Vector Class Reference
[Tensorial fields]

Tensor field of valence 1. More...

#include <vector.h>

Inheritance diagram for Vector:
Tensor Vector_divfree

List of all members.

Public Member Functions

 Vector (const Map &map, int tipe, const Base_vect &triad_i)
 Standard constructor.
 Vector (const Map &map, int tipe, const Base_vect *triad_i)
 Standard constructor with the triad passed as a pointer.
 Vector (const Vector &a)
 Copy constructor.
 Vector (const Tensor &a)
 Constructor from a Tensor .
 Vector (const Map &map, const Base_vect &triad_i, FILE *fich)
 Constructor from a file (see Tensor::sauve(FILE*) ).
virtual ~Vector ()
 Destructor.
virtual void change_triad (const Base_vect &)
 Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.
virtual void operator= (const Vector &a)
 Assignment from a Vector.
virtual void operator= (const Tensor &a)
 Assignment from a Tensor.
void set_vr_eta_mu (const Scalar &vr_i, const Scalar &eta_i, const Scalar &mu_i)
 Defines the components through potentials $\eta$ and $\mu$ (see members p_eta and p_mu ), as well as the $V^r$ component of the vector.
void decompose_div (const Metric &) const
 Makes the Helmholtz decomposition (see documentation of p_potential ) of this with respect to a given Metric , only in the case of contravariant vectors.
const Scalar & potential (const Metric &) const
 Returns the potential in the Helmholtz decomposition.
const Vector_divfree & div_free (const Metric &) const
 Returns the div-free vector in the Helmholtz decomposition.
virtual void exponential_filter_r (int lzmin, int lzmax, int p, double alpha=-16.)
 Applies exponential filters to all components (see Scalar::exponential_filter_r ).
virtual void exponential_filter_ylm (int lzmin, int lzmax, int p, double alpha=-16.)
 Applies exponential filters to all components (see Scalar::exponential_filter_ylm ).
Scalar & set (int)
 Read/write access to a component.
const Scalar & operator() (int) const
 Readonly access to a component.
virtual int position (const Itbl &idx) const
 Returns the position in the Scalar array cmp of a component given by its index.
virtual Itbl indices (int place) const
 Returns the index of a component given by its position in the Scalar array cmp .
virtual void std_spectral_base ()
 Sets the standard spectal bases of decomposition for each component.
virtual void pseudo_spectral_base ()
 Sets the standard spectal bases of decomposition for each component for a pseudo_vector.
virtual const Scalar & eta () const
 Gives the field $\eta$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

virtual const Scalar & mu () const
 Gives the field $\mu$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

virtual const Scalar & A () const
 Gives the field $A$ defined by

\[ A = {\partial \eta \over \partial r} + { \eta \over r} - {V^r \over r} \]

Related to the curl, A is insensitive to the longitudinal part of the vector.

void update_vtvp ()
 Computes the components $V^\theta$ and $V^\varphi$ from the potential $\eta$ and $\mu$, according to:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

const Scalar & divergence (const Metric &) const
 The divergence of this with respect to a Metric .
const Vector_divfree curl () const
 The curl of this with respect to a (flat) Metric .
Vector derive_lie (const Vector &v) const
 Computes the Lie derivative of this with respect to some vector field v.
Sym_tensor ope_killing (const Metric &gam) const
 Computes the Killing operator associated with a given metric.
Sym_tensor ope_killing_conf (const Metric &gam) const
 Computes the conformal Killing operator associated with a given metric.
Vector poisson (double lambda, int method=6) const
 Solves the vector Poisson equation with *this as a source.
Vector poisson (double lambda, const Metric_flat &met_f, int method=6) const
 Solves the vector Poisson equation with *this as a source.
Vector poisson (const double lambda, Param &par, int method=6) const
 Solves the vector Poisson equation with *this as a source and parameters controlling the solution.
void poisson_boundary (double lambda, const Mtbl_cf &limit_vr, const Mtbl_cf &limit_eta, const Mtbl_cf &limit_mu, int num_front, double fact_dir, double fact_neu, Vector &resu) const
 Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.
void poisson_boundary2 (double lam, Vector &resu, Scalar boundvr, Scalar boundeta, Scalar boundmu, double dir_vr, double neum_vr, double dir_eta, double neum_eta, double dir_mu, double neum_mu) const
 Alternative to previous poisson_boundary method for vectors ; this uses method 6 for vectorial solving, updated version (as in the poisson_vector_block routine).
Vector poisson_dirichlet (double lambda, const Valeur &limit_vr, const Valeur &limit_vt, const Valeur &limit_vp, int num_front) const
Vector poisson_neumann (double lambda, const Valeur &limit_vr, const Valeur &limit_vt, const Valeur &limit_vp, int num_front) const
 Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.
Vector poisson_robin (double lambda, const Valeur &limit_vr, const Valeur &limit_vt, const Valeur &limit_vp, double fact_dir, double fact_neu, int num_front) const
 Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.
double flux (double radius, const Metric &met) const
 Computes the flux of the vector accross a sphere r = const.
void poisson_block (double lambda, Vector &resu) const
void visu_arrows (double xmin, double xmax, double ymin, double ymax, double zmin, double zmax, const char *title0=0x0, const char *filename0=0x0, bool start_dx=true, int nx=8, int ny=8, int nz=8) const
 3D visualization via OpenDX.
void visu_streamline (double xmin, double xmax, double ymin, double ymax, double zmin, double zmax, const char *title0=0x0, const char *filename0=0x0, bool start_dx=true, int nx=8, int ny=8, int nz=8) const
virtual void set_etat_nondef ()
 Sets the logical state of all components to ETATNONDEF (undefined state).
virtual void set_etat_zero ()
 Sets the logical state of all components to ETATZERO (zero state).
virtual void set_etat_qcq ()
 Sets the logical state of all components to ETATQCQ (ordinary state).
virtual void allocate_all ()
 Performs the memory allocation of all the elements, down to the double arrays of the Tbl s.
void set_triad (const Base_vect &new_triad)
 Assigns a new vectorial basis (triad) of decomposition.
Scalar & set (const Itbl &ind)
 Returns the value of a component (read/write version).
Scalar & set (int i1, int i2)
 Returns the value of a component for a tensor of valence 2 (read/write version).
Scalar & set (int i1, int i2, int i3)
 Returns the value of a component for a tensor of valence 3 (read/write version).
Scalar & set (int i1, int i2, int i3, int i4)
 Returns the value of a component for a tensor of valence 4 (read/write version).
void annule_domain (int l)
 Sets the Tensor to zero in a given domain.
virtual void annule (int l_min, int l_max)
 Sets the Tensor to zero in several domains.
void annule_extern_cn (int l_0, int deg)
 Performs a smooth (C^n) transition in a given domain to zero.
virtual void std_spectral_base_odd ()
 Sets the standard odd spectal bases of decomposition for each component.
virtual void dec_dzpuis (int dec=1)
 Decreases by dec units the value of dzpuis and changes accordingly the values in the compactified external domain (CED).
virtual void inc_dzpuis (int inc=1)
 Increases by inc units the value of dzpuis and changes accordingly the values in the compactified external domain (CED).
const Tensor & derive_cov (const Metric &gam) const
 Returns the covariant derivative of this with respect to some metric $\gamma$.
const Tensor & derive_con (const Metric &gam) const
 Returns the "contravariant" derivative of this with respect to some metric $\gamma$, by raising the last index of the covariant derivative (cf.
Tensor derive_lie (const Vector &v) const
 Computes the Lie derivative of this with respect to some vector field v.
Tensor up (int ind, const Metric &gam) const
 Computes a new tensor by raising an index of *this.
Tensor down (int ind, const Metric &gam) const
 Computes a new tensor by lowering an index of *this.
Tensor up_down (const Metric &gam) const
 Computes a new tensor by raising or lowering all the indices of *this .
Tensor trace (int ind1, int ind2) const
 Trace on two different type indices.
Tensor trace (int ind1, int ind2, const Metric &gam) const
 Trace with respect to a given metric.
Scalar trace () const
 Trace on two different type indices for a valence 2 tensor.
Scalar trace (const Metric &gam) const
 Trace with respect to a given metric for a valence 2 tensor.
const Map & get_mp () const
 Returns the mapping.
const Base_vect * get_triad () const
 Returns the vectorial basis (triad) on which the components are defined.
int get_valence () const
 Returns the valence.
int get_n_comp () const
 Returns the number of stored components.
int get_index_type (int i) const
 Gives the type (covariant or contravariant) of the index number i .
Itbl get_index_type () const
 Returns the types of all the indices.
int & set_index_type (int i)
 Sets the type of the index number i .
Itbl & set_index_type ()
 Sets the types of all the indices.
const Scalar & operator() (const Itbl &ind) const
 Returns the value of a component (read-only version).
const Scalar & operator() (int i1, int i2) const
 Returns the value of a component for a tensor of valence 2 (read-only version).
const Scalar & operator() (int i1, int i2, int i3) const
 Returns the value of a component for a tensor of valence 3 (read-only version).
const Scalar & operator() (int i1, int i2, int i3, int i4) const
 Returns the value of a component for a tensor of valence 4 (read-only version).
void operator+= (const Tensor &)
 += Tensor
void operator-= (const Tensor &)
 -= Tensor
virtual void sauve (FILE *) const
 Save in a binary file.
virtual void spectral_display (const char *comment=0x0, double threshold=1.e-7, int precision=4, ostream &ostr=cout) const
 Displays the spectral coefficients and the associated basis functions of each component.

Protected Member Functions

virtual void del_deriv () const
 Deletes the derived quantities.
void set_der_0x0 () const
 Sets the pointers on derived quantities to 0x0.
virtual void del_derive_met (int) const
 Logical destructor of the derivatives depending on the i-th element of met_depend in the class Vector.
void set_der_met_0x0 (int) const
 Sets all the i-th components of met_depend in the class Vector (p_potential , etc.
void set_dependance (const Metric &) const
 To be used to describe the fact that the derivatives members have been calculated with met .
int get_place_met (const Metric &) const
 Returns the position of the pointer on metre in the array met_depend .
void compute_derive_lie (const Vector &v, Tensor &resu) const
 Computes the Lie derivative of this with respect to some vector field v (protected method; the public interface is method derive_lie ).

Protected Attributes

Scalar * p_potential [N_MET_MAX]
 The potential $\phi$ giving the gradient part in the Helmholtz decomposition of any 3D vector $\vec{V}: \quad \vec{V} = \vec{\nabla} \phi + \vec{\nabla} \wedge \vec{\psi}$.
Vector_divfree * p_div_free [N_MET_MAX]
 The divergence-free vector $\vec{W} = \vec{\nabla} \wedge \vec{\psi}$ of the Helmholtz decomposition of any 3D vector $\vec{V}: \quad \vec{V} = \vec{\nabla} \phi + \vec{\nabla} *\wedge \vec{\psi}$.
Scalar * p_eta
 Field $\eta$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Scalar * p_mu
 Field $\mu$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Scalar * p_A
 Field $A$ defined by

\[ A = {\partial \eta \over \ partial r} + { \eta \over r} - {V^r \over r} \]

Insensitive to the longitudinal part of the vector, related to the curl.

const Map *const mp
 Mapping on which the numerical values at the grid points are defined.
int valence
 Valence of the tensor (0 = scalar, 1 = vector, etc...).
const Base_vect * triad
 Vectorial basis (triad) with respect to which the tensor components are defined.
Itbl type_indice
 1D array of integers (class Itbl ) of size valence containing the type of each index: COV for a covariant one and CON for a contravariant one.
int n_comp
 Number of stored components, depending on the symmetry.
Scalar ** cmp
 Array of size n_comp of pointers onto the components.
const Metric * met_depend [N_MET_MAX]
 Array on the Metric 's which were used to compute derived quantities, like p_derive_cov , etc.
Tensor * p_derive_cov [N_MET_MAX]
 Array of pointers on the covariant derivatives of this with respect to various metrics.
Tensor * p_derive_con [N_MET_MAX]
 Array of pointers on the contravariant derivatives of this with respect to various metrics.
Tensor * p_divergence [N_MET_MAX]
 Array of pointers on the divergence of this with respect to various metrics.

Friends

class Scalar
class Vector
class Sym_tensor
class Tensor_sym
class Metric
ostream & operator<< (ostream &, const Tensor &)
Scalar operator+ (const Tensor &, const Scalar &)
Scalar operator+ (const Scalar &, const Tensor &)
Scalar operator- (const Tensor &, const Scalar &)
Scalar operator- (const Scalar &, const Tensor &)
Tensor operator* (const Tensor &, const Tensor &)
Tensor_sym operator* (const Tensor &, const Tensor_sym &)
Tensor_sym operator* (const Tensor_sym &, const Tensor &)
Tensor_sym operator* (const Tensor_sym &, const Tensor_sym &)
 Tensorial product of two symmetric tensors.

Detailed Description

Tensor field of valence 1.

()

Definition at line 184 of file vector.h.

Constructor & Destructor Documentation

Vector::Vector ( const Map &  map,
int  tipe,
const Base_vect &  triad_i 
)

Standard constructor.

Parameters:
map the mapping
tipe the type COV for a covariant vector (1-form) and CON for a contravariant one
triad_i vectorial basis (triad) with respect to which the vector components are defined

Definition at line 152 of file vector.C.

References set_der_0x0().

Vector::Vector ( const Map &  map,
int  tipe,
const Base_vect *  triad_i 
)

Standard constructor with the triad passed as a pointer.

Parameters:
map the mapping
tipe the type COV for a covariant vector (1-form) and CON for a contravariant one
triad_i pointer on the vectorial basis (triad) with respect to which the vector components are defined

Definition at line 161 of file vector.C.

References set_der_0x0().

Vector::Vector ( const Vector &  a  ) 

Copy constructor.

Definition at line 169 of file vector.C.

References set_der_0x0(), and Tensor::valence.

Vector::Vector ( const Tensor &  a  ) 

Constructor from a Tensor .

The Tensor must be of valence one.

Definition at line 180 of file vector.C.

References set_der_0x0(), and Tensor::valence.

Vector::Vector ( const Map &  map,
const Base_vect &  triad_i,
FILE *  fich 
)

Constructor from a file (see Tensor::sauve(FILE*) ).

Parameters:
map the mapping
triad_i vectorial basis (triad) with respect to which the tensor components are defined. It will be checked that it coincides with the basis saved in the file.
fich file which has been created by the function sauve(FILE*) .

Definition at line 190 of file vector.C.

References Tensor::n_comp, set_der_0x0(), and Tensor::valence.

Vector::~Vector (  )  [virtual]

Destructor.

Definition at line 204 of file vector.C.

References del_deriv().

Member Function Documentation

const Scalar & Vector::A (  )  const [virtual]

Gives the field $A$ defined by

\[ A = {\partial \eta \over \partial r} + { \eta \over r} - {V^r \over r} \]

Related to the curl, A is insensitive to the longitudinal part of the vector.

Definition at line 125 of file vector_etamu.C.

References Tensor::cmp, Scalar::div_r_dzpuis(), Scalar::dsdr(), eta(), p_A, p_eta, Scalar::set_dzpuis(), and Tensor::triad.

void Tensor::allocate_all (  )  [virtual, inherited]

Performs the memory allocation of all the elements, down to the double arrays of the Tbl s.

This function performs in fact recursive calls to set_etat_qcq() on each element of the chain Scalar -> Valeur -> Mtbl -> Tbl .

Reimplemented in Scalar.

Definition at line 504 of file tensor.C.

References Scalar::allocate_all(), Tensor::cmp, Tensor::del_deriv(), and Tensor::n_comp.

void Tensor::annule ( int  l_min,
int  l_max 
) [virtual, inherited]

Sets the Tensor to zero in several domains.

Parameters:
l_min [input] The Tensor will be set (logically) to zero in the domains whose indices are in the range [l_min,l_max] .
l_max [input] see the comments for l_min .

Note that annule(0,nz-1) , where nz is the total number of domains, is equivalent to set_etat_zero() .

Reimplemented in Scalar.

Definition at line 667 of file tensor.C.

References Scalar::annule(), Tensor::cmp, Tensor::del_deriv(), Map::get_mg(), Mg3d::get_nzone(), Tensor::mp, Tensor::n_comp, and Tensor::set_etat_zero().

void Tensor::annule_domain ( int  l  )  [inherited]

Sets the Tensor to zero in a given domain.

Parameters:
l [input] Index of the domain in which the Tensor will be set (logically) to zero.

Definition at line 662 of file tensor.C.

References Tensor::annule().

void Tensor::annule_extern_cn ( int  l_0,
int  deg 
) [inherited]

Performs a smooth (C^n) transition in a given domain to zero.

Parameters:
l_0 [input] in the domain of index l0 the tensor is multiplied by the right polynomial (of degree 2n+1), to ensure continuty of the function and its n first derivative at both ends of this domain. The tensor is unchanged in the domains l < l_0 and set to zero in domains l > l_0.
deg [input] the degree n of smoothness of the transition.

Definition at line 686 of file tensor.C.

References Scalar::allocate_all(), Scalar::annule(), Itbl::annule_hard(), Tensor::cmp, Tensor::del_deriv(), Map::get_mg(), Mg3d::get_nr(), Mg3d::get_nzone(), Mg3d::get_type_r(), Tensor::mp, Tensor::n_comp, pow(), Map::r, Tbl::set(), Itbl::set(), Scalar::set_domain(), Tbl::set_etat_qcq(), Scalar::std_spectral_base(), and Map::val_r().

void Vector::change_triad ( const Base_vect &  new_triad  )  [virtual]

Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.

Reimplemented from Tensor.

Definition at line 71 of file vector_change_triad.C.

References Tensor::cmp, Map::comp_p_from_cartesian(), Map::comp_r_from_cartesian(), Map::comp_t_from_cartesian(), Map::comp_x_from_spherical(), Map::comp_y_from_spherical(), Map::comp_z_from_spherical(), Base_vect_cart::get_align(), Map::get_bvect_cart(), Map::get_bvect_spher(), Map::get_mg(), Mg3d::get_nt(), Mg3d::get_nzone(), Tensor::mp, and Tensor::triad.

void Tensor::compute_derive_lie ( const Vector &  v,
Tensor &  resu 
) const [protected, inherited]

Computes the Lie derivative of this with respect to some vector field v (protected method; the public interface is method derive_lie ).

Definition at line 335 of file tensor_calculus.C.

References Tensor::cmp, contract(), Scalar::dec_dzpuis(), Tensor::derive_cov(), Map::flat_met_cart(), Map::flat_met_spher(), Scalar::get_dzpuis(), Tensor::get_n_comp(), Tensor::get_triad(), Tensor::indices(), Tensor::mp, Tensor::n_comp, Tensor::operator()(), Tensor::set(), Itbl::set(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

const Vector_divfree Vector::curl (  )  const

The curl of this with respect to a (flat) Metric .

The Vector is assumed to be contravariant.

Definition at line 398 of file vector.C.

References Tensor::cmp, Scalar::div_r(), Scalar::div_tant(), Scalar::dsdt(), Scalar::dsdx(), Scalar::dsdy(), Scalar::dsdz(), Map::flat_met_cart(), Map::flat_met_spher(), Map::get_bvect_cart(), Map::get_bvect_spher(), Base_vect::identify(), Tensor::mp, Scalar::mult_r(), Tensor::set(), Scalar::srstdsdp(), and Tensor::triad.

void Tensor::dec_dzpuis ( int  dec = 1  )  [virtual, inherited]

Decreases by dec units the value of dzpuis and changes accordingly the values in the compactified external domain (CED).

Reimplemented in Scalar.

Definition at line 804 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), and Tensor::n_comp.

void Vector::decompose_div ( const Metric &  metre  )  const

Makes the Helmholtz decomposition (see documentation of p_potential ) of this with respect to a given Metric , only in the case of contravariant vectors.

Definition at line 512 of file vector.C.

References Scalar::annule(), Tensor::annule_domain(), Valeur::base, Valeur::c_cf, Scalar::check_dzpuis(), Tensor::cmp, Valeur::coef(), Tensor::dec_dzpuis(), Scalar::derive_con(), divergence(), Scalar::get_dzpuis(), Scalar::get_etat(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Tensor::get_place_met(), Scalar::get_spectral_va(), Mg3d::get_type_r(), Matrice::inverse(), Tensor::mp, p_div_free, p_potential, Scalar::poisson(), pow(), Map::r, R_CHEB, R_CHEBI, R_CHEBP, Scalar::Scalar(), Mtbl_cf::set(), Tbl::set(), Matrice::set(), Matrice::set_band(), Valeur::set_base_r(), Tensor::set_dependance(), Scalar::set_dzpuis(), Matrice::set_etat_qcq(), Tbl::set_etat_qcq(), Scalar::set_etat_zero(), Matrice::set_lu(), Scalar::set_spectral_va(), Scalar::std_spectral_base(), Tensor::triad, Tensor::type_indice, Valeur::ylm(), and Valeur::ylm_i().

void Vector::del_deriv (  )  const [protected, virtual]

Deletes the derived quantities.

Reimplemented from Tensor.

Reimplemented in Vector_divfree.

Definition at line 215 of file vector.C.

References del_derive_met(), p_A, p_eta, p_mu, and set_der_0x0().

void Vector::del_derive_met ( int  j  )  const [protected, virtual]

Logical destructor of the derivatives depending on the i-th element of met_depend in the class Vector.

Reimplemented from Tensor.

Definition at line 238 of file vector.C.

References Tensor::met_depend, p_div_free, p_potential, and set_der_met_0x0().

const Tensor & Tensor::derive_con ( const Metric &  gam  )  const [inherited]

Returns the "contravariant" derivative of this with respect to some metric $\gamma$, by raising the last index of the covariant derivative (cf.

method derive_cov() ) with $\gamma$.

Reimplemented in Scalar, and Tensor_sym.

Definition at line 1010 of file tensor.C.

References Metric::con(), contract(), Tensor::derive_cov(), Tensor::get_index_type(), Tensor::get_place_met(), Tensor::mp, Tensor::p_derive_con, Itbl::set(), Tensor::set_dependance(), Tensor_sym::sym_index1(), Tensor_sym::sym_index2(), Tensor::Tensor(), Tensor::triad, and Tensor::valence.

const Tensor & Tensor::derive_cov ( const Metric &  gam  )  const [inherited]

Returns the covariant derivative of this with respect to some metric $\gamma$.

$T$ denoting the tensor represented by this and $\nabla T$ its covariant derivative with respect to the metric $\gamma$, the extra index (with respect to the indices of $T$) of $\nabla T$ is chosen to be the last one. This convention agrees with that of MTW (see Eq. (10.17) of MTW). For instance, if $T$ is a 1-form, whose components w.r.t. the triad $e^i$ are $T_i$: $T=T_i \; e^i$, then the covariant derivative of $T$ is the bilinear form $\nabla T$ whose components $\nabla_j T_i$ are such that

\[ \nabla T = \nabla_j T_i \; e^i \otimes e^j \]

Parameters:
gam metric $\gamma$
Returns:
covariant derivative $\nabla T$ of this with respect to the connection $\nabla$ associated with the metric $\gamma$

Reimplemented in Scalar, and Tensor_sym.

Definition at line 998 of file tensor.C.

References Metric::connect(), Tensor::get_place_met(), Connection::p_derive_cov(), Tensor::p_derive_cov, and Tensor::set_dependance().

Tensor Tensor::derive_lie ( const Vector &  v  )  const [inherited]

Computes the Lie derivative of this with respect to some vector field v.

Reimplemented in Scalar, Sym_tensor, and Tensor_sym.

Definition at line 477 of file tensor_calculus.C.

References Tensor::compute_derive_lie(), Tensor::mp, Tensor::triad, Tensor::type_indice, and Tensor::valence.

Vector Vector::derive_lie ( const Vector &  v  )  const

Computes the Lie derivative of this with respect to some vector field v.

Definition at line 388 of file vector.C.

References Tensor::compute_derive_lie(), Tensor::mp, Tensor::triad, and Tensor::type_indice.

const Vector_divfree & Vector::div_free ( const Metric &  metre  )  const

Returns the div-free vector in the Helmholtz decomposition.

It first makes the Helmholtz decomposition (see documentation of p_potential ) of this with respect to a given Metric and then returns $\vec{W}$. Only in the case of contravariant vectors.

Definition at line 500 of file vector.C.

References decompose_div(), Tensor::get_place_met(), p_div_free, and Tensor::set_dependance().

const Scalar & Vector::divergence ( const Metric &  metre  )  const

The divergence of this with respect to a Metric .

The Vector is assumed to be contravariant.

Reimplemented from Tensor.

Definition at line 377 of file vector.C.

Tensor Tensor::down ( int  ind,
const Metric &  gam 
) const [inherited]

Computes a new tensor by lowering an index of *this.

Parameters:
ind index to be lowered, with the following convention :

  • ind1 = 0 : first index of the tensor
  • ind1 = 1 : second index of the tensor
  • and so on... (ind must be of covariant type (CON )).
gam metric used to lower the index (contraction with the twice covariant form of the metric on the index ind ).

Definition at line 261 of file tensor_calculus.C.

References contract(), Metric::cov(), Tensor::indices(), Tensor::mp, Tensor::n_comp, Tensor::set(), Itbl::set(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

const Scalar & Vector::eta (  )  const [virtual]

Gives the field $\eta$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Reimplemented in Vector_divfree.

Definition at line 62 of file vector_etamu.C.

References Tensor::cmp, Scalar::div_tant(), Scalar::dsdt(), p_eta, Scalar::poisson_angu(), Scalar::stdsdp(), and Tensor::triad.

void Vector::exponential_filter_r ( int  lzmin,
int  lzmax,
int  p,
double  alpha = -16. 
) [virtual]

Applies exponential filters to all components (see Scalar::exponential_filter_r ).

Does a loop for Cartesian components, and works in terms of the r-component, $\eta$ and $\mu$ for spherical components.

Reimplemented from Tensor.

Definition at line 846 of file vector.C.

References Tensor::cmp, eta(), Scalar::exponential_filter_r(), Map::get_bvect_cart(), Map::get_bvect_spher(), Base_vect::identify(), Tensor::mp, mu(), Tensor::n_comp, operator()(), set_vr_eta_mu(), and Tensor::triad.

void Vector::exponential_filter_ylm ( int  lzmin,
int  lzmax,
int  p,
double  alpha = -16. 
) [virtual]

Applies exponential filters to all components (see Scalar::exponential_filter_ylm ).

Does a loop for Cartesian components, and works in terms of the r-component, $\eta$ and $\mu$ for spherical components.

Reimplemented from Tensor.

Definition at line 863 of file vector.C.

References Tensor::cmp, eta(), Scalar::exponential_filter_ylm(), Map::get_bvect_cart(), Map::get_bvect_spher(), Base_vect::identify(), Tensor::mp, mu(), Tensor::n_comp, operator()(), set_vr_eta_mu(), and Tensor::triad.

double Vector::flux ( double  radius,
const Metric &  met 
) const

Computes the flux of the vector accross a sphere r = const.

Parameters:
radius radius of the sphere S on which the flux is to be taken; the center of S is assumed to be the center of the mapping (member mp). radius can take the value __infinity (to get the flux at spatial infinity).
met metric $ \gamma $ giving the area element of the sphere
Returns:
$ \oint_S V^i ds_i $, where $ V^i $ is the vector represented by *this and $ ds_i $ is the area element induced on S by $ \gamma $.

Definition at line 803 of file vector.C.

References change_triad(), Map::get_bvect_spher(), Map_af::integrale_surface(), Map_af::integrale_surface_infini(), Tensor::mp, Tensor::triad, Tensor::type_indice, and Vector().

Itbl Tensor::get_index_type (  )  const [inline, inherited]

Returns the types of all the indices.

Returns:
1-D array of integers (class Itbl ) of size valence containing the type of each index, COV for a covariant one and CON for a contravariant one.

Definition at line 892 of file tensor.h.

References Tensor::type_indice.

int Tensor::get_index_type ( int  i  )  const [inline, inherited]

Gives the type (covariant or contravariant) of the index number i .

i must be strictly lower than valence and obey the following convention:

  • i = 0 : first index
  • i = 1 : second index
  • and so on...
Returns:
COV for a covariant index, CON for a contravariant one.

Definition at line 882 of file tensor.h.

References Tensor::type_indice.

const Map& Tensor::get_mp (  )  const [inline, inherited]

Returns the mapping.

Definition at line 857 of file tensor.h.

References Tensor::mp.

int Tensor::get_n_comp (  )  const [inline, inherited]

Returns the number of stored components.

Definition at line 868 of file tensor.h.

References Tensor::n_comp.

int Tensor::get_place_met ( const Metric &  metre  )  const [protected, inherited]

Returns the position of the pointer on metre in the array met_depend .

Definition at line 439 of file tensor.C.

References Tensor::met_depend.

const Base_vect* Tensor::get_triad (  )  const [inline, inherited]

Returns the vectorial basis (triad) on which the components are defined.

Definition at line 862 of file tensor.h.

References Tensor::triad.

int Tensor::get_valence (  )  const [inline, inherited]

Returns the valence.

Definition at line 865 of file tensor.h.

References Tensor::valence.

void Tensor::inc_dzpuis ( int  inc = 1  )  [virtual, inherited]

Increases by inc units the value of dzpuis and changes accordingly the values in the compactified external domain (CED).

Reimplemented in Scalar.

Definition at line 812 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), and Tensor::n_comp.

virtual Itbl Vector::indices ( int  place  )  const [inline, virtual]

Returns the index of a component given by its position in the Scalar array cmp .

Returns:
the index is stored in an 1-D array (Itbl ) of size 1 giving its value for the component located at the position place in the Scalar array cmp . The element of this Itbl corresponds to a spatial index 1, 2 or 3.

Reimplemented from Tensor.

Definition at line 407 of file vector.h.

const Scalar & Vector::mu (  )  const [virtual]

Gives the field $\mu$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Definition at line 94 of file vector_etamu.C.

References Tensor::cmp, Scalar::div_tant(), Scalar::dsdt(), p_mu, Scalar::poisson_angu(), Scalar::stdsdp(), and Tensor::triad.

Sym_tensor Vector::ope_killing ( const Metric &  gam  )  const

Computes the Killing operator associated with a given metric.

The Killing operator is defined by $ D^i V^j + D^j V^i $ for a contravariant vector and by $ D_i V_j + D_j V_i $ for a covariant vector.

Parameters:
gam metric with respect to which the covariant derivative $ D_i $ is defined.

Definition at line 434 of file vector.C.

References Tensor::derive_con(), Tensor::derive_cov(), Tensor::mp, Tensor::set(), Tensor::triad, and Tensor::type_indice.

Sym_tensor Vector::ope_killing_conf ( const Metric &  gam  )  const

Computes the conformal Killing operator associated with a given metric.

The conformal Killing operator is defined by $ D^i V^j + D^j V^i - \frac{2}{3} D_k V^k \, \gamma^{ij} $ for a contravariant vector and by $ D_i V_j + D_j V_i - \frac{2}{3} D^k V_k \, \gamma_{ij}$ for a covariant vector.

Parameters:
gam metric $\gamma_{ij}$ with respect to which the covariant derivative $ D_i $ is defined.

Definition at line 458 of file vector.C.

References Metric::con(), Metric::cov(), Tensor::derive_con(), Tensor::derive_cov(), divergence(), Tensor::mp, Tensor::set(), Tensor::trace(), Tensor::triad, and Tensor::type_indice.

const Scalar & Tensor::operator() ( int  i1,
int  i2,
int  i3,
int  i4 
) const [inherited]

Returns the value of a component for a tensor of valence 4 (read-only version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
i3 value of the third index (1, 2 or 3)
i4 value of the fourth index (1, 2 or 3)
Returns:
reference on the component specified by (i1,i2,i3,i4)

Definition at line 779 of file tensor.C.

References Tensor::cmp, Tensor::position(), Itbl::set(), and Tensor::valence.

const Scalar & Tensor::operator() ( int  i1,
int  i2,
int  i3 
) const [inherited]

Returns the value of a component for a tensor of valence 3 (read-only version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
i3 value of the third index (1, 2 or 3)
Returns:
reference on the component specified by (i1,i2,i3)

Definition at line 767 of file tensor.C.

References Tensor::cmp, Tensor::position(), Itbl::set(), and Tensor::valence.

const Scalar & Tensor::operator() ( int  i1,
int  i2 
) const [inherited]

Returns the value of a component for a tensor of valence 2 (read-only version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
Returns:
reference on the component specified by (i1,i2)

Definition at line 756 of file tensor.C.

References Tensor::cmp, Tensor::position(), Itbl::set(), and Tensor::valence.

const Scalar & Tensor::operator() ( const Itbl &  ind  )  const [inherited]

Returns the value of a component (read-only version).

Parameters:
ind 1-D Itbl of size valence containing the values of each index specifing the component, with the following storage convention:

  • ind(0) : value of the first index (1, 2 or 3)
  • ind(1) : value of the second index (1, 2 or 3)
  • and so on...
Returns:
reference on the component specified by ind

Definition at line 794 of file tensor.C.

References Tensor::cmp, Itbl::get_dim(), Itbl::get_ndim(), Tensor::position(), and Tensor::valence.

const Scalar & Vector::operator() ( int  index  )  const

Readonly access to a component.

Definition at line 301 of file vector.C.

References Tensor::cmp.

void Tensor::operator+= ( const Tensor &  t  )  [inherited]

+= Tensor

Reimplemented in Scalar.

Definition at line 567 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::indices(), Tensor::n_comp, Tensor::position(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

void Tensor::operator-= ( const Tensor &  t  )  [inherited]

-= Tensor

Reimplemented in Scalar.

Definition at line 583 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::indices(), Tensor::n_comp, Tensor::position(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

void Vector::operator= ( const Tensor &  a  )  [virtual]

Assignment from a Tensor.

Reimplemented in Vector_divfree.

Definition at line 275 of file vector.C.

References Tensor::cmp, del_deriv(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

void Vector::operator= ( const Vector &  a  )  [virtual]

Assignment from a Vector.

Reimplemented from Tensor.

Reimplemented in Vector_divfree.

Definition at line 263 of file vector.C.

References Tensor::cmp, del_deriv(), Tensor::triad, and Tensor::type_indice.

Vector Vector::poisson ( const double  lambda,
Param &  par,
int  method = 6 
) const

Solves the vector Poisson equation with *this as a source and parameters controlling the solution.

The equatiopn solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential ), with a flat metric, deduced from the triad.

Parameters:
lambda [input] $\lambda$.
par [input/output] possible parameters
uu [input/output] solution u with the boundary condition u =0 at spatial infinity.

Definition at line 539 of file vector_poisson.C.

References Param::add_cmp_mod(), Param::add_double(), Param::add_int(), Param::add_int_mod(), Cmp::annule_hard(), Tensor::cmp, Tensor::dec_dzpuis(), Scalar::derive_con(), div_free(), Map::get_bvect_cart(), Map::get_bvect_spher(), Param::get_cmp_mod(), Param::get_double(), Param::get_int(), Param::get_int_mod(), Scalar::inc_dzpuis(), Tensor::mp, Vector_divfree::poisson(), Scalar::poisson(), potential(), set(), Tenseur::set(), Tenseur::set_etat_qcq(), Tensor::set_etat_qcq(), Scalar::set_etat_zero(), Tenseur::set_std_base(), and Tensor::triad.

Vector Vector::poisson ( double  lambda,
const Metric_flat &  met_f,
int  method = 6 
) const

Solves the vector Poisson equation with *this as a source.

The equation solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential ), with the flat metric met_f given in argument.

Parameters:
lambda [input] $\lambda$.
met_f [input] the flat metric for the Helmholtz decomposition.
method [input] method used to solve the equation:

  • 0 : It uses the Helmholtz decomposition (see documentation of p_potential ), with the flat metric met_f given in argument (the default).
  • 1 : It solves, first for the divergence (calculated using met_f ), then the r -component, the $\eta$ potential, and fianlly the $\mu$ potential (see documentation of Vector_div_free .
  • 2 : The sources is transformed to cartesian components and the equation is solved using Shibata method (see Granclement et al. JCPH 2001.
  • 6 : Solves for the r -component and $ \eta $ together in a system, and for the $ \mu $ potential (which decouples). The solution is then built from these fields through the method Vector::set_vr_eta_mu(). It is the default method.
Returns:
the solution $N^i$.

Definition at line 126 of file vector_poisson.C.

References Scalar::annule_hard(), Valeur::base, Valeur::c, Valeur::c_cf, Tensor::cmp, Valeur::coef(), Scalar::dec_dzpuis(), Tensor::dec_dzpuis(), Scalar::derive_con(), div_free(), Scalar::div_r_dzpuis(), Scalar::div_sint(), Scalar::dsdr(), Scalar::dsdt(), Map::get_bvect_cart(), Map::get_bvect_spher(), Scalar::get_etat(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Mg3d::get_type_r(), Scalar::lapang(), Tensor::mp, mu(), Scalar::mult_r_dzpuis(), Scalar::mult_sint(), poisson(), Scalar::poisson(), Scalar::poisson_angu(), potential(), Scalar::primr(), set(), Mtbl_cf::set(), Scalar::set_dzpuis(), Tenseur::set_etat_qcq(), Scalar::set_etat_qcq(), Scalar::set_etat_zero(), Tensor::set_etat_zero(), Param_elliptic::set_poisson_vect_eta(), Param_elliptic::set_poisson_vect_r(), Scalar::set_spectral_va(), set_vr_eta_mu(), Scalar::sol_elliptic(), Scalar::stdsdp(), Tensor::triad, Valeur::ylm(), and Valeur::ylm_i().

Vector Vector::poisson ( double  lambda,
int  method = 6 
) const

Solves the vector Poisson equation with *this as a source.

The equation solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential ), with a flat metric, deduced from the triad.

Parameters:
lambda [input] $\lambda$.
method [input] method used to solve the equation (see Vector::poisson(double, Metric_flat, int) for details).
Returns:
the solution $N^i$.

Definition at line 514 of file vector_poisson.C.

References Map::flat_met_cart(), Map::flat_met_spher(), Tensor::mp, and Tensor::triad.

void Vector::poisson_boundary ( double  lambda,
const Mtbl_cf &  limit_vr,
const Mtbl_cf &  limit_eta,
const Mtbl_cf &  limit_mu,
int  num_front,
double  fact_dir,
double  fact_neu,
Vector &  resu 
) const

Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.

The equation solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential )

Parameters:
lambda [input] $\lambda$.
resu [output] the solution $N^i$.

Definition at line 61 of file vector_poisson_boundary.C.

References Tbl::annule_hard(), Mtbl_cf::annule_hard(), Scalar::annule_hard(), Valeur::base, Valeur::c_cf, Tensor::cmp, eta(), Map_af::get_alpha(), Mg3d::get_angu(), Map_af::get_beta(), Scalar::get_etat(), Diff_dsdx::get_matrice(), Diff_xdsdx::get_matrice(), Diff_dsdx2::get_matrice(), Diff_xdsdx2::get_matrice(), Diff_x2dsdx2::get_matrice(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Scalar::get_spectral_va(), Mg3d::get_type_r(), Base_val::give_quant_numbers(), Matrice::inverse(), Tensor::mp, mu(), pow(), Mtbl_cf::set(), Tbl::set(), Matrice::set(), Matrice::set_band(), Valeur::set_etat_cf_qcq(), Scalar::set_etat_qcq(), Tbl::set_etat_qcq(), Matrice::set_etat_qcq(), Matrice::set_lu(), Param_elliptic::set_poisson_vect_r(), Scalar::set_spectral_base(), Scalar::set_spectral_va(), set_vr_eta_mu(), Scalar::sol_elliptic_boundary(), Tbl::t, Tensor::triad, Valeur::ylm(), and Valeur::ylm_i().

void Vector::poisson_boundary2 ( double  lam,
Vector &  resu,
Scalar  boundvr,
Scalar  boundeta,
Scalar  boundmu,
double  dir_vr,
double  neum_vr,
double  dir_eta,
double  neum_eta,
double  dir_mu,
double  neum_mu 
) const

Alternative to previous poisson_boundary method for vectors ; this uses method 6 for vectorial solving, updated version (as in the poisson_vector_block routine).

Boundary arguments are here required as scalar fields.

Definition at line 76 of file vector_poisson_boundary2.C.

References Matrice::annule_hard(), Tbl::annule_hard(), Mtbl_cf::annule_hard(), Scalar::annule_hard(), Valeur::base, Valeur::c_cf, Tensor::cmp, eta(), Map_af::get_alpha(), Map_af::get_beta(), Scalar::get_etat(), Diff_sx2::get_matrice(), Diff_sxdsdx::get_matrice(), Diff_dsdx::get_matrice(), Diff_xdsdx::get_matrice(), Diff_dsdx2::get_matrice(), Diff_xdsdx2::get_matrice(), Diff_x2dsdx2::get_matrice(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nr(), Mg3d::get_nt(), Mg3d::get_nzone(), Scalar::get_spectral_base(), Scalar::get_spectral_va(), Mg3d::get_type_r(), Base_val::give_quant_numbers(), Matrice::inverse(), Tensor::mp, mu(), Mtbl_cf::set(), Tbl::set(), Matrice::set(), Scalar::set_dzpuis(), Valeur::set_etat_cf_qcq(), Scalar::set_etat_qcq(), Tbl::set_etat_qcq(), Matrice::set_etat_qcq(), Matrice::set_lu(), Param_elliptic::set_poisson_vect_r(), Scalar::set_spectral_base(), Scalar::set_spectral_va(), set_vr_eta_mu(), Scalar::sol_elliptic_boundary(), Scalar::std_spectral_base(), Tbl::t, Tensor::triad, Mtbl_cf::val_in_bound_jk(), Mtbl_cf::val_out_bound_jk(), Valeur::ylm(), and Valeur::ylm_i().

Vector Vector::poisson_neumann ( double  lambda,
const Valeur &  limit_vr,
const Valeur &  limit_vt,
const Valeur &  limit_vp,
int  num_front 
) const

Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.

The equation solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential )

Parameters:
lambda [input] $\lambda$.
resu [output] the solution $N^i$.

Definition at line 687 of file vector_poisson_boundary.C.

References Tensor::mp, poisson_robin(), and Tensor::triad.

Vector Vector::poisson_robin ( double  lambda,
const Valeur &  limit_vr,
const Valeur &  limit_vt,
const Valeur &  limit_vp,
double  fact_dir,
double  fact_neu,
int  num_front 
) const

Solves the vector Poisson equation with *this as a source with a boundary condition on the excised sphere.

The equation solved is $\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$. *this must be given with dzpuis = 4. It uses the Helmholtz decomposition (see documentation of p_potential )

Parameters:
lambda [input] $\lambda$.
resu [output] the solution $N^i$.

Definition at line 698 of file vector_poisson_boundary.C.

References Scalar::annule_hard(), Valeur::base, Valeur::c_cf, Tensor::cmp, Valeur::coef(), Scalar::div_tant(), Scalar::dsdt(), Valeur::get_etat(), Map::get_mg(), Mg3d::get_np(), Mg3d::get_nt(), Mg3d::get_nzone(), Tensor::mp, norme(), Scalar::poisson_angu(), poisson_boundary(), Tensor::set(), Valeur::set_base(), Tensor::set_etat_zero(), Scalar::set_grid_point(), Scalar::set_spectral_va(), Scalar::stdsdp(), Tensor::triad, Scalar::val_grid_point(), and Valeur::ylm().

virtual int Vector::position ( const Itbl &  idx  )  const [inline, virtual]

Returns the position in the Scalar array cmp of a component given by its index.

Returns:
position in the Scalar array cmp corresponding to the index given in idx . idx must be a 1-D Itbl of size 1, the element of which must be one of the spatial indices 1, 2 or 3.

Reimplemented from Tensor.

Definition at line 388 of file vector.h.

References Itbl::get_dim(), and Itbl::get_ndim().

const Scalar & Vector::potential ( const Metric &  metre  )  const

Returns the potential in the Helmholtz decomposition.

It first makes the Helmholtz decomposition (see documentation of p_potential ) of this with respect to a given Metric and then returns $\phi$. Only in the case of contravariant vectors.

Definition at line 488 of file vector.C.

References decompose_div(), Tensor::get_place_met(), p_potential, and Tensor::set_dependance().

void Vector::pseudo_spectral_base (  )  [virtual]

Sets the standard spectal bases of decomposition for each component for a pseudo_vector.

Definition at line 342 of file vector.C.

References Tensor::cmp, Map::get_bvect_cart(), Map::get_bvect_spher(), Map::get_mg(), Base_vect::identify(), Tensor::mp, Mg3d::pseudo_base_vect_cart(), Mg3d::pseudo_base_vect_spher(), Scalar::set_spectral_base(), and Tensor::triad.

void Tensor::sauve ( FILE *  fd  )  const [virtual, inherited]

Save in a binary file.

Reimplemented in Scalar, and Tensor_sym.

Definition at line 902 of file tensor.C.

References Tensor::cmp, fwrite_be(), Tensor::n_comp, Base_vect::sauve(), Itbl::sauve(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

Scalar & Tensor::set ( int  i1,
int  i2,
int  i3,
int  i4 
) [inherited]

Returns the value of a component for a tensor of valence 4 (read/write version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
i3 value of the third index (1, 2 or 3)
i4 value of the fourth index (1, 2 or 3)
Returns:
modifiable reference on the component specified by (i1,i2,i3,i4)

Definition at line 633 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::position(), Itbl::set(), and Tensor::valence.

Scalar & Tensor::set ( int  i1,
int  i2,
int  i3 
) [inherited]

Returns the value of a component for a tensor of valence 3 (read/write version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
i3 value of the third index (1, 2 or 3)
Returns:
modifiable reference on the component specified by (i1,i2,i3)

Definition at line 617 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::position(), Itbl::set(), and Tensor::valence.

Scalar & Tensor::set ( int  i1,
int  i2 
) [inherited]

Returns the value of a component for a tensor of valence 2 (read/write version).

Parameters:
i1 value of the first index (1, 2 or 3)
i2 value of the second index (1, 2 or 3)
Returns:
modifiable reference on the component specified by (i1,i2)

Definition at line 602 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::position(), Itbl::set(), and Tensor::valence.

Scalar & Tensor::set ( const Itbl &  ind  )  [inherited]

Returns the value of a component (read/write version).

Parameters:
ind 1-D Itbl of size valence containing the values of each index specifing the component, with the following storage convention:

  • ind(0) : value of the first index (1, 2 or 3)
  • ind(1) : value of the second index (1, 2 or 3)
  • and so on...
Returns:
modifiable reference on the component specified by ind

Definition at line 650 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Itbl::get_dim(), Itbl::get_ndim(), Tensor::position(), and Tensor::valence.

Scalar & Vector::set ( int  index  ) 

Read/write access to a component.

Definition at line 292 of file vector.C.

References Tensor::cmp, and del_deriv().

void Tensor::set_dependance ( const Metric &  met  )  const [protected, inherited]

To be used to describe the fact that the derivatives members have been calculated with met .

First it sets a null element of met_depend to &met and puts this in the list of the dependancies of met .

Definition at line 449 of file tensor.C.

References Tensor::met_depend, and Metric::tensor_depend.

void Vector::set_der_0x0 (  )  const [protected]

Sets the pointers on derived quantities to 0x0.

Reimplemented from Tensor.

Reimplemented in Vector_divfree.

Definition at line 228 of file vector.C.

References p_A, p_eta, p_mu, and set_der_met_0x0().

void Vector::set_der_met_0x0 ( int  i  )  const [protected]

Sets all the i-th components of met_depend in the class Vector (p_potential , etc.

..) to 0x0.

Reimplemented from Tensor.

Definition at line 254 of file vector.C.

References p_div_free, and p_potential.

void Tensor::set_etat_nondef (  )  [virtual, inherited]

Sets the logical state of all components to ETATNONDEF (undefined state).

Reimplemented in Scalar.

Definition at line 485 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::n_comp, and Scalar::set_etat_nondef().

void Tensor::set_etat_qcq (  )  [virtual, inherited]

Sets the logical state of all components to ETATQCQ (ordinary state).

Reimplemented in Scalar.

Definition at line 477 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::n_comp, and Scalar::set_etat_qcq().

void Tensor::set_etat_zero (  )  [virtual, inherited]

Sets the logical state of all components to ETATZERO (zero state).

Reimplemented in Scalar.

Definition at line 493 of file tensor.C.

References Tensor::cmp, Tensor::del_deriv(), Tensor::n_comp, and Scalar::set_etat_zero().

Itbl& Tensor::set_index_type (  )  [inline, inherited]

Sets the types of all the indices.

Returns:
a reference on the 1-D array of integers (class Itbl ) of size valence that can be modified (COV for a covariant one and CON for a contravariant one)

Definition at line 914 of file tensor.h.

References Tensor::type_indice.

int& Tensor::set_index_type ( int  i  )  [inline, inherited]

Sets the type of the index number i .

i must be strictly lower than valence and obey the following convention:

  • i = 0 : first index
  • i = 1 : second index
  • and so on...
Returns:
reference on the type that can be modified (COV for a covariant index, CON for a contravariant one)

Definition at line 905 of file tensor.h.

References Itbl::set(), and Tensor::type_indice.

void Tensor::set_triad ( const Base_vect &  new_triad  )  [inherited]

Assigns a new vectorial basis (triad) of decomposition.

NB: this function modifies only the member triad and leave unchanged the components (member cmp ). In order to change them coherently with the new basis, the function change_triad(const Base_vect& ) must be called instead.

Definition at line 515 of file tensor.C.

References Tensor::triad.

void Vector::set_vr_eta_mu ( const Scalar &  vr_i,
const Scalar &  eta_i,
const Scalar &  mu_i 
)

Defines the components through potentials $\eta$ and $\mu$ (see members p_eta and p_mu ), as well as the $V^r$ component of the vector.

Parameters:
vr_i [input] component $V^r$ of the vector
eta_i [input] angular potential $\eta$
mu_i [input] angular potential $\mu$

Reimplemented in Vector_divfree.

Definition at line 185 of file vector_etamu.C.

References Tensor::cmp, del_deriv(), Tensor::get_mp(), p_eta, p_mu, Tensor::triad, and update_vtvp().

void Tensor::spectral_display ( const char *  comment = 0x0,
double  threshold = 1.e-7,
int  precision = 4,
ostream &  ostr = cout 
) const [virtual, inherited]

Displays the spectral coefficients and the associated basis functions of each component.

This function shows only the values greater than a given threshold.

Parameters:
comment comment to be printed at top of the display (default: 0x0 = nothing printed)
threshold [input] Value above which a coefficient is printed (default: 1.e-7)
precision [input] Number of printed digits (default: 4)
ostr [input] Output stream used for the printing (default: cout)

Reimplemented in Scalar.

Definition at line 870 of file tensor.C.

References Tensor::cmp, Tensor::indices(), Tensor::n_comp, Scalar::spectral_display(), and Tensor::valence.

void Vector::std_spectral_base (  )  [virtual]

Sets the standard spectal bases of decomposition for each component.

Reimplemented from Tensor.

Definition at line 312 of file vector.C.

References Tensor::cmp, Map::get_bvect_cart(), Map::get_bvect_spher(), Map::get_mg(), Base_vect::identify(), Tensor::mp, Scalar::set_spectral_base(), Mg3d::std_base_vect_cart(), Mg3d::std_base_vect_spher(), and Tensor::triad.

void Tensor::std_spectral_base_odd (  )  [virtual, inherited]

Sets the standard odd spectal bases of decomposition for each component.

Currently only implemented for a scalar.

Reimplemented in Scalar.

Definition at line 978 of file tensor.C.

References Tensor::cmp, Scalar::std_spectral_base_odd(), and Tensor::valence.

Scalar Tensor::trace ( const Metric &  gam  )  const [inherited]

Trace with respect to a given metric for a valence 2 tensor.

Parameters:
gam metric used to raise or lower one of the indices, in order to take the trace

Definition at line 193 of file tensor_calculus.C.

References Metric::con(), contract(), Metric::cov(), Tensor::trace(), Tensor::type_indice, and Tensor::valence.

Scalar Tensor::trace (  )  const [inherited]

Trace on two different type indices for a valence 2 tensor.

Definition at line 176 of file tensor_calculus.C.

References Tensor::mp, Tensor::operator()(), Scalar::set_etat_zero(), Tensor::type_indice, and Tensor::valence.

Tensor Tensor::trace ( int  ind1,
int  ind2,
const Metric &  gam 
) const [inherited]

Trace with respect to a given metric.

Parameters:
ind1 first index for the contraction, with the following convention :

  • ind1 = 0 : first index of the tensor
  • ind1 = 1 : second index of the tensor
  • and so on...
ind2 second index for the contraction
gam metric used to raise or lower ind1 in order that it has a opposite type than ind2

Definition at line 149 of file tensor_calculus.C.

References Metric::con(), contract(), Metric::cov(), Tensor::trace(), Tensor::type_indice, and Tensor::valence.

Tensor Tensor::trace ( int  ind1,
int  ind2 
) const [inherited]

Trace on two different type indices.

Parameters:
ind1 first index for the contraction, with the following convention :

  • ind1 = 0 : first index of the tensor
  • ind1 = 1 : second index of the tensor
  • and so on...
ind2 second index for the contraction

Definition at line 90 of file tensor_calculus.C.

References Tensor::cmp, Tensor::get_n_comp(), Tensor::indices(), Tensor::mp, Tensor::position(), Tensor::set(), Itbl::set(), Scalar::set_etat_zero(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

Tensor Tensor::up ( int  ind,
const Metric &  gam 
) const [inherited]

Computes a new tensor by raising an index of *this.

Parameters:
ind index to be raised, with the following convention :

  • ind1 = 0 : first index of the tensor
  • ind1 = 1 : second index of the tensor
  • and so on... (ind must be of covariant type (COV )).
gam metric used to raise the index (contraction with the twice contravariant form of the metric on the index ind ).

Definition at line 221 of file tensor_calculus.C.

References Metric::con(), contract(), Tensor::indices(), Tensor::mp, Tensor::n_comp, Tensor::set(), Itbl::set(), Tensor::triad, Tensor::type_indice, and Tensor::valence.

Tensor Tensor::up_down ( const Metric &  gam  )  const [inherited]

Computes a new tensor by raising or lowering all the indices of *this .

Parameters:
gam metric used to lower the contravariant indices and raising the covariant ones.

Definition at line 301 of file tensor_calculus.C.

References Tensor::down(), Tensor::Tensor(), Tensor::type_indice, Tensor::up(), and Tensor::valence.

void Vector::update_vtvp (  ) 

Computes the components $V^\theta$ and $V^\varphi$ from the potential $\eta$ and $\mu$, according to:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Definition at line 163 of file vector_etamu.C.

References Tensor::cmp, del_deriv(), Scalar::dsdt(), p_eta, p_mu, and Scalar::stdsdp().

void Vector::visu_arrows ( double  xmin,
double  xmax,
double  ymin,
double  ymax,
double  zmin,
double  zmax,
const char *  title0 = 0x0,
const char *  filename0 = 0x0,
bool  start_dx = true,
int  nx = 8,
int  ny = 8,
int  nz = 8 
) const

3D visualization via OpenDX.

Parameters:
xmin [input] defines with xmax the x range of the visualization box
xmax [input] defines with xmin the x range of the visualization box
ymin [input] defines with ymax the y range of the visualization box
ymax [input] defines with ymin the y range of the visualization box
zmin [input] defines with zmax the z range of the visualization box
zmax [input] defines with zmin the z range of the visualization box
title [input] title for the graph (for OpenDX legend)
filename [input] name for the file which will be the input for OpenDX; the default 0x0 is transformed into "vector_arrows"
start_dx [input] determines whether OpenDX must be launched (as a subprocess) to view the field; if set to false , only input files for future usage of OpenDX are created
nx [input] number of points in the x direction (uniform sampling)
ny [input] number of points in the y direction (uniform sampling)
nz [input] number of points in the z direction (uniform sampling)

Definition at line 58 of file vector_visu.C.

References Valeur::c_cf, change_triad(), Scalar::check_dzpuis(), Valeur::coef(), Map::convert_absolute(), Scalar::dec_dzpuis(), Map::get_bvect_cart(), Map::get_bvect_spher(), Scalar::get_dzpuis(), Tensor::mp, operator()(), set(), Tensor::triad, Map::val_lx(), Mtbl_cf::val_point(), and Vector().

Friends And Related Function Documentation

Tensor_sym operator* ( const Tensor_sym &  a,
const Tensor_sym &  b 
) [friend, inherited]

Tensorial product of two symmetric tensors.

NB: the output is an object of class Tensor_sym , with the two symmetric indices corresponding to the symmetric indices of tensor a . This means that the symmetries of tensor b indices are not used in the storage, since there is currently no class in Lorene to manage tensors with more than two symmetric indices.

Definition at line 147 of file tensor_sym_calculus.C.

Member Data Documentation

Scalar** Tensor::cmp [protected, inherited]

Array of size n_comp of pointers onto the components.

Definition at line 311 of file tensor.h.

const Metric* Tensor::met_depend[N_MET_MAX] [mutable, protected, inherited]

Array on the Metric 's which were used to compute derived quantities, like p_derive_cov , etc.

.. The i-th element of this array is the Metric used to compute the i-th element of p_derive_cov , etc..

Definition at line 323 of file tensor.h.

const Map* const Tensor::mp [protected, inherited]

Mapping on which the numerical values at the grid points are defined.

Definition at line 291 of file tensor.h.

int Tensor::n_comp [protected, inherited]

Number of stored components, depending on the symmetry.

Definition at line 308 of file tensor.h.

Scalar* Vector::p_A [mutable, protected]

Field $A$ defined by

\[ A = {\partial \eta \over \ partial r} + { \eta \over r} - {V^r \over r} \]

Insensitive to the longitudinal part of the vector, related to the curl.

Definition at line 237 of file vector.h.

Tensor* Tensor::p_derive_con[N_MET_MAX] [mutable, protected, inherited]

Array of pointers on the contravariant derivatives of this with respect to various metrics.

See the comments of met_depend . See also the comments of method derive_con() for a precise definition of a "contravariant" derivative.

Definition at line 339 of file tensor.h.

Tensor* Tensor::p_derive_cov[N_MET_MAX] [mutable, protected, inherited]

Array of pointers on the covariant derivatives of this with respect to various metrics.

See the comments of met_depend . See also the comments of method derive_cov() for the index convention of the covariant derivation.

Definition at line 331 of file tensor.h.

Vector_divfree* Vector::p_div_free[N_MET_MAX] [mutable, protected]

The divergence-free vector $\vec{W} = \vec{\nabla} \wedge \vec{\psi}$ of the Helmholtz decomposition of any 3D vector $\vec{V}: \quad \vec{V} = \vec{\nabla} \phi + \vec{\nabla} *\wedge \vec{\psi}$.

Only in the case of contravariant vectors.

Definition at line 201 of file vector.h.

Tensor* Tensor::p_divergence[N_MET_MAX] [mutable, protected, inherited]

Array of pointers on the divergence of this with respect to various metrics.

See the comments of met_depend . See also the comments of method divergence() for a precise definition of a the divergence with respect to a given metric.

Definition at line 347 of file tensor.h.

Scalar* Vector::p_eta [mutable, protected]

Field $\eta$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Definition at line 215 of file vector.h.

Scalar* Vector::p_mu [mutable, protected]

Field $\mu$ such that the angular components $(V^\theta, V^\varphi)$ of the vector are written:

\[ V^\theta = {\partial \eta \over \partial\theta} - {1\over\sin\theta} {\partial \mu \over \partial\varphi} \]

\[ V^\varphi = {1\over\sin\theta} {\partial \eta \over \partial\varphi} + {\partial \mu \over \partial\theta} \]

.

Definition at line 229 of file vector.h.

Scalar* Vector::p_potential[N_MET_MAX] [mutable, protected]

The potential $\phi$ giving the gradient part in the Helmholtz decomposition of any 3D vector $\vec{V}: \quad \vec{V} = \vec{\nabla} \phi + \vec{\nabla} \wedge \vec{\psi}$.

Only in the case of contravariant vectors.

Definition at line 194 of file vector.h.

const Base_vect* Tensor::triad [protected, inherited]

Vectorial basis (triad) with respect to which the tensor components are defined.

Definition at line 299 of file tensor.h.

Itbl Tensor::type_indice [protected, inherited]

1D array of integers (class Itbl ) of size valence containing the type of each index: COV for a covariant one and CON for a contravariant one.

Definition at line 306 of file tensor.h.

int Tensor::valence [protected, inherited]

Valence of the tensor (0 = scalar, 1 = vector, etc...).

Definition at line 294 of file tensor.h.

The documentation for this class was generated from the following files:
Generated on 7 Oct 2014 for LORENE by  doxygen 1.6.1