LORENE
et_bin_bhns_extr_velpot.C
1 /*
2  * Method of class Et_bin_bhns_extr to compute the velocity scalar
3  * potential $\psi$ by solving the continuity equation
4  * in the Kerr-Schild background metric or in the conformally flat one
5  * with extreme mass ratio
6  *
7  * (see file et_bin_bhns_extr.h for documentation).
8  *
9  */
10 
11 /*
12  * Copyright (c) 2004-2005 Keisuke Taniguchi
13  *
14  * This file is part of LORENE.
15  *
16  * LORENE is free software; you can redistribute it and/or modify
17  * it under the terms of the GNU General Public License version 2
18  * as published by the Free Software Foundation.
19  *
20  * LORENE is distributed in the hope that it will be useful,
21  * but WITHOUT ANY WARRANTY; without even the implied warranty of
22  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
23  * GNU General Public License for more details.
24  *
25  * You should have received a copy of the GNU General Public License
26  * along with LORENE; if not, write to the Free Software
27  * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
28  *
29  */
30 
31 char et_bin_bhns_extr_velpot_C[] = "$Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_bhns_extr_velpot.C,v 1.3 2014/10/13 08:52:55 j_novak Exp $" ;
32 
33 /*
34  * $Id: et_bin_bhns_extr_velpot.C,v 1.3 2014/10/13 08:52:55 j_novak Exp $
35  * $Log: et_bin_bhns_extr_velpot.C,v $
36  * Revision 1.3 2014/10/13 08:52:55 j_novak
37  * Lorene classes and functions now belong to the namespace Lorene.
38  *
39  * Revision 1.2 2005/02/28 23:17:18 k_taniguchi
40  * Modification to include the case of the conformally flat background metric
41  *
42  * Revision 1.1 2004/11/30 20:51:56 k_taniguchi
43  * *** empty log message ***
44  *
45  *
46  * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_bhns_extr_velpot.C,v 1.3 2014/10/13 08:52:55 j_novak Exp $
47  *
48  */
49 
50 // C headers
51 //#include <math.h>
52 
53 // Lorene headers
54 #include "et_bin_bhns_extr.h"
55 #include "scalar.h"
56 #include "metrique.h"
57 #include "etoile.h"
58 #include "eos.h"
59 #include "param.h"
60 #include "coord.h"
61 #include "unites.h"
62 
63 // Local prototype
64 namespace Lorene {
65 Cmp raccord_c1(const Cmp& uu, int l1) ;
66 
67 double Et_bin_bhns_extr::velocity_pot_extr(const double& mass,
68  const double& sepa,
69  int mermax, double precis,
70  double relax) {
71 
72  using namespace Unites ;
73 
74  if (kerrschild) {
75 
76  if (eos.identify() == 5 || eos.identify() == 4 ||
77  eos.identify() == 3) {
78 
79  cout << "Etoile_bin::velocity_pot_extr" << endl ;
80  cout << "The code has not prepared for this kind of EOS!!!"
81  << endl;
82  abort() ;
83 
84  } // End of strange stars case
85  else {
86 
87  int nzm1 = mp.get_mg()->get_nzone() - 1 ;
88 
89  //--------------------------------
90  // Specific relativistic enthalpy ---> hhh
91  //--------------------------------
92 
93  Tenseur hhh = exp(unsurc2 * ent) ; // = 1 at the Newtonian limit
94  hhh.set_std_base() ;
95 
96  //--------------------------------------------
97  // Computation of W^i = - A^2 h Gamma_n B^i/N
98  //--------------------------------------------
99 
100  Tenseur www = - a_car * hhh * gam_euler * bsn ;
101 
102  www.change_triad( mp.get_bvect_cart() ) ;
103  // components on the mapping Cartesian basis
104 
105  //-------------------------------------------------
106  // Constant value of W^i at the center of the star
107  //-------------------------------------------------
108 
109  Tenseur v_orb(mp, 1, CON, mp.get_bvect_cart()) ;
110 
111  v_orb.set_etat_qcq() ;
112  for (int i=0; i<3; i++) {
113  v_orb.set(i) = www(i)(0, 0, 0, 0) ;
114  }
115 
116  v_orb.set_triad( *(www.get_triad()) ) ;
117  v_orb.set_std_base() ;
118 
119  // Some auxiliary terms
120  // --------------------
121 
122  const Coord& xx = mp.x ;
123  const Coord& yy = mp.y ;
124  const Coord& zz = mp.z ;
125 
126  Tenseur r_bh(mp) ;
127  r_bh.set_etat_qcq() ;
128  r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
129  r_bh.set_std_base() ;
130 
131  Tenseur msr(mp) ;
132  msr = ggrav * mass / r_bh ;
133  msr.set_std_base() ;
134 
135  Tenseur lapse_bh2(mp) ; // lapse_bh * lapse_bh
136  lapse_bh2 = 1. / (1.+2.*msr) ;
137  lapse_bh2.set_std_base() ;
138 
139  Tenseur lapse_bh3(mp) ; // lapse_bh * lapse_bh * lapse_bh
140  lapse_bh3 = 1./pow(1.+2.*msr, 1.5) ;
141  lapse_bh3.set_std_base() ;
142 
143  Tenseur xx_con(mp, 1, CON, mp.get_bvect_cart()) ;
144  xx_con.set_etat_qcq() ;
145  xx_con.set(0) = xx + sepa ;
146  xx_con.set(1) = yy ;
147  xx_con.set(2) = zz ;
148  xx_con.set_std_base() ;
149 
150  Tenseur xsr_con(mp, 1, CON, mp.get_bvect_cart()) ;
151  xsr_con = xx_con / r_bh ;
152  xsr_con.set_std_base() ;
153 
154  Tenseur xx_cov(mp, 1, COV, mp.get_bvect_cart()) ;
155  xx_cov.set_etat_qcq() ;
156  xx_cov.set(0) = xx + sepa ;
157  xx_cov.set(1) = yy ;
158  xx_cov.set(2) = zz ;
159  xx_cov.set_std_base() ;
160 
161  Tenseur xsr_cov(mp, 1, COV, mp.get_bvect_cart()) ;
162  xsr_cov = xx_cov / r_bh ;
163  xsr_cov.set_std_base() ;
164 
165  // X_i/r_bh in the spherical coordinate
166  const Coord& rr = mp.r ;
167  const Coord& st = mp.sint ;
168  const Coord& ct = mp.cost ;
169  const Coord& sp = mp.sinp ;
170  const Coord& cp = mp.cosp ;
171 
172  Tenseur rr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
173  rr_spher_cov.set_etat_qcq() ;
174  rr_spher_cov.set(0) = rr + sepa*st*cp ;
175  rr_spher_cov.set(1) = sepa*ct*cp ;
176  rr_spher_cov.set(2) = - sepa*sp ;
177  rr_spher_cov.set_std_base() ;
178 
179  Tenseur xsr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
180  xsr_spher_cov = rr_spher_cov / r_bh ;
181  xsr_spher_cov.set_std_base() ;
182 
183  // l^j D_j H_ent
184  Tenseur ldent = flat_scalar_prod(xsr_con, ent.gradient()) ;
185 
186  // l^j D_j beta_auto
187  Tenseur ldbeta = flat_scalar_prod(xsr_con, beta_auto.gradient()) ;
188 
189  // l^j D_j psi0
190  Tenseur ldpsi0 = flat_scalar_prod(xsr_con, psi0.gradient()) ;
191 
192  // l^i D_i (l^j D_j psi0)
193  Tenseur ldldpsi0 = flat_scalar_prod(xsr_con, ldpsi0.gradient()) ;
194 
195  //-------------------------------------------------
196  // Source and coefficients a,b for poisson_compact
197  // (idenpendent from psi0)
198  //-------------------------------------------------
199 
200  Cmp dndh_log = eos.der_nbar_ent(ent(), nzet) ;
201 
202  // In order to avoid any division by zero in the computation of
203  // zeta_h the value of dndh_log is set to 1
204  // in the external domains:
205  for (int l=nzet; l <= nzm1; l++) {
206  dndh_log.set(l) = 1 ;
207  }
208 
209  double erreur ;
210 
211  Tenseur zeta_h( ent() / dndh_log ) ;
212  zeta_h.set_std_base() ;
213 
214  Tenseur tmp_zeta = 1 - unsurc2 * zeta_h ;
215  tmp_zeta.set_std_base() ;
216 
217  Tenseur bb = tmp_zeta * (ent.gradient_spher()
218  - 2.*lapse_bh2 * msr * ldent
219  * xsr_spher_cov)
220  + unsurc2 * zeta_h * (beta_auto.gradient_spher()
221  - 2.*lapse_bh2 * msr * ldbeta
222  * xsr_spher_cov)
223  - unsurc2 * 2. * zeta_h * lapse_bh2 * lapse_bh2 * msr / r_bh
224  * (1.+4.*msr) * xsr_spher_cov ;
225 
226  Tenseur entmb = ent - beta_auto ;
227 
228  Tenseur source = flat_scalar_prod(www - v_orb, ent.gradient())
229  + unsurc2*zeta_h*( flat_scalar_prod(v_orb, entmb.gradient())
231  / gam_euler )
232  + 2.*lapse_bh2 * msr * flat_scalar_prod(xsr_cov, v_orb)
233  * flat_scalar_prod(xsr_con, ent.gradient())
234  + unsurc2 * 2. * zeta_h
235  * (lapse_bh2*msr*(ldldpsi0
236  - flat_scalar_prod(xsr_cov, v_orb)
237  * (flat_scalar_prod(xsr_con, entmb.gradient())
238  - lapse_bh2 * (1.+4.*msr) / r_bh))
239  + a_car * hhh * gam_euler * lapse_bh3 * msr * (1.+3.*msr)
240  / r_bh) ;
241 
242  source.annule(nzet, nzm1) ;
243 
244  //----------------------------------------------------
245  // Resolution by means of Map_radial::poisson_compact
246  //----------------------------------------------------
247 
248  Param par ;
249  int niter ;
250  par.add_int(mermax) ;
251  par.add_double(precis, 0) ;
252  par.add_double(relax, 1) ;
253  par.add_int_mod(niter) ;
254 
255  if (psi0.get_etat() == ETATZERO) {
256  psi0.set_etat_qcq() ;
257  psi0.set() = 0 ;
258  }
259 
260  source.set().va.ylm() ;
261 
262  mp.poisson_compact(source(), zeta_h(), bb, par, psi0.set() ) ;
263 
264  //-----------------------
265  // Check of the solution
266  //-----------------------
267 
268  Tenseur bb_dpsi0 = flat_scalar_prod( bb, psi0.gradient_spher() ) ;
269 
270  Cmp oper = zeta_h() * psi0().laplacien() + bb_dpsi0() ;
271 
272  source.set().va.ylm_i() ;
273 
274  erreur = diffrel(oper, source())(0) ;
275 
276  cout << "Check of the resolution of the continuity equation : "
277  << endl ;
278  cout << "norme(source) : " << norme(source())(0)
279  << " diff oper/source : " << erreur << endl ;
280 
281  //--------------------------
282  // Computation of grad(psi)
283  //--------------------------
284 
285  // The computation is done component by component because
286  // psi0.gradient() is a covariant vector, whereas v_orb is a
287  // contravariant one.
288 
289  d_psi.set_etat_qcq() ;
290 
291  for (int i=0; i<3; i++) {
292  d_psi.set(i) = (psi0.gradient())(i) + v_orb(i) ;
293  }
294 
295  d_psi.set_triad( *(v_orb.get_triad()) ) ;
296 
297  // C^1 continuation of d_psi outside the star
298  // (to ensure a smooth enthalpy field accross the stellar surface)
299  // ----------------------------------------------------------------
300 
301  d_psi.annule(nzet, nzm1) ;
302  for (int i=0; i<3; i++) {
303  d_psi.set(i) = raccord_c1(d_psi(i), nzet) ;
304  }
305 
306  assert( d_psi.get_triad() == &(mp.get_bvect_cart()) ) ;
307 
309 
310  return erreur ;
311 
312  }
313 
314  }
315  else {
316 
317  if (eos.identify() == 5 || eos.identify() == 4 ||
318  eos.identify() == 3) {
319 
320  cout << "Etoile_bin::velocity_pot_extr" << endl ;
321  cout << "The code has not prepared for this kind of EOS!!!"
322  << endl;
323  abort() ;
324 
325  } // End of strange stars case
326  else {
327 
328  int nzm1 = mp.get_mg()->get_nzone() - 1 ;
329 
330  //--------------------------------
331  // Specific relativistic enthalpy ---> hhh
332  //--------------------------------
333 
334  Tenseur hhh = exp(unsurc2 * ent) ; // = 1 at the Newtonian limit
335  hhh.set_std_base() ;
336 
337  //--------------------------------------------
338  // Computation of W^i = - A^2 h Gamma_n B^i/N
339  //--------------------------------------------
340 
341  Tenseur www = - a_car * hhh * gam_euler * bsn ;
342 
343  www.change_triad( mp.get_bvect_cart() ) ;
344  // components on the mapping Cartesian basis
345 
346  //-------------------------------------------------
347  // Constant value of W^i at the center of the star
348  //-------------------------------------------------
349 
350  Tenseur v_orb(mp, 1, CON, mp.get_bvect_cart()) ;
351 
352  v_orb.set_etat_qcq() ;
353  for (int i=0; i<3; i++) {
354  v_orb.set(i) = www(i)(0, 0, 0, 0) ;
355  }
356 
357  v_orb.set_triad( *(www.get_triad()) ) ;
358  v_orb.set_std_base() ;
359 
360  // Some auxiliary terms
361  // --------------------
362 
363  const Coord& xx = mp.x ;
364  const Coord& yy = mp.y ;
365  const Coord& zz = mp.z ;
366 
367  Tenseur r_bh(mp) ;
368  r_bh.set_etat_qcq() ;
369  r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
370  r_bh.set_std_base() ;
371 
372  Tenseur msr(mp) ;
373  msr = ggrav * mass / r_bh ;
374  msr.set_std_base() ;
375 
376  Tenseur xx_cov(mp, 1, COV, mp.get_bvect_cart()) ;
377  xx_cov.set_etat_qcq() ;
378  xx_cov.set(0) = xx + sepa ;
379  xx_cov.set(1) = yy ;
380  xx_cov.set(2) = zz ;
381  xx_cov.set_std_base() ;
382 
383  // X_i in the spherical coordinate
384  const Coord& rr = mp.r ;
385  const Coord& st = mp.sint ;
386  const Coord& ct = mp.cost ;
387  const Coord& sp = mp.sinp ;
388  const Coord& cp = mp.cosp ;
389 
390  Tenseur rr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
391  rr_spher_cov.set_etat_qcq() ;
392  rr_spher_cov.set(0) = rr + sepa*st*cp ;
393  rr_spher_cov.set(1) = sepa*ct*cp ;
394  rr_spher_cov.set(2) = - sepa*sp ;
395  rr_spher_cov.set_std_base() ;
396 
397  Tenseur tmp_bh(mp) ;
398  tmp_bh = 0.5 * msr * msr / (1.-0.25*msr*msr) / r_bh / r_bh ;
399  tmp_bh.set_std_base() ;
400 
401  //-------------------------------------------------
402  // Source and coefficients a,b for poisson_compact
403  // (idenpendent from psi0)
404  //-------------------------------------------------
405 
406  Cmp dndh_log = eos.der_nbar_ent(ent(), nzet) ;
407 
408  // In order to avoid any division by zero in the computation of
409  // zeta_h the value of dndh_log is set to 1
410  // in the external domains:
411  for (int l=nzet; l <= nzm1; l++) {
412  dndh_log.set(l) = 1 ;
413  }
414 
415  double erreur ;
416 
417  Tenseur zeta_h( ent() / dndh_log ) ;
418  zeta_h.set_std_base() ;
419 
420  Tenseur tmp_zeta = 1 - unsurc2 * zeta_h ;
421  tmp_zeta.set_std_base() ;
422 
423  Tenseur bb = tmp_zeta * ent.gradient_spher()
424  + unsurc2 * zeta_h * (beta_auto.gradient_spher()
425  + tmp_bh * rr_spher_cov) ;
426 
427  Tenseur entmb = ent - beta_auto ;
428 
429  Tenseur source = flat_scalar_prod(www - v_orb, ent.gradient())
430  + unsurc2*zeta_h*( flat_scalar_prod(v_orb, entmb.gradient())
431  - tmp_bh * flat_scalar_prod(v_orb, xx_cov)
433  / gam_euler ) ;
434 
435  source.annule(nzet, nzm1) ;
436 
437  //----------------------------------------------------
438  // Resolution by means of Map_radial::poisson_compact
439  //----------------------------------------------------
440 
441  Param par ;
442  int niter ;
443  par.add_int(mermax) ;
444  par.add_double(precis, 0) ;
445  par.add_double(relax, 1) ;
446  par.add_int_mod(niter) ;
447 
448  if (psi0.get_etat() == ETATZERO) {
449  psi0.set_etat_qcq() ;
450  psi0.set() = 0 ;
451  }
452 
453  source.set().va.ylm() ;
454 
455  mp.poisson_compact(source(), zeta_h(), bb, par, psi0.set() ) ;
456 
457  //-----------------------
458  // Check of the solution
459  //-----------------------
460 
461  Tenseur bb_dpsi0 = flat_scalar_prod( bb, psi0.gradient_spher() ) ;
462 
463  Cmp oper = zeta_h() * psi0().laplacien() + bb_dpsi0() ;
464 
465  source.set().va.ylm_i() ;
466 
467  erreur = diffrel(oper, source())(0) ;
468 
469  cout << "Check of the resolution of the continuity equation : "
470  << endl ;
471  cout << "norme(source) : " << norme(source())(0)
472  << " diff oper/source : " << erreur << endl ;
473 
474  //--------------------------
475  // Computation of grad(psi)
476  //--------------------------
477 
478  // The computation is done component by component because
479  // psi0.gradient() is a covariant vector, whereas v_orb is a
480  // contravariant one.
481 
482  d_psi.set_etat_qcq() ;
483 
484  for (int i=0; i<3; i++) {
485  d_psi.set(i) = (psi0.gradient())(i) + v_orb(i) ;
486  }
487 
488  d_psi.set_triad( *(v_orb.get_triad()) ) ;
489 
490  // C^1 continuation of d_psi outside the star
491  // (to ensure a smooth enthalpy field accross the stellar surface)
492  // ----------------------------------------------------------------
493 
494  d_psi.annule(nzet, nzm1) ;
495  for (int i=0; i<3; i++) {
496  d_psi.set(i) = raccord_c1(d_psi(i), nzet) ;
497  }
498 
499  assert( d_psi.get_triad() == &(mp.get_bvect_cart()) ) ;
500 
502 
503  return erreur ;
504 
505  }
506 
507  }
508 
509 }
510 }
Component of a tensorial field *** DEPRECATED : use class Scalar instead ***.
Definition: cmp.h:446
Valeur va
The numerical value of the Cmp
Definition: cmp.h:464
Tbl & set(int l)
Read/write of the value in a given domain.
Definition: cmp.h:724
Active physical coordinates and mapping derivatives.
Definition: coord.h:90
virtual int identify() const =0
Returns a number to identify the sub-classe of Eos the object belongs to.
Cmp der_nbar_ent(const Cmp &ent, int nzet, int l_min=0, const Param *par=0x0) const
Computes the logarithmic derivative from the log-enthalpy and extra parameters.
Definition: eos.C:407
bool kerrschild
Indicator of the background metric: true for the Kerr-Shild metric, false for the conformally flat on...
double velocity_pot_extr(const double &mass, const double &sepa, int mermax, double precis, double relax)
Computes the non-translational part of the velocity scalar potential by solving the continuity equat...
Tenseur psi0
Scalar potential of the non-translational part of fluid 4-velocity (in the irrotational case)
Definition: etoile.h:833
Tenseur bsn
3-vector shift, divided by N , of the rotating coordinates, .
Definition: etoile.h:950
Tenseur d_psi
Gradient of (in the irrotational case) (Cartesian components with respect to ref_triad )
Definition: etoile.h:838
const Base_vect & ref_triad
Reference triad ("absolute frame"), with respect to which the components of all the member Tenseur 's...
Definition: etoile.h:828
int nzet
Number of domains of *mp occupied by the star.
Definition: etoile.h:432
const Eos & eos
Equation of state of the stellar matter.
Definition: etoile.h:451
Tenseur gam_euler
Lorentz factor between the fluid and Eulerian observers.
Definition: etoile.h:471
Map & mp
Mapping associated with the star.
Definition: etoile.h:429
Tenseur ent
Log-enthalpy (relativistic case) or specific enthalpy (Newtonian case)
Definition: etoile.h:457
Tenseur beta_auto
Logarithm of the part of the product AN generated principaly by by the star.
Definition: etoile.h:506
Tenseur a_car
Total conformal factor .
Definition: etoile.h:515
double unsurc2
: unsurc2=1 for a relativistic star, 0 for a Newtonian one.
Definition: etoile.h:442
Coord cosp
Definition: map.h:724
Coord y
y coordinate centered on the grid
Definition: map.h:727
Coord sint
Definition: map.h:721
virtual void poisson_compact(const Cmp &source, const Cmp &aa, const Tenseur &bb, const Param &par, Cmp &psi) const =0
Resolution of the elliptic equation in the case where the stellar interior is covered by a single do...
Coord r
r coordinate centered on the grid
Definition: map.h:718
const Mg3d * get_mg() const
Gives the Mg3d on which the mapping is defined.
Definition: map.h:765
const Base_vect_spher & get_bvect_spher() const
Returns the orthonormal vectorial basis associated with the coordinates of the mapping.
Definition: map.h:783
Coord sinp
Definition: map.h:723
Coord z
z coordinate centered on the grid
Definition: map.h:728
Coord x
x coordinate centered on the grid
Definition: map.h:726
const Base_vect_cart & get_bvect_cart() const
Returns the Cartesian basis associated with the coordinates (x,y,z) of the mapping,...
Definition: map.h:791
Coord cost
Definition: map.h:722
int get_nzone() const
Returns the number of domains.
Definition: grilles.h:448
Parameter storage.
Definition: param.h:125
void add_double(const double &x, int position=0)
Adds the the address of a new double to the list.
Definition: param.C:315
void add_int_mod(int &n, int position=0)
Adds the address of a new modifiable int to the list.
Definition: param.C:385
void add_int(const int &n, int position=0)
Adds the address of a new int to the list.
Definition: param.C:246
Tensor handling *** DEPRECATED : use class Tensor instead ***.
Definition: tenseur.h:301
Cmp & set()
Read/write for a scalar (see also operator=(const Cmp&) ).
Definition: tenseur.C:824
void set_etat_qcq()
Sets the logical state to ETATQCQ (ordinary state).
Definition: tenseur.C:636
void annule(int l)
Sets the Tenseur to zero in a given domain.
Definition: tenseur.C:900
const Tenseur & gradient() const
Returns the gradient of *this (Cartesian coordinates)
Definition: tenseur.C:1542
void set_std_base()
Set the standard spectal basis of decomposition for each component.
Definition: tenseur.C:1170
const Tenseur & gradient_spher() const
Returns the gradient of *this (Spherical coordinates) (scalar field only).
Definition: tenseur.C:1548
void change_triad(const Base_vect &new_triad)
Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.
Definition: tenseur.C:668
void set_triad(const Base_vect &new_triad)
Assigns a new vectorial basis (triad) of decomposition.
Definition: tenseur.C:674
const Base_vect * get_triad() const
Returns the vectorial basis (triad) on which the components are defined.
Definition: tenseur.h:704
int get_etat() const
Returns the logical state.
Definition: tenseur.h:707
void ylm()
Computes the coefficients of *this.
Definition: valeur_ylm.C:138
void ylm_i()
Inverse of ylm()
Definition: valeur_ylm_i.C:131
Cmp exp(const Cmp &)
Exponential.
Definition: cmp_math.C:270
Tbl diffrel(const Cmp &a, const Cmp &b)
Relative difference between two Cmp (norme version).
Definition: cmp_math.C:504
Tbl norme(const Cmp &)
Sums of the absolute values of all the values of the Cmp in each domain.
Definition: cmp_math.C:481
Cmp pow(const Cmp &, int)
Power .
Definition: cmp_math.C:348
Tenseur flat_scalar_prod(const Tenseur &t1, const Tenseur &t2)
Scalar product of two Tenseur when the metric is : performs the contraction of the last index of t1 w...
Lorene prototypes.
Definition: app_hor.h:64
Standard units of space, time and mass.