < 0) nG = 1.0;
                      gmm::add(gmm::scaled(VV, -dd / nG), X);
                      dd = (*(list_constraints[jj])).grad(X, G);
                    }
                    if (gmm::abs(dd) > 1e-16*radius_cv) { // brute force dychotomy
                      base_node X1 = msh.points_of_convex(j)[ii];
                      base_node X2 = msh.points_of_convex(j)[prev_point[jj]], X3;
                      scalar_type dd1 = (*(list_constraints[jj]))(X1);
                      scalar_type dd2 = (*(list_constraints[jj]))(X2);
                      if (dd1 > dd2) { std::swap(dd1, dd2); std::swap(X1, X2); }
                      while (gmm::abs(dd1) > 1e-15*radius_cv) {
                        X3 = (X1 + X2) / 2.0;
                        scalar_type dd3 = (*(list_constraints[jj]))(X3);
                        if (dd3 == dd1 || dd3 == dd2) break;
                        if (dd3 > 0) { dd2 = dd3; X2 = X3; }
                        else { dd1 = dd3; X1 = X3; }
                      }
                      X = X1;
                    }
                    
                    size_type kk = mesh_points.add(X);
                    if (!(retained_points[kk])) {
                      retained_points.add(kk);
                      delaunay_again = true;
                      break;
                      // goto delaunay_fin;
                    }
                  }
                  if (signs[jj] == 0.0) { signs[jj] = dd; prev_point[jj] = ii; }
                }
              }
            }
          }
        }
        
        // delaunay_fin :;
        
      }
      if (!h0_is_ok) continue;

      // Produces an order K mesh for K > 1 (with affine element for the moment)
      if (K>1) {
        for (dal::bv_visitor_c j(msh.convex_index()); !j.finished(); ++j) {
          bgeot::pgeometric_trans pgt2 = bgeot::simplex_geotrans(n, K);
          cvpts.resize(pgt2->nb_points());
          for (size_type k=0; k < pgt2->nb_points(); ++k) {
            cvpts[k] = bgeot::simplex_geotrans(n,1)->transform
              (pgt2->convex_ref()->points()[k], msh.points_of_convex(j));
          }
          msh.sup_convex(j);
          /* some new point will be added to the mesh, but the initial ones
             (those on the convex vertices) are kepts */
          size_type jj = msh.add_convex_by_points(pgt2, cvpts.begin());
          assert(jj == j);
        }
      }

      if (noisy) {
        getfem::stored_mesh_slice sl;
        sl.build(msh, getfem::slicer_none(), 1);
        char s[512]; sprintf(s, "totobefore%d.dx", int(cv));
        getfem::dx_export exp(s);
        exp.exporting(sl);
        exp.exporting_mesh_edges();
        exp.write_mesh();
      }

      /* detect the faces lying on level_set boundaries
         (not exact, some other faces maybe be found, use this only for vizualistion) */
      for (dal::bv_visitor i(msh.convex_index()); !i.finished(); ++i) {
        for (short_type f = 0; f <= n; ++f) {
          const mesh::ind_pt_face_ct &fpts
            = msh.ind_points_of_face_of_convex(i, f);
          
          dal::bit_vector cts; bool first = true;
          for (unsigned k=0; k < fpts.size(); ++k) {
            if (fpts[k] >= fixed_points_constraints.size()) {
              assert(K>1);
              continue; /* ignore additional point inserted when K>1 */
            }
            if (first) { cts = fixed_points_constraints[fpts[k]]; first=false; }
            else cts &= fixed_points_constraints[fpts[k]];
          }
          for (dal::bv_visitor ii(cts); !ii.finished(); ++ii) {
            if (ii >= nbeltconstraints)
              ls_border_faces.add(i, f);
          }
        }
      }

      
      if (K > 1) { // To be done only on the level-set ...
        dal::bit_vector ptdone;
        std::vector<size_type> ipts;
        // for (mr_visitor icv(ls_border_faces); !icv.finished(); ++icv) {
        //  size_type i = icv.cv();
        //   short_type f = icv.f();
        for (dal::bv_visitor i(msh.convex_index()); !i.finished(); ++i) {
          for (short_type f = 0; f <= n; ++f) {
            const mesh::ind_pt_face_ct &fpts
              = msh.ind_points_of_face_of_convex(i, f);
            ipts.assign(fpts.begin(), fpts.end());
#ifdef DEBUG_LS
            interpolate_face_chrono.tic();
#endif
            
            interpolate_face(pgt, msh, ptdone, ipts,
                             msh.trans_of_convex(i)->structure()
                             ->faces_structure()[f], fixed_points.size(),
                             fixed_points_constraints, list_constraints);
#ifdef DEBUG_LS
            interpolate_face_chrono.toc();
#endif
          }
        }
      }

      if (noisy) {
        getfem::stored_mesh_slice sl;
        sl.build(msh, getfem::slicer_none(), 12);
        char s[512]; sprintf(s, "toto%d.dx", int(cv));
        getfem::dx_export exp(s);
        exp.exporting(sl);
        exp.exporting_mesh_edges();
        exp.write_mesh();
      }
    
      /* 
       * Step 4 : Building the new integration method.
       */
      base_matrix G;
      bgeot::pgeometric_trans pgt2 = bgeot::simplex_geotrans(n, K);
      papprox_integration
        pai = classical_approx_im(pgt2,dim_type(2*K))->approx_method();
      auto new_approx = std::make_shared<approx_integration>(pgt->convex_ref());
      base_matrix KK(n,n), CS(n,n);
      base_matrix pc(pgt2->nb_points(), n); 
      for (dal::bv_visitor i(msh.convex_index()); !i.finished(); ++i) {
        vectors_to_base_matrix(G, msh.points_of_convex(i));
        bgeot::geotrans_interpolation_context c(msh.trans_of_convex(i),
                                                pai->point(0), G);
        scalar_type sign = 0.0;
        for (size_type j = 0; j < pai->nb_points_on_convex(); ++j) {
          c.set_xref(pai->point(j));
          pgt2->poly_vector_grad(pai->point(j), pc);
          gmm::mult(G,pc,KK);
          scalar_type J = gmm::lu_det(KK);
          if (noisy && J * sign < 0) {
            cout << "CAUTION reversal situation in convex " << i
                 << "sign = " << sign << " J = " << J << endl;
            for (size_type nip = 0; nip < msh.nb_points_of_convex(i); ++nip)
              cout << "Point " << nip << "=" << msh.points_of_convex(i)[nip]
                   << " ";
            cout << endl;
          }
          if (sign == 0 && gmm::abs(J) > 1E-14) sign = J;
          new_approx->add_point(c.xreal(), pai->coeff(j) * gmm::abs(J));
        }
      }

      getfem::mesh_region border_faces;
      getfem::outer_faces_of_mesh(msh, border_faces);
      for (getfem::mr_visitor it(border_faces); !it.finished(); ++it) {
        vectors_to_base_matrix(G, msh.points_of_convex(it.cv()));
        bgeot::geotrans_interpolation_context c(msh.trans_of_convex(it.cv()),
                                                pai->point(0), G);
        for (size_type j = 0; j < pai->nb_points_on_face(it.f()); ++j) {
          c.set_xref(pai->point_on_face(it.f(), j));
          new_approx->add_point(c.xreal(), pai->coeff_on_face(it.f(), j)
                               * gmm::abs(c.J()), it.f() /* faux */);
        }
      }
      new_approx->valid_method();

      /* 
       * Step 5 : Test the validity of produced integration method.
       */
      
      scalar_type error = test_integration_error(new_approx, 1);
      if (noisy) cout << " max monomial integration err: " << error << "\n";
      if (error > 1e-5) {
        if (noisy) cout << "Not Good ! Let us make a finer cut.\n";
        if (dmin > 3*h0) { dmin /= 2.; }
        else { h0 /= 2.0; dmin = 2.*h0; }
        h0_is_ok = false;
      }
 

      if (h0_is_ok && noisy) { // ajout dans global mesh pour visu
        vectors_to_base_matrix(G, linked_mesh().points_of_convex(cv));
        std::vector<size_type> pts(msh.nb_points());
        for (size_type i = 0; i < msh.nb_points(); ++i)
          pts[i] = global_mesh().add_point(pgt->transform(msh.points()[i], G));
        
        for (dal::bv_visitor i(msh.convex_index()); !i.finished(); ++i)
          global_mesh().add_convex(msh.trans_of_convex(i), 
                                   gmm::index_ref_iterator(pts.begin(),
                                   msh.ind_points_of_convex(i).begin()));
      }

    } while (!h0_is_ok);

#ifdef DEBUG_LS
    cout << "Interpolate face: " << interpolate_face_chrono.total()
         << " moyenne: " << interpolate_face_chrono.mean() << "\n";
#endif
  }


  void mesh_level_set::global_cut_mesh(mesh &m) const {
    m.clear();
    base_matrix G;
    for (dal::bv_visitor cv(linked_mesh().convex_index()); 
         !cv.finished(); ++cv) {
      if (is_convex_cut(cv)) {
        const convex_info &ci = (cut_cv.find(cv))->second;
        mesh &msh = *(ci.pmsh);
        bgeot::pgeometric_trans pgt = linked_mesh().trans_of_convex(cv);
        vectors_to_base_matrix(G, linked_mesh().points_of_convex(cv));
        std::vector<size_type> pts(msh.nb_points());
        for (size_type i = 0; i < msh.nb_points(); ++i)
          pts[i] = m.add_point(pgt->transform(msh.points()[i], G));
        
        std::vector<size_type> ic2(msh.nb_allocated_convex());

        for (dal::bv_visitor i(msh.convex_index()); !i.finished(); ++i) {
          ic2[i] = m.add_convex(msh.trans_of_convex(i), 
                                gmm::index_ref_iterator(pts.begin(),
                                msh.ind_points_of_convex(i).begin()));
          
        }
        for (mr_visitor i(ci.ls_border_faces); !i.finished(); ++i) {
          m.region(0).add(ic2[i.cv()], i.f());
        }

      } else {
        m.add_convex_by_points(linked_mesh().trans_of_convex(cv),
                               linked_mesh().points_of_convex(cv).begin());
      }
    }


  }

  void mesh_level_set::update_crack_tip_convexes() {
    crack_tip_convexes_.clear();
     
    for (std::map<size_type, convex_info>::const_iterator it = cut_cv.begin(); 
         it != cut_cv.end(); ++it) {
      size_type cv = it->first;
      mesh &msh = *(it->second.pmsh);      
      for (unsigned ils = 0; ils < nb_level_sets(); ++ils) {
        if (get_level_set(ils)->has_secondary()) {
          pmesher_signed_distance
            mesherls0 = get_level_set(ils)->mls_of_convex(cv, 0, false);
          pmesher_signed_distance
            mesherls1 = get_level_set(ils)->mls_of_convex(cv, 1, false);
          for (dal::bv_visitor ii(msh.convex_index()); !ii.finished(); ++ii) {
            for (unsigned ipt = 0; ipt < msh.nb_points_of_convex(ii); ++ipt) {
              if (gmm::abs((*mesherls0)(msh.points_of_convex(ii)[ipt])) < 1E-10
                  && gmm::abs((*mesherls1)(msh.points_of_convex(ii)[ipt])) < 1E-10) {
                crack_tip_convexes_.add(cv);
                goto next_convex;
              }
            }
          }
        }
      }
    next_convex:
      ;
    }    
  }

  void mesh_level_set::adapt(void) {

    // compute the elements touched by each level set
    // for each element touched, compute the sub mesh
    //   then compute the adapted integration method
    GMM_ASSERT1(linked_mesh_ != 0, "Uninitialized mesh_level_set");
    cut_cv.clear();
    allsubzones.clear();
    zones_of_convexes.clear();
    allzones.clear();

    // noisy = true;

    std::string z;
    for (dal::bv_visitor cv(linked_mesh().convex_index()); 
         !cv.finished(); ++cv) {
      scalar_type radius = linked_mesh().convex_radius_estimate(cv);
      dal::bit_vector prim, sec;
      find_crossing_level_set(cv, prim, sec, z, radius);
      zones_of_convexes[cv] = &(*(allsubzones.insert(z).first));
      if (noisy) cout << "element " << cv << " cut level sets : "
                      << prim << " zone : " << z << endl;
      if (prim.card()) {
        cut_element(cv, prim, sec, radius);
        find_zones_of_element(cv, z, radius);
      }
    }
    if (noisy) {
      getfem::stored_mesh_slice sl;
      sl.build(global_mesh(), getfem::slicer_none(), 6);
      getfem::dx_export exp("totoglob.dx");
      exp.exporting(sl);
      exp.exporting_mesh_edges();
      exp.write_mesh();
    }

    update_crack_tip_convexes();
    is_adapted_ = true;
  }

  // detect the intersection of two or more level sets and the convexes
  // where the intersection occurs. To be  alled after adapt.
  void mesh_level_set::find_level_set_potential_intersections
  (std::vector<size_type> &icv, std::vector<dal::bit_vector> &ils) {

    GMM_ASSERT1(linked_mesh_ != 0, "Uninitialized mesh_level_set");
    std::string z;
    for (dal::bv_visitor cv(linked_mesh().convex_index()); 
         !cv.finished(); ++cv)
      if (is_convex_cut(cv)) {
        scalar_type radius = linked_mesh().convex_radius_estimate(cv);
        dal::bit_vector prim, sec;
        find_crossing_level_set(cv, prim, sec, z, radius);
        if (prim.card() > 1) {
          icv.push_back(cv);
          ils.push_back(prim);
        }
    }
  }
  
  // return +1 if the sub-element is on the one side of the level-set
  //        -1 if the sub-element is on the other side of the level-set
  //         0 if the sub-element is one the positive part of the secundary
  //           level-set if any.
  int mesh_level_set::sub_simplex_is_not_crossed_by(size_type cv,
                                                    plevel_set ls,
                                                    size_type sub_cv,
                                                    scalar_type radius) {
    scalar_type EPS = 1e-7 * radius;
    bgeot::pgeometric_trans pgt = linked_mesh().trans_of_convex(cv);
    convex_info &cvi = cut_cv[cv];
    bgeot::pgeometric_trans pgt2 = cvi.pmsh->trans_of_convex(sub_cv);

    // cout << "cv " << cv << " radius = " << radius << endl;

    pmesher_signed_distance mls0 = ls->mls_of_convex(cv, 0), mls1(mls0);
    if (ls->has_secondary()) mls1 = ls->mls_of_convex(cv, 1);
    int p = 0;
    bool is_cut = false;
    scalar_type d2 = 0, d1 = 1, d0 = 0, d0min = 0;
    for (size_type i = 0; i < pgt2->nb_points(); ++i) {
      d0 = (*mls0)(cvi.pmsh->points_of_convex(sub_cv)[i]);
      if (i == 0) d0min = gmm::abs(d0);
      else d0min = std::min(d0min, gmm::abs(d0));
      if (ls->has_secondary())
        d1 = std::min(d1, (*mls1)(cvi.pmsh->points_of_convex(sub_cv)[i]));
     
      int p2 = ( (d0 < -EPS) ? -1 : ((d0 > EPS) ? +1 : 0));
      if (p == 0) p = p2;
      if (gmm::abs(d0) > gmm::abs(d2)) d2 = d0;
      if (!p2 || p*p2 < 0) is_cut = true;
    }
    // cout << "d0min = " << d0min << " d1 = " << d1 << " iscut = "
    //      << is_cut << endl;
    if (is_cut && ls->has_secondary() && d1 >= -radius*0.0001) return 0;
    // if (d0min < EPS && ls->has_secondary() && d1 >= -EPS) return 0;
    return (d2 < 0.) ? -1 : 1;
  }

  int mesh_level_set::is_not_crossed_by(size_type cv, plevel_set ls,
                                        unsigned lsnum, scalar_type radius) {
    const mesh_fem &mf = ls->get_mesh_fem();
    GMM_ASSERT1(!mf.is_reduced(), "Internal error");
    const mesh_fem::ind_dof_ct &dofs = mf.ind_basic_dof_of_element(cv);
    pfem pf = mf.fem_of_element(cv);
    int p = -2;
    scalar_type EPS = 1e-8 * radius;

    /* easy cases: 
     - different sign on the dof nodes => intersection for sure
     - min value of the levelset greater than the radius of the convex
     => no intersection
    */ 
    for (mesh_fem::ind_dof_ct::const_iterator it=dofs.begin();
         it != dofs.end(); ++it) {
      scalar_type v = ls->values(lsnum)[*it];
      int p2 = ( (v < -EPS) ? -1 : ((v > EPS) ? +1 : 0));
      if (p == -2) p=p2;
      if (!p2 || p*p2 < 0) return 0;
    }

    pmesher_signed_distance mls1 = ls->mls_of_convex(cv, lsnum, false);
    base_node X(pf->dim()), G(pf->dim());
    gmm::fill_random(X); X *= 1E-2;
    scalar_type d = mls1->grad(X, G);
    if (gmm::vect_norm2(G)*2.5 < gmm::abs(d)) return p;

    bgeot::pgeometric_trans pgt = linked_mesh().trans_of_convex(cv);
    pmesher_signed_distance ref_element = new_ref_element(pgt);
    
    gmm::fill_random(X); X *= 1E-2;
    mesher_intersection mi1(ref_element, mls1);
    if (!try_projection(mi1, X)) return p;
    if ((*ref_element)(X) > 1E-8) return p;
    
    gmm::fill_random(X); X *= 1E-2;
    pmesher_signed_distance mls2 = ls->mls_of_convex(cv, lsnum, true);
    mesher_intersection mi2(ref_element, mls2);
    if (!try_projection(mi2, X)) return p;
    if ((*ref_element)(X) > 1E-8) return p;
   
    return 0;
  }

  void mesh_level_set::find_crossing_level_set(size_type cv,
                                               dal::bit_vector &prim,
                                               dal::bit_vector &sec,
                                               std::string &z,
                                               scalar_type radius) {
    prim.clear(); sec.clear();
    z = std::string(level_sets.size(), '*');
    unsigned lsnum = 0;
    for (size_type k = 0; k < level_sets.size(); ++k, ++lsnum) {
      if (noisy) cout << "testing cv " << cv << " with level set "
                      << k << endl;
      int s = is_not_crossed_by(cv, level_sets[k], 0, radius);
      if (!s) {
        if (noisy) cout << "is cut \n";
        if (level_sets[k]->has_secondary()) {
          s = is_not_crossed_by(cv, level_sets[k], 1, radius);
          if (!s) { sec.add(lsnum); prim.add(lsnum); }
          else if (s < 0) prim.add(lsnum); else z[k] = '0';
        }
        else prim.add(lsnum);
      }
      else z[k] = (s < 0) ? '-' : '+';
    }
  }


}  /* end of namespace getfem.                                             */