path: root/doc/IGE335/Section3.07.tex

\subsection{The {\tt FLU:} module}\label{sect:FLUData}

The \moc{FLU:} module is used to solve the linear system of multigroup collision
probability or response matrix equations in DRAGON. Different types of solution are
available, such as fixed source problem, fixed source eigenvalue problem (GPT type) or
different types of eigenvalue problems. The calling specifications are:

\begin{DataStructure}{Structure \dstr{FLU:}}
\dusa{FLUNAM} \moc{:=} \moc{FLU:} $[~\{$ \dusa{FLUNAM} $|$ \dusa{FLUDSA} $\}~]$ \dusa{PIJNAM} 
\dusa{LIBNAM}  \dusa{TRKNAM} $[$ \dusa{TRKFIL} $]$ \\
$~~~~[~\{$ \dusa{TRKFLP} \dusa{TRKGPT} $|$ \dusa{SOUNAM} $\}~]$ \moc{::} \dstr{descflu}
\end{DataStructure}

\noindent where
\begin{ListeDeDescription}{mmmmmmmm}

\item[\dusa{FLUNAM}] {\tt character*12} name of the \dds{fluxunk} data structure
containing the solution ({\tt L\_FLUX} signature). If \dusa{FLUNAM} appears on
the RHS, the solution previously stored in \dusa{FLUNAM} (flux and buckling) is used to initialize
the new iterative process; otherwise, a uniform unknown vector and a zero buckling
are used.

\item[\dusa{FLUDSA}] {\tt character*12} name of the \dds{fluxunk} data structure
containing an initial approximation of the solution ({\tt L\_FLUX} signature). This solution
corresponds to a DSA-type simplified
calculation compatible with \dusa{FLUNAM}. This option is only available with a \moc{SNT:} tracking.

\item[\dusa{PIJNAM}] {\tt character*12} name of the \dds{asmpij} data
structure containing the group-dependent system
matrices ({\tt L\_PIJ} signature, see \Sect{ASMData}).

\item[\dusa{LIBNAM}] {\tt character*12} name of the \dds{macrolib} or \dds{microlib} data structure that contains the
macroscopic cross sections ({\tt L\_MACROLIB} or {\tt L\_LIBRARY} signature, see \Sectand{MACData}{LIBData}).
Module {\tt FLU:} is performing a {\sl direct} or {\sl adjoint} calculation, depending if the adjoint flag
is set to {\tt .false.} or {\tt .true.} in the {\tt STATE-VECTOR} record of the \dds{macrolib}.

\item[\dusa{TRKNAM}] {\tt character*12} name of the \dds{tracking} data
structure containing the tracking ({\tt L\_TRACK} signature, see \Sect{TRKData}).

\item[\dusa{TRKFIL}] {\tt character*12} name of the sequential binary tracking
file used to store the tracks lengths. This file is given if and only if it was
required in the previous tracking module call (see \Sect{TRKData}).

\item[\dusa{TRKFLP}] {\tt character*12} name of the \dds{fluxunk} data structure containing the
unperturbed flux used to decontaminate the GPT solution ({\tt L\_FLUX} signature). This object is
mandatory if and only if ``{\tt TYPE P}" is selected.

\item[\dusa{TRKGPT}] {\tt character*12} name of the \dds{source} data structure
containing the GPT fixed sources ({\tt L\_SOURCE} signature). This object is
mandatory if and only if ``{\tt TYPE P}" is selected.

\item[\dusa{SOUNAM}] {\tt character*12} name of the \dds{source} data structure
containing the fixed sources ({\tt L\_SOURCE} signature) used for a ``{\tt TYPE S}" calculation.
By default, piecewise-constant fixed sources available in the \dds{macrolib} (or \dds{microlib}) \dusa{LIBNAM}
are used.

\item[\dstr{descflu}] structure containing the input data to this module (see
\Sect{descflu}).

\end{ListeDeDescription}

\clearpage

\subsubsection{Data input for module {\tt FLU:}}\label{sect:descflu}

\begin{DataStructure}{Structure \dstr{descflu}}
$[$ \moc{EDIT} \dusa{iprint} $]$ \\
$[$ \moc{INIT} $\{$ \moc{OFF} $|$ \moc{ON} $|$ \moc{DSA} $\}~]$ \\
\moc{TYPE} $\{$ \moc{N} $|$ \moc{S} $|$ \moc{F} $|$ \moc{P} $|$ \moc{K} $[$ \dstr{descleak} $]$ $|$
$\{$\moc{B} $|$ \moc{L} $\}$ \dstr{descleak} $\}$ $]$  \\
$[$ \moc{EXTE} $[$ \dusa{maxout} $]~~[$ \dusa{epsout} $]~]$ \\
$[$ \moc{THER} $[$ \dusa{maxthr} $]~~[$ \dusa{epsthr} $]~]~~[$ \moc{REBA} $[$ \moc{OFF} $]~]$ \\
$[$ \moc{UNKT} $[$ \dusa{epsunk} $]~]$ \\
$[$ \moc{ACCE} \dusa{nlibre} \dusa{naccel}  $]$ \\
{\tt ;}
\end{DataStructure}

\goodbreak
\noindent where
\begin{ListeDeDescription}{mmmmmmm}

\item[\moc{EDIT}] keyword used to modify the print level \dusa{iprint}.

\item[\dusa{iprint}] index used to control the printing of this operator. The
amount of output produced by this operator will vary substantially
depending on the print level specified. 

\item[\moc{OFF}] keyword to specify that the neutron flux
is to be initialized with a flat distribution (default option).

\item[\moc{ON}] keyword to specify that the initial neutron flux distribution
is to be recovered from \dusa{FLUNAM} if present in the RHS arguments.

\item[\moc{DSA}] keyword to specify that the initial neutron flux distribution
is to be recovered from the DSA compatible data structure \dusa{FLUDSA} if present in the RHS arguments.
This option is only available with a \moc{SNT:} tracking.

\item[\moc{TYPE}] keyword to specify the type of solution used in the flux
operator.

\item[\moc{N}] keyword to specify that no flux calculation is to be performed.
This option is usually activated when one simply wishes to initialize the
neutron flux distribution and to store this information in \dusa{FLUNAM}.

\item[\moc{S}]  keyword to specify that a fixed source problem is to be
treated. Such problem can also include fission source contributions.

\item[\moc{F}] keyword to specify that a 1D Fourier analysis calculation in $S_n$ is to be treated. This is similar to a fixed source problem, but the calculation stopped early to compute an L2 error norm in the flux. This yields a numerical estimate of the eigenvalue for the scattering source equation.

\item[\moc{P}]  keyword to specify that a fixed source eigenvalue problem (GPT type) is to be
treated. Such problem includes fission source contributions in addition of GPT sources.

\item[\moc{K}] keyword to specify that a fission source eigenvalue problem is
to be treated. The eigenvalue is then the effective multiplication factor $K_{\rm eff}$ with a
fixed buckling $B^2$. In this case, the fixed sources, if any is present on the
\dds{macrolib} or \dds{microlib} data structure, are not used.  

\item[\moc{B}] keyword to specify that a fission source eigenvalue problem is
to be treated. The eigenvalue in this case is the critical buckling $B^2$ with a fixed
effective multiplication factor $K_{\rm eff}$. The buckling eigenvalue has meaning only in the
case of a cell without boundary leakages (see the structure \dstr{descBC} in
\Sect{descBC}). It is also possible to use an open geometry with
\moc{VOID} boundary conditions  provided it is closed by the \moc{ASM:} module
(see \Sect{descasm}) using the keywords \moc{NORM} or \moc{ALSB}. {\sl Note:} \moc{TYPE~B}
cannot be used if no fission occurs in the system.

\item[\moc{L}] keyword to specify that a critical medium eigenvalue problem, with or without
fission sources, is to be treated. The eigenvalue in this case is the critical buckling $B^2$,
with or without a fixed effective multiplication factor $K_{\rm eff}$. The buckling eigenvalue has meaning only
in the case of a cell without boundary leakages (see the structure \dstr{descBC} in
\Sect{descBC}). It is also possible to use an open geometry with
\moc{VOID} boundary conditions  provided it is closed by the \moc{ASM:} module
(see \Sect{descasm}) using the keywords \moc{NORM} or \moc{ALSB}. {\sl Note:} \moc{TYPE~L}
cannot be used if no positive or negative $dB^2$ leakage occurs in the system.

\item[\dstr{descleak}] structure describing the general leakage parameters
options (see \Sect{descleak}). This information is mandatory for producing the
diffusion coefficients.

\item[\moc{EXTE}] keyword to specify that the control parameters for the
external iteration are to be modified. 

\item[\dusa{maxout}] maximum number of external iterations. The fixed default
value for a case with no leakage model is \dusa{maxout}=$2\times n_{f}-1$ where
$n_{f}$ is the number of regions containing fuel. The fixed default value for a
case with a leakage model is \dusa{maxout}=$10\times n_{f}-1$.

\item[\dusa{epsout}] convergence criterion for the external iterations. The
fixed default value is \dusa{epsout}=$5.0\times 10^{-5}$.

\item[\moc{THER}] keyword to specify that the control parameters for the
thermal iterations are to be modified.

\item[\dusa{maxthr}] maximum number of thermal iterations. The fixed default
value is \dusa{maxthr}=2$\times$\dusa{ngroup}-1 (using scattering modified CP)
or \dusa{maxthr}=4$\times$\dusa{ngroup}-1 (using standard CP).

\item[\dusa{epsthr}] convergence criterion for the thermal iterations. The
fixed default value is \dusa{epsthr}=$5.0\times 10^{-5}$.

\item[\moc{UNKT}] keyword to specify the flux error tolerance in
the outer iteration.

\item[\dusa{epsunk}] convergence criterion for flux components in the outer
iteration. The fixed default value is \dusa{epsunk}=\dusa{epsthr}.

\item[\moc{REBA}] keyword used to specify that the flux rebalancing option is
to be turned on or off in the thermal iteration. By default (floating default)
the flux rebalancing option is initially activated. This keyword is required to
toggle between the on and off position of the flux rebalancing option. 

\item[\moc{OFF}] keyword used to deactivate the flux rebalancing option. When
this keyword is absent the flux rebalancing option is reactivated.

\item[\moc{ACCE}] keyword used to modify the variational acceleration
parameters. This option is active by default (floating default) with
\dusa{nlibre}=3 free iterations followed by \dusa{naccel}=3 accelerated
iterations. 

\item[\dusa{nlibre}] number of free iterations per cycle of
\dusa{nlibre}+\dusa{naccel} iterations. 

\item[\dusa{naccel}] number of accelerated iterations per cycle of
\dusa{nlibre}+\dusa{naccel} iterations. Variational acceleration may be
deactivated by using \dusa{naccel}=0.

\end{ListeDeDescription}
\clearpage

\subsubsection{Leakage model specification structure}\label{sect:descleak}

Without leakage model, the multigroup flux $\vec\phi_g$ of the collision
probability method is obtained from equation

\begin{equation}
\vec\phi_g={\bf W}_g \vec Q^\diamond_g
\label{eq:eq3.64}
\end{equation}

\noindent where ${\bf W}_g$ is the scattering reduced collision probability matrix
and $ Q^\diamond_g$ is the fission and out-of-group scattering source. This equation is
modified by the leakage model. The leakage models \moc{PNLR}, \moc{PNL}, \moc{SIGS}
(default model), \moc{HETE} and \moc{ECCO} can be used with any solutions technique of
the Boltzmann transport equation. The leakage model \moc{TIBERE} can be used with the collision
probability method and with the method of characteristics.

\vskip 0.2cm

A leakage model can be set in {\sl fundamental mode condition} if all boundary conditions are
conservative (such as \moc{REFL}, \moc{SYME}, \moc{SSYM}, \moc{DIAG}, \moc{ALBE 1.0}). If a boundary condition is
non-conservative (such as \moc{VOID}), it is nevertheless possible to set a simplified leakage model based on the
Todorova approximation with option \moc{HETE}. The \dstr{descleak} structure allows the following
information to be specified:

\begin{DataStructure}{Leakage structure \dstr{descleak}}
$\{$ \moc{LKRD} $|$ \moc{RHS} $|$ \moc{P0} $|$ \moc{P1} $|$ \moc{P0TR} $|$ \moc{B0} $|$ \moc{B1} $|$ \moc{B0TR} $\}$ \\
$[~\{$ \moc{PNLR} $|$ \moc{PNL} $|$ \moc{SIGS} $|$ \moc{ALBS} $|$ \moc{HETE} $[$ (\dusa{imergl}(ii),ii=1,nbmix) $]~|$ \moc{ECCO} $|$ \moc{TIBERE}
$[$ $\{$ \moc{G} $|$ \moc{R} $|$ \moc{Z} $|$ \moc{X} $|$ \moc{Y} $\}~]~\}~]$ \\
$[$ $\{$ \moc{BUCK} $\{$ \dusa{valb2} $|$ $[$ \moc{G} \dusa{valb2} $]$  
$[$ \moc{R} \dusa{valbr2} $]$ $[$ \moc{Z} \dusa{valbz2} $]$ 
$[$ \moc{X} \dusa{valbx2} $]$
$[$ \moc{Y} \dusa{valby2} $]$ $\}$
$|$ \moc{KEFF} \dusa{valk} $|$ \moc{IDEM} $\}$
$]$  \end{DataStructure}

\begin{ListeDeDescription}{mmmmmmm}

\item[\moc{LKRD}] keyword used to specify that the leakage coefficients are
recovered from data structure named \dusa{LIBNAM}. The \moc{LKRD} option is not
available with the \moc{ECCO} and \moc{TIBERE} leakage models.

\item[\moc{RHS}] keyword used to specify that the leakage coefficients are
recovered from RHS flux data structure named \dusa{FLUNAM}. The \moc{RHS} option is not
available with the \moc{ECCO} and \moc{TIBERE} leakage models. If the flux calculation is
an adjoint calculation, the energy group ordering of the leakage coefficients is permuted.

\item[\moc{P0}] keyword used to specify that the leakage coefficients are
calculated using a $P_0$ model.

\item[\moc{P1}] keyword used to specify that the leakage coefficients are
calculated using a $P_1$ model. 

\item[\moc{P0TR}] keyword used to specify that the leakage coefficients are
calculated using a $P_0$ model with transport correction.

\item[\moc{B0}] keyword used to specify that the leakage coefficients are
calculated using a $B_0$ model. This is the default value when a buckling
calculation is required (\moc{B}).

\item[\moc{B1}] keyword used to specify that the leakage coefficients are
calculated using a $B_1$ model.

\item[\moc{B0TR}] keyword used to specify that the leakage coefficients are
calculated using a $B_0$ model with transport correction.

\item[\moc{PNLR}] keyword used to specify that the elements of the scattering
modified collision probability matrix
are multiplied by the adequate non-leakage homogeneous buckling dependent
factor.\cite{ALSB1}. The non-leakage
factor $P_{{\rm NLR},g}$ is defined as

\begin{equation}
P_{{\rm NLR},g}={\bar\Sigma_g-\bar\Sigma_{{\rm s0},g \gets g}\over{\bar\Sigma_g-\bar\Sigma_{{\rm s0},g \gets g}+d_g(B) \ B^2}}
\end{equation}

\noindent where transport-corrected total
cross sections are used to compute the ${\bf W}_g$ matrix. $\bar\Sigma_{{\rm s0},g \gets g}$ is the average
transport-corrected macroscopic within-group scattering cross section in group $g$,
homogenized over the lattice and transport corrected. \eq(eq3.64) is then replaced by
 
\begin{equation}
\vec\phi_g=P_{{\rm NLR},g} {\bf W}_g \vec Q^\diamond_g \ \ \ .
\label{eq:eq5.32}
\end{equation}

\item[\moc{PNL}] keyword used to specify that the elements of the collision
probability matrix are multiplied by the adequate non-leakage homogeneous buckling
dependent factor.\cite{ALSB1}. The non-leakage factor $P_{{\rm NL},g}$ is defined as

\begin{equation}
P_{{\rm NL},g}={\bar\Sigma_g\over{\bar\Sigma_g+d_g(B) \ B^2}}
\end{equation}

\noindent where $\bar\Sigma_g$ is the average transport-corrected macroscopic total cross section
in group $g$, homogenized over the lattice and transport corrected. \eq(eq3.64) is then replaced by

\begin{equation}
\vec\phi_g={\bf W}_g \left[ P_{{\rm NL},g} \vec Q^\diamond_g -(1-P_{{\rm NL},g}) {\bf \Sigma}_{{\rm s0},g\gets g} \ \vec\phi_g \right]
\label{eq:eq5.33b}
\end{equation}

\noindent where ${\bf \Sigma}_{{\rm s0},g\gets g}={\rm diag} \{ \Sigma_{{\rm s0},i,g \gets g}\> ;\> \forall i \}$
and the total cross sections used to compute the ${\bf W}_g$ matrix are also
transport-corrected.

\vskip 0.02cm

\noindent It is important to note that that the \moc{PNLR} option reduces to the \moc{PNL} option in
cases where no scattering reduction is performed. Scattering reduction can be avoided in module
\moc{ASM:} by setting {\tt PIJ SKIP} (See \Sect{descasm}).

\item[\moc{SIGS}] keyword used to specify that an homogeneous buckling
correction is to be applied on the diffusion cross section ($\Sigma_{s} -
dB^{2}$). \eq(eq3.64) is then replaced by

\begin{equation}
\vec\phi_g={\bf W}_g\left[ \vec Q^\diamond_g-d_g(B) \ B^2 \ \vec\phi_g\right]
\label{eq:eq5.33}
\end{equation}

\noindent where transport-corrected total
cross sections are used to compute the ${\bf W}_g$ matrix. This is the so called
{\sl DIFFON method} used in the APOLLO-family of thermal lattice codes. The \moc{SIGS} option is
the default option when a buckling calculation is required (\moc{TYPE B} or \moc{TYPE L}) or a
fission source eigenvalue problem (\moc{TYPE K}) with imposed buckling is considered.

\item[\moc{ALBS}] keyword used to specify that an homogeneous buckling
contribution is introduced by a group dependent correction of the
albedo.\cite{ALSB2} This leakage model is restricted to the collision probability
method. It is then necessary to define the geometry with an
external boundary condition of type \moc{VOID} (see \Sect{descBC}) and to close
the region in module \moc{ASM:} using the \moc{ALBS} option (see
\Sect{descasm}). \eq(eq3.64) is then replaced by

\begin{equation}
\vec\phi_g={\bf W}_g \ \vec Q^\diamond_g-\left[ {\bf I}+{\bf W}_g{\bf \Sigma}_{{\rm s0},g\gets g}\right] d_g(B) \ B^2 
\ \gamma \ {\bf P}_{{\rm iS},g}
\label{eq:eq5.34}
\end{equation}

\noindent where ${\bf P}_{{\rm iS},g}=\{P_{{\rm iS},g} \ ; \ i=1,I \}$ is the array of escape
probabilities in the open geometry and where

\begin{equation}
\gamma={\sum\limits_j V_j \phi_{j,g} \over \sum\limits_j V_j \phi_{j,g} P_{{\rm jS},g}} \ \ \ .
\label{eq:eq5.35}
\end{equation}

\item[\moc{HETE}] keyword used to perform a simplified heterogeneous leakage calculation, over one or many leakage zones, based
on the Todorova approximation.\cite{todorova} A leakage zone is a set of material mixtures where the leakage coefficient $d_{i,g}$ is
forced to be uniform in each energy group. Such a model is usefull to represent axial leakage in a {\tt TYPE~K} calculation or to
perform colorset calculations with more than one leakage zone. The \moc{HETE} leakage model can be used as an homogeneous model
assuming uniform leakage across the complete domain or as an heterogeneous model with more leakage zones defined using $\dusa{imergl}$
information. If a boundary condition is non-conservative (such as \moc{VOID}), it is nevertheless possible to use the \moc{HETE} option
with a $P_n$ or $B_n$ leakage model.

\item[\dusa{imergl}] array of homogenized leakage zone indices to which are associated the material mixtures. \dusa{nbmix} is the
total number of material mixtures. By default, a unique leakage zone is set. In this case, option $\moc{HETE}$ reduces to option $\moc{SIGS}$.

The simplified heterogeneous leakage model is based on a generalization of \eq(eq5.33), now written as
\begin{equation}
\vec\phi_g={\bf W}_g\left[ \vec Q^\diamond_g-B^2 \ \vec J_g\right]
\label{eq:eq5.36a}
\end{equation}

\noindent where each component of vector $\vec J_g$ is defined in term of heterogeneous leakage coefficients $d_{i,g}$ as
\begin{equation}
J_{i,g}=d_{i,g} \phi_{i,g}.
\label{eq:eq5.36b}
\end{equation}

A leakage zone index $m$ is assigned to each region $i$ using \dusa{imergl} information. In a colorset calculation, leakage zones 1 and 2
are assigned to black and red assemblies, respectively. In the $P_0$ and $B_0$ cases, the heterogeneous leakage coefficients in each leakage zone $m$
are obtained using the {\sl outscatter} approximation as
\begin{equation}
d_{m,g} = {1\over 3\gamma(B,\bar\Sigma_{m,g})}\left[ {\left<\phi_g\right>_m\over \left<\Sigma_g\phi_g\right>_m}\right]={1\over 3\gamma(B,\bar\Sigma_{m,g})\bar\Sigma_{m,g}}
\label{eq:eq5.36c}
\end{equation}
\noindent where $\left<\phi_g\right>_m$ is the integrated flux in leakage zone $m$ and $\left<\Sigma_g\phi_g\right>_m$ is a reaction rate
in zone $m$. The $\gamma(B,\bar\Sigma_{m,g})$ factor is equal to one with $P_n$ leakage models or to a leakage-zone dependent value with $B_n$
leakage models.\cite{PIP2009} Here, $\bar\Sigma_{m,g}$ is the leakage-zone averaged macroscopic total cross section in group $g$ defined as
\begin{equation}
\bar\Sigma_{m,g}={\left<\Sigma_g\phi_g\right>_m \over \left<\phi_g\right>_m}.
\label{eq:eq5.36d}
\end{equation}
\
In the $P_1$ and $B_1$ cases, the leakage coefficients are given as the solution of the following implicit equation, known as the {\sl inscatter} approximation:
\begin{equation}
d_{m,g}\left<\Sigma_g\phi_g\right>_m = {1\over \gamma(B,\bar\Sigma_{m,g})}\left[ {\left<\phi_g\right>_m\over 3}+
\sum_{h=1}^G \, d_{m,h} \left<\Sigma_{{\rm s1},g \leftarrow h}\phi_h\right>_m\right] .
\label{eq:eq5.36e}
\end{equation}

In transport-corrected $P_0$ and $B_0$ cases, we use the micro-reversibility principle, written as
\begin{equation}
\sum_{h=1}^G \Sigma_{{\rm s1},i,g \leftarrow h} J_{i,h} =\sum_{h=1}^G \Sigma_{{\rm s1},i,h \leftarrow g} J_{i,g}=\Sigma_{{\rm s1},i,g} J_{i,g} .
\label{eq:eq5.36f}
\end{equation}

Substitution of \eq(eq5.36f) into \eq(eq5.36e) leads to
\begin{equation}
d_{m,g} = {1\over 3}\left[ {\left<\phi_g\right>_m\over \gamma(B,\bar\Sigma_{m,g})\left<\Sigma_g\phi_g\right>_m-\left<\Sigma_{{\rm s1},g}\phi_g\right>_m}\right]=
{1\over 3\left[\gamma(B,\bar\Sigma_{m,g})\bar\Sigma_{m,g}-\bar\Sigma_{{\rm s1},m,g}\right]} .
\label{eq:eq5.36c}
\end{equation}

\item[\moc{ECCO}] keyword used to perform an ECCO--type leakage
calculation taking into account isotropic streaming effects. This method
introduces an heterogeneous buckling contribution as a group dependent correction
to the source term.\cite{ecco,rimpault} It is then necessary to set the keyword \moc{ECCO}
in module \moc{ASM:} (see \Sect{descasm}). In the $P_1$ non--consistent case,
\eq(eq3.64) is then replaced by

\vskip -0.3cm

\begin{eqnarray}
\vec\varphi_g&=& {\bf W}_g \left(\vec Q^\diamond_g - B^2 \ {i\vec{\cal J}_g\over B}\right)
\label{eq:eq5.37flux} \\
{i\vec{\cal J}_g\over B} &=& {\bf X}_g \left[{1 \over 3}
\ \vec\varphi_g + \sum_{h\not= g} {\bf \Sigma}_{{\rm s1},g \gets h} \
{i\vec{\cal J}_h\over B} \right]
\label{eq:eq5.37cour}
\end{eqnarray}

\noindent where $i\vec{\cal J}_{j,g}/B$ is the multigroup fundamental current, ${\bf \Sigma}_{{\rm s1},g \gets h}={\rm diag}\{ \Sigma_{{\rm s1},i,g \gets h}\> ;\> \forall i \}$ and where

\begin{equation}
{\bf X}_g=[{\bf I}-{\bf p}_g \ {\bf\Sigma}_{{\rm s}1,g\gets g}]^{-1} {\bf p}_g \ \ \ .
\label{eq:eq5.37ter}
\end{equation}

\item[\moc{TIBERE}] keyword used to perform a TIB\`ERE--type leakage
calculation taking into account anisotropic streaming effects. This method
introduces an heterogeneous buckling contribution as a group dependent correction
to the source term.\cite{PIJK0,PIJK} The heterogeneous buckling contribution is
introduced in the $B_n$ model using directional collision probabilities (PIJK method).
It is then necessary to set the keyword
\moc{PIJK} in module \moc{ASM:} (see \Sect{descasm}).

\item[\moc{G}] keyword used to specify that the buckling search will assume
all directional buckling to be identical (floating default option).

\item[\moc{R}] keyword used to specify that a radial buckling search will be
considered assuming an imposed $z$-direction buckling.

\item[\moc{Z}] keyword used to specify that a $z$-direction buckling search
will be considered  assuming an imposed $x$-direction and $y$-direction
buckling.

\item[\moc{X}] keyword used to specify that a $x$-direction buckling search
will be considered  assuming an imposed $y$-direction and $z$-direction
buckling.

\item[\moc{Y}] keyword used to specify that a $y$-direction buckling search
will be considered  assuming an imposed $x$-direction and $z$-direction
buckling.

\item[\moc{BUCK}] keyword used to specify the initial (for a buckling
eigenvalue problem) or fixed (for a effective multiplication factor eigenvalue
problem) buckling. 

\item[\moc{G}] keyword used to specify that the buckling in the $x$-direction,
$y$-direction and $z$-direction are to be initialized to \dusa{valb2}/3
(floating default).

\item[\moc{R}] keyword used to specify that the buckling in the $x$-direction,
and $y$-direction are to be initialized to \dusa{valbr2}/2.

\item[\moc{Z}] keyword used to specify that the buckling in the $z$-direction,
is to be initialized to \dusa{valbz2}.

\item[\moc{X}] keyword used to specify that the buckling in the $x$-direction,
is to be initialized to \dusa{valbx2}.

\item[\moc{Y}] keyword used to specify that the buckling in the $y$-direction,
is to be initialized to \dusa{valby2}.

\item[\dusa{valb2}] value of the fixed or initial total buckling in $cm^{-2}$.
The floating default value is
$${\it valb2}={\it valbx2}+{\it valby2}+{\it valbz2}.$$

\item[\dusa{valbr2}] value of the fixed or initial radial buckling in
$cm^{-2}$. The floating default value is
$${\it valbr2}={\it valbx2}+{\it valby2}.$$

\item[\dusa{valbz2}] value of the fixed or initial $z$-direction buckling in
$cm^{-2}$. The floating default value is \dusa{valbz2}=0.0 $cm^{-2}$. If
\dusa{valb2} is specified then \dusa{valbz2}=\dusa{valb2}/3.

\item[\dusa{valbx2}] value of the fixed or initial $z$-direction buckling in
$cm^{-2}$. The floating default value is \dusa{valbx2}=0.0 $cm^{-2}$. If
\dusa{valb2} is specified then \dusa{valbx2}=\dusa{valb2}/3. If \dusa{valbr2} is
specified then \dusa{valbx2}=\dusa{valbr2}/2.

\item[\dusa{valby2}] value of the fixed or initial $z$-direction buckling in
$cm^{-2}$. The floating default value is \dusa{valby2}=0.0 $cm^{-2}$. If
\dusa{valb2} is specified then \dusa{valby2}=\dusa{valb2}/3. If \dusa{valbr2} is
specified then \dusa{valby2}=\dusa{valbr2}/2.

\item[\moc{KEFF}] keyword used to specify the fixed (for a buckling eigenvalue
problem) effective multiplication factor. 

\item[\dusa{valk}] value of the fixed effective multiplication factor $K_{\rm eff}$. The
fixed default value is \dusa{valk}=1.0.

\item[\moc{IDEM}] keyword used to specify that the initial (for a buckling
eigenvalue problem) or fixed (for a effective multiplication factor eigenvalue
problem) buckling is to be read from the data structure \dusa{LIBNAM}. 

\end{ListeDeDescription}
\eject