Op_Conv_EF_Stab_PolyMAC_MPFA_Face.cpp

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#include <Op_Conv_EF_Stab_PolyMAC_MPFA_Face.h>
#include <Op_Grad_PolyMAC_MPFA_Face.h>
#include <Champ_Face_PolyMAC_MPFA.h>
#include <Masse_ajoutee_base.h>
#include <Domaine_Cl_PolyMAC_family.h>
#include <Schema_Temps_base.h>
#include <Domaine_Poly_base.h>
#include <Option_PolyMAC_family.h>
#include <Pb_Multiphase.h>
#include <Matrix_tools.h>
#include <Array_tools.h>
#include <TRUSTLists.h>
#include <Dirichlet.h>
#include <Param.h>
#include <cmath>
#include <Dirichlet_homogene.h>
#include <Symetrie.h>
 
Implemente_instanciable( Op_Conv_EF_Stab_PolyMAC_MPFA_Face, "Op_Conv_EF_Stab_PolyMAC_MPFA_Face", Op_Conv_EF_Stab_PolyMAC_HFV_Face ) ;
Implemente_instanciable(Op_Conv_Amont_PolyMAC_MPFA_Face, "Op_Conv_Amont_PolyMAC_MPFA_Face", Op_Conv_EF_Stab_PolyMAC_MPFA_Face);
Implemente_instanciable(Op_Conv_Centre_PolyMAC_MPFA_Face, "Op_Conv_Centre_PolyMAC_MPFA_Face", Op_Conv_EF_Stab_PolyMAC_MPFA_Face);
 
// XD Op_Conv_EF_Stab_PolyMAC_MPFA_Face interprete Op_Conv_EF_Stab_PolyMAC_MPFA_Face BRACE Class
// XD_CONT Op_Conv_EF_Stab_PolyMAC_MPFA_Face
Sortie& Op_Conv_EF_Stab_PolyMAC_MPFA_Face::printOn(Sortie& os) const { return Op_Conv_PolyMAC_CDO_base::printOn(os); }
Sortie& Op_Conv_Amont_PolyMAC_MPFA_Face::printOn(Sortie& os) const { return Op_Conv_PolyMAC_CDO_base::printOn(os); }
Sortie& Op_Conv_Centre_PolyMAC_MPFA_Face::printOn(Sortie& os) const { return Op_Conv_PolyMAC_CDO_base::printOn(os); }
 
Entree& Op_Conv_EF_Stab_PolyMAC_MPFA_Face::readOn(Entree& is) { return Op_Conv_EF_Stab_PolyMAC_CDO_Face::readOn(is); }
 
Entree& Op_Conv_Amont_PolyMAC_MPFA_Face::readOn(Entree& is)
{
  alpha_ = 1.0;
  return Op_Conv_PolyMAC_CDO_base::readOn(is);
}
Entree& Op_Conv_Centre_PolyMAC_MPFA_Face::readOn(Entree& is)
{
  alpha_ = 0.0;
  return Op_Conv_PolyMAC_CDO_base::readOn(is);
}
 
void Op_Conv_EF_Stab_PolyMAC_MPFA_Face::completer()
{
  Op_Conv_EF_Stab_PolyMAC_HFV_Face::completer();
  ref_cast(Champ_Face_PolyMAC_MPFA, vitesse_.valeur()).init_auxiliary_variables();
}
 
/*! @brief Computes the stability time step for the convection operator.
 *
 * This method calculates the maximum allowable time step based on the CFL stability condition
 * for the convection operator. The computation considers incoming fluxes through
 * each face of every element and determines the most restrictive time step constraint across the
 * entire domain. For multiphase flows, elements with negligible phase presence are excluded from
 * the time step calculation.
 *
 * @return The global minimum stability time step across all MPI processes
 *
 * @note Only incoming convective fluxes are considered in the stability analysis.
 * @note For multiphase problems, elements with phase fraction below 1e-3 are ignored.
 * @note In axisymmetric calculations, boundary faces with specific conditions may be excluded.
 * @note The final time step is synchronized across all MPI processes using the minimum value.
 * @note Flux contributions account for face porosity and surface area weighting.
 */
double Op_Conv_EF_Stab_PolyMAC_MPFA_Face::calculer_dt_stab() const
{
  double dt = 1e10;
  const Domaine_Poly_base& domaine = le_dom_poly_.valeur();
  const DoubleVect& fs = domaine.face_surfaces(), &pf = equation().milieu().porosite_face(), &ve = domaine.volumes(),
                    &pe = equation().milieu().porosite_elem();
  const DoubleTab& vit = vitesse_->valeurs(),
                   *alp = sub_type(Pb_Multiphase, equation().probleme()) ? &ref_cast(Pb_Multiphase, equation().probleme()).equation_masse().inconnue().passe() : nullptr;
  const IntTab& e_f = domaine.elem_faces(), &f_e = domaine.face_voisins();
  const IntTab *fcl = polymac_flica5 ? nullptr : &ref_cast(Champ_Face_PolyMAC_MPFA, equation().inconnue()).fcl();
  const int N = vit.line_size();
  DoubleTrav flux(N); //sum of fluxes pf * |f| * vf, minimum volume of elements/faces affected by this flux
 
  for (int e = 0; e < domaine.nb_elem(); e++)
    {
      const double vol = pe(e) * ve(e);
      flux = 0.;
 
      for (int i = 0; i < e_f.dimension(1); i++)
        {
          const int f = e_f(e, i);
          if (f < 0) continue;
          if (fcl && Option_PolyMAC_family::TRAITEMENT_AXI && (*fcl)(f, 0) == 2) continue;
 
          for (int n = 0; n < N; n++)
            flux(n) += pf(f) * fs(f) * std::max((e == f_e(f, 1) ? 1 : -1) * vit(f, n), 0.);
        }
 
      for (int n = 0; n < N; n++)
        if ((!alp || (*alp)(e, n) > 1e-3) && std::abs(flux(n)) > 1e-12)
          dt = std::min(dt, vol / flux(n));
    }
 
  return Process::mp_min(dt);
}
 
/*! @brief Dimensions the matrix blocks for the PolyMAC_MPFA Face convection operator with EF stabilization.
 *
 * This method constructs the sparsity pattern and allocates memory for the system matrix by analyzing
 * face-element connectivity and establishing the stencil relationships between degrees of freedom.
 * The resulting matrix structure accounts for face-face and element-element contributions based on
 * the PolyMAC_MPFA Face discretization scheme.
 *
 * @param matrices Map containing the system matrices indexed by unknown field names
 * @param semi_impl Map of semi-implicit terms indexed by unknown field names
 *
 * @note The method returns early if no diagonal block exists or if semi-implicit treatment is requested.
 * @note For multiphase problems with added mass correlation, cross-coupling terms between phases are included.
 * @note The stencil construction handles face equivalence relationships and boundary conditions appropriately.
 * @note Duplicate entries in the stencil are automatically removed before matrix allocation.
 */
void Op_Conv_EF_Stab_PolyMAC_MPFA_Face::dimensionner_blocs(matrices_t matrices, const tabs_t& semi_impl) const
{
  const Domaine_Poly_base& domaine = le_dom_poly_.valeur();
  const Champ_Face_PolyMAC_MPFA& ch = ref_cast(Champ_Face_PolyMAC_MPFA, equation().inconnue());
 
  const std::string& nom_inco = ch.le_nom().getString();
  if (!matrices.count(nom_inco) || semi_impl.count(nom_inco))
    return; //no diagonal block or semi-implicit -> nothing to do
 
  const IntTab& f_e = domaine.face_voisins(), &e_f = domaine.elem_faces(), &fcl = ch.fcl(), &equiv = domaine.equiv();
  const DoubleTab& nf = domaine.face_normales();
  const DoubleVect& fs = domaine.face_surfaces();
 
  const Pb_Multiphase *pbm = sub_type(Pb_Multiphase, equation().probleme()) ? &ref_cast(Pb_Multiphase, equation().probleme()) : nullptr;
  const Masse_ajoutee_base *corr = pbm && pbm->has_correlation("masse_ajoutee") ? &ref_cast(Masse_ajoutee_base, pbm->get_correlation("masse_ajoutee")) : nullptr;
  Matrice_Morse& mat = *matrices.at(nom_inco), mat2;
 
  const int ne_tot = domaine.nb_elem_tot(), nf_tot = domaine.nb_faces_tot(), N = equation().inconnue().valeurs().line_size(), D = dimension;
 
  Stencil stencil(0, 2);
 
  // Loop over all faces of the domain
  for (int f = 0; f < domaine.nb_faces_tot(); f++)
    if (f_e(f, 0) >= 0 && (f_e(f, 1) >= 0 || fcl(f, 0) == 3))
      {
        // Loop over the two adjacent elements of the face
        for (int i = 0; i < 2; i++)
          {
            const int e = f_e(f, i);
            if (e < 0) continue; // virtual elem
 
            for (int j = 0; j < 2; j++)
              {
                const int eb = f_e(f, j);
                // Contribution of faces connected to the element
                if (eb < 0) continue; // virtual elem
 
                for (int k = 0; k < e_f.dimension(1); k++)
                  {
                    const int fb = e_f(e, k);
                    if (fb < 0) continue;
 
                    if (fb < domaine.nb_faces() && fcl(fb, 0) < 2)
                      {
                        int fc = equiv(f, i, k);
                        if (fc >= 0) // real face
                          {
                            // Equivalence case: face -> face
                            for (int n = 0; n < N; n++)
                              for (int m = (corr ? 0 : n); m < (corr ? N : n + 1); m++)
                                stencil.append_line(N * fb + n, N * fc + m);
                          }
                        else if (f_e(f, 1) >= 0) // boundary
                          {
                            // No equivalence: element -> face
                            for (int d = 0; d < D; d++)
                              if (std::fabs(nf(fb, d)) > 1e-6 * fs(fb))
                                for (int n = 0; n < N; n++)
                                  for (int m = (corr ? 0 : n); m < (corr ? N : n + 1); m++)
                                    stencil.append_line(N * fb + n, N * (nf_tot + D * eb + d) + m);
                          }
                      }
                  }
 
                // Contribution element -element
                for (int d = 0; d < D; d++)
                  for (int n = 0; n < N; n++)
                    for (int m = (corr ? 0 : n); m < (corr ? N : n + 1); m++)
                      stencil.append_line(N * (nf_tot + D * e + d) + n, N * (nf_tot + D * eb + d) + m);
              }
          }
      }
 
  // Remove duplicates and allocate the matrix
  tableau_trier_retirer_doublons(stencil);
  Matrix_tools::allocate_morse_matrix(N * (nf_tot + D * ne_tot), N * (nf_tot + D * ne_tot), stencil, mat2);
  mat.nb_colonnes() ? mat += mat2 : mat = mat2;
}
 
/*! @brief Adds convection contributions to the system matrix and right-hand side vector.
 *
 * This method implements the convection operator with EF stabilization for PolyMAC_MPFA Face discretization.
 * It computes face-based convection fluxes using upwind stabilization and assembles the corresponding
 * matrix coefficients and source terms. The method handles both face-face and element-element contributions,
 * accounting for face equivalence relationships and boundary conditions.
 *
 * @param matrices Map containing the system matrices indexed by unknown field names
 * @param secmem Right-hand side vector to be updated with convection contributions
 * @param semi_impl Map of semi-implicit field values indexed by field names
 *
 * @note The convection flux computation uses a blended upwind scheme with stabilization parameter alpha_.
 * @note For multiphase flows, added mass correlations are included in the mass matrix contributions.
 * @note The method distinguishes between compressible and incompressible flow formulations.
 * @note Dirichlet boundary conditions are enforced through direct substitution in the source term.
 * @note Face equivalence relationships are handled to maintain consistency across mesh interfaces.
 * @note Performance statistics are automatically tracked during execution.
 */
void Op_Conv_EF_Stab_PolyMAC_MPFA_Face::ajouter_blocs(matrices_t matrices, DoubleTab& secmem, const tabs_t& semi_impl) const
{
  if (polymac_flica5)
    {
      const Domaine_Poly_base& domaine = le_dom_poly_.valeur();
      const Champ_Face_PolyMAC_MPFA& ch = ref_cast(Champ_Face_PolyMAC_MPFA, le_champ_inco ? le_champ_inco.valeur() : equation().inconnue());
      const Conds_lim& cls = la_zcl_poly_->les_conditions_limites();
      const IntTab& f_e = domaine.face_voisins(), &e_f = domaine.elem_faces(), &fcl = ch.fcl(), &equiv = domaine.equiv();
      const DoubleTab& vit = vitesse_->valeurs(), &nf = domaine.face_normales(), &vfd = domaine.volumes_entrelaces_dir();
      const DoubleVect& fs = domaine.face_surfaces(), &pe = porosite_e, &pf = porosite_f, &ve = domaine.volumes();
 
      /* a_r : alpha_rho product if Pb_Multiphase -> via semi-implicit, or by retrieving the conserved field from the mass equation */
      const std::string& nom_inco = ch.le_nom().getString();
      const DoubleTab& inco = semi_impl.count(nom_inco) ? semi_impl.at(nom_inco) : ch.valeurs();
      Matrice_Morse *mat = matrices.count(nom_inco) && !semi_impl.count(nom_inco) ? matrices.at(nom_inco) : nullptr;
 
      const int nf_tot = domaine.nb_faces_tot(), N = inco.line_size(), D = dimension;
 
      DoubleTrav dfac(2, N, N), masse(N, N);
      // Loop over all faces of the domain
      for (int f = 0; f < domaine.nb_faces_tot(); f++)
        {
          if (f_e(f, 0) >= 0 && (f_e(f, 1) >= 0 || fcl(f, 0) == 1 || fcl(f, 0) == 3))
            {
              // Compute face contributions
              dfac = 0.;
              for (int i = 0; i < 2; i++)
                {
                  // Mass: diagonal + correction
                  masse(0, 0) = std::fabs(vit[f]) > 1e-10 ? inco(f) / vit[f] : 1.0;
 
                  // Contribution to dfac
                  const int eb = f_e(f, i);
                  for (int n = 0; n < N; n++)
                    for (int m = 0; m < N; m++)
                      {
                        const double signe = (vit(f, m) * (i ? -1 : 1) >= 0) ? 1. : (vit(f, m) ? -1. : 0.);
                        dfac((fcl(f, 0) == 1) ? 0 : i, n, m) += fs(f) * inco[f] * pe(eb >= 0 ? eb : f_e(f, 0)) * (1. + signe * alpha_) / 2;
                      }
                }
 
              // Compute contributions to the matrix and right-hand side
              for (int i = 0; i < 2; i++)
                {
                  const int e = f_e(f, i);
                  if (e < 0) continue; // virtual elem
 
                  // partie "faces"
                  for (int k = 0; k < e_f.dimension(1); k++)
                    {
                      const int fb = e_f(e, k);
                      if (fb < 0) continue; // face in-existante (polyhedre)
 
                      if (fb < domaine.nb_faces() && fcl(fb, 0) < 2)
                        {
                          // Equivalence case: face -> face
                          int fc = equiv(f, i, k);
                          if (fc >= 0 || f_e(f, 1) < 0)
                            {
                              for (int j = 0; j < 2; j++)
                                {
                                  int eb = f_e(f, j);
                                  const int fd = (j == i) ? fb : fc; // Source face
                                  //multiplier to convert from vf to ve
                                  double mult = (fd < 0 || domaine.dot(&nf(fb, 0), &nf(fd, 0)) > 0) ? 1 : -1;
                                  mult *= (fd >= 0) ? pf(fd) / pe(eb) : 1;
 
                                  for (int n = 0; n < N; n++)
                                    for (int m = 0; m < N; m++)
                                      if (dfac(j, n, m))
                                        {
                                          const double fac = (i ? -1 : 1) * vfd(fb, e != f_e(fb, 0)) * dfac(j, n, m) / ve(e);
 
                                          // Update the right-hand side
                                          if (fd >= 0)
                                            secmem(fb, n) -= fac * mult * vit[fd];
                                          else
                                            // Dirichlet boundary condition
                                            for (int d = 0; d < D; d++)
                                              secmem(fb, n) -= fac * nf(fb, d) / fs(fb) * ref_cast(Dirichlet, cls[fcl(f, 1)].valeur()).val_imp(fcl(f, 2), N * d + m) / masse(0, 0);
 
                                          if (!incompressible_)
                                            secmem(fb, n) += fac * vit[fb];
 
                                          // Update the matrix
                                          if (mat && fac)
                                            {
                                              if (fd >= 0 && fcl(fd, 0) < 2)
                                                (*mat)(N * fb + n, N * fd + m) += fac * mult;
                                              if (!incompressible_)
                                                (*mat)(N * fb + n, N * fb + m) -= fac;
                                            }
                                        }
                                }
                            }
                          // No equivalence: connected elements
                          else
                            {
                              for (int j = 0; j < 2; j++)
                                {
                                  const int eb = f_e(f, j);
                                  for (int d = 0; d < D; d++)
                                    if (std::fabs(nf(fb, d)) > 1e-6 * fs(fb))
                                      for (int n = 0; n < N; n++)
                                        for (int m = 0; m < N; m++)
                                          if (dfac(j, n, m))
                                            {
                                              const double fac = (i ? -1 : 1) * vfd(fb, e != f_e(fb, 0)) * dfac(j, n, m) / ve(e) * nf(fb, d) / fs(fb);
 
                                              // Update of the right-hand side
                                              secmem(fb, n) -= fac * vit[nf_tot + D * eb + d];
                                              if (!incompressible_)
                                                secmem(fb, n) += fac * vit[nf_tot + D * e + d];
 
                                              // Update of the matrix
                                              if (mat && fac)
                                                {
                                                  (*mat)(N * fb + n, N * (nf_tot + D * eb + d) + m) += fac;
                                                  if (!incompressible_)
                                                    (*mat)(N * fb + n, N * (nf_tot + D * e + d) + m) -= fac;
                                                }
                                            }
                                }
                            }
                        }
                    }
 
                  // "elem" part
                  for (int j = 0; j < 2; j++)
                    {
                      const int eb = f_e(f, j);
                      for (int d = 0; d < D; d++)
                        for (int n = 0; n < N; n++)
                          for (int m = 0; m < N; m++)
                            if (dfac(j, n, m))
                              {
                                const double fac = (i ? -1 : 1) * dfac(j, n, m);
 
                                // Update the right-hand side
                                secmem(nf_tot + D * e + d, n) -= fac * (eb >= 0 ? vit[nf_tot + D * eb + d] : ref_cast(Dirichlet, cls[fcl(f, 1)].valeur()).val_imp(fcl(f, 2), N * d + m));
                                if (!incompressible_)
                                  secmem(nf_tot + D * e + d, n) += fac * vit[nf_tot + D * e + d]; //part v div(alpha rho v)
 
                                if (mat && fac)
                                  {
                                    if (eb >= 0)
                                      (*mat)(N * (nf_tot + D * e + d) + n, N * (nf_tot + D * eb + d) + m) += fac;
                                    if (!incompressible_)
                                      (*mat)(N * (nf_tot + D * e + d) + n, N * (nf_tot + D * e + d) + m) -= fac;
                                  }
                              }
                    }
                }
            }
        }
      return;
    }
 
  const Domaine_Poly_base& domaine = le_dom_poly_.valeur();
  const Champ_Face_PolyMAC_MPFA& ch = ref_cast(Champ_Face_PolyMAC_MPFA, equation().inconnue());
  const Conds_lim& cls = la_zcl_poly_->les_conditions_limites();
  const IntTab& f_e = domaine.face_voisins(), &e_f = domaine.elem_faces(), &fcl = ch.fcl(), &equiv = domaine.equiv();
  const DoubleTab& vit = ch.passe(), &nf = domaine.face_normales(), &vfd = domaine.volumes_entrelaces_dir();
  const DoubleVect& fs = domaine.face_surfaces(), &pe = porosite_e, &pf = porosite_f, &ve = domaine.volumes();
 
  /* a_r : alpha_rho product if Pb_Multiphase -> via semi-implicit, or by retrieving the conserved field from the mass equation */
  const std::string& nom_inco = ch.le_nom().getString();
  const Pb_Multiphase *pbm = sub_type(Pb_Multiphase, equation().probleme()) ? &ref_cast(Pb_Multiphase, equation().probleme()) : nullptr;
  const Masse_ajoutee_base *corr = pbm && pbm->has_correlation("masse_ajoutee") ? &ref_cast(Masse_ajoutee_base, pbm->get_correlation("masse_ajoutee")) : nullptr;
  const DoubleTab& inco = semi_impl.count(nom_inco) ? semi_impl.at(nom_inco) : ch.valeurs(),
                   *a_r = !pbm ? nullptr : semi_impl.count("alpha_rho") ? &semi_impl.at("alpha_rho") : &pbm->equation_masse().champ_conserve().valeurs(),
                    *alp = pbm ? &pbm->equation_masse().inconnue().passe() : nullptr, &rho = equation().milieu().masse_volumique().passe();
  Matrice_Morse *mat = matrices.count(nom_inco) && !semi_impl.count(nom_inco) ? matrices.at(nom_inco) : nullptr;
 
  const int nf_tot = domaine.nb_faces_tot(), N = inco.line_size(), D = dimension;
 
  DoubleTrav dfac(2, N, N), masse(N, N);
  // Loop over all faces of the domain
  for (int f = 0; f < domaine.nb_faces_tot(); f++)
    {
      if (f_e(f, 0) >= 0 && (f_e(f, 1) >= 0 || fcl(f, 0) == 1 || fcl(f, 0) == 3))
        {
          // Compute face contributions
          dfac = 0.;
          for (int i = 0; i < 2; i++)
            {
              const int e = f_e(f, (f_e(f, i) >= 0) ? i : 0);
 
              // Mass: diagonal + correction
              masse = 0.;
              for (int n = 0; n < N; n++)
                masse(n, n) = a_r ? (*a_r)(e, n) : 1;
 
              if (corr)
                corr->ajouter(&(*alp)(e, 0), &rho(e, 0), masse);
 
              // Contribution to dfac
              const int eb = f_e(f, i);
              for (int n = 0; n < N; n++)
                for (int m = 0; m < N; m++)
                  {
                    const double signe = (vit(f, m) * (i ? -1 : 1) >= 0) ? 1. : (vit(f, m) ? -1. : 0.);
                    dfac((fcl(f, 0) == 1) ? 0 : i, n, m) += fs(f) * vit(f, m) * pe(eb >= 0 ? eb : f_e(f, 0)) * masse(n, m) * (1. + signe * alpha_) / 2;
                  }
            }
 
          // Compute contributions to the matrix and right-hand side
          for (int i = 0; i < 2 ; i++)
            {
              const int e = f_e(f, i);
              if (e < 0) continue; // virtual elem
 
              // "faces" part
              for (int k = 0; k < e_f.dimension(1); k++)
                {
                  const int fb = e_f(e, k);
                  if (fb < 0) continue; // face in-existante (polyhedre)
 
                  if (fb < domaine.nb_faces() && fcl(fb, 0) < 2)
                    {
                      // Equivalence case: face -> face
                      int fc = equiv(f, i, k);
                      if (fc >= 0 || f_e(f, 1) < 0)
                        {
                          for (int j = 0; j < 2; j++)
                            {
                              int eb = f_e(f, j);
                              const int fd = (j == i) ? fb : fc; // Source face
                              //multiplier to convert from vf to ve
                              double mult = (fd < 0 || domaine.dot(&nf(fb, 0), &nf(fd, 0)) > 0) ? 1 : -1;
                              mult *= (fd >= 0) ? pf(fd) / pe(eb) : 1;
 
                              for (int n = 0; n < N; n++)
                                for (int m = 0; m < N; m++)
                                  if (dfac(j, n, m))
                                    {
                                      const double fac = (i ? -1 : 1) * vfd(fb, e != f_e(fb, 0)) * dfac(j, n, m) / ve(e);
 
                                      // Update the right-hand side
                                      if (fd >= 0)
                                        secmem(fb, n) -= fac * mult * inco(fd, m);
                                      else
                                        // Dirichlet boundary condition
                                        for (int d = 0; d < D; d++)
                                          secmem(fb, n) -= fac * nf(fb, d) / fs(fb) * ref_cast(Dirichlet, cls[fcl(f, 1)].valeur()).val_imp(fcl(f, 2), N * d + m);
 
                                      if (!incompressible_)
                                        secmem(fb, n) += fac * inco(fb, m);
 
                                      // Update the matrix
                                      if (mat && fac)
                                        {
                                          if (fd >= 0 && fcl(fd, 0) < 2)
                                            (*mat)(N * fb + n, N * fd + m) += fac * mult;
                                          if (!incompressible_)
                                            (*mat)(N * fb + n, N * fb + m) -= fac;
                                        }
                                    }
                            }
                        }
                      // No equivalence: connected elements
                      else
                        {
                          for (int j = 0; j < 2; j++)
                            {
                              const int eb = f_e(f, j);
                              for (int d = 0; d < D; d++)
                                if (std::fabs(nf(fb, d)) > 1e-6 * fs(fb))
                                  for (int n = 0; n < N; n++)
                                    for (int m = 0; m < N; m++)
                                      if (dfac(j, n, m))
                                        {
                                          const double fac = (i ? -1 : 1) * vfd(fb, e != f_e(fb, 0)) * dfac(j, n, m) / ve(e) * nf(fb, d) / fs(fb);
 
                                          // Update the right-hand side
                                          secmem(fb, n) -= fac * inco(nf_tot + D * eb + d, m);
                                          if (!incompressible_)
                                            secmem(fb, n) += fac * inco(nf_tot + D * e + d, m);
 
                                          // Update the matrix
                                          if (mat && fac)
                                            {
                                              (*mat)(N * fb + n, N * (nf_tot + D * eb + d) + m) += fac;
                                              if (!incompressible_)
                                                (*mat)(N * fb + n, N * (nf_tot + D * e + d) + m) -= fac;
                                            }
                                        }
                            }
                        }
                    }
                }
 
              // "elem" part
              for (int j = 0; j < 2; j++)
                {
                  const int eb = f_e(f, j);
                  for (int d = 0; d < D; d++)
                    for (int n = 0; n < N; n++)
                      for (int m = 0; m < N; m++)
                        if (dfac(j, n, m))
                          {
                            const double fac = (i ? -1 : 1) * dfac(j, n, m);
 
                            // Update the right-hand side
                            secmem(nf_tot + D * e + d, n) -= fac * (eb >= 0 ? inco(nf_tot + D * eb + d, m) : ref_cast(Dirichlet, cls[fcl(f, 1)].valeur()).val_imp(fcl(f, 2), N * d + m));
                            if (!incompressible_)
                              secmem(nf_tot + D * e + d, n) += fac * inco(nf_tot + D * e + d, m); //part v div(alpha rho v)
 
                            if (mat && fac)
                              {
                                if (eb >= 0)
                                  (*mat)(N * (nf_tot + D * e + d) + n, N * (nf_tot + D * eb + d) + m) += fac;
                                if (!incompressible_)
                                  (*mat)(N * (nf_tot + D * e + d) + n, N * (nf_tot + D * e + d) + m) -= fac;
                              }
                          }
                }
            }
        }
    }
}