A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux...
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![Page 1: A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations A. Beaudoin, J.-R. De Dreuzy and J. Erhel.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649ea95503460f94bacd36/html5/thumbnails/1.jpg)
A comparison between
a direct and a multigrid sparse linear solvers
for highly heterogeneous flux computations
A. Beaudoin, J.-R. De Dreuzy and J. ErhelA. Beaudoin, J.-R. De Dreuzy and J. Erhel ECCOMAS CFD 06, Egmond aan Zee, the Netherlands, September 2006
![Page 2: A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations A. Beaudoin, J.-R. De Dreuzy and J. Erhel.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649ea95503460f94bacd36/html5/thumbnails/2.jpg)
2D Heterogeneous 2D Heterogeneous permeability fieldpermeability fieldStochastic model Y = ln(K)Stochastic model Y = ln(K)with correlation functionwith correlation function
2( ) expY YY
C
rr
31 Y
Physical flow modelPhysical flow model
Q = - K*grad (h)
div (Q) = 0
Fix
ed
head
Fix
ed
head
Nul flux
Nul flux
![Page 3: A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations A. Beaudoin, J.-R. De Dreuzy and J. Erhel.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649ea95503460f94bacd36/html5/thumbnails/3.jpg)
Examples of simulationsExamples of simulationsσσ=0.5 and =0.5 and σσ=3=3
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Numerical method for 2D heterogeneous Numerical method for 2D heterogeneous porous mediumporous medium
Finite Volume Method with a regular mesh
Large sparse structured matrix of order N with 5 entries per row
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Sparse direct solverSparse direct solver
memory size and CPU time with memory size and CPU time with PSPASESPSPASES
Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5)
![Page 6: A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations A. Beaudoin, J.-R. De Dreuzy and J. Erhel.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649ea95503460f94bacd36/html5/thumbnails/6.jpg)
Multigrid sparse solverMultigrid sparse solver
convergence and CPU time with convergence and CPU time with HYPRE/SMGHYPRE/SMG
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Parallel architectureParallel architecturedistributed memorydistributed memory
2 nodes of 32 bi – processors 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go (Proc AMD Opteron 2Ghz with 2Go
of RAM)of RAM)
Parallel architectureParallel architecture
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Direct and multigrid solversDirect and multigrid solvers
Parallel CPU times for various sizesParallel CPU times for various sizes
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Direct and multigrid solversDirect and multigrid solvers
Speed-ups for various sizesSpeed-ups for various sizes
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Direct solverDirect solver
Scalability analysis with PSPASES : Scalability analysis with PSPASES : isoefficiencyisoefficiency
PTp
TE S
PTp
NR
P N Tp R
2 0.26 106 5.60 1.20 106
8 1.05 106 11.33 1.18 106
32 4.19 106 25.70 1,04 106
4 0.26 106 2.92 1.15 106
16 1.05 106 6.06 1.11 106
64 4.19 106 13.08 1,05 106
5.1
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Multigrid solverMultigrid solver
Impact of permeability standard deviation Impact of permeability standard deviation and system sizeand system size
Convergence and CPU timeConvergence and CPU time
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Multigrid solverMultigrid solver
Impact of permeability standard deviation Impact of permeability standard deviation and system sizeand system size
Convergence and CPU timeConvergence and CPU time
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Direct and multigrid solversDirect and multigrid solvers
Impact of permeability standard Impact of permeability standard deviationdeviation
![Page 14: A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations A. Beaudoin, J.-R. De Dreuzy and J. Erhel.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649ea95503460f94bacd36/html5/thumbnails/14.jpg)
Direct and multigrid solversDirect and multigrid solvers
SummarySummary
• PSPASES is more efficient for small matrices• PSPASES is scalable and is more efficient with many processors• HYPRE requires less memory• HYPRE is more efficient for large matrices • HYPRE is very sensitive to the permeability variance
• Another method for large matrices and large variance ?