From 7dfcc480ba1e19bd3232349fc733caef94034292 Mon Sep 17 00:00:00 2001 From: stainer_t Date: Mon, 8 Sep 2025 13:48:49 +0200 Subject: Initial commit from Polytechnique Montreal --- doc/IGE335/Section3.34.tex | 171 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 171 insertions(+) create mode 100644 doc/IGE335/Section3.34.tex (limited to 'doc/IGE335/Section3.34.tex') diff --git a/doc/IGE335/Section3.34.tex b/doc/IGE335/Section3.34.tex new file mode 100644 index 0000000..7816581 --- /dev/null +++ b/doc/IGE335/Section3.34.tex @@ -0,0 +1,171 @@ +\subsection{The {\tt BREF:} module}\label{sect:BREFData} + +This module compute a {\sc macrolib} for a 1D {\sl equivalent reflector} based on various models. One or many fine-group and +fine-mesh reference calculations (using the $S_n$ method) are first performed so as to produce coarse-group and +coarse-mesh Macrolibs stored within output {\sc edition} objects (\dusa{EDIT\_SN}), compatible with the selected +reflector model. Module {\tt BREF:} recovers the $S_n$ {\sc geometry}, depicted in Fig.~\ref{fig:bref}, from object \dusa{GEOM\_SN}. +The $S_n$ {\sc geometry} must have a reflective ({\tt REFL} or {\tt ALBE 1.0}) boundary condition on its left ({\tt X-}) boundary. +Module {\tt BREF:} recovers the following information from each \dusa{EDIT\_SN} object: +\begin{itemize} +\item Coarse group surfacic fluxes between the nodes using averaged flux values recovered into {\sl gap} volumes, corresponding to +tiny meshes in the reflector zones. +\item Coarse group net currents between the nodes obtained from a balance relation, assuming reflection on the left +boundary. +\item Averaged macroscopic cross sections and diffusion coefficients within the {\sl no-gap} homogenized nodes. +\end{itemize} + +\begin{figure}[h!] +\begin{center} +\epsfxsize=11cm +\centerline{ \epsffile{bref_geom.eps}} +\parbox{14cm}{\caption{Definition of geometries used by the {\tt BREF:} module} +\label{fig:bref}} +\end{center} +\end{figure} + +At output, a {\sc macrolib} object is produced with equivalent macroscopic cross sections, diffusion coefficients, discontinuity factors and +albedos. A verification calculation performed over the {\tt BREF:} geometry is depicted in Fig.~\ref{fig:brefVerif}, + +\begin{figure}[h!] +\begin{center} +\epsfxsize=15cm +\centerline{ \epsffile{ErmBeavrsPwrRefl.eps}} +\parbox{14cm}{\caption{Equivalent ERM-NEM reflector for the BEAVRS benchmark} +\label{fig:brefVerif}} +\end{center} +\end{figure} + +\vskip 0.02cm + +Nodal expansion base functions are used to represent the flux with the {\tt DF-NEM} and {\tt ERM-NEM} +methods. By default, polynomials defined over $(-0.5,0.5)$ are used as base functions:\cite{nestle} +\begin{eqnarray} +\nonumber P_0(u)\negthinspace &=&\negthinspace 1\\ +\nonumber P_1(u)\negthinspace &=&\negthinspace u\\ +\nonumber P_2(u)\negthinspace &=&\negthinspace 3u^2-{1\over 4}\\ +\nonumber P_3(u)\negthinspace &=&\negthinspace \left( u^2-{1\over 4}\right)u\\ +P_4(u)\negthinspace &=&\negthinspace \left( u^2-{1\over 4}\right)\left( u^2-{1\over 20}\right) +\end{eqnarray} +There is the option of using hyperbolic functions in some energy groups: +\begin{eqnarray} +\nonumber P_3(u)\negthinspace &=&\negthinspace \sinh(\zeta_g u)\\ +P_4(u)\negthinspace &=&\negthinspace \cosh(\zeta_g u)-{2\over \zeta}\, \sinh(\zeta_g/2) +\end{eqnarray} +\noindent where +\begin{equation} +\zeta_g=\Delta x\sqrt{\Sigma_{{\rm r},g} \over D_g} +\end{equation} +\noindent where $\Delta x$, $\Sigma_{{\rm r},g}$ and $D_g$ are the node width (cm), the macroscopic removal cross section (cm$^{-1}$) +and the diffusion coefficient (cm) in group $g$, respectively. + +\vskip 0.02cm + +Other equivalence techniques are known as {\tt DF-ANM} and {\tt ERM-ANM}. These techniques are similar to {\tt DF-NEM} and +{\tt ERM-NEM} where the nodal expansion method (NEM) is replaced by an analytic solution of the $G-$group diffusion equation. + +\goodbreak + +The calling specifications are: + +\begin{DataStructure}{Structure \dstr{BREF:}} +\dusa{GEOM} \dusa{MACRO}~\moc{:=}~\moc{BREF:}~\dusa{GEOM\_SN} $[[$~\dusa{EDIT\_SN}~$]]$ \moc{::} \dstr{BREF\_data} \\ +\end{DataStructure} + +\noindent where +\begin{ListeDeDescription}{mmmmmmm} + +\item[\dusa{GEOM}] {\tt character*12} name of the nodal {\sc geometry} (type {\tt L\_GEOM}) object open creation mode. This geometry can be used for +performing a verification calculation over the 1D nodal geometry. + +\item[\dusa{MACRO}] {\tt character*12} name of the nodal {\sc macrolib} (type {\tt L\_MACROLIB}) object open in creation mode. + +\item[\dusa{GEOM\_SN}] {\tt character*12} name of the $S_n$ {\sc geometry} (type {\tt L\_GEOM}) object open in read-only mode. + +\item[\dusa{EDIT\_SN}] {\tt character*12} name of one or many $S_n$ {\sc edition} (type {\tt L\_EDIT}) object, containing coarse-group and +coarse-mesh {\sc macrolib} for the edition {\sc macro-geometry} with gaps, corresponding to one or many reference $S_n$ calculations. + +\item[\dusa{BREF\_data}] input data structure containing specific data (see \Sect{descBREF}). + +\end{ListeDeDescription} +\clearpage + +\subsubsection{Data input for module {\tt BREF:}}\label{sect:descBREF} + +\begin{DataStructure}{Structure \dstr{BREF\_data}} +$[$~\moc{EDIT} \dusa{iprint}~$]$ \\ +$[$~\moc{HYPE} \dusa{igmax}~$]$ \\ +\moc{MIX} $[[$ \dusa{imix} $]]$ \moc{GAP} $[[$ \dusa{igap} $]]$ \\ +\moc{MODE} $\{$~\moc{LEFEBVRE-LEB}~$|$~\moc{KOEBKE}~$|$~\moc{DF-NEM}~$|$~\moc{DF-ANM}~$|$~\moc{DF-RT} \dusa{ielem} \dusa{icol} +$[$~\moc{SPN} $[$ \moc{DIFF} $]$ \dusa{nlf} $]~|$ \\ +~~~~\moc{ERM-NEM}~$|$~\moc{ERM-ANM}~$\}$ \\ +$[~\{$~\moc{SPH}~$|$~\moc{NOSP}~$\}~]~[~\{$~\moc{ALBE}~$|$~\moc{NOAL}~$\}~]$ \\ +$[$~\moc{NGET}~$[$~(\dusa{adf}($g$), $g$=1,$N_g$) $]~]$ \\ +{\tt ;} +\end{DataStructure} + +\noindent where +\begin{ListeDeDescription}{mmmmmmmm} + +\item[\moc{EDIT}] keyword used to set \dusa{iprint}. + +\item[\dusa{iprint}] index used to control the printing in module {\tt BREF:}. =0 for no print; =1 for minimum printing (default value). + +\item[\moc{HYPE}] keyword used to specify the type of nodal expansion base functions with the {\tt DF-NEM} and {\tt ERM-NEM} +methods. By default, polynomial base functions are used in all energy groups. + +\item[\dusa{igmax}] hyperbolic base functions are used for coarse energy groups with indices $\ge$ \dusa{igmax}. + +\item[\moc{MIX}] keyword used to set the nodal mixture indices \dusa{imix}. + +\item[\dusa{imix}] index of a mixture index within object \dusa{EDIT\_SN} corresponding to a node. In Fig.~\ref{fig:bref}, this data +is set as {\tt MIX 0 3} for {\tt LEFEBVRE-LEB} and {\tt KOEBKE} geometries and set as {\tt MIX 1 3} for other geometries. + +\item[\moc{GAP}] keyword used to set the gap mixture indices \dusa{igap} where the surfacic fluxes are recovered. + +\item[\dusa{igap}] index of a mixture index within object \dusa{EDIT\_SN} corresponding to a gap. In Fig.~\ref{fig:bref}, this data +is set as {\tt GAP 2 0} for {\tt LEFEBVRE-LEB} and {\tt KOEBKE} geometries and set as {\tt GAP 2 4} for other geometries. + +\item[\moc{MODE}] keyword used to select a specific reflector equivalence model. The character*12 name of this model is chosen among the following values: + +\item[{\tt LEFEBVRE-LEB}] Lefebvre-Lebigot equivalence model. Two \dusa{EDIT\_SN} objects are expected at RHS.\cite{LLB,Frohlicher} +\item[{\tt KOEBKE}] Koebke equivalence model. Two \dusa{EDIT\_SN} objects are expected at RHS.\cite{Koebke,Frohlicher} +\item[{\tt DF-NEM}] Pure discontinuity-factor model based on the nodal expansion method (NEM). Only one \dusa{EDIT\_SN} object is expected at RHS. +\item[{\tt DF-ANM}] Pure discontinuity-factor model based on the analytic nodal method (ANM). Only one \dusa{EDIT\_SN} object is expected at RHS. +\item[{\tt DF-RT}] Pure discontinuity-factor model based on the Raviart-Thomas finite element method in diffusion or $SP_n$ theory. Only one \dusa{EDIT\_SN} object is expected at RHS. +\item[{\tt ERM-NEM}] {\sl Equivalent reflector model} based on {\sl matrix discontinuity factors} and nodal expansion method (NEM). +Two or more \dusa{EDIT\_SN} objects are expected at RHS. +\item[{\tt ERM-ANM}] {\sl Equivalent reflector model} based on {\sl matrix discontinuity factors} and analytic nodal method (ANM). +Two or more \dusa{EDIT\_SN} objects are expected at RHS. + +\item[\dusa{ielem}] order of the Raviart-Thomas finite element method. The values +permitted are 1 (linear polynomials), 2 (parabolic polynomials), 3 (cubic polynomials) or 4 (quartic polynomials). + +\item[\dusa{icol}] type of quadrature used to integrate the mass matrices in the Raviart-Thomas finite element method. The +values permitted are 1 (analytical integration), 2 (Gauss-Lobatto quadrature) or 3 (Gauss-Legendre quadrature). + +\item[{\tt SPN}] keyword to define a simplified spherical harmonics ($SP_n$) macro calculation. By default, diffusion theory is used. + +\item[\moc{DIFF}] keyword to force using $1/3D^{g}$ as $\Sigma_1^{g}-\Sigma_{{\rm s}1}^{g}$ cross sections with the $SP_n$ approximation. A $SP_1$ method +will therefore behave as diffusion theory. + +\item[\dusa{nlf}] order of the $SP_n$ expansion (odd number). + +\item[\moc{SPH}] keyword used to include discontinuity factors within cross sections and diffusion coefficients. This option is not available +with models {\tt ERM-NEM} and {\tt ERM-ANM}. This is the default option with model {\tt DF-RT}. + +\item[\moc{NOSP}] keyword used to store discontinuity factors in {\sc macrolib} \dusa{MACRO} (default option, except for model {\tt DF-RT}). + +\item[\moc{ALBE}] keyword used to compute an equivalent albedo in each coarse energy group with {\tt DF-NEM}, {\tt DF-ANM}, {\tt DF-RT}, +{\tt ERM-NEM} and {\tt ERM-ANM} models (default option). + +\item[\moc{NOAL}] keyword used to desactivate equivalent albedo calculation. + +\item[\moc{NGET}] keyword used to force the value of the fuel assembly discontinuity factor at the fuel-reflector interface, as used +by the NGET normalization. By default, this value is not modified by NGET normalization. + +\item[\dusa{adf}] value of the assembly discontinuity factor (ADF) on the fuel-reflector interface in group $g\le N_g$. If keyword \moc{NGET} is set and +\dusa{adf} values are not given, the ADF values are recovered from \dusa{EDIT\_SN}. + +\end{ListeDeDescription} + +\eject -- cgit v1.2.3