Fluid Mechanics and Flight Mechanics

Discrete Adjoint Optimization Method for 3D Unstructured Grid

  • LI Bin ,
  • DENG Youqi ,
  • TANG Jing ,
  • LV Hongying
Expand
  • 1. Computational Aerodynamic Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China;
    2. Xi'an Advanced Control Technology Institute, Xi'an 710065, China

Received date: 2013-05-06

  Revised date: 2013-09-06

  Online published: 2013-09-17

Abstract

Because solving adjoint equation does not depend on the number of design variables, there is no relationship between the calculation amount of iterative optimization and the number of design variables. Based on the unstructured grid, a discrete adjoint solver is developed for a 3D Reynolds-averaged Navier-Stokes solver. Free-form deformation (FFD) technology is implemented to modify the mesh. Discrete adjoint equation can be acquired directly by formula derivation method and solved through LU-SGS iteration. The adjoint code is verified by comparing flux Jacobian and objective function gradient with finite differences. The design system is successfully applied to ONERA M6 wing transonic shape optimization design with the purpose of reducing drag, and the influence of volume constraint is also studied in this paper. It shows that the optimization method established is effective with a better application prospect.

Cite this article

LI Bin , DENG Youqi , TANG Jing , LV Hongying . Discrete Adjoint Optimization Method for 3D Unstructured Grid[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2014 , 35(3) : 674 -686 . DOI: 10.7527/S1000-6893.2013.0383

References

[1] Elliott J, Peraire J. Practical three-dimensional aerodynamic design and optimization using unstructured grids[J]. AIAA Journal, 1997, 35(9): 1479-1485.

[2] Nielsen E J, Anderson W K. Implementation of a parallel framework for aerodynamic design optimization on unstructured meshes[C]//Proceedings of Parallel Computational Fluid Dynamics, 1999.

[3] Nielsen E J, Anderson W K. Recent improvements in aerodynamic design optimization on unstructured meshes[J]. AIAA Journal, 2002, 40(6): 1155-1163.

[4] Nielsen E J, Kleb B. Efficient construction of discrete adjoint operators on unstructured grids by using complex variables, AIAA-2005-0324[R]. Reston: AIAA, 2005.

[5] Mavriplis D J. Formulation and multigrid solution of the discrete adjoint for optimization problems on unstructured meshes, AIAA-2005-0319[R]. Reston: AIAA, 2005.

[6] Mavriplis D J. A discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, AIAA-2006-0050[R]. Reston: AIAA, 2006.

[7] Brezillon J, Dwight R P. Aerodynamic shape optimization using the discrete adjoint of the Navier-Stokes equations: applications towards complex 3D configurations[C]//KATnet Ⅱ Conference on Key Aerodynamic Technologies, 2009.

[8] Dwight R P, Brezillon J. Effect of various approximations of the discrete adjoint on gradient-based optimization, AIAA-2006-0690[R]. Reston: AIAA, 2006.

[9] Kim H J, Nakahashi K. Discrete adjoint method for unstructured Navier-Stokes solver, AIAA-2005-0449[R]. Reston: AIAA, 2005.

[10] Carpentieri G, Koren B. Development of the discrete adjoint for a 3D unstructured Euler solver, AIAA-2007-3954[R]. Reston: AIAA, 2007.

[11] Giles M B, Duta M C. Algorithm developments for discrete adjoint methods[J]. AIAA Journal, 2003, 41(2): 198-205.

[12] Lee B J, Kim C. Strategies for robust convergence characteristics of discrete adjoint method[J]. Computational Fluid Dynamics, 2009, 33: 633-639.

[13] Huang Y, Chen Z B, Liu G. Aerodynamic design method based on N-S equations and discrete adjoint approach[J]. Acta Aerodynamica Sinica, 1999, 17(4): 413-433. (in Chinese) 黄勇, 陈作斌, 刘刚. 基于伴随方程的翼型数值优化设计方法研究[J]. 空气动力学学报, 1999, 17(4): 413-433.

[14] Zhou Z, Chen Z B. Aerodynamic design method based on N-S equations and discrete adjoint approach[J]. Acta Aerodynamica Sinica, 2002, 20(2): 141-149. (in Chinese) 周铸, 陈作斌. 基于N-S方程的翼型气动优化设计[J]. 空气动力学学报, 2002, 20(2): 141-149.

[15] Xiong J T, Qiao Z D, Yang X D, et al. Optimum aerodynamic design of transonic wing based on viscous adjoint method[J]. Acta Aeronautica et Astronautica Sinica, 2007, 28(2): 281-285. (in Chinese) 熊俊涛, 乔志德, 杨旭东, 等. 基于黏性伴随方法的跨声速机翼气动优化设计[J]. 航空学报, 2007, 28(2): 281-285.

[16] Du L, Ning F F. Aerodynamic inverse design method for low Mach number airfoils[J]. Acta Aeronautica et Astronautica Sinica, 2011, 32(7): 1180-1188.(in Chinese) 杜磊, 宁方飞. 低速叶型气动反问题设计方法[J]. 航空学报, 2011, 32(7): 1180-1188.

[17] Du L, Ning F F. An approximate method for viscous inverse design based on adjoint equations[J]. Acta Aeronautica et Astronautica Sinica, 2012, 33(4): 597-606.(in Chinese) 杜磊, 宁方飞. 一种基于共轭方程法求解黏性反问题的简化方法[J]. 航空学报, 2012, 33(4): 597-606.

[18] Zuo Y T, Gao Z Z, Xia L. Aerodynamic design based on Euler equations and discrete adjoint approach[J]. Chinese Journal of Applied Mechanics, 2009, 26(1): 22-26. (in Chinese) 左英桃, 高正红, 夏露. 基于Euler 方程和离散共轭方法的气动外形优化设计[J]. 应用力学学报, 2009, 26(1): 22-26.

[19] Zuo Y T, Gao Z H, Zhan H. Aerodynamic design method based on N-S equations and discrete adjoint approach[J]. Acta Aerodynamica Sinica, 2009, 27(1): 67-72. (in Chinese) 左英桃, 高正红, 詹浩. 基于N-S方程和离散共轭方法的气动设计方法研究[J]. 空气动力学学报, 2009, 27(1): 67-72.

[20] Zuo Y T, Gao Z H, He J. Aerodynamic design method based on N-S equations and discrete adjoint approach[J]. Acta Aerodynamica Sinica, 2010, 28(5): 509-512. (in Chinese) 左英桃, 高正红, 何俊. 基于N-S 方程和离散共轭方法的气动外形设计[J]. 空气动力学学报, 2010, 28(5): 509-512.

[21] Wu W H, Fan Z L, Qin N. Adjoint operator method based aerodynamics optimization of airliner[C]//Computational Technologies for Commercial Aircraft and High Resolution, 2010: 309-317. (in Chinese) 吴文华, 范召林, 覃宁.基于伴随算子的大飞机布局多参数高精度优化[C]//大型客机与高精度计算方法学术研讨会, 2010: 309-317.

[22] Ma X Y, Fan Z L, Wu W H, et al. Aerodynamic shape optimization for wing based on NURBS[J]. Acta Aeronautica et Astronautica Sinica, 2011, 32(9): 1616-1621. (in Chinese) 马晓永, 范召林, 吴文华, 等. 基于NURBS方法的机翼气动外形优化[J]. 航空学报, 2011, 32(9): 1616-1621.

[23] Liu X Q, Qin N. Design of transonic wing based on adjoint method[C]//Computational Technologies for Commercial Aircraft and High Resolution, 2010: 256-265. (in Chinese) 刘学强, 覃宁. 基于伴随算子的跨音速机翼优化设计[C]//大型客机与高精度计算方法学术研讨会, 2010: 256-265.

[24] Guan J. Aerodynamic shape optimization for 2D airfoil based on adjoint equations[D]. Nanjing: College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 2011. (in Chinese) 关键. 基于伴随方程的二维翼型气动外形优化设计[D]. 南京: 南京航空航天大学航空宇航学院, 2011.

[25] Sharov D, Nakahashi K. Reordering of 3D hybrid unstrucrured grids for vectorized LU-SGS Navier-Stokes computations, AIAA-1998-0614[R]. Reston: AIAA, 1998.

[26] Venkatakrishnan V. On the accuracy of limiters and convergence to steady state solutions, AIAA-1993-0880[R]. Reston: AIAA, 1993.

[27] Van Leer B. Flux vector splitting for the Euler equations[J]. Lecture Notes in Physics, 1982, 170: 507-512.

[28] Spalart P R, Allmaras S R. A one-equation turbulence model for aerodynamic flows, AIAA-1992-0439[R]. Reston: AIAA, 1992.

[29] Moigne A L, Qin N. Variable-fidelity aerodynamic optimization for turbulent flows using a discrete adjoint formulation, AIAA-2004-1234[R]. Reston: AIAA, 2004.

[30] Amoiralis E I, Nikolos I K. Freeform deformation versus B-spline representation in inverse airfoil design[J]. Journal of Computing and Information Science in Engineering, 2008, 8(2): 13.

[31] Lamousin H J, Waggenspack W N. NURBS-based free-form deformations [J]. Computer Graphics and Applications, 1994, 14(6): 59-65.

Outlines

/