| 10 |
%% o automatically inserted at \section{Reference} |
%% o automatically inserted at \section{Reference} |
| 11 |
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| 12 |
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| 13 |
\section{exch2: Extended Cubed Sphere Exchange} |
\section{exch2: Extended Cubed Sphere \mbox{Topology}} |
| 14 |
|
\label{sec:exch2} |
|
\subsection{Introduction} |
|
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|
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|
The exch2 package is an extension to the original cubed sphere exchanges |
|
|
to allow more flexible domain decomposition and parallelization. Cube faces |
|
|
(subdomain) may be divided into whatever number of tiles that divide evenly |
|
|
into the grid point dimensions of the subdomain. Furthermore, the individual |
|
|
tiles may be run on different processors in any combination, (tone this down |
|
|
a bit), and whether exchanges between particular tiles occur between different |
|
|
processors is decided at runtime. |
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|
| 15 |
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| 16 |
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| 17 |
|
\subsection{Introduction} |
| 18 |
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|
| 19 |
|
The \texttt{exch2} package is an extension to the original cubed |
| 20 |
|
sphere topological configuration that allows more flexible domain |
| 21 |
|
decomposition and parallelization. Cube faces (also called |
| 22 |
|
subdomains) may be divided into any number of tiles that divide evenly |
| 23 |
|
into the grid point dimensions of the subdomain. Furthermore, the |
| 24 |
|
individual tiles may be run on separate processors in different |
| 25 |
|
combinations, and whether exchanges between particular tiles occur |
| 26 |
|
between different processors is determined at runtime. This |
| 27 |
|
flexibility provides for manual compile-time load balancing across a |
| 28 |
|
relatively arbitrary number of processors. \\ |
| 29 |
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|
| 30 |
|
The exchange parameters are declared in |
| 31 |
|
\filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h} |
| 32 |
|
and assigned in |
| 33 |
|
\filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The |
| 34 |
|
validity of the cube topology depends on the \file{SIZE.h} file as |
| 35 |
|
detailed below. Both files are generated by Matlab scripts in |
| 36 |
|
\file{utils/exch2/matlab-topology-generator}; see Section |
| 37 |
|
\ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2} |
| 38 |
|
for details on creating alternate topologies. The default files |
| 39 |
|
provided in the release configure a cubed sphere topology of six |
| 40 |
|
tiles, one per subdomain, each with 32$\times$32 grid points, all |
| 41 |
|
running on a single processor. Pregenerated examples of these files |
| 42 |
|
with alternate topologies are provided under |
| 43 |
|
\file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h} |
| 44 |
|
file for single-processor execution. |
| 45 |
|
|
| 46 |
|
\subsection{Invoking exch2} |
| 47 |
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|
| 48 |
|
To use exch2 with the cubed sphere, the following conditions must be |
| 49 |
|
met: \\ |
| 50 |
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|
| 51 |
|
$\bullet$ The exch2 package is included when \file{genmake2} is run. |
| 52 |
|
The easiest way to do this is to add the line \code{exch2} to the |
| 53 |
|
\file{profile.conf} file -- see Section |
| 54 |
|
\ref{sect:buildingCode}\sectiontitle{Building the code} for general |
| 55 |
|
details. \\ |
| 56 |
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|
| 57 |
|
$\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and |
| 58 |
|
\file{w2\_e2setup.F} must reside in a directory containing code |
| 59 |
|
linked when \file{genmake2} runs. The safest place to put these |
| 60 |
|
is the directory indicated in the \code{-mods=DIR} command line |
| 61 |
|
modifier (typically \file{../code}), or the build directory. The |
| 62 |
|
default versions of these files reside in \file{pkg/exch2} and are |
| 63 |
|
linked automatically if no other versions exist elsewhere in the |
| 64 |
|
link path, but they should be left untouched to avoid breaking |
| 65 |
|
configurations other than the one you intend to modify.\\ |
| 66 |
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|
| 67 |
|
$\bullet$ Files containing grid parameters, named |
| 68 |
|
\file{tile???.mitgrid} where \file{???} is \file{001} through |
| 69 |
|
\file{006} (one per subdomain), must be in the working directory |
| 70 |
|
when the MITgcm executable is run. These files are provided in the |
| 71 |
|
example experiments for cubed sphere configurations with |
| 72 |
|
32$\times$32 cube sides and are non-trivial to generate -- please |
| 73 |
|
contact MITgcm support if you want to generate files for other |
| 74 |
|
configurations. \\ |
| 75 |
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|
| 76 |
|
$\bullet$ As always when compiling MITgcm, the file \file{SIZE.h} |
| 77 |
|
must be placed where \file{genmake2} will find it. In particular |
| 78 |
|
for the exch2, the domain decomposition specified in \file{SIZE.h} |
| 79 |
|
must correspond with the particular configuration's topology |
| 80 |
|
specified in \file{W2\_EXCH2\_TOPOLOGY.h} and |
| 81 |
|
\file{w2\_e2setup.F}. Domain decomposition issues particular to |
| 82 |
|
exch2 are addressed in Section \ref{sec:topogen} \sectiontitle{Generating |
| 83 |
|
Topology Files for exch2}; a more general background on the subject |
| 84 |
|
relvant to MITgcm is presented in Section |
| 85 |
|
\ref{sect:specifying_a_decomposition}\sectiontitle{Specifying a |
| 86 |
|
decomposition}.\\ |
| 87 |
|
|
| 88 |
|
As of the time of writing the following examples use exch2 and may be |
| 89 |
|
used for guidance: |
| 90 |
|
|
| 91 |
|
\begin{verbatim} |
| 92 |
|
verification/adjust_nlfs.cs-32x32x1 |
| 93 |
|
verification/adjustment.cs-32x32x1 |
| 94 |
|
verification/aim.5l_cs |
| 95 |
|
verification/global_ocean.cs32x15 |
| 96 |
|
verification/hs94.cs-32x32x5 |
| 97 |
|
\end{verbatim} |
| 98 |
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|
| 99 |
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|
| 100 |
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|
| 101 |
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|
| 102 |
|
\subsection{Generating Topology Files for exch2} |
| 103 |
|
\label{sec:topogen} |
| 104 |
|
|
| 105 |
|
Alternate cubed sphere topologies may be created using the Matlab |
| 106 |
|
scripts in \file{utils/exch2/matlab-topology-generator}. Running the |
| 107 |
|
m-file \file{driver.m} from the Matlab prompt (there are no parameters |
| 108 |
|
to pass) generates exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} |
| 109 |
|
and \file{w2\_e2setup.F} in the working directory and displays a |
| 110 |
|
figure of the topology via Matlab. The other m-files in the directory |
| 111 |
|
are subroutines of \file{driver.m} and should not be run except for |
| 112 |
|
development purposes. \\ |
| 113 |
|
|
| 114 |
|
The parameters that determine the dimensions and topology of the |
| 115 |
|
generated configuration are \code{nr}, \code{nb}, \code{ng}, |
| 116 |
|
\code{tnx} and \code{tny}, and all are assigned early in the script. |
| 117 |
|
|
| 118 |
|
The first three determine the size of the subdomains (cube faces) and |
| 119 |
|
hence the size of the overall domain. Each one determines the number |
| 120 |
|
of grid points, and therefore the resolution, along the subdomain |
| 121 |
|
sides in a ``great circle'' around each axis of the cube. At the time |
| 122 |
|
of this writing MITgcm requires these three parameters to be equal, |
| 123 |
|
but they provide for future releases of MITgcm to accomodate different |
| 124 |
|
resolutions around the axes to allow (for example) greater resolution |
| 125 |
|
around the equator.\\ |
| 126 |
|
|
| 127 |
|
The parameters \code{tnx} and \code{tny} determine the dimensions of |
| 128 |
|
the tiles into which the subdomains are decomposed, and must evenly |
| 129 |
|
divide the integer assigned to \code{nr}, \code{nb} and \code{ng}. |
| 130 |
|
The result is a rectangular tiling of the subdomain. Figure |
| 131 |
|
\ref{fig:24tile} shows one possible topology for a twenty-four tile |
| 132 |
|
cube, and figure \ref{fig:12tile} shows one for twelve tiles. \\ |
| 133 |
|
|
| 134 |
|
\begin{figure} |
| 135 |
|
\begin{center} |
| 136 |
|
\resizebox{4in}{!}{ |
| 137 |
|
\includegraphics{part6/s24t_16x16.ps} |
| 138 |
|
} |
| 139 |
|
\end{center} |
| 140 |
|
\caption{Plot of cubed sphere topology with a 32$\times$32 grid and |
| 141 |
|
twenty-four tiles (\code{tnx=16, tny=16}) |
| 142 |
|
} \label{fig:24tile} |
| 143 |
|
\end{figure} |
| 144 |
|
|
| 145 |
|
\begin{figure} |
| 146 |
|
\begin{center} |
| 147 |
|
\resizebox{4in}{!}{ |
| 148 |
|
\includegraphics{part6/s12t_16x32.ps} |
| 149 |
|
} |
| 150 |
|
\end{center} |
| 151 |
|
\caption{Plot of cubed sphere topology with a 32$\times$32 grid and |
| 152 |
|
twelve tiles (\code{tnx=16, tny=32}) |
| 153 |
|
} \label{fig:12tile} |
| 154 |
|
\end{figure} |
| 155 |
|
|
| 156 |
|
Tiles can be selected from the topology to be omitted from being |
| 157 |
|
allocated memory and processors. This kind otuning is useful in |
| 158 |
|
ocean modeling for omitting tiles that fall entirely on land. The |
| 159 |
|
tiles omitted are specified in the file \file{blanklist.txt} by |
| 160 |
|
their tile number in the topology, separated by a newline. \\ |
| 161 |
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|
| 162 |
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|
| 163 |
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|
| 164 |
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|
| 165 |
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|
| 166 |
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|
| 167 |
|
\subsection{Key Variables} |
| 168 |
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|
| 169 |
|
The descriptions of the variables are divided up into scalars, |
| 170 |
|
one-dimensional arrays indexed to the tile number, and two and three |
| 171 |
|
dimensional arrays indexed to tile number and neighboring tile. This |
| 172 |
|
division actually reflects the functionality of these variables: the |
| 173 |
|
scalars are common to every part of the topology, the tile-indexed |
| 174 |
|
arrays to individual tiles, and the arrays indexed to tile and |
| 175 |
|
neighbor to relationships between tiles and their neighbors. |
| 176 |
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|
| 177 |
|
\subsubsection{Scalars} |
| 178 |
|
|
| 179 |
|
The number of tiles in a particular topology is set with the parameter |
| 180 |
|
\texttt{NTILES}, and the maximum number of neighbors of any tiles by |
| 181 |
|
\texttt{MAX\_NEIGHBOURS}. These parameters are used for defining the |
| 182 |
|
size of the various one and two dimensional arrays that store tile |
| 183 |
|
parameters indexed to the tile number.\\ |
| 184 |
|
|
| 185 |
|
The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt} |
| 186 |
|
and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number |
| 187 |
|
of tiles in the x and y global indices. For example, the default |
| 188 |
|
setup of six tiles has \texttt{exch2\_domain\_nxt=6} and |
| 189 |
|
\texttt{exch2\_domain\_nyt=1}. A topology of twenty-four square (in |
| 190 |
|
gridpoints) tiles, four (2x2) per subdomain, will have |
| 191 |
|
\texttt{exch2\_domain\_nxt=12} and \texttt{exch2\_domain\_nyt=2}. |
| 192 |
|
Note that these parameters express the tile layout to allow global |
| 193 |
|
data files that are tile-layout-neutral and have no bearing on the |
| 194 |
|
internal storage of the arrays. The tiles are internally stored in a |
| 195 |
|
range from \texttt{1,bi} (in the x axis) and y-axis variable |
| 196 |
|
\texttt{bj} is generally ignored within the package. |
| 197 |
|
|
| 198 |
|
\subsubsection{Arrays Indexed to Tile Number} |
| 199 |
|
|
| 200 |
|
The following arrays are of size \texttt{NTILES}, are indexed to the |
| 201 |
|
tile number, and the indices are omitted in their descriptions. |
| 202 |
|
|
| 203 |
|
The arrays \varlink{exch2\_tnx}{exch2_tnx} and |
| 204 |
|
\varlink{exch2\_tny}{exch2_tny} express the x and y dimensions of each |
| 205 |
|
tile. At present for each tile \texttt{exch2\_tnx=sNx} and |
| 206 |
|
\texttt{exch2\_tny=sNy}, as assigned in \texttt{SIZE.h}. Future |
| 207 |
|
releases of MITgcm are to allow varying tile sizes. |
| 208 |
|
|
| 209 |
|
The location of the tiles' Cartesian origin within a subdomain are |
| 210 |
|
determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and |
| 211 |
|
\varlink{exch2\_tbasey}{exch2_tbasey}. These variables are used to |
| 212 |
|
relate the location of the edges of the tiles to each other. As an |
| 213 |
|
example, in the default six-tile topology (the degenerate case) each |
| 214 |
|
index in these arrays are set to 0. The twenty-four, 32x32 cube face |
| 215 |
|
case discussed above will have values of 0 or 16, depending on the |
| 216 |
|
quadrant the tile falls within the subdomain. The array |
| 217 |
|
\varlink{exch2\_myFace}{exch2_myFace} contains the number of the |
| 218 |
|
cubeface/subdomain of each tile, numbered 1-6 in the case of the |
| 219 |
|
standard cube topology. |
| 220 |
|
|
| 221 |
|
The arrays \varlink{exch2\_txglobalo}{exch2_txglobalo} and |
| 222 |
|
\varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to |
| 223 |
|
\varlink{exch2\_tbasex}{exch2_tbasex} and |
| 224 |
|
\varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the |
| 225 |
|
global address space, similar to that used by global files. |
| 226 |
|
|
| 227 |
|
The arrays \varlink{exch2\_isWedge}{exch2_isWedge}, |
| 228 |
|
\varlink{exch2\_isEedge}{exch2_isEedge}, |
| 229 |
|
\varlink{exch2\_isSedge}{exch2_isSedge}, and |
| 230 |
|
\varlink{exch2\_isNedge}{exch2_isNedge} are set to 1 if the indexed |
| 231 |
|
tile lies on the edge of a subdomain, 0 if not. The values are used |
| 232 |
|
within the topology generator to determine the orientation of |
| 233 |
|
neighboring tiles and to indicate whether a tile lies on the corner of |
| 234 |
|
a subdomain. The latter case indicates special exchange and numerical |
| 235 |
|
handling for the singularities at the eight corners of the cube. |
| 236 |
|
\varlink{exch2\_nNeighbours}{exch2_nNeighbours} contains a count of |
| 237 |
|
how many neighboring tiles each tile has, and is used for setting |
| 238 |
|
bounds for looping over neighboring tiles. |
| 239 |
|
\varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each |
| 240 |
|
tile, and is used in interprocess communication. |
| 241 |
|
|
| 242 |
|
\subsubsection{Arrays Indexed to Tile Number and Neighbor} |
| 243 |
|
|
| 244 |
|
The following arrays are all of size \texttt{MAX\_NEIGHBOURS} $\times$ |
| 245 |
|
\texttt{NTILES} and describe the orientations between the the tiles. |
| 246 |
|
|
| 247 |
|
The array \texttt{exch2\_neighbourId(a,T)} holds the tile number for |
| 248 |
|
each of the $n$ neighboring tiles. The neighbor tiles are indexed |
| 249 |
|
\texttt{(1,MAX\_NEIGHBOURS} in the order right to left on the north |
| 250 |
|
then south edges, and then top to bottom on the east and west edges. |
| 251 |
|
Maybe throw in a fig here, eh? |
| 252 |
|
|
| 253 |
|
The \texttt{exch2\_opposingSend\_record(a,T)} array holds the index c |
| 254 |
|
in \texttt{exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T. |
| 255 |
|
In other words, |
| 256 |
|
\begin{verbatim} |
| 257 |
|
exch2_neighbourId( exch2_opposingSend_record(a,T), |
| 258 |
|
exch2_neighbourId(a,T) ) = T |
| 259 |
|
\end{verbatim} |
| 260 |
|
and this provides a back-reference from the neighbor tiles. |
| 261 |
|
|
| 262 |
|
The arrays \varlink{exch2\_pi}{exch2_pi}, |
| 263 |
|
\varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi}, |
| 264 |
|
\varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and |
| 265 |
|
\varlink{exch2\_oj\_f}{exch2_oj_f} specify the transformations in |
| 266 |
|
exchanges between the neighboring tiles. The dimensions of |
| 267 |
|
\texttt{exch2\_pi(t,N,T)} and \texttt{exch2\_pj(t,N,T)} are the |
| 268 |
|
neighbor ID \textit{N} and the tile number \textit{T} as explained |
| 269 |
|
above, plus the transformation vector {\em t }, of length two. The |
| 270 |
|
first element of the transformation vector indicates the factor by |
| 271 |
|
which variables representing the same vector component of a tile will |
| 272 |
|
be multiplied, and the second element indicates the transform to the |
| 273 |
|
variable in the other direction. As an example, |
| 274 |
|
\texttt{exch2\_pi(1,N,T)} holds the transform of the i-component of a |
| 275 |
|
vector variable in tile \texttt{T} to the i-component of tile |
| 276 |
|
\texttt{T}'s neighbor \texttt{N}, and \texttt{exch2\_pi(2,N,T)} hold |
| 277 |
|
the component of neighbor \texttt{N}'s j-component. |
| 278 |
|
|
| 279 |
|
Under the current cube topology, one of the two elements of |
| 280 |
|
\texttt{exch2\_pi} or \texttt{exch2\_pj} for a given tile \texttt{T} |
| 281 |
|
and neighbor \texttt{N} will be 0, reflecting the fact that the vector |
| 282 |
|
components are orthogonal. The other element will be 1 or -1, |
| 283 |
|
depending on whether the components are indexed in the same or |
| 284 |
|
opposite directions. For example, the transform dimension of the |
| 285 |
|
arrays for all tile neighbors on the same subdomain will be [1,0], |
| 286 |
|
since all tiles on the same subdomain are oriented identically. |
| 287 |
|
Vectors that correspond to the orthogonal dimension with the same |
| 288 |
|
index direction will have [0,1], whereas those in the opposite index |
| 289 |
|
direction will have [0,-1]. |
| 290 |
|
|
| 291 |
|
|
| 292 |
|
{\footnotesize |
| 293 |
|
\begin{verbatim} |
| 294 |
|
C exch2_pi :: X index row of target to source permutation |
| 295 |
|
C :: matrix for each neighbour entry. |
| 296 |
|
C exch2_pj :: Y index row of target to source permutation |
| 297 |
|
C :: matrix for each neighbour entry. |
| 298 |
|
C exch2_oi :: X index element of target to source |
| 299 |
|
C :: offset vector for cell-centered quantities |
| 300 |
|
C :: of each neighbor entry. |
| 301 |
|
C exch2_oj :: Y index element of target to source |
| 302 |
|
C :: offset vector for cell-centered quantities |
| 303 |
|
C :: of each neighbor entry. |
| 304 |
|
C exch2_oi_f :: X index element of target to source |
| 305 |
|
C :: offset vector for face quantities |
| 306 |
|
C :: of each neighbor entry. |
| 307 |
|
C exch2_oj_f :: Y index element of target to source |
| 308 |
|
C :: offset vector for face quantities |
| 309 |
|
C :: of each neighbor entry. |
| 310 |
|
\end{verbatim} |
| 311 |
|
} |
| 312 |
|
|
| 313 |
|
|
| 314 |
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|
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