CFD数学模型的线性化方法及其应用
收稿日期: 2014-11-17
修回日期: 2015-01-30
网络出版日期: 2015-03-20
基金资助
国家“973”计划 (6132400101)
Method of linearizing computational fluid dynamics model and its applications
Received date: 2014-11-17
Revised date: 2015-01-30
Online published: 2015-03-20
Supported by
National Basic Research Program of China (6132400101)
计算流体力学(CFD)方法不仅仅起到数值模拟的作用,它本身是一个复杂的非线性系统。在流动稳定性分析、气动弹性分析、优化设计以及流动控制等领域,从系统的角度出发,对CFD数学模型线性化后,可以对模型的系统矩阵进行定量分析获得更多的系统特性。但是CFD数学模型往往非常复杂且阶数很高,因此其线性化系统矩阵的获得比较困难。鉴于此,采用人工编程和自动微分相结合,构造有限体积法并行CFD模型的线性化系统矩阵。其中自动微分只被用来得到每个界面通量的局部雅可比矩阵,而采用人工编程方法来实现并行环境下的稀疏雅可比矩阵的组装。线性化系统的并行求解采用了块雅可比预处理的广义最小残量法,每个并行进程内部则采用零填充不完全LU分解预处理。为了验证这种线性化方法,上述方法被用于:① NACA 0012翼型的非定常绕流线性系统构造与求解;② NACA 0012翼型稳态流动的伴随方程构造与求解;③ AGARD wing 445.6机翼颤振问题降阶建模。上述三个算例的结果与CFD模拟的吻合一致。
屈崑 , 李记超 , 蔡晋生 . CFD数学模型的线性化方法及其应用[J]. 航空学报, 2015 , 36(10) : 3218 -3227 . DOI: 10.7527/S1000-6893.2015.0035
Computational fluid dynamics (CFD) is not just a simulation method, but a kind of complicated mathematical model for fluid flows. In the fields like flow stability analysis, aeroelastic analysis, aerodynamic optimization and flow control, from the viewpoint of dynamics system, the system matrix of a CFD model can be constructed for quantitative analysis, obtaining more systematic information about the CFD model. However, CFD model is a complicated high order nonlinear system. It is difficult to construct the system matrix directly. In this paper, automatic differential method is cooperated with manual coding to construct the Jacobian of a parallel finite volume CFD solver based on multiblock structured grid. Automatic differential is applied to obtaining the local Jacobian of the flux across each interface. And by means of manual coding, each local Jacobian is assembled into the global distributed sparse Jacobian. In order to solve the linearized system, preconditioned GMRES method is adopted. In the parallel environment, the block Jacobi preconditioner is used while ILU(0) preconditioner is applied to each parallel thread. In the numerical tests, this procedure is applied to ① constructing and solving the linear system of an unsteady flow around NACA0012 airfoil; ② sensitivity analysis based on the adjoint equation for a steady flow of NACA0012 airfoil; ③ reduced order modeling for the aeroelastic problem of AGARD wing 445.6. The results agree excellently with the data of CFD simulations.
[1] Anderson W K, Venkatakrishnan V. Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, AIAA-1997-0643[R]. Reston: AIAA,1997.
[2] Burgreen G W, Baysal O. Three-dimensional aerodynamic shape optimiza-tion of wings using sensitivity analysis, AIAA-1994-0094[R]. Reston: AIAA, 1994.
[3] Elliot J, Peraire J. Aerodynamic design using unstructured meshes, AIAA-1996-1941[R]. Reston: AIAA,1996.
[4] Gill P E, Murray W, Wright M H. Practical optimization[M]. Vol. 5. London: Academic Press Inc., 1981: 127.
[5] Lyness J N, Moler C B. Numerical differentiation of analytic functions[J]. SIAM Journal on Numerical Analysis, 1967, 4(2): 202-210.
[6] Lyness J N. Numerical algorithms based on the theory of complex variable[C]//Rosenthal S. Proceedings of the 1967 22nd National Conference. New York: ACM, 1967: 125-133.
[7] Li B, Deng Y Q, Tang J L, et al. Discrete adjoint optimization method for 3D unstructured grid[J]. Acta Aeronautica et Astronautica Sinica, 2014, 35(3): 674-686 (in Chinese). 李彬, 邓有奇, 唐静吕, 等. 基于三维非结构混合网格的离散伴随优化方法[J]. 航空学报, 2014, 35(3): 674-686.
[8] Li B, Tang J, Deng Y Q, et al. Application of parallel multigrid algorithm to discrete adjoint optimization[J]. Acta Aeronautica et Astronautica Sinica, 2014, 35(8): 2091-2101 (in Chinese). 李彬, 唐静, 邓有奇, 等. 并行的多重网格方法在离散伴随优化中的应用[J]. 航空学报, 2014, 35(8): 2091-2101.
[9] Kahrimanian H G. Analytical differentiation by a digital computer[D]. Philadelphia: Temple University, 1953.
[10] Nolan J F. Analytical differentiation on a digital computer[D]. Massachusetts: Massachusetts Institute of Technology, 1953.
[11] Rall L B, Corliss G F. An introduction to automatic differentiation[M]. Berz M, Bischof C H, Corliss G F, et al. Computational Differentiation: Techniques, Applications, and Tools. Philadelphia: SIAM, 1996: 1-17.
[12] Zuo Y T, Su W, Gao Z H, et al. Aerodynamic configuration optimization design of hypersonic missile based on discrete adjoint method[J]. Chinese Journal of Computational Mechanics, 2012, 29(2): 284-289 (in Chinese). 左英桃, 苏伟, 高正红, 等. 基于离散共轭方法的高超声速导弹气动外形优化设计[J]. 计算力学学报, 2012, 29(2): 284-289.
[13] Lesoinne M, Sarkis M, Hetmaniuk U, et al. A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes[J]. Computer Methods in Applied Mechanics and Engineering, 2001, 190(24-25): 3121-3146.
[14] Hascoet L, Pascual V. The Tapenade automatic differen-tiation tool: principles, model, and specification[J]. ACM Transactions on Mathematical Software, 2013, 39(3): 1-20.
[15] Balay S, Adams M F, Brown J, et al. PETSc web page[EB/OL]. Chicago: Argonne National Laboratory, 2014. [2014-12-22]. http://www.mcs.anl.gov/petsc.
[16] Balay S, Adams M F, Brown J, et al. PETSc users manual, ANL-95/11 - Revision 3.4[EB/OL]. Chicago: Argonne National Laboratory [2014-12-22]. http://www. mcs. anl. gov/petsc.
[17] Balay S, Gropp W D, McInnes L C, et al. Efficient management of parallelism in object oriented numerical software libraries[C]//Arge E, Bruaset A M, Langtangen H P. Modern Software Tools in Scientific Computing. Basel: Birkhäuser Press, 1997:163-202.
[18] Falgout R D, Yang U M. Hypre: A library of high performance preconditioners[M]//Sloot P M A, Hoekstra A G, Tan C J K, et al. Computational Science—ICCS 2002. Berlin: Springer Berlin Heidelberg, 2002: 632-641.
[19] Davis T A. Algorithm 832: UMFPACK V43---an unsymmetric-pattern multifrontal method[J]. ACM Transactions on Mathematical Software (TOMS), 2004, 30(2): 196-199.
[20] Li X S. An Overview of SuperLU: algorithms, implementation, and user interface[J]. ACM Transactions on Mathematical Software, 2005, 31(3): 302-325.
[21] Xu J, Qu K, Cai J S. Flow simulations for NASA CRM wing-body-tail with implicit hole cutting method[J]. Applied Mechanics and Materials, 2013, 378: 355-361.
[22] Xu J, Liu Q, Cai J. Numerical simulations for DLR-F6 wing/body/nacelle/pylon with enhanced implicit hole cutting method[J]. Parallel Computational Fluid Dynamics Communications in Computer and Information Science, 2014, 405: 185-194.
[23] Chen S Y, Chen Y C, Xia Z H, et al. Constrained large-eddy simulation and detached eddy simulation of flow past a commercial aircraft at 14 degrees angle of attack[J]. Science China Physics, Mechanics and Astron-Omy, 2013, 56(2): 270-276.
[24] Landon R H. NACA0012, Oscillatory and transient pitching, compendium of unsteady aerodynamics measurements, AGARD-R-702[R]. Neuilly sur Seine (France): AGARD, 1982.
[25] Thomas J P, Dowell E H, Hall K C. Three-dimensional transonic aeroelasticity using proper orthogonal decom-position-based reduced-order models[J]. Journal of Air-craft, 2003, 40(3): 544-551.
[26] Yates Jr E C. AGARD standard aeroelastic configura-tions for dynamic response. Candidate configuration I.-wing 445.6, Technical Report NASA-TM-100492[R]. Hampton, VA: NASA Langley Research Center, 1987.
[27] Yates Jr E C. AGARD standard aeroelastic configura-tions for dynamic response I-wing 445.6, AGARD-R-765[R]. Neuilly sur Seine (France): AGARD, 1988.
[28] Liu F, Cai J, Zhu Y, et al. Calculation of wing flutter by a coupled fluid-structure method[J]. Journal of Aircraft, 2001, 38(2): 334-342.
[29] Silva W A. Simultaneous excitation of multiple-input/multiple-output CFD-Based unsteady aeroelastic systems[J]. Journal of Aircraft, 2008, 45(4): 1267-1274.
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