domain_bispheric_chi_first.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 "bispheric.hpp"
#include "param.hpp"
#include "val_domain.hpp"
 
namespace Kadath {
double eta_lim_chi(double, double, double, double) ;
 
// Standard constructor
Domain_bispheric_chi_first::Domain_bispheric_chi_first (int num, int ttype, double a, double etalim, double air, double chim, const Dim_array& nbr) :  Domain(num, ttype, nbr), aa(a), eta_lim(etalim), r_ext(air), chi_max(chim), bound_eta(nullptr), 
bound_eta_der(nullptr), p_eta(nullptr), p_chi(nullptr), p_phi(nullptr),
p_detadx(nullptr), p_detady(nullptr), p_detadz(nullptr), p_dchidx(nullptr), p_dchidy(nullptr), p_dchidz(nullptr), p_dphidy(nullptr), p_dphidz(nullptr), p_dsint(nullptr) {
 
     assert (nbr.get_ndim()==3) ;
     eta_c = log((1+r_ext/aa)/(r_ext/aa-1)) ;
     do_coloc() ;
}
 
// Constructor by copy
Domain_bispheric_chi_first::Domain_bispheric_chi_first (const Domain_bispheric_chi_first& so) : Domain(so), aa(so.aa), eta_lim(so.eta_lim), r_ext(so.r_ext), chi_max(so.chi_max) {
 
    bound_eta = (so.bound_eta!=nullptr) ? new Val_domain(*so.bound_eta) : nullptr ;
    bound_eta_der = (so.bound_eta_der!=nullptr) ? new Val_domain(*so.bound_eta_der) : nullptr ;
    p_eta = (so.p_eta!=nullptr) ? new Val_domain(*so.p_eta) : nullptr ;
    p_chi = (so.p_chi!=nullptr) ? new Val_domain(*so.p_chi) : nullptr ;
    p_phi = (so.p_phi!=nullptr) ? new Val_domain(*so.p_phi) : nullptr ;
    p_detadx = (so.p_detadx!=nullptr) ? new Val_domain(*so.p_detadx) : nullptr ;
    p_detady = (so.p_detady!=nullptr) ? new Val_domain(*so.p_detady) : nullptr ;
    p_detadz = (so.p_detadz!=nullptr) ? new Val_domain(*so.p_detadz) : nullptr ;
    p_dchidx = (so.p_dchidx!=nullptr) ? new Val_domain(*so.p_dchidx) : nullptr ;
    p_dchidy = (so.p_dchidy!=nullptr) ? new Val_domain(*so.p_dchidy) : nullptr ;
    p_dchidz = (so.p_dchidz!=nullptr) ? new Val_domain(*so.p_dchidz) : nullptr ;
    p_dphidy = (so.p_dphidy!=nullptr) ? new Val_domain(*so.p_dphidy) : nullptr ;
    p_dphidz = (so.p_dphidz!=nullptr) ? new Val_domain(*so.p_dphidz) : nullptr ;
    p_dsint = (so.p_dsint!=nullptr) ? new Val_domain(*so.p_dsint) : nullptr ;
}
 
Domain_bispheric_chi_first::Domain_bispheric_chi_first (int num, FILE* fd) : Domain(num, fd) {
    fread_be (&aa, sizeof(double), 1, fd) ;
    fread_be (&eta_lim, sizeof(double), 1, fd) ;
    fread_be (&r_ext, sizeof(double), 1, fd) ;
    fread_be (&chi_max, sizeof(double), 1, fd) ;
    fread_be (&eta_c, sizeof(double), 1, fd) ;
 
    bound_eta = nullptr ;
    bound_eta_der = nullptr ;
    p_eta = nullptr ;
    p_chi = nullptr ;
    p_phi = nullptr ;
    p_detadx = nullptr ;
    p_detady = nullptr ;
    p_detadz = nullptr ;
    p_dchidx = nullptr ;
    p_dchidy = nullptr ;
    p_dchidz = nullptr ;
    p_dphidy = nullptr ;
    p_dphidz = nullptr ;
    p_dsint = nullptr ;
    do_coloc() ;
}
 
// Destructor
Domain_bispheric_chi_first::~Domain_bispheric_chi_first() {
    del_deriv() ;
}
 
void Domain_bispheric_chi_first::save (FILE* fd) const {
    nbr_points.save(fd) ;
    nbr_coefs.save(fd) ;
    fwrite_be (&ndim, sizeof(int), 1, fd) ;
    fwrite_be (&type_base, sizeof(int), 1, fd) ;
    fwrite_be (&aa, sizeof(double), 1, fd) ;
    fwrite_be (&eta_lim, sizeof(double), 1, fd) ;
    fwrite_be (&r_ext, sizeof(double), 1, fd) ;
    fwrite_be (&chi_max, sizeof(double), 1, fd) ;
    fwrite_be (&eta_c, sizeof(double), 1, fd) ;
}
 
// Deletes the derived members.
void Domain_bispheric_chi_first::del_deriv()  {
    for (int l=0 ; l<ndim ; l++) {
        safe_delete(coloc[l]);
        safe_delete(cart[l]);
    }
    safe_delete(radius);
    safe_delete(bound_eta);
    safe_delete(bound_eta_der);
    safe_delete(p_eta);
    safe_delete(p_chi);
    safe_delete(p_phi);
    safe_delete(p_detadx);
    safe_delete(p_detady);
    safe_delete(p_detadz);
    safe_delete(p_dchidx);
    safe_delete(p_dchidy);
    safe_delete(p_dchidz);
    safe_delete(p_dphidy);
    safe_delete(p_dphidz);
    safe_delete(p_dsint);
}
 
// Display
ostream& Domain_bispheric_chi_first::print (ostream& o) const {
  o << "Bispherical domain, eta fonction of chi" << endl ;
  o << "aa      = " << aa << endl ;
  o << "Radius   = " << r_ext << endl ;
  o << " 0 < chi < " << chi_max << endl ;
  o << "Nbr pts = " << nbr_points << endl ;
  o << endl ;
  return o ;
}
 
// Computes the function giving the bound for eta.
void Domain_bispheric_chi_first::do_bound_eta() const {
 
    assert (p_chi!=nullptr) ;
    assert (bound_eta==nullptr) ;
    assert (bound_eta_der==nullptr) ;
    bound_eta = new Val_domain (this) ;
    bound_eta->allocate_conf() ;
    Index index(nbr_points) ;
    int signe = (eta_lim<0) ? -1 : 1 ;
    do {
        bound_eta->set(index) = signe*eta_lim_chi((*p_chi)(index), r_ext, aa, eta_c) ;
    }
    while (index.inc()) ;
 
    bound_eta->std_base() ;
    bound_eta_der = new Val_domain (bound_eta->der_var(2)) ;
}
 
// Computes eta from eta star
void Domain_bispheric_chi_first::do_eta() const {
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    assert (p_eta==nullptr) ;
    assert (bound_eta!=nullptr) ;
    p_eta= new Val_domain(this) ;
    p_eta->allocate_conf() ;
    Index index (nbr_points) ;
    do 
       p_eta->set(index) = ((*bound_eta)(index)-eta_lim)/2.*((*coloc[0])(index(0))) 
                + ((*bound_eta)(index)+eta_lim)/2. ;
    while (index.inc()) ;
}
 
// Computes chi from chi star
void Domain_bispheric_chi_first::do_chi() const {
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    assert (p_chi==nullptr) ;
    p_chi= new Val_domain(this) ;
    p_chi->allocate_conf() ;
    Index index (nbr_points) ;
    do 
       p_chi->set(index) =  chi_max*(*coloc[1])(index(1)) ;
    while (index.inc()) ;
}
 
// Computes phi from phi star
void Domain_bispheric_chi_first::do_phi() const {
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    assert (p_phi==nullptr) ;
    p_phi= new Val_domain(this) ;
    p_phi->allocate_conf() ;
    Index index (nbr_points) ;
    do
       p_phi->set(index) = ((*coloc[2])(index(2))) ;
    while (index.inc()) ;
}
 
const Val_domain & Domain_bispheric_chi_first::get_chi() const {
    if (p_chi==nullptr)
        do_chi() ;
    return *p_chi ;
}
 
const Val_domain & Domain_bispheric_chi_first::get_eta() const {
    if (p_eta==nullptr)
        do_eta() ;
    return *p_eta ;
}
 
// Computes the Cartesian coordinates
void Domain_bispheric_chi_first::do_absol () const  {
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    for (int i=0 ; i<3 ; i++)
       assert (absol[i] == nullptr) ;
    for (int i=0 ; i<3 ; i++) {
       absol[i] = new Val_domain(this) ;
       absol[i]->allocate_conf() ;
    }
       
    Index index (nbr_points) ;
    if (p_chi==nullptr)
        do_chi() ;
    if (p_phi==nullptr)
        do_phi() ;
    if (bound_eta==nullptr) 
        do_bound_eta() ;
    if (p_eta==nullptr)
        do_eta() ;
    
    do  {   
        absol[0]->set(index) = aa*sinh((*p_eta)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        absol[1]->set(index) = 
   aa*sin((*p_chi)(index))*cos((*p_phi)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        absol[2]->set(index) = 
   aa*sin((*p_chi)(index))*sin((*p_phi)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        }
    while (index.inc())  ;
 
    // Basis
    absol[0]->std_base() ;
    absol[1]->std_base() ;
    absol[2]->std_anti_base() ;
 
}
 
// Computes the generalized radius
void Domain_bispheric_chi_first::do_radius () const  {
 
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    assert (radius == nullptr) ;
    radius = new Val_domain(sqrt (get_cart(1)*get_cart(1)+get_cart(2)*get_cart(2)+get_cart(3)*get_cart(3))) ;
}
 
// Computes the Cartesian coordinates
void Domain_bispheric_chi_first::do_cart () const  {
    for (int i=0 ; i<3 ; i++)
       assert (coloc[i] != nullptr) ;
    for (int i=0 ; i<3 ; i++)
       assert (cart[i] == nullptr) ;
    for (int i=0 ; i<3 ; i++) {
       cart[i] = new Val_domain(this) ;
       cart[i]->allocate_conf() ;
    }
       
    Index index (nbr_points) ;
    if (p_chi==nullptr)
        do_chi() ;
    if (p_phi==nullptr)
        do_phi() ;
    if (bound_eta==nullptr) 
        do_bound_eta() ;
    if (p_eta==nullptr)
        do_eta() ;
    
    do  {   
        cart[0]->set(index) = aa*sinh((*p_eta)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        cart[1]->set(index) = 
   aa*sin((*p_chi)(index))*cos((*p_phi)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        cart[2]->set(index) = 
   aa*sin((*p_chi)(index))*sin((*p_phi)(index))/(cosh((*p_eta)(index))-cos((*p_chi)(index))) ;
        }
    while (index.inc())  ;
 
    // Basis
    cart[0]->std_base() ;
    cart[1]->std_base() ;
    cart[2]->std_anti_base() ;
 
}
 
void Domain_bispheric_chi_first::do_dsint () const {
    double rr = aa/sinh(fabs(eta_lim)) ;
    double xc = aa*cosh(eta_lim)/sinh(eta_lim) ;
 
    Val_domain rho2 (get_cart(2)*get_cart(2)+get_cart(3)*get_cart(3)) ;
    Val_domain xx (this) ;
    xx = (xc<0) ? xc-get_cart(1) : get_cart(1)-xc ;
 
    assert (p_dsint==nullptr) ;
    p_dsint = new Val_domain ((0.5*rho2.der_var(2)*xx - xx.der_var(2)*rho2)/rr) ;
}
 
// Check if a point belongs to this domain
bool Domain_bispheric_chi_first::is_in (const Point& abs, double prec) const {
    assert (abs.get_ndim()==3) ;
        
    double xx = abs(1) ;
    double yy = abs(2) ;
    double zz = abs(3) ;
    double air = sqrt (xx*xx+yy*yy+zz*zz) ;
 
    double chi ;
    double rho = sqrt(yy*yy+zz*zz) ;
    
    double x_out = aa*cosh(eta_lim)/sinh(eta_lim) ;
    
    if (rho<prec)
        chi = (fabs(abs(1))<fabs(x_out)) ? M_PI : 0 ; 
    else    
        chi = atan (2*aa*rho/(air*air-aa*aa)) ;
 
    if (chi<0)
        chi += M_PI ;
        
    double eta = 0.5*log((1+(2*aa*xx)/(aa*aa+air*air))/(1-(2*aa*xx)/(aa*aa+air*air))) ;
    
    bool res = true ;
    
    if (chi-prec>chi_max)
        res = false ;
    if (fabs(eta)-prec>fabs(eta_lim))
        res = false ;
    if (res) {
    int signe = (eta_lim<0) ? -1 : 1 ;
    double eta_bound = signe*eta_lim_chi(chi, r_ext, aa, eta_c) ;
    if (fabs(eta)<fabs(eta_bound)-prec)
        res = false ;
    if (eta*eta_lim<0)
        res = false ;
    }
    return res ;
}
 
 
// Converts the absolute coordinates to the numerical ones.
const Point Domain_bispheric_chi_first::absol_to_num(const Point& abs) const {
 
    assert (is_in(abs, 1e-12)) ;
    Point num(3) ;
        
    double xx = abs(1) ;
    double yy = abs(2) ;
    double zz = abs(3) ;
    double air = sqrt (xx*xx+yy*yy+zz*zz) ;
 
    num.set(3) = atan2 (zz, yy) ;
    
    if (num(3)<0)
        num.set(3) += 2*M_PI ;
    
    double chi ;
    double rho = sqrt(yy*yy+zz*zz) ;
    
    if (rho<PRECISION)
        chi = 0 ; 
    else    
        chi = atan (2*aa*rho/(air*air-aa*aa)) ;
 
    if (chi<0)
        chi += M_PI ;
            
    double eta = 0.5*log((1+(2*aa*xx)/(aa*aa+air*air))/(1-(2*aa*xx)/(aa*aa+air*air))) ;
    int signe = (eta<0) ? -1 : 1 ;
    double eta_bound = signe*eta_lim_chi(chi, r_ext, aa, eta_c) ;
    
    num.set(1) = (eta-(eta_bound+eta_lim)/2.)*2/(eta_bound-eta_lim) ;
    num.set(2) = chi/chi_max ;  
 
    return num ;
}
 
// Converts the absolute coordinates to the numerical ones.
const Point Domain_bispheric_chi_first::absol_to_num_bound(const Point& abs, int bound) const {
 
    assert (is_in(abs, 1e-3)) ;
    Point num(3) ;
        
    double xx = abs(1) ;
    double yy = abs(2) ;
    double zz = abs(3) ;
    double air = sqrt (xx*xx+yy*yy+zz*zz) ;
 
    num.set(3) = atan2 (zz, yy) ;
    
    if (num(3)<0)
        num.set(3) += 2*M_PI ;
    
    double chi ;
    double rho = sqrt(yy*yy+zz*zz) ;
    
    if (rho<PRECISION)
        chi = 0 ; 
    else    
        chi = atan (2*aa*rho/(air*air-aa*aa)) ;
 
    if (chi<0)
        chi += M_PI ;
            
    double eta = 0.5*log((1+(2*aa*xx)/(aa*aa+air*air))/(1-(2*aa*xx)/(aa*aa+air*air))) ;
    int signe = (eta<0) ? -1 : 1 ;
    double eta_bound = signe*eta_lim_chi(chi, r_ext, aa, eta_c) ;
    
    num.set(1) = (eta-(eta_bound+eta_lim)/2.)*2/(eta_bound-eta_lim) ;
    num.set(2) = chi/chi_max ;  
 
    switch (bound)  {
      case INNER_BC  :
        num.set(1) = -1 ;
        break ;
      case OUTER_BC :
          num.set(1) = 1 ;
          break ;
      case CHI_ONE_BC :
        num.set(2) = 1 ;
        break ;
      default :
        cerr << "Unknown case in Domain_bispheric_chi_first::absol_to_num_bound" << endl ;
        abort() ;
    }
    
    return num ;
}
 
double coloc_leg(int,int) ;
double coloc_leg_parity(int,int) ;
void Domain_bispheric_chi_first::do_coloc () {
 
    switch (type_base) {
        case CHEB_TYPE:
            nbr_coefs = nbr_points ;            
            del_deriv() ;
            for (int i=0 ; i<ndim ; i++)
                coloc[i] = new Array<double> (nbr_points(i)) ;
            for (int i=0 ; i<nbr_points(0) ; i++)
                coloc[0]->set(i) = -cos(M_PI*i/(nbr_points(0)-1)) ;
            for (int j=0 ; j<nbr_points(1) ; j++)
                coloc[1]->set(j) = sin(M_PI*j/2./(nbr_points(1)-1)) ;
            for (int k=0 ; k<nbr_points(2) ; k++)
                coloc[2]->set(k) = M_PI*k/(nbr_points(2)-1) ;
            do_phi() ;  
            do_chi() ;
            do_bound_eta() ;
            do_eta() ;
            break ;
        case LEG_TYPE:
            nbr_coefs = nbr_points ;
            del_deriv() ;
            for (int i=0 ; i<ndim ; i++) 
                coloc[i] = new Array<double> (nbr_points(i)) ;
            for (int i=0 ; i<nbr_points(0) ; i++)
                coloc[0]->set(i) = coloc_leg(i,nbr_points(0)) ;
            for (int j=0 ; j<nbr_points(1) ; j++)
                coloc[1]->set(j) = coloc_leg_parity(j,nbr_points(1)) ;
            for (int k=0 ; k<nbr_points(2) ; k++)
                coloc[2]->set(k) = M_PI*k/(nbr_points(2)-1) ;
            do_phi() ;
            do_chi() ;
            do_bound_eta() ;
            do_eta() ;
            break ;
        default : 
            cerr << "Unknown type of basis in Domain_bispheric_rect::do_coloc" << endl ;
            abort() ;
    }
}
 
// standard basis for Chebyshev
void Domain_bispheric_chi_first::set_cheb_base(Base_spectral& base) const {
    assert (type_base == CHEB_TYPE) ;
    base.allocate(nbr_coefs) ;
    
    Index index(base.bases_1d[0]->get_dimensions()) ;
    
    base.def=true ;
    base.bases_1d[2]->set(0) = COS ;
    for (int k=0 ; k<nbr_coefs(2) ; k++) {
        base.bases_1d[1]->set(k) = (k%2==0) ? CHEB_EVEN : CHEB_ODD ;
        for (int j=0 ; j<nbr_coefs(1) ; j++) {
            index.set(0) = j ; index.set(1) = k ;
            base.bases_1d[0]->set(index) = CHEB ;
         }
    }
}
 
// Standard basis for Legendre
void Domain_bispheric_chi_first::set_legendre_base(Base_spectral& base) const {
    assert (type_base == LEG_TYPE) ;
    base.allocate(nbr_coefs) ;
    
    Index index(base.bases_1d[0]->get_dimensions()) ;
    
    base.def=true ;
    base.bases_1d[2]->set(0) = COS ;
    for (int k=0 ; k<nbr_coefs(2) ; k++) {
        base.bases_1d[1]->set(k) = (k%2==0) ? LEG_EVEN : LEG_ODD ;
        for (int j=0 ; j<nbr_coefs(1) ; j++) {
            index.set(0) = j ; index.set(1) = k ;
            base.bases_1d[0]->set(index) = LEG ;
         }
    }
}
 
// Antisymetric basis for Chebyshev
void Domain_bispheric_chi_first::set_anti_cheb_base(Base_spectral& base) const {
 
    assert (type_base == CHEB_TYPE) ;
    base.allocate(nbr_coefs) ;
    
    Index index(base.bases_1d[0]->get_dimensions()) ;
    
    base.def=true ;
    base.bases_1d[2]->set(0) = SIN ;
    for (int k=0 ; k<nbr_coefs(2) ; k++) {
        base.bases_1d[1]->set(k) = (k%2==0) ? CHEB_EVEN : CHEB_ODD ;
        for (int j=0 ; j<nbr_coefs(1) ; j++) {
            index.set(0) = j ; index.set(1) = k ;
            base.bases_1d[0]->set(index) = CHEB ;
         }
    }
}
 
// Antisymetric basis for Legendre
void Domain_bispheric_chi_first::set_anti_legendre_base(Base_spectral& base) const {
    assert (type_base == LEG_TYPE) ;
    base.allocate(nbr_coefs) ;
    
    Index index(base.bases_1d[0]->get_dimensions()) ;
    
    base.def=true ;
    base.bases_1d[2]->set(0) = SIN ;
    for (int k=0 ; k<nbr_coefs(2) ; k++) {
        base.bases_1d[1]->set(k) = (k%2==0) ? LEG_EVEN : LEG_ODD ;
        for (int j=0 ; j<nbr_coefs(1) ; j++) {
            index.set(0) = j ; index.set(1) = k ;
            base.bases_1d[0]->set(index) = LEG ;
         }
    }
}
 
// Computes the derivatives with respect to the absolute coordinates with respect of the numerical ones.
void Domain_bispheric_chi_first::do_der_abs_from_der_var(const Val_domain *const *const der_var, Val_domain **const der_abs) const {
 
    if (p_detadx==nullptr)
        do_for_der() ;
 
    // d/dx :
    der_abs[0] = new Val_domain((*der_var[0])*(*p_detadx) + (*der_var[1])*(*p_dchidx)) ;
 
    // d/dy :
    Val_domain auchi_y ((*der_var[2])*(*p_dphidy)) ;
    Val_domain part_y_phi (auchi_y.div_sin_chi()) ;
 
    der_abs[1] = new Val_domain((*der_var[0])*(*p_detady) + (*der_var[1])*(*p_dchidy) + part_y_phi) ;
    // d/dz :
    Val_domain auchi_z ((*der_var[2])*(*p_dphidz)) ;
    Val_domain part_z_phi (auchi_z.div_sin_chi()) ;
    der_abs[2] = new Val_domain((*der_var[0])*(*p_detadz) + (*der_var[1])*(*p_dchidz) + part_z_phi) ;   
}
 
// Computes the various partial derivatives of the numerical coordinates with respect to the Cartesian ones.
void Domain_bispheric_chi_first::do_for_der() const {
 
    if (cart[0]==nullptr)
        do_cart() ;
    if (radius==nullptr)
        do_radius() ;
 
    // Partial derivatives of chi 
    Val_domain rho (sqrt((*cart[1])*(*cart[1])+(*cart[2])*(*cart[2]))) ;
    Val_domain denom_chi (chi_max*(((*radius)*(*radius)-aa*aa)*((*radius)*(*radius)-aa*aa) + 
                    4.*aa*aa*rho*rho)) ;
    p_dchidx = new Val_domain (-4.*aa*(*cart[0])*rho/denom_chi) ;
    
    Val_domain chiyz (2.*aa*((*radius)*(*radius)-aa*aa-2*rho*rho)/denom_chi) ;
    
    // Partial derivatives of phi :
    Val_domain phiyz (((exp(*p_eta)+exp(-*p_eta))/2.-cos(*p_chi))/aa) ;
 
    chiyz.std_base() ;
    phiyz.std_base() ;
    p_dchidx->base = chiyz.der_var(2).get_base() ;
    
    p_dchidy = new Val_domain (chiyz.mult_cos_phi()) ;
    p_dchidz = new Val_domain (chiyz.mult_sin_phi()) ;
    p_dphidy = new Val_domain (-phiyz.mult_sin_phi()) ;
    p_dphidz = new Val_domain (phiyz.mult_cos_phi()) ;
 
    // Derivative eta_bound :
    Val_domain dfdx ((*bound_eta_der)*(*p_dchidx)) ;
    Val_domain dfdy ((*bound_eta_der)*(*p_dchidy)) ;
    Val_domain dfdz ((*bound_eta_der)*(*p_dchidz)) ;
 
    // Partial derivatives of eta 
    Val_domain denom_eta 
        ((aa*aa+(*radius)*(*radius))*((aa*aa+(*radius)*(*radius))) -
                     4.*aa*aa*(*cart[0])*(*cart[0])) ;
    Val_domain detadx ((2.*aa*(aa*aa+(*radius)*(*radius)-2.*(*cart[0])*(*cart[0])))/denom_eta) ;
    Val_domain detady (-4.*aa*(*cart[0])*(*cart[1])/denom_eta) ;
    Val_domain detadz (-4.*aa*(*cart[0])*(*cart[2])/denom_eta) ;
 
    p_detadx = new Val_domain ((detadx-0.5*dfdx)*(2./((*bound_eta)-eta_lim))
                + ((*p_eta) - ((*bound_eta)+eta_lim)/2.)*
                    (-2.*dfdx/pow((*bound_eta)-eta_lim, 2.))) ;
    p_detady = new Val_domain ((detady-0.5*dfdy)*(2./((*bound_eta)-eta_lim))
                + ((*p_eta) - ((*bound_eta)+eta_lim)/2.)*
                    (-2.*dfdy/pow((*bound_eta)-eta_lim, 2.))) ;
    p_detadz = new Val_domain  ((detadz-0.5*dfdz)*(2./((*bound_eta)-eta_lim))
                + ((*p_eta) - ((*bound_eta)+eta_lim)/2.)*
                    (-2.*dfdz/pow((*bound_eta)-eta_lim, 2.))) ;
 
    // The basis :
    p_detadx->std_base() ;
    p_detady->std_base() ;
    p_detadz->std_anti_base() ;
}
 
// Multiplication rules for the basis.
Base_spectral Domain_bispheric_chi_first::mult (const Base_spectral& a, const Base_spectral& b) const {
 
    assert (a.ndim==3) ;
    assert (b.ndim==3) ;
    
    Base_spectral res(3) ;
    bool res_def = true ;
 
    if (!a.def)
        res_def=false ;
    if (!b.def)
        res_def=false ;
        
    if (res_def) {
    // Base in phi :
    res.bases_1d[2] = new Array<int> (a.bases_1d[2]->get_dimensions()) ;
    switch ((*a.bases_1d[2])(0)) {
        case COS:
            switch ((*b.bases_1d[2])(0)) {
                case COS:
                    res.bases_1d[2]->set(0) = COS ;
                    break ;
                case SIN:
                    res.bases_1d[2]->set(0) = SIN ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        case SIN:
            switch ((*b.bases_1d[2])(0)) {
                case COS:
                    res.bases_1d[2]->set(0) = SIN ;
                    break ;
                case SIN:
                    res.bases_1d[2]->set(0) = COS ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        default:
            res_def = false ;
            break ;
    }
 
    // Bases in chi :
    // On check l'alternance :
    Index index_1 (a.bases_1d[1]->get_dimensions()) ;
    res.bases_1d[1] = new Array<int> (a.bases_1d[1]->get_dimensions()) ;
    do {
    switch ((*a.bases_1d[1])(index_1)) {
        case CHEB_EVEN:
            switch ((*b.bases_1d[1])(index_1)) {
                case CHEB_EVEN:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? CHEB_EVEN : CHEB_ODD ;
                    break ;
                case CHEB_ODD:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? CHEB_ODD : CHEB_EVEN ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        case CHEB_ODD:
            switch ((*b.bases_1d[1])(index_1)) {
                case CHEB_EVEN:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? CHEB_ODD : CHEB_EVEN ;
                    break ;
                case CHEB_ODD:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? CHEB_EVEN : CHEB_ODD ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        case LEG_EVEN:
            switch ((*b.bases_1d[1])(index_1)) {
                case LEG_EVEN:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? LEG_EVEN : LEG_ODD ;
                    break ;
                case LEG_ODD:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? LEG_ODD : LEG_EVEN ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        case LEG_ODD:
            switch ((*b.bases_1d[1])(index_1)) {
                case LEG_EVEN:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? LEG_ODD : LEG_EVEN ;
                    break ;
                case LEG_ODD:
                    res.bases_1d[1]->set(index_1) = (index_1(0)%2==0) ? LEG_EVEN : LEG_ODD ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        default:
            res_def = false ;
            break ;
        }
    }
    while (index_1.inc()) ;
 
    // Base in eta :
    Index index_0 (a.bases_1d[0]->get_dimensions()) ;
    res.bases_1d[0] = new Array<int> (a.bases_1d[0]->get_dimensions()) ;
    do {
    switch ((*a.bases_1d[0])(index_0)) {
        case CHEB:
            switch ((*b.bases_1d[0])(index_0)) {
                case CHEB:
                    res.bases_1d[0]->set(index_0) = CHEB ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        case LEG:
            switch ((*b.bases_1d[0])(index_0)) {
                case LEG:
                    res.bases_1d[0]->set(index_0) = LEG ;
                    break ;
                default:
                    res_def = false ;
                    break ;
                }
            break ;
        default:
            res_def = false ;
            break ;
        }
    }
    while (index_0.inc()) ;
    }
        
    if (!res_def) 
        for (int dim=0 ; dim<a.ndim ; dim++)
            if (res.bases_1d[dim]!= nullptr) {
                delete res.bases_1d[dim] ;
                res.bases_1d[dim] = nullptr ;
                }
    res.def = res_def ;
    return res ;
}
 
 
Val_domain Domain_bispheric_chi_first::der_normal (const Val_domain& so, int bound) const { 
 
    double rr, xc ;
    switch (bound) {
        case OUTER_BC :
            return Val_domain(so.der_abs(1)*get_cart(1)/r_ext + so.der_abs(2)*get_cart(2)/r_ext +
                 so.der_abs(3)*get_cart(3)/r_ext) ;
        case INNER_BC :
            xc = aa*cosh(eta_lim)/sinh(eta_lim) ;
            rr = aa/sinh(fabs(eta_lim)) ;
 
            return Val_domain(so.der_abs(1)*(get_cart(1)-xc)/rr + so.der_abs(2)*get_cart(2)/rr +
                 so.der_abs(3)*get_cart(3)/rr) ;
        case CHI_ONE_BC : {
            Val_domain res (so.der_var(2) + so.der_var(1) * (*bound_eta_der) * 
    (-2./(*bound_eta-eta_lim)/(*bound_eta-eta_lim)*(get_eta()-(*bound_eta+eta_lim)/2.) - 
        1./(*bound_eta -eta_lim))) ;
            res /= chi_max ;
            res.set_base() = so.der_var(2).get_base() ;
            return res ;
            }
        default:
            cerr << "Unknown boundary case in Domain_bispheric_chi_first::der_normal" << endl ;
            abort() ;
        }
}
    
double Domain_bispheric_chi_first::integ (const Val_domain& so, int bound) const {
 
    if (bound!=INNER_BC) {
      cerr << "Domain_bispheric_chi_first::integ only defined for inner boundary yet.." ;
      abort() ;
    }
 
    double res = 0 ;
 
    int basep = (*so.base.bases_1d[2]) (0) ;
    if (basep == SIN) {
        // Odd function
        return res ;
    }
    else {
        // Multiply by the surface element :
        if (p_dsint==nullptr)
            do_dsint() ;
 
        Val_domain auxi (so*(*p_dsint)) ;
        auxi.coef() ;
 
        Index pos (get_nbr_coefs()) ;
        pos.set(2) = 0 ;
 
        if (type_base==CHEB_TYPE) {
            for (int j=0 ; j<nbr_coefs(1) ; j++) {
                pos.set(1) = j ;
                int base_chi = (*auxi.base.bases_1d[1]) (0) ;
                double val_cheb ;
                switch (base_chi) {
                    case CHEB_EVEN :
                        if (j==0)
                            val_cheb = 1. ;
                        else
                            val_cheb = double(2*j)/double(4*j*j-1) -1./double(2*j-1) ;
                        break ;
                    case CHEB_ODD :
                        if (j==0) 
                            val_cheb = 1./2. ;
                        else
                            if (j%2==0)
                                val_cheb = 2*double(2*j+1)/double((2*j+1)*(2*j+1)-1) - 1./double(2*j) ;
                            else
                                val_cheb = -1./double(2*j) ;
                        break ;
                    default:
                        cerr << "Unknown basis in Domain_bispheric_chi_first::integ" << endl ;
                        abort() ;
                }
 
                for (int i=0 ; i<nbr_coefs(0) ; i++) {
                    pos.set(0) = i ;
                    if (i%2==0)
                        res += val_cheb * (*auxi.cf)(pos) ;
                    else
                        res -= val_cheb * (*auxi.cf)(pos) ;
                }
            }
        }
    
        if (type_base==LEG_TYPE) {
            int base_chi = (*auxi.base.bases_1d[1]) (0) ;
 
            double val_leg = 1. ;
            double val_m1 = 1. ;
            double val_m = -0.5 ;
 
            for (int j=0 ; j<nbr_coefs(1) ; j++) {
                pos.set(1) = j ;
            
            switch (base_chi) {
                    case LEG_EVEN :
                        val_leg = 0. ;
                        break ;
                    case LEG_ODD :
                        val_leg = (val_m1 - val_m)/double(4*j+3) ;
                        val_m *= -double(2*j+3)/double(2*j+4) ;
                        val_m1 *= -double(2*j+1)/double(2*j+2) ;
                        break ;
                    default:
                        cerr << "Unknown basis in Domain_bispheric_chi_first::integ" << endl ;
                        abort() ;
                }
        
            for (int i=0 ; i<nbr_coefs(0) ; i++) {
                pos.set(0) = i ;
                if (i%2==0)
                    res += val_leg*(*auxi.cf)(pos) ;
                else
                    res -= val_leg*(*auxi.cf)(pos) ;
            }
        }
    }
    return res*M_PI*2 ;
    }
}
 
int Domain_bispheric_chi_first::give_place_var (char* p) const {
    int res = -1 ;
    if (strcmp(p,"ETA ")==0)
    res = 0 ;
    if (strcmp(p,"CHI ")==0)
    res = 1 ;
    if (strcmp(p,"P ")==0)
    res = 2 ;
    return res ;
}
}