numerical simulation of laminar diffusion flames problem ... - MGNet

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The nonlinear residual calculation uses the same vector decomposition used in thematrix{vector and Jacobian routines. Hence, no data needs to be transferred betweenprocessors to start this procedure.One aspect of the Jacobians having so many nonzeroes is that we can aord to storethe complete solution vector on a single processor without using much extra memory.In our case, our parallel computers had from 128 Mb to 2 Gb of local memory on eachnode. Hence, if some operation is quick and does not parallelize well (or at all), we canactually gather all of the data, do the operation on a single node, and then scatter databack to the remaining processors. This was particularly useful while parallelizing thecode one major operation at a time and for debugging purposes.The general computing methodology used was that there was one processor thatdirected all of the processors to do various computing tasks. These included havingprocessors do their part of the parallel iterative procedures, compute a Jacobian, orevaluate a nonlinear residual. Large tasks are the typical operation, not small tasks likerequiring all processors to do an inner product.The communications library we used, EUIH [6], allowed us to produce one executablefor all of the following IBM computers: SP1, SP2, and clusters of IBM RS-6000's. No relinking or recompiling was necessary in order to use the fast switches.Hence, only one copy of the source les and one executable were maintained for all ofthese environments.NUMERICAL EXPERIMENTSThe ame sheet model provides a test problem of a moderate computational costwith which to probe the eciency of solution algorithms that can then be used totackle detailed chemistry diusion problems. Such a study is presented in [2] [3]. Theame sheet that is considered is adequately resolved on a grid with up to 210 4 nodes,with 20% of the mesh nodes clustered in a region covering 0.1% of the computationaldomain. This requires up to 100 megabytes of work space. The adaptive mesh multigridprocedure outlined above is particularly ecient for solving ame sheet problems. Incomparison to traditional solution procedures, the total execution times drop by a factorof 10. Speedups as high as 166 in the time relaxation phase have been seen on a singleprocessor. For three dimensional problems, speedups much greater than 10 should beobtained. At the beginning of this study, wespentasmuch as96minutes on the amesheet phase using an IBM RISC System/6000 model 560 workstation. Recently, wespent as little as 45 seconds using an eight node SP2.Solving a nite rate chemistry ame problem instead of a ame sheet problemis a quite challenging step which involves a dramatic increase in the diculty of theproblem. This is mostly attributable to the much larger number of dependent unknowns,the nonlinear uid dynamics-thermochemistry coupling, and the disparate length scalesthat must be resolved in the computed solution. In particular, excellent resolution ofa ame sheet problem requires up to 128 megabytes, while a reasonable solution ofa diusion ame with 50 chemical species may require up to a gigabyte. Traditionalsolvers for two-dimensional laminar diusion ames used to require more than a hundred8
CPU hours on a supercomputer to get a solution. Thanks to recent developments ofcomputational algorithms and computer capabilities, this is no longer the case.The ame conguration is a methane-air lifted laminar diusion ame with a tripleame structure at its base, as illustrated in Figure 1. Although extremely dicult tocompute numerically, this ame conguration is chosen in the present study since experimentaland numerical results are already available for these ames [8]. The numericalsolution that is considered rst includes 16 chemical species engaged in a C1-chain reactionmechanism, i.e., only molecules with at most one carbon atom are considered. Inaddition to the 16 chemical species, 4 unknowns are associated with each mesh point.Hence, there are a total of 20 unknowns per mesh point. The ame is appropriatelyresolved using 510 3 mesh nodes, and very good agreement with previous experimental(2:510 5 data points) and numerical data is obtained.Consider a one way nonlinear multigrid with Bi{CGSTAB/GS as the solver on alllevels. On the nest level, these are the (wall clock) times we see to converge on the89 85 ne grid.Machine Processors MinutesRISC System/6000{580 1 602.59SP1 4 137.218 70.3712 58.8916 53.85SP2 4 47.416 30.05Peak oating point is about 130 Mops for the SP1 and RISC System/6000{580, butabout 266 for the SP2.The chemical mechanism just considered yields adequate resolution for the temperatureand several chemical species. More detailed investigation of the ame structurerequires additional chemical species in the model. For instance, the study of pollutantformation, such as nitric oxide, requires up to 50 chemical species. While thememory requirements are too large for current serial computers, such problems can beadequately solved on parallel computers with a large amount of memory per node. Thediusion ame just described can then be conveniently used to initialize diusion amecalculations with more detailed chemistry submodels. Such work is in progress.REFERENCES[1] C. C. Douglas, Implementing abstract multigrid or multilevel methods, in Sixth Copper MountainConference on Multigrid Methods, N. D. Melson, T. A. Manteuel, and S. F. McCormick, eds.,vol. CP 3224, Hampton, VA, 1993, NASA, pp. 127{141.[2] C. C. Douglas and A. Ern, Numerical solution of ame sheet problems with and withoutmultigrid methods, in Sixth Copper Mountain Conference on Multigrid Methods, N. D. Melson,T. A. Manteuel, and S. F. McCormick, eds., vol. CP 3224, Hampton, VA, 1993, NASA,pp. 143{157.9
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