Post on 12-Jan-2016
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
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
Examples of simulationsExamples of simulationsσσ=0.5 and =0.5 and σσ=3=3
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
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)
Multigrid sparse solverMultigrid sparse solver
convergence and CPU time with convergence and CPU time with HYPRE/SMGHYPRE/SMG
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
Direct and multigrid solversDirect and multigrid solvers
Parallel CPU times for various sizesParallel CPU times for various sizes
Direct and multigrid solversDirect and multigrid solvers
Speed-ups for various sizesSpeed-ups for various sizes
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
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
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
Direct and multigrid solversDirect and multigrid solvers
Impact of permeability standard Impact of permeability standard deviationdeviation
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 ?