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-revision 1.8 by edhill,
Tue Feb 17 21:58:56 2004 UTC
+revision 1.15 by afe,
Thu Mar 18 14:56:25 2004 UTC
 Line 10
  %%    o automatically inserted at \section{Reference}
- \section{Extended Cubed Sphere Exchange}
+ \section{exch2: Extended Cubed Sphere \mbox{Topology}}
  \label{sec:exch2}
  \subsection{Introduction}
- The \texttt{exch2} package is an extension to the original cubed
+ The \texttt{exch2} package extends the original cubed
- sphere exchanges to allow more flexible domain decomposition and
+ sphere topology configuration to allow more flexible domain
- parallelization.  Cube faces (subdomains) may be divided into whatever
+ decomposition and parallelization.  Cube faces (also called
- number of tiles that divide evenly into the grid point dimensions of
+ subdomains) may be divided into any number of tiles that divide evenly
- the subdomain.  Furthermore, the individual tiles may be run on
+ into the grid point dimensions of the subdomain.  Furthermore, the
- separate processors in different combinations, and whether exchanges
+ individual tiles can run on separate processors in different
- between particular tiles occur between different processors is
+ combinations, and whether exchanges between particular tiles occur
- determined at runtime.
+ between different processors is determined at runtime.  This
+ flexibility provides for manual compile-time load balancing across a
+ relatively arbitrary number of processors. \\
  The exchange parameters are declared in
  \filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h}
  and assigned in
- \filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}, both in
+ \filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The
- the \texttt{pkg/exch2} directory.  The validity of the cube topology
+ validity of the cube topology depends on the \file{SIZE.h} file as
- depends on the \texttt{SIZE.h} file as detailed below.  Both files are
+ detailed below.  The default files provided in the release configure a
- generated by Matlab scripts and should not be edited.  The default
+ cubed sphere topology of six tiles, one per subdomain, each with
- files provided in the release set up a cube sphere arrangement of six
+$\times$32 grid points, all running on a single processor.  Both
- tiles, one per subdomain, each with 32x32 grid points, running on a
+ files are generated by Matlab scripts in
- single processor.
+ \file{utils/exch2/matlab-topology-generator}; see Section
+ \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
+ for details on creating alternate topologies.  Pregenerated examples
+ of these files with alternate topologies are provided under
+ \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
+ file for single-processor execution.
+ \subsection{Invoking exch2}
+ To use exch2 with the cubed sphere, the following conditions must be
+ met: \\
+ $\bullet$ The exch2 package is included when \file{genmake2} is run.
+   The easiest way to do this is to add the line \code{exch2} to the
+   \file{profile.conf} file -- see Section
+   \ref{sect:buildingCode} \sectiontitle{Building the code} for general
+   details. \\
+ $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
+   \file{w2\_e2setup.F} must reside in a directory containing code
+   linked when \file{genmake2} runs.  The safest place to put these
+   is the directory indicated in the \code{-mods=DIR} command line
+   modifier (typically \file{../code}), or the build directory.  The
+   default versions of these files reside in \file{pkg/exch2} and are
+   linked automatically if no other versions exist elsewhere in the
+   link path, but they should be left untouched to avoid breaking
+   configurations other than the one you intend to modify.\\
+ $\bullet$ Files containing grid parameters, named
+   \file{tile00$n$.mitgrid} where $n$=\code{(1:6)} (one per subdomain),
+   must be in the working directory when the MITgcm executable is run.
+   These files are provided in the example experiments for cubed sphere
+   configurations with 32$\times$32 cube sides and are non-trivial to
+   generate -- please contact MITgcm support if you want to generate
+   files for other configurations. \\
+ $\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} must
+   be placed where \file{genmake2} will find it.  In particular for
+   exch2, the domain decomposition specified in \file{SIZE.h} must
+   correspond with the particular configuration's topology specified in
+   \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
+   decomposition issues particular to exch2 are addressed in Section
+   \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
+   and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more
+   general background on the subject relevant to MITgcm is presented in
+   Section \ref{sect:specifying_a_decomposition}
+   \sectiontitle{Specifying a decomposition}.\\
+ As of the time of writing the following examples use exch2 and may be
+ used for guidance:
+ \begin{verbatim}
+ verification/adjust_nlfs.cs-32x32x1
+ verification/adjustment.cs-32x32x1
+ verification/aim.5l_cs
+ verification/global_ocean.cs32x15
+ verification/hs94.cs-32x32x5
+ \end{verbatim}
+ \subsection{Generating Topology Files for exch2}
+ \label{sec:topogen}
+ Alternate cubed sphere topologies may be created using the Matlab
+ scripts in \file{utils/exch2/matlab-topology-generator}. Running the
+ m-file
+ \filelink{driver.m}{utils-exch2-matlab-topology-generator_driver.m}
+ from the Matlab prompt (there are no parameters to pass) generates
+ exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
+ \file{w2\_e2setup.F} in the working directory and displays a figure of
+ the topology via Matlab.  The other m-files in the directory are
+ subroutines of \file{driver.m} and should not be run ``bare'' except
+ for development purposes. \\
+ The parameters that determine the dimensions and topology of the
+ generated configuration are \code{nr}, \code{nb}, \code{ng},
+ \code{tnx} and \code{tny}, and all are assigned early in the script. \\
+ The first three determine the size of the subdomains and
+ hence the size of the overall domain.  Each one determines the number
+ of grid points, and therefore the resolution, along the subdomain
+ sides in a ``great circle'' around an axis of the cube.  At the time
+ of this writing MITgcm requires these three parameters to be equal,
+ but they provide for future releases  to accomodate different
+ resolutions around the axes to allow (for example) greater resolution
+ around the equator.\\
+ The parameters \code{tnx} and \code{tny} determine the dimensions of
+ the tiles into which the subdomains are decomposed, and must evenly
+ divide the integer assigned to \code{nr}, \code{nb} and \code{ng}.
+ The result is a rectangular tiling of the subdomain.  Figure
+ \ref{fig:24tile} shows one possible topology for a twentyfour-tile
+ cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\
+ \begin{figure}
+ \begin{center}
+  \resizebox{4in}{!}{
+   \includegraphics{part6/s24t_16x16.ps}
+  }
+ \end{center}
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ divided into six 32$\times$32 subdomains, each of which is divided into four tiles
+ (\code{tnx=16, tny=16}) for a total of twentyfour tiles.
+ } \label{fig:24tile}
+ \end{figure}
+ \begin{figure}
+ \begin{center}
+  \resizebox{4in}{!}{
+   \includegraphics{part6/s12t_16x32.ps}
+  }
+ \end{center}
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ divided into six 32$\times$32 subdomains of two tiles each
+  (\code{tnx=16, tny=32}).
+ } \label{fig:12tile}
+ \end{figure}
+ \begin{figure}
+ \begin{center}
+  \resizebox{4in}{!}{
+   \includegraphics{part6/s6t_32x32.ps}
+  }
+ \end{center}
+ \caption{Plot of a cubed sphere topology with a 32$\times$192 domain
+ divided into six 32$\times$32 subdomains with one tile each
+ (\code{tnx=32, tny=32}).  This is the default configuration.
+   }
+ \label{fig:6tile}
+ \end{figure}
+ Tiles can be selected from the topology to be omitted from being
+ allocated memory and processors.  This tuning is useful in ocean
+ modeling for omitting tiles that fall entirely on land.  The tiles
+ omitted are specified in the file
+ \filelink{blanklist.txt}{utils-exch2-matlab-topology-generator_blanklist.txt}
+ by their tile number in the topology, separated by a newline. \\
+ \subsection{exch2, SIZE.h, and multiprocessing}
+ \label{sec:exch2mpi}
+ Once the topology configuration files are created, the Fortran
+ \code{PARAMETER}s in \file{SIZE.h} must be configured to match.
+ Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
+ a decomposition} provides a general description of domain
+ decomposition within MITgcm and its relation to \file{SIZE.h}. The
+ current section specifies certain constraints the exch2 package
+ imposes as well as describes how to enable parallel execution with
+ MPI. \\
+ As in the general case, the parameters \varlink{sNx}{sNx} and
+ \varlink{sNy}{sNy} define the size of the individual tiles, and so
+ must be assigned the same respective values as \code{tnx} and
+ \code{tny} in \file{driver.m}.\\
+ The halo width parameters \varlink{OLx}{OLx} and \varlink{OLy}{OLy}
+ have no special bearing on exch2 and may be assigned as in the general
+ case. The same holds for \varlink{Nr}{Nr}, the number of vertical
+ levels in the model.\\
+ The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
+ \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
+ tiles and how they are distributed on processors.  When using exch2,
+ the tiles are stored in single dimension, and so
+ \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
+ configured by exch2 cannot be split up accross processors without
+ regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\
+ The number of tiles MITgcm allocates and how they are distributed
+ between processors depends on \varlink{nPx}{nPx} and
+ \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
+ processor and \varlink{nPx}{nPx} the number of processors.  The total
+ number of tiles in the topology minus those listed in
+ \file{blanklist.txt} must equal \code{nSx*nPx}. \\
+ The following is an example of \file{SIZE.h} for the twelve-tile
+ configuration illustrated in figure \ref{fig:12tile} running on
+ one processor: \\
+ \begin{verbatim}
+       PARAMETER (
+      &           sNx =  16,
+      &           sNy =  32,
+      &           OLx =   2,
+      &           OLy =   2,
+      &           nSx =  12,
+      &           nSy =   1,
+      &           nPx =   1,
+      &           nPy =   1,
+      &           Nx  = sNx*nSx*nPx,
+      &           Ny  = sNy*nSy*nPy,
+      &           Nr  =   5)
+ \end{verbatim}
+ The following is an example for the twentyfour-tile topology in figure
+ \ref{fig:24tile} running on six processors:
+ \begin{verbatim}
+       PARAMETER (
+      &           sNx =  16,
+      &           sNy =  16,
+      &           OLx =   2,
+      &           OLy =   2,
+      &           nSx =   4,
+      &           nSy =   1,
+      &           nPx =   6,
+      &           nPy =   1,
+      &           Nx  = sNx*nSx*nPx,
+      &           Ny  = sNy*nSy*nPy,
+      &           Nr  =   5)
+ \end{verbatim}
  \subsection{Key Variables}
  The descriptions of the variables are divided up into scalars,
  one-dimensional arrays indexed to the tile number, and two and three
  dimensional arrays indexed to tile number and neighboring tile.  This
- division actually reflects the functionality of these variables: the
+ division reflects the functionality of these variables: The
  scalars are common to every part of the topology, the tile-indexed
- arrays to individual tiles, and the arrays indexed to tile and
+ arrays to individual tiles, and the arrays indexed by tile and
- neighbor to relationships between tiles and their neighbors.
+ neighbor to relationships between tiles and their neighbors. \\
  \subsubsection{Scalars}
  The number of tiles in a particular topology is set with the parameter
- \texttt{NTILES}, and the maximum number of neighbors of any tiles by
+ \code{NTILES}, and the maximum number of neighbors of any tiles by
- \texttt{MAX\_NEIGHBOURS}.  These parameters are used for defining the
+ \code{MAX\_NEIGHBOURS}.  These parameters are used for defining the
  size of the various one and two dimensional arrays that store tile
- parameters indexed to the tile number.
+ parameters indexed to the tile number and are assigned in the files
+ generated by \file{driver.m}.\\
  The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt}
  and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number
- of tiles in the x and y global indices.  For example, the default
+ of tiles in the $x$ and $y$ global indices.  For example, the default
- setup of six tiles has \texttt{exch2\_domain\_nxt=6} and
+ setup of six tiles (Fig. \ref{fig:6tile}) has
- \texttt{exch2\_domain\_nyt=1}.  A topology of twenty-four square (in
+ \code{exch2\_domain\_nxt=6} and \code{exch2\_domain\_nyt=1}.  A
- gridpoints) tiles, four (2x2) per subdomain, will have
+ topology of twenty-four square tiles, four per subdomain (as in figure
- \texttt{exch2\_domain\_nxt=12} and \texttt{exch2\_domain\_nyt=2}.
+ \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and
- Note that these parameters express the tile layout to allow global
+ \code{exch2\_domain\_nyt=2}.  Note that these parameters express the
- data files that are tile-layout-neutral and have no bearing on the
+ tile layout to allow global data files that are tile-layout-neutral
- internal storage of the arrays.  The tiles are internally stored in a
+ and have no bearing on the internal storage of the arrays.  The tiles
- range from \texttt{1,bi} (in the x axis) and y-axis variable
+ are internally stored in a range from \code{(1:\varlink{bi}{bi})} the
- \texttt{bj} is generally ignored within the package.
+ $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is generally
+ ignored within the package. \\
  \subsubsection{Arrays Indexed to Tile Number}
- The following arrays are of size \texttt{NTILES}, are indexed to the
+ The following arrays are of length \code{NTILES}and are indexed to the
- tile number, and the indices are omitted in their descriptions.
+ tile number, which is indicated in the diagrams with the notation
+ \textsf{t}$n$.  The indices are omitted in the descriptions. \\
  The arrays \varlink{exch2\_tnx}{exch2_tnx} and
- \varlink{exch2\_tny}{exch2_tny} express the x and y dimensions of each
+ \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
- tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
+ each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
- \texttt{exch2\_tny=sNy}, as assigned in \texttt{SIZE.h}.  Future
+ \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
- releases of MITgcm are to allow varying tile sizes.
+ section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
+ multiprocessing}.  Future releases of MITgcm are to allow varying tile
+ sizes. \\
  The location of the tiles' Cartesian origin within a subdomain are
  determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
  \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to
- relate the location of the edges of the tiles to each other.  As an
+ relate the location of the edges of different tiles to each other.  As
- example, in the default six-tile topology (the degenerate case) each
+ an example, in the default six-tile topology (Fig. \ref{fig:6tile})
- index in these arrays are set to 0.  The twenty-four, 32x32 cube face
+ each index in these arrays is set to \code{0} since a tile occupies
- case discussed above will have values of 0 or 16, depending on the
+ its entire subdomain.  The twentyfour-tile case discussed above will
- quadrant the tile falls within the subdomain.  The array
+ have values of \code{0} or \code{16}, depending on the quadrant the
- \varlink{exch2\_myFace}{exch2_myFace} contains the number of the
+ tile falls within the subdomain.  The elements of the arrays
- cubeface/subdomain of each tile, numbered 1-6 in the case of the
+ \varlink{exch2\_txglobalo}{exch2_txglobalo} and
- standard cube topology.
- The arrays \varlink{exch2\_txglobalo}{exch2_txglobalo} and
  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
  \varlink{exch2\_tbasex}{exch2_tbasex} and
  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the
- global address space, similar to that used by global files.
+ global address space, similar to that used by global files. \\
+ The array \varlink{exch2\_myFace}{exch2_myFace} contains the number of
+ the subdomain of each tile, in a range \code{(1:6)} in the case of the
+ standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
+ figures \ref{fig:12tile} and
+ \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
+ contains a count the  neighboring tiles each tile has, and is
+ used for setting bounds for looping over neighboring tiles.
+ \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
+ tile, and is used in interprocess communication.  \\
  The arrays \varlink{exch2\_isWedge}{exch2_isWedge},
  \varlink{exch2\_isEedge}{exch2_isEedge},
  \varlink{exch2\_isSedge}{exch2_isSedge}, and
- \varlink{exch2\_isNedge}{exch2_isNedge} are set to 1 if the indexed
+ \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
- tile lies on the edge of a subdomain, 0 if not.  The values are used
+ indexed tile lies on the respective edge of a subdomain, \code{0} if
- within the topology generator to determine the orientation of
+ not.  The values are used within the topology generator to determine
- neighboring tiles and to indicate whether a tile lies on the corner of
+ the orientation of neighboring tiles, and to indicate whether a tile
- a subdomain.  The latter case indicates special exchange and numerical
+ lies on the corner of a subdomain.  The latter case requires special
- handling for the singularities at the eight corners of the cube.
+ exchange and numerical handling for the singularities at the eight
- \varlink{exch2\_nNeighbours}{exch2_nNeighbours} contains a count of
+ corners of the cube. \\
- how many neighboring tiles each tile has, and is used for setting
- bounds for looping over neighboring tiles.
- \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
- tile, and is used in interprocess communication.
- \subsubsection{Arrays Indexed to Tile Number and Neighbor}
- The following arrays are all of size \texttt{MAX\_NEIGHBOURS} $\times$
+ \subsubsection{Arrays Indexed to Tile Number and Neighbor}
- \texttt{NTILES} and describe the orientations between the the tiles.
- The array \texttt{exch2\_neighbourId(a,T)} holds the tile number for
+ The following arrays are all of size
- each of the $n$ neighboring tiles.  The neighbor tiles are indexed
+ \code{MAX\_NEIGHBOURS}$\times$\code{NTILES} and describe the
- \texttt{(1,MAX\_NEIGHBOURS} in the order right to left on the north
+ orientations between the the tiles. \\
- then south edges, and then top to bottom on the east and west edges.
- Maybe throw in a fig here, eh?
+ The array \code{exch2\_neighbourId(a,T)} holds the tile number
+ \code{Tn} for each of the tile number \code{T}'s neighboring tiles
- The \texttt{exch2\_opposingSend\_record(a,T)} array holds the index c
+ \code{a}.  The neighbor tiles are indexed
- in \texttt{exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T.
+ \code{(1:exch2\_NNeighbours(T))} in the order right to left on the
- In other words,
+ north then south edges, and then top to bottom on the east and west
+ edges.  Maybe throw in a fig here, eh?  \\
+ \sloppy The \code{exch2\_opposingSend\_record(a,T)} array holds the
+ index \code{b} of the element in \texttt{exch2\_neighbourId(b,Tn)}
+ that holds the tile number \code{T}, given
+ \code{Tn=exch2\_neighborId(a,T)}.  In other words,
  \begin{verbatim}
     exch2_neighbourId( exch2_opposingSend_record(a,T),
                        exch2_neighbourId(a,T) ) = T
  \end{verbatim}
- and this provides a back-reference from the neighbor tiles.
+ This provides a back-reference from the neighbor tiles. \\
+ The arrays \varlink{exch2\_pi}{exch2_pi} and
+ \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices
+ in exchanges between the neighboring tiles.  These transformations are
+ necessary in exchanges between subdomains because the array index in
+ one dimension may map to the other index in an adjacent subdomain, and
+ may be have its indexing reversed. This swapping arises from the
+ ``folding'' of two-dimensional arrays into a three-dimensional cube.
+ The dimensions of \code{exch2\_pi(t,N,T)} and \code{exch2\_pj(t,N,T)}
+ are the neighbor ID \code{N} and the tile number \code{T} as explained
+ above, plus a vector of length \code{2} containing transformation
+ factors \code{t}.  The first element of the transformation vector
+ holds the factor to multiply the index in the same axis, and the
+ second element holds the the same for the orthogonal index.  To
+ clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis
+ index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor
+ \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s
+ $x$ index to the neighbor \code{N}'s $y$ index. \\
+ One of the two elements of \code{exch2\_pi} or \code{exch2\_pj} for a
+ given tile \code{T} and neighbor \code{N} will be \code{0}, reflecting
+ the fact that the two axes are orthogonal.  The other element will be
+ \code{1} or \code{-1}, depending on whether the axes are indexed in
+ the same or opposite directions.  For example, the transform vector of
+ the arrays for all tile neighbors on the same subdomain will be
+ \code{(1,0)}, since all tiles on the same subdomain are oriented
+ identically.  An axis that corresponds to the orthogonal dimension
+ with the same index direction in a particular tile-neighbor
+ orientation will have \code{(0,1)}.  Those in the opposite index
+ direction will have \code{(0,-1)} in order to reverse the ordering. \\
- The arrays \varlink{exch2\_pi}{exch2_pi},
+ The arrays \varlink{exch2\_oi}{exch2_oi},
- \varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi},
  \varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and
- \varlink{exch2\_oj\_f}{exch2_oj_f} specify the transformations in
+ \varlink{exch2\_oj\_f}{exch2_oj_f} are indexed to tile number and
- exchanges between the neighboring tiles.  The dimensions of
+ neighbor and specify the relative offset within the subdomain of the
- \texttt{exch2\_pi(t,N,T)} and \texttt{exch2\_pj(t,N,T)} are the
+ array index of a variable going from a neighboring tile $N$ to a local
- neighbor ID \textit{N} and the tile number \textit{T} as explained
+ tile $T$.  Consider the six-tile case (Fig. \ref{fig:6tile}), where
- above, plus the transformation vector {\em t }, of length two.  The
+ \code{exch2\_oi(1,1)=33}, \code{exch2\_oi(2,1)=0},
- first element of the transformation vector indicates the factor by
+ \code{exch2\_oi(3,1)=32}, and \code{exch2\_oi(4,1)=-32}.  Each of these
- which variables representing the same vector component of a tile will
+ indicates the offset in the $x$ direction \\
- be multiplied, and the second element indicates the transform to the
- variable in the other direction.  As an example,
+ Finally, \varlink{exch2\_itlo\_c}{exch2_itlo_c},
- \texttt{exch2\_pi(1,N,T)} holds the transform of the i-component of a
+ \varlink{exch2\_ithi\_c}{exch2_ithi_c},
- vector variable in tile \texttt{T} to the i-component of tile
+ \varlink{exch2\_jtlo\_c}{exch2_jtlo_c} and
- \texttt{T}'s neighbor \texttt{N}, and \texttt{exch2\_pi(2,N,T)} hold
+ \varlink{exch2\_jthi\_c}{exch2_jthi_c} hold the location and index
- the component of neighbor \texttt{N}'s j-component.
+ bounds of the edge segment of the neighbor tile \code{N}'s subdomain
+ that gets exchanged with the local tile \code{T}.  To take the example
- Under the current cube topology, one of the two elements of
+ of tile \code{T=2} in the twelve-tile topology
- \texttt{exch2\_pi} or \texttt{exch2\_pj} for a given tile \texttt{T}
+ (Fig. \ref{fig:12tile}): \\
- and neighbor \texttt{N} will be 0, reflecting the fact that the vector
- components are orthogonal.  The other element will be 1 or -1,
+ \begin{verbatim}
- depending on whether the components are indexed in the same or
+        exch2_itlo_c(4,2)=17
- opposite directions.  For example, the transform dimension of the
+        exch2_ithi_c(4,2)=17
- arrays for all tile neighbors on the same subdomain will be [1,0],
+        exch2_jtlo_c(4,2)=0
- since all tiles on the same subdomain are oriented identically.
+        exch2_jthi_c(4,2)=33
- Vectors that correspond to the orthogonal dimension with the same
+ \end{verbatim}
- index direction will have [0,1], whereas those in the opposite index
- direction will have [0,-1].
+ Here \code{N=4}, indicating the western neighbor, which is \code{Tn=1}.
+ \code{Tn=1} resides on the same subdomain as \code{T=2}, so the tiles
+ have the same orientation and the same $x$ and $y$ axes.  The $i$
- {\footnotesize
+ component is orthogonal to the western edge and the tile is 16 points
- \begin{verbatim}
+ wide, so \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} indicate the
- C      exch2_pi          :: X index row of target to source permutation
+ column beyond \code{Tn=1}'s eastern edge, in that tile's halo
- C                        :: matrix for each neighbour entry.
+ region. Since the border of the tiles extends through the entire
- C      exch2_pj          :: Y index row of target to source permutation
+ height of the subdomain, the $y$ axis bounds \code{exch2\_jtlo\_c} to
- C                        :: matrix for each neighbour entry.
+ \code{exch2\_jthi\_c} cover the height, plus 1 in either direction to
- C      exch2_oi          :: X index element of target to source
+ cover part of the halo. \\
- C                        :: offset vector for cell-centered quantities
- C                        :: of each neighbor entry.
+ For the north edge of the same tile \code{T=2} where \code{N=1} and
- C      exch2_oj          :: Y index element of target to source
+ the neighbor tile is \code{Tn=5}:
- C                        :: offset vector for cell-centered quantities
- C                        :: of each neighbor entry.
+ \begin{verbatim}
- C      exch2_oi_f        :: X index element of target to source
+        exch2_itlo_c(1,2)=0
- C                        :: offset vector for face quantities
+        exch2_ithi_c(1,2)=0
- C                        :: of each neighbor entry.
+        exch2_jtlo_c(1,2)=0
- C      exch2_oj_f        :: Y index element of target to source
+        exch2_jthi_c(1,2)=17
- C                        :: offset vector for face quantities
- C                        :: of each neighbor entry.
  \end{verbatim}
- }
+ \code{T}'s northern edge is parallel to the $x$ axis, but since
+ \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis,
+ \code{T}'s northern edge exchanges with \code{Tn}'s western edge.
+ The western edge of the tiles corresponds to the lower bound of the
+ $x$ axis, so \code{exch2\_itlo\_c} \code{exch2\_ithi\_c} are \code{0}. The
+ range of \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the
+ width of \code{T}'s northern edge, plus the halo. \\
+ This needs some diagrams. \\

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