From 7dfcc480ba1e19bd3232349fc733caef94034292 Mon Sep 17 00:00:00 2001
From: stainer_t <thomas.stainer@oecd-nea.org>
Date: Mon, 8 Sep 2025 13:48:49 +0200
Subject: Initial commit from Polytechnique Montreal

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+\section{Contents of a 
+\dir{fluxunk} directory}\label{sect:fluxunkdir}
+
+This directory contains the main flux calculations results, including the multigroup flux, the
+eigenvalue for the problem and the diffusion coefficients when computed. The following types of
+equations can be solved:
+\begin{enumerate}
+\item Fixed source problem
+\begin{equation}
+\bf{A} \ \vec\Phi = \vec S
+\label{eq:flux1}
+\end{equation}
+\noindent where $\bf{A}$ is the coefficient matrix, $\vec S$ is the source vector and
+$\vec\Phi$ is the unknown vector.
+
+\item Direct eigenvalue problem
+\begin{equation}
+\bf{A} \ \vec\Phi_\alpha + {1 \over K_{{\rm eff},\alpha}} \ \bf{B} \ \vec\Phi_\alpha = \vec 0
+\label{eq:flux2}
+\end{equation}
+\noindent where $\bf{B}$ is the second coefficient matrix and where (${1 \over K_{{\rm eff},\alpha}}$
+,$\vec\Phi_\alpha$) is the eigensolution corresponding to the $\alpha$--th eigenvalue
+or harmonic mode. Generally, only the eigensolution corresponding to the maximum value of $K_{{\rm eff},\alpha}$ is found (the fundamental mode).
+
+\item Adjoint eigenvalue problem
+\begin{equation}
+\bf{A}^\top \ \vec\Phi_\alpha^* + {1 \over K_{{\rm eff},\alpha}} \ \bf{B}^\top \ \vec\Phi_\alpha^* = \vec 0
+\label{eq:flux3}
+\end{equation}
+\noindent where matrices $\bf{A}$ and $\bf{B}$ are transposed.
+
+\item Fixed source direct eigenvalue equation (direct GPT)
+\begin{equation}
+\bf{A} \ \vec\Gamma_\alpha + {1 \over K_{{\rm eff},\alpha}} \ \bf{B} \ \vec\Gamma_\alpha = \vec S
+\ \ \ \ {\rm where} \ \ \ \ \left<\Phi_\alpha^*, \ \vec S \right>=0
+\label{eq:flux4}
+\end{equation}
+\noindent where the direct source vector $\vec S$ is orthogonal to the adjoint flux.
+
+\item Fixed source adjoint eigenvalue equation (adjoint GPT)
+\begin{equation}
+\bf{A}^\top \ \vec\Gamma_\alpha^* + {1 \over K_{{\rm eff},\alpha}} \ \bf{B}^\top \ \vec\Gamma_\alpha^* = \vec S^*
+\ \ \ \ {\rm where} \ \ \ \ \left<\Phi_\alpha, \ \vec S^* \right>=0
+\label{eq:flux5}
+\end{equation}
+\noindent where the adjoint source vector $\vec S^*$ is orthogonal to the direct flux.
+
+\end{enumerate}
+
+\subsection{State vector content for the \dir{fluxunk} data structure}\label{sect:fluxunkstate}
+
+The dimensioning parameters for this data structure, which are stored in the state vector
+$\mathcal{S}^{f}_{i}$, represent:
+
+\begin{itemize}
+
+\item The number of energy groups $N_{G}=\mathcal{S}^{f}_{1}$
+
+\item The number of unknowns per energy group $N_{U}=\mathcal{S}^{f}_{2}$
+
+\item The type of equation considered $ I_{e} = \mathcal{S}^{f}_{3} = \alpha_1 + 10 \ \alpha_2 + 100 \ \alpha_3 + 1000 \ \alpha_4 $ where
+\vskip -0.45cm
+
+\begin{eqnarray}
+\nonumber \alpha_1 &=& 0/1\textrm{:} \ \ \textrm{Fixed source (\Eq{flux1}) or \keff{} (\Eq{flux2}) direct eigenvalue equation} \\
+\nonumber &~&\textrm{absent/present} \\
+\nonumber \alpha_2 &=& 0/1\textrm{:} \ \ \textrm{Adjoint eigenvalue equation (\Eq{flux3}) absent/present} \\
+\nonumber \alpha_3 &=& 0/1\textrm{:} \ \ \textrm{Direct fixed source eigenvalue equation -- or GPT equation (\Eq{flux4})} \\
+\nonumber &~&\textrm{absent/present} \\
+\nonumber \alpha_4 &=& 0/1\textrm{:} \ \ \textrm{Adjoint fixed source eigenvalue equation -- or GPT equation (\Eq{flux5})} \\
+\nonumber &~&\textrm{absent/present}
+\end{eqnarray}
+
+\item The number of harmonics considered $N_{h}=\mathcal{S}^{f}_{4}$ where
+
+\begin{displaymath}
+N_{h} = \left\{
+\begin{array}{rl}
+ 0 & \textrm{the harmonic calculation is not enabled} \\
+ \ge 1 & \textrm{the harmonic calculation is enabled. $N_{h}$ is the number of harmonics.} \\
+\end{array} \right.
+\end{displaymath}
+
+\item The number of specific GPT equations considered $N_{\rm gpt}=\mathcal{S}^{f}_{5}$ where
+
+\begin{displaymath}
+N_{\rm gpt} = \left\{
+\begin{array}{rl}
+ 0 & \textrm{the GPT calculation is not enabled} \\
+ \ge 1 & \textrm{the GPT calculation is enabled. $N_{\rm gpt}$ is the number of specific GPT} \\
+  & \textrm{equations.} \\
+\end{array} \right.
+\end{displaymath}
+
+\item The type of $B_n$ solution considered $I_{s}=\mathcal{S}^{f}_{6}$ where
+
+\begin{displaymath}
+I_{s} = \left\{
+\begin{array}{rl}
+-2 & \textrm{1D Fourier analysis, fixed source problem, no eigenvalue}\\
+-1 & \textrm{No flux calculation, fluxes taken from input file}\\
+ 0 & \textrm{Fixed source problem, no eigenvalue} \\
+ 1 & \textrm{fixed source eigenvalue problem (GPT type) with fission} \\
+ 2 & \textrm{\keff{} eigenvalue problem with fission and without leakage} \\
+ 3 & \textrm{\keff{} eigenvalue problem with fission and leakage } \\
+ 4 & \textrm{Buckling eigenvalue problem with fission and leakage} \\
+ 5 & \textrm{Buckling eigenvalue problem without fission but with leakage} 
+\end{array} \right.
+\end{displaymath}
+
+\item The type of leakage model $I_{l}=\mathcal{S}^{f}_{7}$ where
+
+\begin{displaymath}
+I_{l} = \left\{
+\begin{array}{rl}
+ 0 & \textrm{No leakage model} \\
+ 1 & \textrm{Homogeneous \moc{PNLR} calculation} \\
+ 2 & \textrm{Homogeneous \moc{PNL} calculation} \\
+ 3 & \textrm{Homogeneous \moc{SIGS} calculation} \\
+ 4 & \textrm{Homogeneous \moc{ALSB} calculation} \\
+ 5 & \textrm{Leakage with isotropic streaming effects -- Todorova simplified model} \\
+ 6 & \textrm{Leakage with isotropic streaming effects -- ECCO model} \\
+17 & \textrm{Leakage with anisotropic streaming effects -- imposed buckling} \\
+27 & \textrm{Leakage with anisotropic streaming effects -- X-Buckling search} \\
+37 & \textrm{Leakage with anisotropic streaming effects -- Y-Buckling search} \\
+47 & \textrm{Leakage with anisotropic streaming effects -- Z-Buckling search} \\
+57 & \textrm{Leakage with anisotropic streaming effects -- radial Buckling search} \\
+67 & \textrm{Leakage with anisotropic streaming effects -- total Buckling search} \\
+\end{array} \right.
+\end{displaymath}
+
+\item Number of free iteration per variational acceleration cycle $N_{f}=\mathcal{S}^{f}_{8}$
+ 
+\item Number of accelerated iteration per variational acceleration cycle $N_{a}=\mathcal{S}^{f}_{9}$ 
+
+\item Thermal rebalancing option $I_{r}=\mathcal{S}^{f}_{10}$ where
+
+\begin{displaymath}
+I_{r} = \left\{
+\begin{array}{rl}
+ 0 & \textrm{No thermal iteration rebalancing} \\
+ 1 & \textrm{Thermal iteration rebalancing activated} \\
+\end{array} \right.
+\end{displaymath}
+
+\item Maximum number of thermal (up-scattering) iterations $M_{\rm in}=\mathcal{S}^{f}_{11}$
+
+\item Maximum number of outer iterations $M_{\rm out}=\mathcal{S}^{f}_{12}$
+
+\item Initial number of ADI iterations in Trivac $M_{\rm adi}=\mathcal{S}^{f}_{13}$
+
+\item Block size of the Arnoldi Hessenberg matrix with the implicit restarted Arnoldi method (IRAM) ($=0$ if the symmetrical variational acceleration technique (SVAT) is used) $N_{\rm blsz}=\mathcal{S}^{f}_{14}$
+
+\item Number of iterations before restarting with the GMRES(m) acceleration method for solving the ADI-preconditionned linear systems in Trivac ($=0$ if $M_{\rm adi}$ free iterations are used) $N_{\rm gmr1}=\mathcal{S}^{f}_{15}$
+
+\item Number of iterations before restarting with the GMRES(m) acceleration method for solving a multigroup fixed-source problem ($=0$ if the variational acceleration technique is used) $N_{\rm gmr2}=\mathcal{S}^{f}_{16}$
+
+\item Number of material mixtures $N_m=\mathcal{S}^{f}_{17}$
+
+\item Number of leakage zones $N_{\rm leak}=\mathcal{S}^{f}_{18}$. Set to zero if no leakage zones are defined.
+
+\end{itemize}
+
+\subsection{The main \dir{fluxunk} directory}\label{sect:fluxunkdirmain}
+
+On its first level, the
+following records and sub-directories will be found in the \dir{fluxunk} directory:
+
+\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{fluxunk}}{8.0cm}
+\CharEnr
+  {SIGNATURE\blank{3}}{$*12$}
+  {Signature of the data structure ($\mathsf{SIGNA}=${\tt L\_FLUX\blank{6}})}
+\IntEnr
+  {STATE-VECTOR}{$40$}
+  {Vector describing the various parameters associated with this data structure $\mathcal{S}^{f}_{i}$,
+  as defined in \Sect{fluxunkstate}.}
+\CharEnr
+  {TRACK-TYPE\blank{2}}{$*12$}
+  {Type of tracking considered ($\mathsf{CDOOR}$). Allowed values are:
+  {\tt 'EXCELL'}, {\tt 'SYBIL'}, {\tt 'MCCG'}, {\tt 'SN'}, {\tt 'BIVAC'} and {\tt 'TRIVAC'}.}
+\CharEnr
+  {OPTION\blank{6}}{$*4$}
+  {Type of leakage coefficients ({\tt 'LKRD'}: recover leakage coefficients in Macrolib; {\tt 'RHS'}: recover
+  leakage coefficients in RHS flux object; {\tt 'B0'}: $B_0$; {\tt 'P0'}: $P_0$; {\tt 'B1'}: $B_1$; {\tt 'P1'}:
+   $P_1$; {\tt 'B0TR'}: $B_0$ with transport correction; {\tt 'P0TR'}: $P_0$ with transport correction).}
+\RealEnr
+  {EPS-CONVERGE}{$5$}{}
+  {Convergence parameters $\Delta_i^\epsilon$}
+\IntEnr
+  {KEYFLX\blank{6}}{$\mathcal{S}^{t}_{1}$}
+  {Location in unknown vector of averaged regional flux $I_{r}$}
+\OptRealEnr
+  {K-EFFECTIVE\blank{1}}{$1$}{$\mathcal{S}^{f}_{6}\ge 1$}{}
+  {Computed or imposed effective multiplication factor for direct eigenvalue problem,
+  corresponding to the fundamental mode}
+\OptRealEnr
+  {AK-EFFECTIVE}{$1$}{${\mathcal{S}^{f}_{3}\over 10} \bmod 10 = 1$}{}
+  {Computed effective multiplication factor for adjoint eigenvalue problem,
+  corresponding to the fundamental mode.
+  The theoretical value is equal
+  to {\tt 'K-EFFECTIVE'} but difference may occurs for numerical reasons.}
+\OptRealEnr
+  {K-INFINITY\blank{2}}{$1$}{$\mathcal{S}^{f}_{6}\ge 2$}{}
+  {Computed infinite multiplication constant for eigenvalue problem,
+  corresponding to the fundamental mode}
+\OptRealEnr
+  {B2\blank{2}B1HOM\blank{3}}{$1$}{$\mathcal{S}^{f}_{6}\ge 1$}{cm$^{-2}$}
+  {Homogeneous buckling $B^{2}$,
+  corresponding to the fundamental mode}
+\OptRealEnr
+  {SPEC-RADIUS\blank{1}}{$1$}{$\mathcal{S}^{f}_{6}= -2$}{cm}
+  {Spectral radius}
+\OptRealEnr
+  {DIFFHET\blank{5}}{$N_{\rm leak}\times G$}{$\mathcal{S}^{f}_{18}\ge 1$}{cm}
+  {Multigroup leakage coefficients in each leakage zone and energy group $D_l^g$}
+\OptIntEnr
+  {IMERGE-LEAK\blank{1}}{$N_m$}{$\mathcal{S}^{f}_{18}\ge 1$}
+  {Leakage zone index assigned to each material mixture $L_m^g$}
+\OptRealEnr
+  {B2\blank{2}HETE\blank{4}}{$3$}{$\mathcal{S}^{f}_{7} \ge 6$}{cm$^{-2}$}
+  {Directional buckling components $B^{2}_{i}$,
+  corresponding to the fundamental mode}
+\OptRealEnr
+  {GAMMA\blank{7}}{$G$}{$\mathcal{S}^{f}_{7}\ge 5$}{}
+  {Gamma factors used with $B_n$--type streaming models.}
+\DirlEnr
+  {FLUX\blank{8}}{$\mathcal{S}^{f}_{1}$}
+  {List of real arrays. Each component of this list is a real array of dimension $\mathcal{S}^{f}_{2}$
+  containing the solution of a fixed source (\Eq{flux1}) or of a direct eigenvalue (\Eq{flux2}) equation,
+  corresponding to the fundamental mode.}
+\DirlEnr
+  {SOUR\blank{8}}{$\mathcal{S}^{f}_{1}$}
+  {List of real arrays. Each component of this list is a real array of dimension $\mathcal{S}^{f}_{2}$
+  containing the RHS source distributions corresponding to the {\tt FLUX} records.}
+\OptDirlEnr
+  {AFLUX\blank{7}}{$\mathcal{S}^{f}_{1}$}{${\mathcal{S}^{f}_{3}\over 10} \bmod 10 = 1$}
+  {List of real arrays. Each component of this list is a real array of dimension $\mathcal{S}^{f}_{2}$
+  containing the solution of an adjoint eigenvalue (\Eq{flux3}) equation,
+  corresponding to the fundamental mode.}
+\OptDirlEnr
+  {MODE\blank{8}}{$\mathcal{S}^{f}_{4}$}{$\mathcal{S}^{f}_{4}\ge 1$}
+  {List of {\sl harmonic mode} sub-directories. Each component of this list follows
+  the specification presented in \Sect{mode_spec}.}
+\OptDirlEnr
+  {DFLUX\blank{7}}{$\mathcal{S}^{f}_{5}$}{$\mathcal{S}^{f}_{3}=100$}
+  {List of direct (explicit) GPT sub-directories. Each component of this list is a multigroup list of
+  dimension $\mathcal{S}^{f}_{1}$. Each component of the multigroup list is a real array of dimension
+  $\mathcal{S}^{f}_{2}$ containing the solution of a fixed source direct eigenvalue equation similar to \Eq{flux4}.}
+\OptDirlEnr
+  {ADFLUX\blank{6}}{$\mathcal{S}^{f}_{5}$}{$\mathcal{S}^{f}_{3}=1000$}
+  {List of adjoint (implicit) GPT sub-directories. Each component of this list is a multigroup list of
+  dimension $\mathcal{S}^{f}_{1}$. Each component of the multigroup list is a real array of dimension
+  $\mathcal{S}^{f}_{2}$ containing the solution of a fixed source adjoint eigenvalue equation similar to \Eq{flux5}.}
+\OptDirlEnr
+  {DRIFT\blank{7}}{$\mathcal{S}^{f}_{1}$}{$\mathcal{S}^{t}_{12}=6$}
+  {Drift coefficients used in nodal correction iterations. Each component of the multigroup list is a real array of dimension
+  $6\times \mathcal{S}^{t}_{1}$.}
+\end{DescriptionEnregistrement}
+
+The convergence parameters $\Delta_i^\epsilon$ represent:
+\begin{itemize}
+\item $\Delta_1^\epsilon$ is the thermal (up-scattering) iteration flux convergence parameter
+\item $\Delta_2^\epsilon$ is the outer iteration eigenvalue convergence parameter
+\item $\Delta_3^\epsilon$ is the outer iteration flux convergence parameter
+\item $\Delta_4^\epsilon$ is the GMRES convergence parameter used at inner iteration
+\item $\Delta_5^\epsilon$ is the relaxation factor of the flux used in multiphysics applications. $\Delta_5^\epsilon=1$ is equivalent to no
+relaxation.
+\end{itemize}
+\goodbreak
+
+\subsection{The harmonic mode sub-directories in \dir{fluxunk}}\label{sect:mode_spec}
+
+Each component of the list named {\tt 'MODE'} contains the information relative to a specific
+harmonic mode.
+
+\begin{DescriptionEnregistrement}{Component of the harmonic mode directory}{7.5cm}
+\RealEnr
+  {K-EFFECTIVE\blank{1}}{$1$}{}
+  {Computed effective multiplication factor for eigenvalue problem,
+  corresponding to the $\alpha$--th mode}
+\DirlEnr
+  {FLUX\blank{8}}{$\mathcal{S}^{f}_{1}$}
+  {List of real arrays. Each component of this list is a real array of dimension $\mathcal{S}^{f}_{2}$
+  containing the solution of the $\alpha$--th mode of a direct eigenvalue (\Eq{flux2}) equation.}
+\OptDirlEnr
+  {AFLUX\blank{7}}{$\mathcal{S}^{f}_{1}$}{${\mathcal{S}^{f}_{3}\over 10} \bmod 10 = 1$}
+  {List of real arrays. Each component of this list is a real array of dimension $\mathcal{S}^{f}_{2}$
+  containing the solution of the $\alpha$--th mode of an adjoint eigenvalue (\Eq{flux3}) equation.}
+\end{DescriptionEnregistrement}
+
+\eject
-- 
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