efficient parallel computation to simulate blood flow - CiteSeerX

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Chapter 2 Optimum design of a parallel solver for 2D CFD codes 2.1 Introduction and motivations Solving efficiently a linear system is paramount for many scientific applications such as CFD codes, since it is usually the most time consuming part of the code. Depending on the size of the problem, between 50% and 97% of the elapsed time of the whole code is dedicated to the resolution of the linear system of an incompressible Navier Stokes code [60]. Our study is focused only on finite differences and finite volume scheme on a mesh topologically equivalent to Cartesian mesh and for elliptic solver. This context allows us to deal easily with domain decomposition and to combine it with the 19
penalty method [8] to execute realistic simulations with real geometries. This will be detailed in the third chapter. The goal of this chapter is to present a numerical software interface that can be integrated easily in a CFD or heat transfer code and allows the systematic investigation of the efficiency of a broad class of solvers to optimize the code. In this chapter, the study is focused on two dimensional space applications. We systematically investigate the performance of solvers with different test cases and we show for each test case that the choice of the best solver may depend critically on the grid size, the aspect ratio of the grid, and further the physical parameters of the problem and the architecture of the processor. We have developed an interface that allows us to include easily in an existing CFD or heat transfer code any of the elliptic solvers available in Lapack, Sparskit and Hypre. This interface has the simplicity of a Matlab command but keeps the efficiency of the original Fortran or C library. This interface can help us to systematically investigate what is the best solver as a preprocessing procedure. This will be used for a domain decomposition technique in order to get the fastest solver method for the resolution of a subdomain. In this chapter, we will first describe the interface solver with a brief overview on the different libraries used and a performance analysis on different test cases. Then, we present a domain decomposition technique called Aitken Schwarz for distributing a computational domain and solving the problem with the fastest subdomain solver. Finally, we analyze the results and output the performances on a two dimensional 20
Page 1 and 2: EFFICIENT PARALLEL COMPUTATION TO S
Page 3 and 4: Acknowledgments First, I would like
Page 5 and 6: and the Tunisian community and its
Page 7 and 8: Abstract Hemodynamic simulation is
Page 9 and 10: 2.3.2 Overview on the AS method . .
Page 11 and 12: List of Figures 1.1 Steps of the bl
Page 13 and 14: 3.15 Speedup of the Navier Stokes c
Page 15 and 16: List of Tables 2.1 Descriptions of
Page 17 and 18: process called aneurysm, or a steno
Page 19 and 20: plaques, which induces the thicknes
Page 21 and 22: 1.2 Incompressible Navier Stokes In
Page 23 and 24: with some synchronization. Parallel
Page 25 and 26: 1.4 Domain decomposition As describ
Page 27 and 28: Another technique to accelerate the
Page 29 and 30: have been developed to solve effici
Page 31 and 32: 1.6 Objective of the dissertation T
Page 33: technique has been studied and comp
Page 37 and 38: in Lapack, Sparskit, and Hypre. •
Page 39 and 40: this dissertation, we address only
Page 41 and 42: • options2: stands for more advan
Page 43 and 44: Figure 2.1: Comparison between BICG
Page 45 and 46: Figure 2.4: Comparison between BICG
Page 47 and 48: Figure 2.6: Comparison of the elaps
Page 49 and 50: Figure 2.10: Surface Response on Hy
Page 51 and 52: 2.3.1 Introduction and Motivations
Page 53 and 54: only neighboring subdomains. The ra
Page 55 and 56: u n+1 1|Γ 1 = u n 2|Γ 1 , u n+1 2
Page 57 and 58: • step 5: Apply the Aitken accele
Page 59 and 60: Let us denote λ k = 4 sin(kπhy/2)
Page 61 and 62: 2.3.3.3 Nonhomogeneous Dirichlet bo
Page 63 and 64: to solve the linear system in each
Page 65 and 66: 18 16 160−160 pts 240−240 pts 3
Page 67 and 68: Table 2.3: Performance of different
Page 69 and 70: Table 2.4: LU and GMRES elapsed tim
Page 71 and 72: Figure 2.20: Surface response with
Page 73 and 74: 18 16 14 12 Speedup 10 8 6 Aikten
Page 75 and 76: SuperLU contains a collection of th
Page 77 and 78: architecture. Indeed, this method p
Page 79 and 80: Chapter 3 Fast Parallel 3D Image-Ba
Page 81 and 82: and formulation of the NS problem.
Page 83 and 84: Considering an immersed boundary ap
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3.2.2 Integration of the Image Anal
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is much smaller than the area outsi
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Figure 3.3: Spatial discretization
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These two steps of the NS solver ha
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measurement. It can be noted that t
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Figure 3.4: Domain Decomposition wi
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Figure 3.5: Benchmark problem Figur
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8 7 6 size 150x50 size 225x75 size
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9 8 Speedup for a 4000x400 NS 2D pr
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3.4.1.2 Verification of the three d
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Figure 3.22: Velocity v Figure 3.23
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0.9 Problem of a size 100−50−50
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approximately by 100 step time, we
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3.4.4 Test cases Figure 3.35 repres
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Figure 3.39: Velocity inside the ca
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e very time-consuming. As a matter
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Figure 4.1: Database linked to the
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sensitivity studies and/or artery g
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5 4.5 4 3.5 Elapsed time 3 2.5 2 1.
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Figure 4.5: Scalability Performance
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Chapter 5 Conclusion This chapter s
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5.2 Perspectives and suggestions fo
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the tools needed to solve efficient
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[11] I. N. Bankman. Handbook of med
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[38] M. Gander. Optimized schwarz m
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[60] D. Keyes. Technologies and too
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[82] F. Nataf. Interface connection
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[105] W. Shyy. Moving boundaries in
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