（節(jié)選）外文翻譯--船閘灌輸系統(tǒng)建模的牛頓-克雷洛夫并行求解器（原文）

1、Procedia Computer Science 00 (2009) 000–000Procedia Computer Science www.elsevier.com/locate/procediaInternational Conference on Computational Science, ICCS 2010Parallel Newton-Krylov solvers for modeling of a navigat

2、ion lock filling systemHung V. Nguyen1*, Jing-Ru C. Cheng1 , E. Allen Hammack2, and Robert S. Maier11U.S. Army Engineer Research and Development Center (ERDC) Information Technology Laboratory (ITL), Vicksburg, MS, 391

3、80 2U.S. Army Engineer Research and Development Center (ERDC) Coastal and Hydraulics Laboratory (CHL), Vicksburg, MS, 39180AbstractThe Galerkin least-squares finite element method for solving the Reynolds-averaged incomp

4、ressible turbulent 3-D Navier-Stokes equations is employed to simulate a navigation lock filling system in the numerical code Adaptive Hydraulics (ADH). The linear system is solved at each nonlinear iteration within ev

5、ery time-step using biconjugate gradient stabilized (BiCGstab) in combination with block-Jacobi (bjacobi) preconditioners, as it failed to solve the linear system because of dramatic changes in flow velocity and pressu

6、re early in the simulation. To overcome this problem, we used the Portable Extensible Toolkit for Scientific Computation (PETSc), a numerical library that provides multiple types of linear solvers. PETSc has been incor

7、porated into the ADH code. The ADH-PETSc interface helps to systematically investigate the best linear solver for an ADH simulation. We found that a variant, known as enhanced BiCGstab(l) in combination with the additi

8、ve Schwarz method (ASM), made it possible to simulate the John Day lock filling system. The BiCGstab(l) solver improved the rate of convergence because of a more reliable update strategy for the residuals. In addition,

9、 the simulation was run with various numbers of processors. The result shows good scaling of solution time as the number of processors increasesKeywords: Navigation lock, iterative solvers, ADH, PETSc, and turbulent flo

10、w.1. IntroductionA numerical model capable of simulating free-surface flow in complex, 3-D structures is vital for detailed evaluation of navigation locks and lock components [12]. The Adaptive Hydraulics (ADH) code is

11、a model that can simulate saturated and unsaturated groundwater, overland flow, 2-D and 3-D shallow-water problems, and the 3-D Navier-Stokes problems such as 3-D flow in navigation locks. ADH employs the Galerkin least

12、-squares finite element method for solving the Reynolds-averaged incompressible turbulent 3-D Navier-Stokes equations. Turbulence is modeled with an adverse pressure gradient eddy viscosity technique. ADH uses the Newt

13、on algorithm to solve the nonlinear problem, and the resulting linear system is nonsymmetric. A significant part of ADH computation time is spent solving the linear system. Therefore, the performance of linear solvers i

14、s of great interest.* Corresponding author. Tel.: +01-601-634-3607; fax: +01-601-634-2324. E-mail address: Hung.V.Nguyen@usace.army.mil.c ? 2012 Published by Elsevier Ltd.Procedia Computer Science 1 (2012) 699–707www.els

15、evier.com/locate/procedia1877-0509 c ? 2012 Published by Elsevier Ltd.doi:10.1016/j.procs.2010.04.075Open access under CC BY-NC-ND license.Open access under CC BY-NC-ND license.Author name / Procedia Computer Science 00

16、(2010) 000–000are included in the inflow and outflow. Free surface boundary conditions were applied to the valve well and in the upstream bulkhead. ADH calculates the water-surface location using a moving mesh method du

17、ring simulation. The 3-D mesh contains 213,391 nodes, each with four degrees of freedom and 1,095,587 tetrahedral elements. The elements have sides ranging from 13 mm (on valve surface) to about 0.76 m (far from valve)

18、as shown in Fig. 1(b). Fig.1 (a) Flow domain; (b) Unstructured mesh with fine mesh size near valve2. Numerical Results2.1. Numerical Results for Lock Filling SystemThe ADH uses its own Newton solver routine; currently we

19、 only write the interface for ADH to use the PETSc KSP linear solver and PC preconditioner to solve the linear system at each Newton step. The ADH matrices were converted into a BlockAIJ (BAIJ) format because of four de

20、grees of freedom (pressure p, u, v, and w velocities) at each node.The numerical model for the John Day lock filling system simulates a 100-second physical time. The time-steps are 0.1 and 1.0 second for periods of 0 t

21、o 10 seconds and 10 to 100 seconds, respectively. The time-steps were chosen above because of a dramatic change of pressure and velocity at the early state of the simulation. However, ADH employs an adaptive time-step,

22、based upon a local error estimator. This error estimator helps to increase model efficiency while maintaining a given accuracy. The simulation was run with 32 processors on the Cray XT4 system, using BiCGstab(l) in comb

23、ination with an additive Schawarz (ASM) preconditioner. The stopping criteria are based on the l2-norm of the preconditioned residual or maximum number of iterations (maxits). Convergence is detected at k iterations if

24、 || rk||2 < ? ||b||2, where r denotes the residual, b the right-hand side vector, ?= 5.0x10-5 the relative tolerance, and maxits = 5,000. Fig. 2 (a) shows the BiCGstab(l) algorithm while Fig. 2 (b) shows the converge

25、nce rate for BiCGstab and BiCGstab(l) with l = 2, 4, and 8 at simulation time t = 0.475 second. For l = 1, this algorithm coincides with BiCGstab. If l ? (l = 1) is close to zero, then stagnation or even break down mi

26、ght occur. The BiCGstab convergence rate depicts that the true residual norm suddenly increases after 300 iterations and the BiCGstab divergences after 400 iterations. This explains why the BiCGstab fails to simulate t

27、he lock filling system. The convergence rates of BiCGstab(l) with l = 2, 4, and 8 are followed by the same pattern, and suddenly the rates abruptly increase and quickly satisfy the convergence criteria. In general, the

28、 average computation cost per iteration is higher with respect to the number of inner product and vector updates when l is larger. However, as can be seen from Fig. 2 (b), the larger values of l give a smaller number o

29、f iterations and greater performance since the solver times are 432.187, 354.394, and 287.45 seconds for l = 2, 4, and 8, respectively. The BiCGstab(l) is the efficient linear solver because the auxiliary polynomial can

30、 be used to gain efficiency and to improve residual reduction. In addition, Sleijpen and van der Vorst [11] show that occasionally replacing the preconditioned residual norm with the true residual norm H.V. Nguyen et al