domain_shell_outer_adapted_term_eq.cpp

/*
    Copyright 2017 Philippe Grandclement
 
    This file is part of Kadath.
 
    Kadath 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 3 of the License, or
    (at your option) any later version.
 
    Kadath 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 Kadath.  If not, see <http://www.gnu.org/licenses/>.
*/
 
#include "headcpp.hpp"
#include "utilities.hpp"
#include "adapted.hpp"
#include "point.hpp"
#include "array_math.hpp"
#include "scalar.hpp"
#include "tensor_impl.hpp"
#include "vector.hpp"
 
namespace Kadath {
Term_eq Domain_shell_outer_adapted::dr_term_eq (const Term_eq& so) const {
  return derive_r(so) ;
}
  
Term_eq Domain_shell_outer_adapted::derive_r (const Term_eq& so) const {
  
  assert (so.get_dom() == num_dom) ;
  
  Tensor val_res(*so.val_t, false) ;
  
  for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
    Array<int>ind (val_res.indices(cmp)) ;
    val_res.set(ind).set_domain(num_dom) = (*so.val_t)(ind)(num_dom).der_var(1) / (*der_rad_term_eq->val_t)()(num_dom)  ;
  }
  
  bool doder = ((so.der_t==0x0) || (der_rad_term_eq->der_t==0x0)) ? false : true ;
 
  if (doder) {
      Tensor der_res (*so.der_t, false) ;
    for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
      Array<int>ind (val_res.indices(cmp)) ;
      der_res.set(ind).set_domain(num_dom) = ((*so.der_t)(ind)(num_dom).der_var(1)*(*der_rad_term_eq->val_t)()(num_dom) 
          - (*so.val_t)(ind)(num_dom).der_var(1)*(*der_rad_term_eq->der_t)()(num_dom)) / 
          ((*der_rad_term_eq->val_t)()(num_dom) * (*der_rad_term_eq->val_t)()(num_dom))  ; 
    }
    return Term_eq (num_dom, val_res, der_res) ;
  }
  else
    return Term_eq (num_dom, val_res) ;
}
 
Term_eq Domain_shell_outer_adapted::derive_t (const Term_eq& so) const {
 
   assert (so.get_dom() == num_dom) ;
  Term_eq* dtprime ;
  Tensor val_res(*so.val_t, false) ;
  
  for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
    Array<int>ind (val_res.indices(cmp)) ;
    val_res.set(ind).set_domain(num_dom) = (*so.val_t)(ind)(num_dom).der_var(2)  ;
  }
  bool doder = (so.der_t==0x0) ? false : true ;
 
  if (doder) {
    Tensor der_res (*so.der_t, false) ;
     for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
       Array<int>ind (val_res.indices(cmp)) ;
       der_res.set(ind).set_domain(num_dom) = (*so.der_t)(ind)(num_dom).der_var(2)  ; 
     }
    dtprime = new Term_eq (num_dom, val_res, der_res) ;
  }
  else
    dtprime = new Term_eq (num_dom, val_res) ;
  
  
  Term_eq res (*dtprime - (*dt_rad_term_eq) * derive_r(so)) ;
  delete dtprime ;
  return res ;
  
}
 
Term_eq Domain_shell_outer_adapted::derive_p (const Term_eq& so) const {
  
  Term_eq* dpprime ;
  Tensor val_res(*so.val_t, false) ;
  
  for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
    Array<int>ind (val_res.indices(cmp)) ;
    val_res.set(ind).set_domain(num_dom) = (*so.val_t)(ind)(num_dom).der_var(3)  ;
  }
  bool doder = (so.der_t==0x0) ? false : true ;
 
  if (doder) {
    Tensor der_res (*so.der_t, false) ;
     for (int cmp = 0 ; cmp<so.val_t->get_n_comp() ; cmp++) {
       Array<int>ind (val_res.indices(cmp)) ;
       der_res.set(ind).set_domain(num_dom) = (*so.der_t)(ind)(num_dom).der_var(3)  ; 
     }
    dpprime = new Term_eq (num_dom, val_res, der_res) ;
  }
  else
    dpprime = new Term_eq (num_dom, val_res) ;
  
  
  Term_eq res (*dpprime - (*dp_rad_term_eq) * derive_r(so)) ;
  delete dpprime ;
  return res ;
  
}
 
 
Term_eq Domain_shell_outer_adapted::flat_grad_spher (const Term_eq& so) const {
  
  assert (so.get_dom() == num_dom) ;
  
  int valso = so.get_val_t().get_valence() ;
  Array<int> type_ind (valso+1) ; 
  type_ind.set(0) = COV ;
  for (int i=0 ; i<valso ; i++)
    type_ind.set(i+1) = so.get_val_t().get_index_type(i) ;
  
  Base_tensor basis (sp) ;
  basis.set_basis(num_dom) =  SPHERICAL_BASIS ;
  Tensor res (sp, valso+1 , type_ind, basis) ;
  
  Term_eq comp_r (derive_r(so)) ;
  Term_eq comp_t (derive_t(so)/(*rad_term_eq)) ;
  Term_eq comp_p (do_comp_by_comp(derive_p(so), &Domain::div_sin_theta) / (*rad_term_eq)) ;
  
  
  // Loop on cmp :
  for (int nc=0 ; nc<so.get_val_t().get_n_comp() ; nc++) {
    
    Array<int> ind (so.get_val_t().indices(nc)) ;
    Array<int> indtarget (valso+1) ;
    for (int i=0 ; i<valso ; i++)
    indtarget.set(i+1) = ind(i) ;
    
    // R comp :
    indtarget.set(0) = 1 ;
    res.set(indtarget).set_domain(num_dom) = comp_r.get_val_t()(ind)(num_dom);
    // theta comp :
    indtarget.set(0) = 2 ;
    res.set(indtarget).set_domain(num_dom) = comp_t.get_val_t()(ind)(num_dom) ;
     // Phi comp :
    indtarget.set(0) = 3 ;
    res.set(indtarget).set_domain(num_dom) = comp_p.get_val_t()(ind)(num_dom) ;
  }
  bool doder = ((so.der_t==0x0) || (outer_radius_term_eq->der_t==0x0)) ? false : true ;
  
  if (!doder) {
    return Term_eq (num_dom, res) ;
  }
  else {
    
    Tensor resder (sp, valso+1 , type_ind, basis) ;
     // Loop on cmp :
  for (int nc=0 ; nc<so.get_val_t().get_n_comp() ; nc++) {
    
    Array<int> ind (so.get_val_t().indices(nc)) ;
    Array<int> indtarget (valso+1) ;
    for (int i=0 ; i<valso ; i++)
    indtarget.set(i+1) = ind(i) ;
   
    // R comp :
    indtarget.set(0) = 1 ;
    resder.set(indtarget).set_domain(num_dom) = comp_r.get_der_t()(ind)(num_dom);
    // theta comp :
    indtarget.set(0) = 2 ;
    resder.set(indtarget).set_domain(num_dom) = comp_t.get_der_t()(ind)(num_dom) ;
     // Phi comp :
    indtarget.set(0) = 3 ;
    resder.set(indtarget).set_domain(num_dom) = comp_p.get_der_t()(ind)(num_dom) ;
    }
   return Term_eq (num_dom, res, resder) ; 
  }
}
 
void Domain_shell_outer_adapted::do_normal_spher () const {
  
    Term_eq grad (flat_grad_spher(*rad_term_eq - *outer_radius_term_eq)) ;
 
    Scalar val_norme (sp) ;
    val_norme.set_domain(num_dom) = sqrt((*grad.val_t)(1)(num_dom)*(*grad.val_t)(1)(num_dom)
                  + (*grad.val_t)(2)(num_dom)*(*grad.val_t)(2)(num_dom)
                 + (*grad.val_t)(3)(num_dom)*(*grad.val_t)(3)(num_dom)) ;
    bool doder = ((grad.der_t==0x0) || (outer_radius_term_eq->der_t==0x0)) ? false : true ;
    if (doder) {
       Scalar der_norme (sp) ;
       der_norme.set_domain(num_dom) = ((*grad.der_t)(1)(num_dom)*(*grad.val_t)(1)(num_dom)
                  + (*grad.der_t)(2)(num_dom)*(*grad.val_t)(2)(num_dom)
                  + (*grad.der_t)(3)(num_dom)*(*grad.val_t)(3)(num_dom)) / val_norme(num_dom) ;               
    Term_eq norme (num_dom, val_norme, der_norme) ;
    if (normal_spher==0x0)
      normal_spher = new Term_eq (grad / norme) ;
    else
      *normal_spher = Term_eq (grad/norme) ;
    }
    else {
      Term_eq norme (num_dom, val_norme) ;
      if (normal_spher==0x0)
    normal_spher = new Term_eq (grad / norme)  ;
      else
    *normal_spher = Term_eq (grad/norme) ;
    }
}
 
 
 
void Domain_shell_outer_adapted::do_normal_cart () const {
  
  
    do_normal_spher() ;
    
    Base_tensor basis (sp) ;
    basis.set_basis(num_dom) =  CARTESIAN_BASIS ;
    Vector val (sp, CON, basis) ;
    
    val.set(1).set_domain(num_dom) = mult_cos_phi(mult_sin_theta((*normal_spher->val_t)(1)(num_dom)) + mult_cos_theta((*normal_spher->val_t)(2)(num_dom)))
      - mult_sin_phi((*normal_spher->val_t)(3)(num_dom)) ;
    val.set(2).set_domain(num_dom) = mult_sin_phi(mult_sin_theta((*normal_spher->val_t)(1)(num_dom)) + mult_cos_theta((*normal_spher->val_t)(2)(num_dom)))
      + mult_cos_phi((*normal_spher->val_t)(3)(num_dom)) ;
    val.set(3).set_domain(num_dom) = mult_cos_theta((*normal_spher->val_t)(1)(num_dom)) - mult_sin_theta((*normal_spher->val_t)(2)(num_dom)) ;
    
    bool doder = (normal_spher->der_t==0x0) ? false : true ;
    if (doder) {
       Vector der (sp, CON, basis) ;
    
    der.set(1).set_domain(num_dom) = mult_cos_phi(mult_sin_theta((*normal_spher->der_t)(1)(num_dom)) + mult_cos_theta((*normal_spher->der_t)(2)(num_dom)))
      - mult_sin_phi((*normal_spher->der_t)(3)(num_dom)) ;
    der.set(2).set_domain(num_dom) = mult_sin_phi(mult_sin_theta((*normal_spher->der_t)(1)(num_dom)) + mult_cos_theta((*normal_spher->der_t)(2)(num_dom)))
      + mult_cos_phi((*normal_spher->der_t)(3)(num_dom)) ;
    der.set(3).set_domain(num_dom) = mult_cos_theta((*normal_spher->der_t)(1)(num_dom)) - mult_sin_theta((*normal_spher->der_t)(2)(num_dom)) ;
      
    if (normal_cart==0x0)
      normal_cart = new Term_eq (num_dom, val, der) ;
    else
      *normal_cart = Term_eq(num_dom, val,der) ;
    }
    else {
     if (normal_cart==0x0)
      normal_cart = new Term_eq (num_dom, val) ;
    else
      *normal_cart = Term_eq(num_dom, val) ;
    }
}
 
Term_eq Domain_shell_outer_adapted::der_normal_term_eq (const Term_eq& so, int bound) const {
  
  
  switch (bound) {
    case OUTER_BC : {
      // Deformed surface
      if (normal_spher == 0x0)
    do_normal_spher() ;
      
      int valso = so.get_val_t().get_valence() ;
      
      Term_eq grad (flat_grad_spher(so)) ;
  
      Tensor res (so.get_val_t(), false) ;
      
      // Loop on cmp :
      for (int nc=0 ; nc<so.get_val_t().get_n_comp() ; nc++) {
    
    Array<int> ind (so.get_val_t().indices(nc)) ;
    Array<int> indgrad (valso+1) ;
    for (int i=0 ; i<valso ; i++)
        indgrad.set(i+1) = ind(i) ;
    
    indgrad.set(0) = 1 ;
    res.set(ind).set_domain(num_dom) = (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->val_t)(1)(num_dom) ;
    indgrad.set(0) = 2 ;
    res.set(ind).set_domain(num_dom) += (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->val_t)(2)(num_dom) ;
    indgrad.set(0) = 3 ;
    res.set(ind).set_domain(num_dom) += (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->val_t)(3)(num_dom) ;
      }
              
      bool doder = ((grad.der_t == 0x0) || (normal_spher->der_t == 0x0)) ? false : true ;
      if (doder) {
    
      Tensor der (so.get_val_t(), false) ;
      
      // Loop on cmp :
      for (int nc=0 ; nc<so.get_val_t().get_n_comp() ; nc++) {
    
    Array<int> ind (so.get_val_t().indices(nc)) ;
    Array<int> indgrad (valso+1) ;
    for (int i=0 ; i<valso ; i++)
        indgrad.set(i+1) = ind(i) ;
    
    indgrad.set(0) = 1 ;
    der.set(ind).set_domain(num_dom) = (*grad.der_t)(indgrad)(num_dom) * (*normal_spher->val_t)(1)(num_dom) 
                    + (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->der_t)(1)(num_dom) ;
    indgrad.set(0) = 2 ;
    der.set(ind).set_domain(num_dom) += (*grad.der_t)(indgrad)(num_dom) * (*normal_spher->val_t)(2)(num_dom) 
                    + (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->der_t)(2)(num_dom) ;
    indgrad.set(0) = 3 ;
    der.set(ind).set_domain(num_dom) += (*grad.der_t)(indgrad)(num_dom) * (*normal_spher->val_t)(3)(num_dom) 
                    + (*grad.val_t)(indgrad)(num_dom) * (*normal_spher->der_t)(3)(num_dom) ;
      }
             
              
    return Term_eq (num_dom, res, der) ;
      }
      else 
    return Term_eq (num_dom, res) ;
    }
    case INNER_BC : {
    return derive_r(so) ;
    }
    default : 
    cerr << "Unknown boundary in Domain_shell_outer_adapted::der_normal" << endl ;
    abort() ;
  }
}
 
Term_eq Domain_shell_outer_adapted::lap_term_eq (const Term_eq& so, int mm) const {
#ifndef REMOVE_ALL_CHECKS
    if (mm!=0) {
      cerr << "Domain_shell_outer_adapted::lap_term_eq not defined for m != 0 (for now)" << endl ;
      abort() ;
  }
   
   if (so.get_val_t().get_valence() != 0) {
      cerr << "Domain_shell_outer_adapted::lap_term_eq only defined for scalars" << endl ;
      abort() ;
  }
#endif
  
  // Angular part :
  Term_eq dert (derive_t(so));
  Term_eq p1 (derive_t(dert)) ;
  Term_eq p2 (do_comp_by_comp(do_comp_by_comp(dert, &Domain::mult_cos_theta), &Domain::div_sin_theta)) ;
  Term_eq der2p (derive_p(derive_p(so))) ;
  Term_eq p3 (do_comp_by_comp(do_comp_by_comp(der2p, &Domain::div_sin_theta), &Domain::div_sin_theta)) ;
  
  Term_eq dr(derive_r(so)) ;
  
  Term_eq res (derive_r(dr) + 2 *dr / (*rad_term_eq) + (p1+p2+p3)/(*rad_term_eq)/(*rad_term_eq));    
 
  
  return res ;
}
 
Term_eq Domain_shell_outer_adapted::mult_r_term_eq (const Term_eq& so) const {
 
  
  return so * (*rad_term_eq) ;
}
 
void Domain_shell_outer_adapted::update_term_eq (Term_eq* so) const {
  
  Tensor der (*so->val_t, false) ;
  for (int cmp=0 ; cmp<so->val_t->get_n_comp() ; cmp ++) {
    
    
    
    Val_domain derr ((*(*so->val_t).cmp[cmp])(num_dom).der_var(1) / (*der_rad_term_eq->val_t)()(num_dom)) ;
    
    if (!derr.check_if_zero()) { 
      Val_domain res(this) ;
      res.allocate_conf() ;
      Index pos (nbr_points) ;     
      do {
    res.set(pos) =   (*(*outer_radius_term_eq->der_t).cmp[0])(num_dom)(pos) / 2. * (1+((*coloc[0])(pos(0)))) * derr(pos) ;
      }
    while (pos.inc()) ;
  res.set_base() = (*(*so->val_t).cmp[cmp])(num_dom).get_base() ;
  der.cmp[cmp]->set_domain(num_dom) = res ;
    }
    else
      der.cmp[cmp]->set_domain(num_dom).set_zero() ;
  }
  
  so->set_der_t (der) ;
}
 
Term_eq Domain_shell_outer_adapted::partial_spher (const Term_eq& so) const {
   int dom = so.get_dom() ;
  assert (dom == num_dom) ;
  
  int valence = so.val_t->get_valence() ;
 
  Array<int> type_ind (valence+1) ;
  type_ind.set(0) = COV ;
  for (int i=1 ; i<valence+1 ; i++)
      type_ind.set(i) = so.val_t->get_index_type(i-1) ;
  
  Base_tensor basis (so.get_val_t().get_space()) ;
  basis.set_basis(dom) = SPHERICAL_BASIS ;
 
  Term_eq comp_r (derive_r(so)) ;
  Term_eq comp_t (derive_t(so) / (*rad_term_eq)) ;
  Term_eq comp_p (do_comp_by_comp((derive_p(so)), &Domain::div_sin_theta) / (*rad_term_eq)) ;
 
  Tensor val_res (so.get_val_t().get_space(), valence+1, type_ind, basis) ;
  {
  Index pos_so (*so.val_t) ;
  Index pos_res (val_res) ;
  do {
    for (int i=1 ; i<valence+1 ; i++)
      pos_res.set(i) = pos_so(i-1) ;
    // R part
    pos_res.set(0) = 0 ;
    val_res.set(pos_res).set_domain(num_dom) = (*comp_r.val_t)(pos_so)(num_dom) ;
    // Theta part
    pos_res.set(0) = 1 ;
    val_res.set(pos_res).set_domain(num_dom) = (*comp_t.val_t)(pos_so)(num_dom) ;
    // Phi part
    pos_res.set(0) = 2 ;
    val_res.set(pos_res).set_domain(num_dom) = (*comp_p.val_t)(pos_so)(num_dom) ;
  }
  while (pos_so.inc()) ;
}
  
  bool doder = ((so.der_t==0x0) || (outer_radius_term_eq->der_t==0x0)) ? false : true ;
  if (doder) {
     Tensor der_res (so.get_val_t().get_space(), valence+1, type_ind, basis) ;
     Index pos_so (*so.der_t) ;
     Index pos_res (val_res) ;
  do {
    for (int i=1 ; i<valence+1 ; i++)
      pos_res.set(i) = pos_so(i-1) ;
    // R part
    pos_res.set(0) = 0 ;
    der_res.set(pos_res).set_domain(num_dom) = (*comp_r.der_t)(pos_so)(num_dom) ;
    // Theta part
    pos_res.set(0) = 1 ;
    der_res.set(pos_res).set_domain(num_dom) = (*comp_t.der_t)(pos_so)(num_dom) ;
    // Phi part
    pos_res.set(0) = 2 ;
    der_res.set(pos_res).set_domain(num_dom) = (*comp_p.der_t)(pos_so)(num_dom) ;
  }
  while (pos_so.inc()) ;
    
    return Term_eq (num_dom, val_res, der_res) ;
  }
  else
      return Term_eq (num_dom, val_res) ;
}
 
 
Term_eq Domain_shell_outer_adapted::connection_spher (const Term_eq& so) const {
  
  int dom = so.get_dom() ;
  assert (dom == num_dom) ;
  
  int valence = so.val_t->get_valence() ;
  int val_res = so.val_t->get_valence() + 1 ;
 
  Array<int> type_ind (val_res) ;
  type_ind.set(0) = COV ;
  for (int i=1 ; i<val_res ; i++)
      type_ind.set(i) = so.val_t->get_index_type(i-1) ;
  
  Base_tensor basis (so.get_val_t().get_space()) ;
  basis.set_basis(dom) = SPHERICAL_BASIS ;
    
  Tensor auxi_val (so.get_val_t().get_space(), val_res, type_ind, basis) ;
  for (int cmp=0 ; cmp<auxi_val.get_n_comp() ; cmp ++)
    auxi_val.set(auxi_val.indices(cmp)) = 0 ;
  
  for (int ind_sum=0 ; ind_sum<valence ; ind_sum++) {
 
      //Loop on the components :
      Index pos_auxi(auxi_val) ;
      Index pos_so (*so.val_t) ;
 
      do {
          for (int i=0 ; i<valence ; i++)
              pos_so.set(i) = pos_auxi(i+1) ; 
          // Different cases of the derivative index :
          switch (pos_auxi(0)) {
              case 0 : 
            // Dr nothing
            break ;
              case 1 :
            // Dtheta
            // Different cases of the source index
            switch (pos_auxi(ind_sum+1)) {
                case 0 :
                  //Dtheta S_r 
                  pos_so.set(ind_sum) = 1 ;
                  auxi_val.set(pos_auxi).set_domain(dom) -= (*so.val_t)(pos_so)(dom).div_r() ;
                  break ;
                case 1 :
                  // Dtheta S_theta
                  pos_so.set(ind_sum) = 0 ;
                  auxi_val.set(pos_auxi).set_domain(dom) += (*so.val_t)(pos_so)(dom).div_r() ;
                  break ;
                case 2 :
                  //Dtheta S_phi 
                  break ;
                default :
                  cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
                  abort() ;
            }
            break ;
              case 2 :
            // Dphi
            // Different cases of the source index
            switch (pos_auxi(ind_sum+1)) {
                case 0 :
                  //Dphi S_r 
                  pos_so.set(ind_sum) = 2 ;
                  auxi_val.set(pos_auxi).set_domain(dom) -= (*so.val_t)(pos_so)(dom).div_r() ;
                  break ;
                case 1 :
                  // Dphi S_theta
                  pos_so.set(ind_sum) = 2 ;
                  auxi_val.set(pos_auxi).set_domain(dom) -= (*so.val_t)(pos_so)(dom).div_r().mult_cos_theta().div_sin_theta() ;
                  break ;
                case 2 :
                  //Dphi S_phi
                  pos_so.set(ind_sum) = 0 ;
                  auxi_val.set(pos_auxi).set_domain(dom) += (*so.val_t)(pos_so)(dom).div_r() ;
                  pos_so.set(ind_sum) = 1 ;
                  auxi_val.set(pos_auxi).set_domain(dom) += (*so.val_t)(pos_so)(dom).div_r().mult_cos_theta().div_sin_theta() ;
                  break ;
                default :
                  cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
                  abort() ;
            }
            break ;
              default :
            cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
            abort() ;
          }
      }
      while (pos_auxi.inc()) ;
    }
    
    
    if ((so.der_t==0x0) || (outer_radius_term_eq->der_t==0x0)) {
        // No need for derivative :
        return Term_eq (dom, auxi_val) ;
    }
    else {
    
      // Need to compute the derivative :
      // Tensor for der
      Tensor auxi_der (so.get_val_t().get_space(), val_res, type_ind, basis) ;
      for (int cmp=0 ; cmp<auxi_der.get_n_comp() ; cmp ++)
        auxi_der.set(auxi_der.indices(cmp)) = 0 ;
 
        // Loop indice summation on connection symbols 
    for (int ind_sum=0 ; ind_sum<valence ; ind_sum++) {
 
      //Loop on the components :
      Index pos_auxi_der(auxi_der) ;
      Index pos_so (*so.der_t) ;
 
      do {
          for (int i=0 ; i<valence ; i++)
              pos_so.set(i) = pos_auxi_der(i+1) ;
          // Different cases of the derivative index :
          switch (pos_auxi_der(0)) {
              case 0 : 
            // Dr nothing
            break ;
              case 1 :
            // Dtheta
            // Different cases of the source index
            switch (pos_auxi_der(ind_sum+1)) {
                case 0 :
                  //Dtheta S_r 
                  pos_so.set(ind_sum) = 1 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) -= 
                ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r() ;
                  break ;
                case 1 :
                  // Dtheta S_theta
                  pos_so.set(ind_sum) = 0 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) += 
                ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r() ;
                  break ;
                case 2 :
                  //Dtheta S_phi 
                  break ;
                default :
                  cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
                  abort() ;
            }
            break ;
              case 2 :
            // Dphi
            // Different cases of the source index
            switch (pos_auxi_der(ind_sum+1)) {
                case 0 :
                  //Dphi S_r 
                  pos_so.set(ind_sum) = 2 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) -= 
                  ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r() ;
                  break ;
                case 1 :
                  // Dphi S_theta
                  pos_so.set(ind_sum) = 2 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) -= 
                  ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r().mult_cos_theta().div_sin_theta() ;
                  break ;
                case 2 :
                  //Dphi S_phi
                  pos_so.set(ind_sum) = 0 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) += 
                  ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r() ;
                  pos_so.set(ind_sum) = 1 ;
                  auxi_der.set(pos_auxi_der).set_domain(dom) += 
                  ((*so.der_t)(pos_so)(dom) -(*so.val_t)(pos_so)(dom)*(*rad_term_eq->der_t)()(dom)/(*rad_term_eq->val_t)()(dom)).div_r().mult_cos_theta().div_sin_theta() ;
                  break ;
                default :
                  cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
                  abort() ;
            }
            break ;
              default :
            cerr << "Bad indice in Domain_shell_outer_adapted::connection_spher" << endl  ;
            abort() ;
          }
      }
      while (pos_auxi_der.inc()) ;
    }
 
    return Term_eq (dom, auxi_val, auxi_der) ;
    }
}
 
Term_eq Domain_shell_outer_adapted::partial_cart (const Term_eq& so) const {
  int dom = so.get_dom() ;
  assert (dom == num_dom) ;
  
  int valence = so.val_t->get_valence() ;
 
  Array<int> type_ind (valence+1) ;
  type_ind.set(0) = COV ;
  for (int i=1 ; i<valence+1 ; i++)
      type_ind.set(i) = so.val_t->get_index_type(i-1) ;
  
  Base_tensor basis (so.get_val_t().get_space()) ;
  basis.set_basis(dom) = CARTESIAN_BASIS ;
 
  Term_eq comp_r (derive_r(so)) ;
  Term_eq comp_t (derive_t(so) / (*rad_term_eq)) ;
  Term_eq comp_p (do_comp_by_comp((derive_p(so)/(*rad_term_eq)), &Domain::div_sin_theta)) ;
 
  Tensor val_res (so.get_val_t().get_space(), valence+1, type_ind, basis) ;
  {
  Index pos_so (*so.val_t) ;
  Index pos_res (val_res) ;
  do {
    for (int i=1 ; i<valence+1 ; i++)
      pos_res.set(i) = pos_so(i-1) ;
   
    Val_domain auxi (mult_sin_theta((*comp_r.val_t)(pos_so)(num_dom)) + mult_cos_theta((*comp_t.val_t)(pos_so)(num_dom))) ;
    // X part
    pos_res.set(0) = 0 ;
    val_res.set(pos_res).set_domain(num_dom) = mult_cos_phi(auxi) - mult_sin_phi((*comp_p.val_t)(pos_so)(num_dom)) ;
    // Y part
    pos_res.set(0) = 1 ;
    val_res.set(pos_res).set_domain(num_dom) =  mult_sin_phi(auxi) + mult_cos_phi((*comp_p.val_t)(pos_so)(num_dom)) ;
    // Z part
    pos_res.set(0) = 2 ;
    val_res.set(pos_res).set_domain(num_dom) = 
      mult_cos_theta((*comp_r.val_t)(pos_so)(num_dom)) - mult_sin_theta((*comp_t.val_t)(pos_so)(num_dom)) ;
  }
  while (pos_so.inc()) ;
}
  
  bool doder = ((so.der_t==0x0) || (rad_term_eq->der_t==0x0)) ? false : true ;
  if (doder) {
     Tensor der_res (so.get_val_t().get_space(), valence+1, type_ind, basis) ;
     Index pos_so (*so.der_t) ;
     Index pos_res (val_res) ;
  do {
    for (int i=1 ; i<valence+1 ; i++)
      pos_res.set(i) = pos_so(i-1) ;
    
    Val_domain auxi (mult_sin_theta((*comp_r.der_t)(pos_so)(num_dom)) + mult_cos_theta((*comp_t.der_t)(pos_so)(num_dom))) ;
    // X part
    pos_res.set(0) = 0 ;
    der_res.set(pos_res).set_domain(num_dom) =  mult_cos_phi(auxi) - mult_sin_phi((*comp_p.der_t)(pos_so)(num_dom)) ;
    // Y part
    pos_res.set(0) = 1 ;
    der_res.set(pos_res).set_domain(num_dom) = mult_sin_phi(auxi) + mult_cos_phi((*comp_p.der_t)(pos_so)(num_dom)) ;
    // Z part
    pos_res.set(0) = 2 ;
    der_res.set(pos_res).set_domain(num_dom) = 
      mult_cos_theta((*comp_r.der_t)(pos_so)(num_dom)) - mult_sin_theta((*comp_t.der_t)(pos_so)(num_dom)) ;
  }
  while (pos_so.inc()) ;
    
    return Term_eq (num_dom, val_res, der_res) ;
  }
  else
      return Term_eq (num_dom, val_res) ;
}
 
const Term_eq* Domain_shell_outer_adapted::give_normal (int bound, int tipe) const {
   assert (bound==OUTER_BC) ;
   switch (tipe) {
    case CARTESIAN_BASIS : 
      if (normal_cart==0x0)
    do_normal_cart() ;
      return normal_cart ;
    
    case SPHERICAL_BASIS : 
      if (normal_spher==0x0)
    do_normal_spher() ;
      return normal_spher ;
      
    default: 
      cerr << "Unknown type of tensorial basis in Domain_shell_inner_adapted::give_normal" << endl ;
      abort() ;
  }
}
 
double integral_1d (int, const Array<double>&) ;
Term_eq Domain_shell_outer_adapted::integ_volume_term_eq (const Term_eq& target) const {
 
    int dom = target.get_dom() ;
    // Check it is a tensor
#ifndef REMOVE_ALL_CHECKS
    if (target.type_data != TERM_T) {
        cerr << "Ope_int_volume only defined with respect for a tensor" << endl ;
        abort() ;
    }
 
    if (target.val_t->get_n_comp() != 1) {
        cerr << "Ope_int_volume only defined with respect to a scalar" << endl ;
        abort() ;
    }
#endif
 
    Term_eq integrant (do_comp_by_comp(mult_r_term_eq(mult_r_term_eq(target))*(*der_rad_term_eq), &Domain::mult_sin_theta)) ;
  
    // The value
    Array<int> ind (target.val_t->indices(0)) ;
    Val_domain value ((*integrant.val_t)(ind)(dom)) ;
    double resval = 0 ;
    if (value.check_if_zero()) 
        resval= 0. ;
    else {
          int baset = (*value.get_base().bases_1d[1])(0) ;
          assert(baset==SIN_ODD) ;
          Index pos (nbr_coefs) ;
          for (int j=0 ; j<nbr_coefs(1) ; j++) {
        pos.set(1) = j ;
          int baser = (*value.get_base().bases_1d[0]) (j, 0) ;
          assert (baser==CHEB) ;
 
          Array<double> cf (nbr_coefs(0)) ;
          for (int i=0 ; i<nbr_coefs(0) ; i++) {
        pos.set(0) = i ;
        cf.set(i) = value.get_coef(pos) ;
          }
          resval += 2./(2.*j+1) * integral_1d(CHEB, cf) ;
        }
      resval *= 2*M_PI ;
    }
    
    // The der
    if (integrant.der_t!=0x0) {
        Val_domain derval ((*integrant.der_t)(ind)(dom)) ;
        double resder = 0 ;
          if (derval.check_if_zero()) 
        resder= 0. ;
    else {
          int baset = (*derval.get_base().bases_1d[1]) (0) ;
          assert(baset==SIN_ODD) ;
          Index pos (nbr_coefs) ;
          for (int j=0 ; j<nbr_coefs(1) ; j++) {
        pos.set(1) = j ;
          int baser = (*derval.get_base().bases_1d[0]) (j, 0) ;
          assert (baser==CHEB) ;
 
          Array<double> cf (nbr_coefs(0)) ;
          for (int i=0 ; i<nbr_coefs(0) ; i++) {
        pos.set(0) = i ;
        cf.set(i) = derval.get_coef(pos) ;
          }
          resder += 2./(2.*j+1) * integral_1d(CHEB, cf) ;
        }
      resder *= 2*M_PI ;
    }
    return Term_eq (dom, resval, resder) ;
    }
    else {
        return Term_eq (dom, resval) ;
    }
}
 
 
Term_eq Domain_shell_outer_adapted::integ_term_eq (const Term_eq&, int) const {
  cerr << "Domain_shell_outer_adapted::integ_term_eq not implemented yet" << endl ;
  abort() ;
}
}