et_bfrot_equilibre.C

/*
 * Method of class Et_rot_bifluid to compute a static spherical configuration.
 *
 * (see file etoile.h for documentation).
 *
 */

/*
 *   Copyright (c) 2001 Jerome Novak
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation; either version 2 of the License, or
 *   (at your option) any later version.
 *
 *   LORENE 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 General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */


 

/*
 * $Id: et_bfrot_equilibre.C,v 1.22 2017/10/06 12:36:34 a_sourie Exp $
 * $Log: et_bfrot_equilibre.C,v $
 * Revision 1.22  2017/10/06 12:36:34  a_sourie
 * Cleaning of tabulated 2-fluid EoS class + superfluid rotating star model.
 *
 * Revision 1.21  2016/12/05 16:17:52  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.20  2015/06/10 14:39:17  a_sourie
 * New class Eos_bf_tabul for tabulated 2-fluid EoSs and associated functions for the computation of rotating stars with such EoSs.
 *
 * Revision 1.19  2014/10/13 08:52:54  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.18  2014/10/06 15:13:07  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.17  2006/03/13 10:02:27  j_novak
 * Added things for triaxial perturbations.
 *
 * Revision 1.16  2004/09/01 10:56:05  r_prix
 * added option of converging baryon-mass to equilibrium_bi()
 *
 * Revision 1.15  2004/08/30 09:54:20  r_prix
 * experimental version of Kepler-limit finder for 2-fluid stars
 *
 * Revision 1.14  2004/03/25 10:29:03  j_novak
 * All LORENE's units are now defined in the namespace Unites (in file unites.h).
 *
 * Revision 1.13  2003/12/11 12:43:35  r_prix
 * activated adaptive grid for 2-fluid star (taken from Etoile_rot)
 *
 * Revision 1.12  2003/12/04 14:28:26  r_prix
 * allow for the case of "slow-rot-style" EOS inversion, in which we need to adapt
 * the inner domain to n_outer=0 instead of mu_outer=0 ...
 * (this should only be used for comparison to analytic slow-rot solution!)
 *
 * Revision 1.11  2003/11/25 12:49:44  j_novak
 * Modified headers to compile on IRIX. Changed Mapping to be Map_af (speed
 * enhancement).
 *
 * Revision 1.10  2003/11/20 14:01:25  r_prix
 * changed member names to better conform to Lorene coding standards:
 * J_euler -> j_euler, EpS_euler -> enerps_euler, Delta_car -> delta_car
 *
 * Revision 1.9  2003/11/19 22:01:57  e_gourgoulhon
 * -- Relaxation on logn and dzeta performed only if mer >= 10.
 * -- err_grv2 is now evaluated also in the Newtonian case.
 *
 * Revision 1.8  2003/11/18 18:38:11  r_prix
 * use of new member EpS_euler: matter sources in equilibrium() and global quantities
 * no longer distinguish Newtonian/relativistic, as all terms should have the right limit...
 *
 * Revision 1.7  2003/11/17 13:49:43  r_prix
 * - moved superluminal check into hydro_euler()
 * - removed some warnings
 *
 * Revision 1.6  2003/11/13 12:14:35  r_prix
 * *) removed all use of etoile-type specific u_euler, press
 *   and use 3+1 components of Tmunu instead
 *
 * Revision 1.5  2002/10/16 14:36:35  j_novak
 * Reorganization of #include instructions of standard C++, in order to
 * use experimental version 3 of gcc.
 *
 * Revision 1.4  2002/04/05 09:09:36  j_novak
 * The inversion of the EOS for 2-fluids polytrope has been modified.
 * Some errors in the determination of the surface were corrected.
 *
 * Revision 1.3  2002/01/16 15:03:28  j_novak
 * *** empty log message ***
 *
 * Revision 1.2  2002/01/03 15:30:28  j_novak
 * Some comments modified.
 *
 * Revision 1.1.1.1  2001/11/20 15:19:28  e_gourgoulhon
 * LORENE
 *
 * Revision 1.1  2001/06/22  15:40:06  novak
 * Initial revision
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bfrot_equilibre.C,v 1.22 2017/10/06 12:36:34 a_sourie Exp $
 *
 */

// Headers C
#include <cmath>

// Headers Lorene
#include "et_rot_bifluid.h"
#include "param.h"
#include "unites.h"     

#include "graphique.h"
#include "utilitaires.h"

namespace Lorene {

//-------------------------------------------------------------------------
//-------------------------------------------------------------------------

//                        Axial Equilibrium

//-------------------------------------------------------------------------
//-------------------------------------------------------------------------

void Et_rot_bifluid::equilibrium_bi
(double ent_c, double ent2_c, double omega0, double omega20, 
 const Tbl& ent_limit, const Tbl& ent2_limit, const Itbl& icontrol, 
 const Tbl& control, Tbl& diff,
 int mer_mass, double mbar1_wanted, double mbar2_wanted, double aexp_mass) 
{
                 
  // Fundamental constants and units
  // -------------------------------
  using namespace Unites ;

  // For the display 
  // ---------------
  char display_bold[]="x[1m" ; display_bold[0] = 27 ;
  char display_normal[] = "x[0m" ; display_normal[0] = 27 ;

  // Grid parameters
  // ---------------
    
  const Mg3d* mg = mp.get_mg() ; 
  int nz = mg->get_nzone() ;        // total number of domains

  // The following is required to initialize mp_prev as a Map_af
  Map_et& mp_et = dynamic_cast<Map_et&>(mp) ;  // reference
    
  // Index of the point at phi=0, theta=pi/2 at the surface of the star:
  assert(mg->get_type_t() == SYM) ; 
  int l_b = nzet - 1 ; 
  int i_b = mg->get_nr(l_b) - 1 ; 
  int j_b = mg->get_nt(l_b) - 1 ; 
  int k_b = 0 ; 
    
  // Value of the enthalpies defining the surface of each fluid
  double ent1_b = ent_limit(nzet-1)  ;
  double ent2_b = ent2_limit(nzet-1) ;

  // This value is chosen so that the grid contain both fluids
//       double ent_b = (ent1_b > ent2_b ? ent1_b : ent2_b) ; 
    
  // Parameters to control the iteration
  // -----------------------------------
    
  int mer_max = icontrol(0) ; 
  int mer_rot = icontrol(1) ;
  int mer_change_omega = icontrol(2) ; 
  int mer_fix_omega = icontrol(3) ; 
  int mermax_poisson = icontrol(4) ; 
  int nzadapt = icontrol(5);        // number of domains for adaptive grid
  int kepler_fluid = icontrol(6);   // fluid-index for kepler-limit (0=none,3=both)
  int kepler_wait_steps = icontrol(7);
  int mer_triax = icontrol(8) ;
  
  int niter ;

  // Protections:
  if (mer_change_omega < mer_rot) {
    cout << "Et_rot_bifluid::equilibrium: mer_change_omega < mer_rot !" << endl ;
    cout << " mer_change_omega = " << mer_change_omega << endl ; 
    cout << " mer_rot = " << mer_rot << endl ; 
    abort() ; 
  }
  if (mer_fix_omega < mer_change_omega) {
    cout << "Et_rot_bifluid::equilibrium: mer_fix_omega < mer_change_omega !" 
     << endl ;
    cout << " mer_fix_omega = " << mer_fix_omega << endl ; 
    cout << " mer_change_omega = " << mer_change_omega << endl ; 
    abort() ; 
  }

  double precis = control(0) ; 
  double omega_ini = control(1) ;
  double omega2_ini = control(2) ;
  double relax = control(3) ;
  double relax_prev = double(1) - relax ;  
  double relax_poisson = control(4) ; 
  // some additional stuff for adaptive grid:
  double thres_adapt = control(5) ; 
  double precis_adapt = control(6) ; 
  double kepler_factor = control(7);
  if (kepler_factor <= 1.0)
    {
      cout << "ERROR: Kepler factor has to be greater than 1!!\n";
      abort();
    }
  double ampli_triax = control(8) ;

  // Error indicators
  // ----------------
    
  diff.set_etat_qcq() ; 
  double diff_ent ;
  double& diff_ent1 = diff.set(0) ; 
  double& diff_ent2 = diff.set(1) ; 
  double& diff_nuf = diff.set(2) ; 
  double& diff_nuq = diff.set(3) ; 
  //    double& diff_dzeta = diff.set(4) ; 
  //    double& diff_ggg = diff.set(5) ; 
  double& diff_shift_x = diff.set(6) ; 
  double& diff_shift_y = diff.set(7) ; 
  double& vit_triax = diff.set(8) ;  

  // Parameters for the function Map_et::adapt
  // -----------------------------------------
    
  Param par_adapt ; 
  int nitermax = 100 ;  
  int adapt_flag = 1 ;    //  1 = performs the full computation, 
                //  0 = performs only the rescaling by 
                //      the factor alpha_r
  int nz_search = nzet + 1 ;  // Number of domains for searching the enthalpy
                //  isosurfaces
  double alpha_r ; 
  double reg_map = 1. ; // 1 = regular mapping, 0 = contracting mapping

  par_adapt.add_int(nitermax, 0) ; // maximum number of iterations to
                     // locate zeros by the secant method
  par_adapt.add_int(nzadapt, 1) ; // number of domains where the adjustment 
                    // to the isosurfaces of ent is to be 
                    // performed
  par_adapt.add_int(nz_search, 2) ; // number of domains to search for
                    // the enthalpy isosurface
  par_adapt.add_int(adapt_flag, 3) ; //  1 = performs the full computation, 
                       //  0 = performs only the rescaling by 
                       //      the factor alpha_r
  par_adapt.add_int(j_b, 4) ; //  theta index of the collocation point 
                    //  (theta_*, phi_*)
  par_adapt.add_int(k_b, 5) ; //  theta index of the collocation point 
                    //  (theta_*, phi_*)

  par_adapt.add_int_mod(niter, 0) ;  //  number of iterations actually used in 
                       //  the secant method
    
  par_adapt.add_double(precis_adapt, 0) ; // required absolute precision in 
                         // the determination of zeros by 
                         // the secant method
  par_adapt.add_double(reg_map, 1)  ;  // 1. = regular mapping, 0 = contracting mapping
    
  par_adapt.add_double(alpha_r, 2) ;    // factor by which all the radial 
                       // distances will be multiplied 
           
  par_adapt.add_tbl(ent_limit, 0) ; // array of values of the field ent 
                        // to define the isosurfaces.              


  // Parameters for the function Map_et::poisson for nuf
  // ----------------------------------------------------

  double precis_poisson = 1.e-16 ;     

  Param par_poisson_nuf ; 
  par_poisson_nuf.add_int(mermax_poisson,  0) ;  // maximum number of iterations
  par_poisson_nuf.add_double(relax_poisson,  0) ; // relaxation parameter
  par_poisson_nuf.add_double(precis_poisson, 1) ; // required precision
  par_poisson_nuf.add_int_mod(niter, 0) ;  //  number of iterations actually used 
  par_poisson_nuf.add_cmp_mod( ssjm1_nuf ) ; 
                       
  Param par_poisson_nuq ; 
  par_poisson_nuq.add_int(mermax_poisson,  0) ;  // maximum number of iterations
  par_poisson_nuq.add_double(relax_poisson,  0) ; // relaxation parameter
  par_poisson_nuq.add_double(precis_poisson, 1) ; // required precision
  par_poisson_nuq.add_int_mod(niter, 0) ;  //  number of iterations actually used 
  par_poisson_nuq.add_cmp_mod( ssjm1_nuq ) ; 
                       
  Param par_poisson_tggg ; 
  par_poisson_tggg.add_int(mermax_poisson,  0) ;  // maximum number of iterations
  par_poisson_tggg.add_double(relax_poisson,  0) ; // relaxation parameter
  par_poisson_tggg.add_double(precis_poisson, 1) ; // required precision
  par_poisson_tggg.add_int_mod(niter, 0) ;  //  number of iterations actually used 
  par_poisson_tggg.add_cmp_mod( ssjm1_tggg ) ; 
  double lambda_tggg ;
  par_poisson_tggg.add_double_mod( lambda_tggg ) ; 
    
  Param par_poisson_dzeta ; 
  double lbda_grv2 ;
  par_poisson_dzeta.add_double_mod( lbda_grv2 ) ; 
                       
  // Parameters for the function Tenseur::poisson_vect
  // -------------------------------------------------

  Param par_poisson_vect ; 

  par_poisson_vect.add_int(mermax_poisson,  0) ;  // maximum number of iterations
  par_poisson_vect.add_double(relax_poisson,  0) ; // relaxation parameter
  par_poisson_vect.add_double(precis_poisson, 1) ; // required precision
  par_poisson_vect.add_cmp_mod( ssjm1_khi ) ; 
  par_poisson_vect.add_tenseur_mod( ssjm1_wshift ) ; 
  par_poisson_vect.add_int_mod(niter, 0) ;   

                       
    // Initializations
    // ---------------

    // Initial angular velocities
  omega = 0 ; 
  omega2 = 0 ;
  
  double accrois_omega = (omega0 - omega_ini) / 
    double(mer_fix_omega - mer_change_omega) ; 
  double accrois_omega2 = (omega20 - omega2_ini) / 
    double(mer_fix_omega - mer_change_omega) ; 


  update_metric() ; // update of the metric coefficients

  equation_of_state() ; // update of the densities, pressure, etc...
    
  hydro_euler() ;   // update of the hydro quantities relative to the 
  //  Eulerian observer

    // Quantities at the previous step :    

  Map_et mp_prev = mp_et;
  Tenseur ent_prev = ent ;  
  Tenseur ent2_prev = ent2 ;
  Tenseur logn_prev = logn ;        
  Tenseur dzeta_prev = dzeta ;      
    
    // Creation of uninitialized tensors:
  Tenseur source_nuf(mp) ;    // source term in the equation for nuf
  Tenseur source_nuq(mp) ;    // source term in the equation for nuq
  Tenseur source_dzf(mp) ;  // matter source term in the eq. for dzeta
  Tenseur source_dzq(mp) ;  // quadratic source term in the eq. for dzeta
  Tenseur source_tggg(mp) ; // source term in the eq. for tggg
  Tenseur source_shift(mp, 1, CON, mp.get_bvect_cart()) ;  
  // source term for shift
  Tenseur mlngamma(mp) ;    // centrifugal potential
  Tenseur mlngamma2(mp) ;   // centrifugal potential

  Tenseur *outer_ent_p;     // pointer to the enthalpy field of the outer fluid
    
  // Preparations for the Poisson equations:
  // --------------------------------------
  if (nuf.get_etat() == ETATZERO) {
    nuf.set_etat_qcq() ; 
    nuf.set() = 0 ; 
  }
    
  if (relativistic) {
    if (nuq.get_etat() == ETATZERO) {
      nuq.set_etat_qcq() ; 
      nuq.set() = 0 ; 
    }

    if (tggg.get_etat() == ETATZERO) {
      tggg.set_etat_qcq() ; 
      tggg.set() = 0 ; 
    }
    
    if (dzeta.get_etat() == ETATZERO) {
      dzeta.set_etat_qcq() ; 
      dzeta.set() = 0 ; 
    }
  }
            
  ofstream fichconv("convergence.d") ;    // Output file for diff_ent
  fichconv << "#     diff_ent     GRV2        max_triax      vit_triax" << endl ; 
    
  ofstream fichfreq("frequency.d") ;    // Output file for  omega
  fichfreq << "#       f1 [Hz]     f2 [Hz]" << endl ; 
    
  ofstream fichevol("evolution.d") ;    // Output file for various quantities
  fichevol << "#           r_pole/r_eq       ent_c    ent2_c" << endl ; 
    
  diff_ent = 1 ; 
  double err_grv2 = 1 ; 
  double max_triax_prev = 0 ;       // Triaxial amplitude at previous step
    
    //=========================================================================
    //          Start of iteration
    //=========================================================================

  for(int mer=0 ; (diff_ent > precis) && (mer<mer_max) ; mer++ ) {

    cout << "-----------------------------------------------" << endl ;
    cout << "step: " << mer << endl ;
    cout << "diff_ent = " << display_bold << diff_ent << display_normal
     << endl ;    
    cout << "err_grv2 = " << err_grv2 << endl ;    
    fichconv << mer ;
    fichfreq << mer ;
    fichevol << mer ;
      
    if (mer >= mer_rot) {
    
      if (mer < mer_change_omega) {
    omega = omega_ini ; 
    omega2 = omega2_ini ;
      }
      else {
    if (mer <= mer_fix_omega) {
      omega = omega_ini + accrois_omega * 
        (mer - mer_change_omega) ;
      omega2 = omega2_ini + accrois_omega2 * 
        (mer - mer_change_omega) ;
    }
      }
    
    }

    //-----------------------------------------------
    //  Sources of the Poisson equations
    //-----------------------------------------------
    
    // Source for nu
    // -------------
    Tenseur beta = log(bbb) ; 
    beta.set_std_base() ; 

    // common source term for relativistic and Newtonian ! (enerps_euler has the right limit)
    source_nuf =  qpig * a_car * enerps_euler;

    if (relativistic) 
      source_nuq = ak_car - flat_scalar_prod(logn.gradient_spher(),logn.gradient_spher() + beta.gradient_spher()) ; 
    else 
      source_nuq = 0 ; 

    source_nuf.set_std_base() ;     
    source_nuq.set_std_base() ;     

    if (relativistic) {
      // Source for dzeta
      // ----------------
      source_dzf = 2 * qpig * a_car * sphph_euler;
      source_dzf.set_std_base() ; 
      
      source_dzq = 1.5 * ak_car - flat_scalar_prod(logn.gradient_spher(),logn.gradient_spher() ) ;      
      source_dzq.set_std_base() ;   
      
      // Source for tggg
      // ---------------
      
      source_tggg = 2 * qpig * nnn * a_car * bbb * (s_euler - sphph_euler);
      source_tggg.set_std_base() ; 
      
      (source_tggg.set()).mult_rsint() ; 
      
      
      // Source for shift
      // ----------------
      
      // Matter term 
      source_shift = (-4*qpig) * nnn * a_car * j_euler;
      
      // Quadratic terms:
      Tenseur vtmp =  3 * beta.gradient_spher() - logn.gradient_spher() ;
      vtmp.change_triad(mp.get_bvect_cart()) ; 
      
      Tenseur squad  = 2 * nnn * flat_scalar_prod(tkij, vtmp) ;     
      
      // The addition of matter terms and quadratic terms is performed
      //  component by component because u_euler is contravariant,
      //  while squad is covariant. 
      
      if (squad.get_etat() == ETATQCQ) {
    for (int i=0; i<3; i++) {
      source_shift.set(i) += squad(i) ; 
    }
      }
      
      source_shift.set_std_base() ;     
    }
    //----------------------------------------------
    // Resolution of the Poisson equation for nuf 
    //----------------------------------------------
    
    source_nuf().poisson(par_poisson_nuf, nuf.set()) ; 
    
//     cout << "Test of the Poisson equation for nuf :" << endl ; 
//     Tbl err = source_nuf().test_poisson(nuf(), cout, true) ; 
//     diff_nuf = err(0, 0) ; 
    diff_nuf = 0 ;
    
    //---------------------------------------
    // Triaxial perturbation of nuf
    //---------------------------------------
    
    if (mer == mer_triax) {
    
    if ( mg->get_np(0) == 1 ) {
        cout << 
        "Et_rot_bifluid::equilibrium: np must be stricly greater than 1"
         << endl << " to set a triaxial perturbation !" << endl ; 
        abort() ; 
    }
    
    const Coord& phi = mp.phi ; 
    const Coord& sint = mp.sint ; 
    Cmp perturb(mp) ; 
    perturb = 1 + ampli_triax * sint*sint * cos(2*phi) ;
    nuf.set() = nuf() * perturb ;  
        
    nuf.set_std_base() ;    // set the bases for spectral expansions
                            //  to be the standard ones for a 
                            //  scalar field

    }
    
    // Monitoring of the triaxial perturbation
    // ---------------------------------------
    
    Valeur& va_nuf = nuf.set().va ; 
    va_nuf.coef() ;     // Computes the spectral coefficients
    double max_triax = 0 ; 

    if ( mg->get_np(0) > 1 ) {

    for (int l=0; l<nz; l++) {  // loop on the domains
        for (int j=0; j<mg->get_nt(l); j++) {
        for (int i=0; i<mg->get_nr(l); i++) {
        
            // Coefficient of cos(2 phi) : 
            double xcos2p = (*(va_nuf.c_cf))(l, 2, j, i) ; 

            // Coefficient of sin(2 phi) : 
            double xsin2p = (*(va_nuf.c_cf))(l, 3, j, i) ; 
            
            double xx = sqrt( xcos2p*xcos2p + xsin2p*xsin2p ) ; 
            
            max_triax = ( xx > max_triax ) ? xx : max_triax ;           
        }
        } 
    }
    }
    cout << "Triaxial part of nuf : " << max_triax << endl ; 

    if (relativistic) {
      
      //----------------------------------------------
      // Resolution of the Poisson equation for nuq 
      //----------------------------------------------

      source_nuq().poisson(par_poisson_nuq, nuq.set()) ; 
        
//        cout << "Test of the Poisson equation for nuq :" << endl ; 
//        err = source_nuq().test_poisson(nuq(), cout, true) ;
//        diff_nuq = err(0, 0) ; 
      diff_nuq = 0 ;
    
      //---------------------------------------------------------
      // Resolution of the vector Poisson equation for the shift
      //---------------------------------------------------------

      if (source_shift.get_etat() != ETATZERO) {

    for (int i=0; i<3; i++) {
      if(source_shift(i).dz_nonzero()) {
        assert( source_shift(i).get_dzpuis() == 4 ) ; 
      }
      else{
        (source_shift.set(i)).set_dzpuis(4) ; 
      }
    }

      }

      double lambda_shift = double(1) / double(3) ; 

      if ( mg->get_np(0) == 1 ) {
    lambda_shift = 0 ; 
      }
    
      source_shift.poisson_vect(lambda_shift, par_poisson_vect, 
                shift, w_shift, khi_shift) ;      
        
//        cout << "Test of the Poisson equation for shift_x :" << endl ; 
//        err = source_shift(0).test_poisson(shift(0), cout, true) ;
//        diff_shift_x = err(0, 0) ; 
      diff_shift_x = 0 ; 
    
//        cout << "Test of the Poisson equation for shift_y :" << endl ; 
//        err = source_shift(1).test_poisson(shift(1), cout, true) ;
//        diff_shift_y = err(0, 0) ; 
      diff_shift_y = 0 ;
        
      // Computation of tnphi and nphi from the Cartesian components
      //  of the shift
      // -----------------------------------------------------------
        
      fait_nphi() ; 
    
    }


    //----------------------------------------
    // Shall we search for the Kepler limit?
    //----------------------------------------
    bool kepler = false;
    bool too_fast = false;

    if ( (kepler_fluid > 0) && (mer > mer_fix_omega + kepler_wait_steps) )
      {
    if (kepler_fluid & 0x01)
      omega *= kepler_factor;
    if (kepler_fluid & 0x02)
      omega2 *= kepler_factor;
      }


    // ============================================================
    kepler = true;
    while (kepler)  
      { 

    // New computation of delta_car, gam_euler, enerps_euler etc...
    // ------------------------------------------------------
    hydro_euler() ; 
    

    //------------------------------------------------------
    //  First integral of motion 
    //------------------------------------------------------
    
    // Centrifugal potential : 
    if (relativistic) {
      mlngamma = - log( gam_euler ) ;
      mlngamma2 = - log( gam_euler2) ;
    }
    else {
      mlngamma = - 0.5 * uuu*uuu ;
      mlngamma2 = -0.5 * uuu2*uuu2 ;
    }
    
    // Central values of various potentials :
    double nuf_c = nuf()(0,0,0,0) ; 
    double nuq_c = nuq()(0,0,0,0) ; 

    // Scale factor to ensure that the enthalpy is equal to ent_b at 
    //  the equator for the "outer" fluid
    double alpha_r2 = 0;
    
    int j=j_b;
    
    // Boundary values of various potentials :
    double nuf_b  = nuf()(l_b, k_b, j, i_b) ; 
    double nuq_b  = nuq()(l_b, k_b, j, i_b) ; 
    double mlngamma_b  = mlngamma()(l_b, k_b, j, i_b) ; 
    double mlngamma2_b  = mlngamma2()(l_b, k_b, j, i_b) ; 


    // RP: "hack": adapt the radius correctly if using "slow-rot-style" EOS inversion
    // 
    if ( eos.identify() == 2 ) // only applies to Eos_bf_poly_newt
      {
        const Eos_bf_poly_newt &eos0 = dynamic_cast<const Eos_bf_poly_newt&>(eos);
        if (eos0.get_typeos() == 5)
          {
        double vn_b = uuu()(l_b, k_b, j, i_b);
        double vp_b = uuu2()(l_b, k_b, j, i_b);
        double D2_b = (vp_b - vn_b)*(vp_b - vn_b);
        double kdet = eos0.get_kap3() + eos0.get_beta()*D2_b;
        double kaps1 = kdet / ( eos0.get_kap2() - kdet );
        double kaps2 = kdet / ( eos0.get_kap1() - kdet );
        
        ent1_b = kaps1 * ( ent2_c - ent_c - mlngamma2_b + mlngamma_b );
        ent2_b = kaps2 * ( ent_c - ent2_c - mlngamma_b + mlngamma2_b );
        
        cout << "**********************************************************************\n";
        cout << "DEBUG: Rescaling domain for slow-rot-style EOS inversion \n";
        cout << "DEBUG: ent1_b = " << ent1_b << "; ent2_b = " << ent2_b << endl;
        cout << "**********************************************************************\n";
        
        adapt_flag = 0; // don't do adaptive-grid if using slow-rot-style inversion!
          }
      }
    
    double alpha1_r2 = ( ent_c - ent1_b - mlngamma_b + nuq_c - nuq_b) / ( nuf_b - nuf_c  ) ;
    double alpha2_r2 = ( ent2_c - ent2_b  - mlngamma2_b + nuq_c - nuq_b) / ( nuf_b - nuf_c  ) ;
    
    cout << "DEBUG: j= "<< j<<" ; alpha1 = " << alpha1_r2 <<" ; alpha2 = " << alpha2_r2 << endl;

    int outer_fluid = (alpha1_r2 > alpha2_r2) ? 1 : 2;  // index of 'outer' fluid (at equator!)
    
    outer_ent_p = (outer_fluid == 1) ? (&ent) : (&ent2);
    
    alpha_r2 = (outer_fluid == 1) ? alpha1_r2 : alpha2_r2 ;
    
    alpha_r = sqrt(alpha_r2);
    
    cout << "alpha_r = " << alpha_r << endl ; 
    
    // Readjustment of nu :
    // -------------------
    
    logn = alpha_r2 * nuf + nuq ;
    double nu_c =  logn()(0,0,0,0) ;
    
    
    // First integral   --> enthalpy in all space
    //-----------------
    
    ent = (ent_c + nu_c) - logn - mlngamma ;
    ent2 = (ent2_c + nu_c) - logn - mlngamma2 ;


    // now let's try to figure out if we have overstepped the Kepler-limit
    // (FIXME) we assume that the enthalpy of the _outer_ fluid being negative
    // inside the star 
    kepler = false;
    for (int l=0; l<nzet; l++) {
      int imax = mg->get_nr(l) - 1 ;
      if (l == l_b) imax-- ;    // The surface point is skipped
      for (int i=0; i<imax; i++) { 
        if ( (*outer_ent_p)()(l, 0, j_b, i) < 0. ) {
          kepler = true;
          cout << "(outer) ent < 0 for l, i : " << l << "  " << i 
           << "   ent = " << (*outer_ent_p)()(l, 0, j_b, i) << endl ;  
        } 
      }
    }
    
    if ( kepler ) 
      {
        cout << "**** KEPLERIAN VELOCITY REACHED ****" << endl ; 
        if (kepler_fluid & 0x01)
          omega /= kepler_factor ;    // Omega is decreased
        if (kepler_fluid & 0x02)
          omega2 /= kepler_factor;

        cout << "New rotation frequencies : " 
         << "Omega = " << omega/(2.*M_PI) * f_unit << " Hz; " 
         << "Omega2 = " << omega2/(2.*M_PI) * f_unit << endl ; 

        too_fast = true;
      }

      } /* while kepler */

    
    if ( too_fast ) 
      { // fact_omega is decreased for the next step 
    kepler_factor = sqrt( kepler_factor ) ; 
    cout << "**** New fact_omega : " << kepler_factor << endl ; 
      }
    // ============================================================


    // Cusp-check: shall the adaptation still be performed?
    // ------------------------------------------
    double dent_eq = (*outer_ent_p)().dsdr()(l_b, k_b, j_b, i_b) ; 
    double dent_pole = (*outer_ent_p)().dsdr()(l_b, k_b, 0, i_b) ;
    double rap_dent = fabs( dent_eq / dent_pole ) ; 
    cout << "| dH/dr_eq / dH/dr_pole | = " << rap_dent << endl ; 
    
    if ( rap_dent < thres_adapt ) {
      adapt_flag = 0 ;  // No adaptation of the mapping 
      cout << "******* FROZEN MAPPING  *********" << endl ; 
    }
    
    // Rescaling of the grid and adaption to (outer) enthalpy surface
    //---------------------------------------
    if (adapt_flag  && (nzadapt > 0) ) 
      {
    mp_prev = mp_et ; 
    
    mp.adapt( (*outer_ent_p)(), par_adapt) ; 
    
    mp_prev.homothetie(alpha_r) ;
    
    mp.reevaluate(&mp_prev, nzet+1, ent.set()) ; 
    mp.reevaluate(&mp_prev, nzet+1, ent2.set()) ; 
      }
    else
      mp.homothetie (alpha_r);


    //----------------------------------------------------
    // Equation of state  
    //----------------------------------------------------
    
    equation_of_state() ;   // computes new values for nbar1,2 , ener (e) 
                // and press (p) from the new ent,ent2 
    
    //---------------------------------------------------------
    // Matter source terms in the gravitational field equations 
    //---------------------------------------------------------

    //## Computation of tnphi and nphi from the Cartesian components
    //  of the shift for the test in hydro_euler():
        
    fait_nphi() ; 

    hydro_euler() ;     // computes new values for ener_euler (E), 
                // s_euler (S) and u_euler (U^i)

    if (relativistic) {

      //-------------------------------------------------------
      //    2-D Poisson equation for tggg
      //-------------------------------------------------------

      mp.poisson2d(source_tggg(), mp.cmp_zero(), par_poisson_tggg, tggg.set()) ; 
        
      //-------------------------------------------------------
      //    2-D Poisson equation for dzeta
      //-------------------------------------------------------

      mp.poisson2d(source_dzf(), source_dzq(), par_poisson_dzeta, dzeta.set()) ; 
        
      err_grv2 = lbda_grv2 - 1; 
      cout << "GRV2: " << err_grv2 << endl ; 
        
    }
    else {
      err_grv2 = grv2() ; 
    }


    //---------------------------------------
    // Computation of the metric coefficients (except for N^phi)
    //---------------------------------------

    // Relaxations on nu and dzeta :  

    if (mer >= 10) {
        logn = relax * logn + relax_prev * logn_prev ;

        dzeta = relax * dzeta + relax_prev * dzeta_prev ; 
    }

    // Update of the metric coefficients N, A, B and computation of K_ij :

    update_metric() ; 
    
    //-----------------------
    //  Informations display
    //-----------------------

    //    partial_display(cout) ; 
    fichfreq << "  " << omega / (2*M_PI) * f_unit ; 
    fichfreq << "  " << omega2 / (2*M_PI) * f_unit ; 
    fichevol << "  " << ray_pole() / ray_eq() ; 
    fichevol << "  " << ent_c ; 
    fichevol << "  " << ent2_c ; 


    //-----------------------------------------
    // Convergence towards given baryon masses  (if mer_mass > 0)
    //-----------------------------------------
    

    cout << "DEBUG MODE : mbar1_wanted : " << mbar1_wanted/msol << endl ;
    cout << "DEBUG MODE : mbar2_wanted : " << mbar2_wanted/msol << endl ; 
    
    // If we want to impose baryonic masses for both fluids. 
    // Be careful, the code acts on mu_n and mu_p (at the center)
    // -> beta equilibrium can be not verified
    // CV towards Mbn and Mbp (without beta equilibrium at the center)
    if (mbar2_wanted > 0 )          
    {
      if ((mer_mass>0) && (mer > mer_mass)) {
      
    double xx, xprog, ax, fact; 

    // fluid 1
    xx = mass_b1() / mbar1_wanted - 1. ;
    cout << "Discrep. baryon mass1 <-> wanted bar. mass1 : " << xx << endl ; 

    xprog = ( mer > 2*mer_mass) ? 1. : double(mer - mer_mass)/double(mer_mass) ; 
    xx *= xprog ; 
    ax = 0.5 * ( 2. + xx ) / (1. + xx ) ; 
    fact = pow(ax, aexp_mass) ; 
    cout << "Fluid1:  xprog, xx, ax, fact : " << xprog << "  " << xx << "  " << ax << "  " << fact << endl ; 
    ent_c *= fact ; 

    // fluid 2
    xx = mass_b2() / mbar2_wanted - 1. ;
    cout << "Discrep. baryon mass2 <-> wanted bar. mass2 : " << xx << endl ; 

    xprog = ( mer > 2*mer_mass) ? 1. : double(mer - mer_mass)/double(mer_mass) ; 
    xx *= xprog ; 
    ax = 0.5 * ( 2. + xx ) / (1. + xx ) ; 
    fact = pow(ax, aexp_mass) ; 
    cout << "Fluid2: xprog, xx, ax, fact : " << xprog << "  " << xx << "  " << ax << "  " << fact << endl ; 
    ent2_c *= fact ; 
    cout << "H1c = " << ent_c << "   H2c = " << ent2_c << endl ;
      }
    }

    // CV towards a given GRAVITATIONAL mass (with beta-eq at the center)
    else if (mbar2_wanted/msol == -1. ) {
      
      if ((mer_mass>0) && (mer > mer_mass)) {
      
    double xx, xprog, ax, fact; 

    // total mass
    xx = mass_g() / mbar1_wanted - 1. ; // mbar1_wanted = "mgrav_wanted"
    cout << "Discrep. baryon mass <-> wanted bar. mass : " << xx << endl ; 

    xprog = ( mer > 2*mer_mass) ? 1. : double(mer - mer_mass)/double(mer_mass) ; 
    xx *= xprog ; 
    ax = 0.5 * ( 2. + xx ) / (1. + xx ) ; 
    fact = pow(ax, aexp_mass) ; 
    cout << "Fluid1:  xprog, xx, ax, fact : " << xprog << "  " << xx << "  " << ax << "  " << fact << endl ; 
    ent_c *= fact ; 
    
    double m1 = eos.get_m1() ; 
    double m2 = eos.get_m2() ; 
    cout << "m1 = " << m1 << "    m2 = " << m2 << endl;
    ent2_c = ent_c + log(m1/m2); // to ensure beta_equilibrium  at the center
    cout << "DEBUG MODE : ent_c " << ent_c << endl ;
    cout << "DEBUG MODE : ent2_c " << ent2_c << endl ;  
    cout << "H1c = " << ent_c << "   H2c = " << ent2_c << endl ;
      
      }      
    }
    // If we want to impose Mb_tot and beta equilibrium (at the center)
    // In this case : mbar1_wanted = total baryonic mass wanted
    //               mbar2_wanted should be set to 0 and is not used.   
    else {

      if ((mer_mass>0) && (mer > mer_mass)) {
      
    double xx, xprog, ax, fact; 
    
    // total mass
    xx = mass_b() / mbar1_wanted - 1. ; // mbar1_wanted = " mbar_wanted"
    cout << "Discrep. baryon mass <-> wanted bar. mass : " << xx << endl ; 

    xprog = ( mer > 2*mer_mass) ? 1. : double(mer - mer_mass)/double(mer_mass) ; 
    xx *= xprog ; 
    ax = 0.5 * ( 2. + xx ) / (1. + xx ) ; 
    fact = pow(ax, aexp_mass) ; 
    cout << "Fluid1:  xprog, xx, ax, fact : " << xprog << "  " << xx << "  " << ax << "  " << fact << endl ; 
    ent_c *= fact ; 
    
    double m1 = eos.get_m1() ; 
    double m2 = eos.get_m2() ; 
    cout << "m1 = " << m1 << "    m2 = " << m2 << endl;
    ent2_c = ent_c + log(m1/m2); // to ensure beta_equilibrium  at the center
    cout << "DEBUG MODE : ent_c " << ent_c << endl ;
    cout << "DEBUG MODE : ent2_c " << ent2_c << endl ;  
    cout << "H1c = " << ent_c << "   H2c = " << ent2_c << endl ;
      
      }
      
    } /* if mer > mer_mass */
    

    //-------------------------------------------------------------
    //  Relative change in enthalpies with respect to previous step 
    //-------------------------------------------------------------

    Tbl diff_ent_tbl = diffrel( ent(), ent_prev() ) ; 
    diff_ent1 = diff_ent_tbl(0) ; 
    for (int l=1; l<nzet; l++) {
      diff_ent1 += diff_ent_tbl(l) ; 
    }
    diff_ent1 /= nzet ; 
    diff_ent_tbl = diffrel( ent2(), ent2_prev() ) ;
    diff_ent2 = diff_ent_tbl(0) ; 
    for (int l=1; l<nzet; l++) {
      diff_ent2 += diff_ent_tbl(l) ; 
    }
    diff_ent2 /= nzet ;
    diff_ent = 0.5*(diff_ent1 + diff_ent2) ; 
    
    fichconv << "  " << log10( fabs(diff_ent) + 1.e-16 ) ;
    fichconv << "  " << log10( fabs(err_grv2) + 1.e-16 ) ;
    fichconv << "  " << log10( fabs(max_triax) + 1.e-16 ) ;

    vit_triax = 0 ;
    if ( (mer > mer_triax+1) && (max_triax_prev > 1e-13) ) {
    vit_triax = (max_triax - max_triax_prev) / max_triax_prev ; 
    }
    
    fichconv << "  " << vit_triax ;
    
    //------------------------------
    //  Recycling for the next step
    //------------------------------
    
    ent_prev = ent ; 
    ent2_prev = ent2 ; 
    logn_prev = logn ; 
    dzeta_prev = dzeta ; 
    max_triax_prev = max_triax ; 
    
    fichconv << endl ;
    fichfreq << endl ;
    fichevol << endl ;
    fichconv.flush() ; 
    fichfreq.flush() ; 
    fichevol.flush() ; 

  } // End of main loop

  //=========================================================================
  //            End of iteration
  //=========================================================================

  fichconv.close() ; 
  fichfreq.close() ; 
  fichevol.close() ; 
    

}

}