Aerodynamic shape optimization using first and second order adjoint and direct approaches. (English) Zbl 1170.76348

Summary: This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.


76N25 Flow control and optimization for compressible fluids and gas dynamics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
Full Text: DOI


[1] Lighthill MJ (1945) A new method of two-dimensional aerodynamic design. Aeronautical Research Council
[2] McFadden GB (1979) An artificial viscosity method for the design of supercritical airfoils. New York University Report C00-3077-158
[3] Davis L (1991) Handbook of genetic algorithms. Van Nostrand Reinhold, New York
[4] Michalewicz Z (1994) Genetic algorithms + data structures = evolution programs, 2nd edn. Springer, Berlin · Zbl 0818.68017
[5] Bäck T (1996) Evolutionary algorithms in theory and practice. Evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, Oxford · Zbl 0877.68060
[6] Bäck T (1996) Evolutionary algorithms in theory and practice. Oxford University Press, Oxford · Zbl 0877.68060
[7] Bertsekas DP (1996) Constrained optimization and Lagrange multiplier methods, 1st edn. Athena Scientific, Nashua
[8] Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Nashua · Zbl 1015.90077
[9] Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic, New York
[10] Luenberger DG (2003) Linear and nonlinear programming, 2nd edn. Kluwer Academic, Dordrecht
[11] Fletcher R (1988) Practical methods of optimization, 2nd edn. Wiley, New York · Zbl 0651.20029
[12] Jin Y (2003) A comprehensive survey of fitness approximation in evolutionary computation. Soft Comput 9:3–12 · Zbl 05036836
[13] Wang GG, Shan S (2006) Review of metamodelling techniques in support of engineering design optimization. Trans ASME, J Mech Des 129(4):370–380
[14] El-Beltagy MA, Nair PB, Keane AJ (1999) Metamodeling techniques for evolutionary optimization of computationally expensive problems: Promises and limitations. In: GECCO99, genetic and evolutionary computation conference, Orlando, July 1999
[15] Giannakoglou K (2002) Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence. Int Rev J Prog Aerosp Sci 38:43–76
[16] Lions JL (1971) Optimal control of systems governed by partial differential equations. Springer, New York · Zbl 0203.09001
[17] Pironneau O (1974) On optimum design in fluid mechanics. J Fluid Mech 64:97–110 · Zbl 0281.76020
[18] Pironneau O (1984) Optimal shape design for elliptic systems. Springer, New York · Zbl 0534.49001
[19] Jameson A (1988) Aerodynamic design via control theory. J Sci Comput 3:233–260 · Zbl 0676.76055
[20] Jameson A, Reuther J (1994) Control theory based airfoil design using the Euler equations. AIAA Paper 94-4272
[21] Jameson A (1995) Optimum aerodynamic design using CFD and control theory. AIAA Paper 95-1729 · Zbl 0875.76497
[22] Jameson A, Pierce N, Martinelli L (1998) Optimum aerodynamic design using the Navier-Stokes equations. Theor Comput Fluid Dyn 10:213–237 · Zbl 0912.76067
[23] Anderson WK, Nielsen E (2001) Sensitivity analysis for Navier-Stokes equations on unstructured grids using complex variables. AIAA J 39(31):56–63
[24] Lyness JN, Moler CB (1967) Numerical differentiation of analytic functions. In: ACM 22nd national conference · Zbl 0155.48003
[25] Martins R, Kroo IM, Alonso J (2000) An automated method for sensitivity analysis using complex variables. AIAA Paper 2000-0689
[26] Squire W, Trapp G (1998) Using complex variables to estimate derivatives of real functions. SIAM Rev 10(1):110–112 · Zbl 0913.65014
[27] Newman JC, Anderson WK, Whitfield DL (1998) Multidisciplinary sensitivity derivatives using complex variables. Tech Rep MSSU-COE-ERC-98-08
[28] Nielsen EJ, Kleb WL (2005) Efficient construction of discrete adjoint operators on unstructured grids by using complex variables. AIAA Paper 2005-0324
[29] Courty F, Dervieux A, Koobus B, Hascoët L (2003) Reverse automatic differentiation for optimum design: from adjoint state assembly to gradient computation. Optim Methods Softw 18(5):615–627 · Zbl 1142.90524
[30] Griewank A (1989) On automatic differentiation. In: Mathematical programming: recent developments and applications. Kluwer Academic, Dordrecht
[31] Hovland P, Mohammadi B, Bischof C (1997) Automatic differentiation of Navier–Stokes computations. Technical Report MCS-P687-0997, Argonne National Laboratory · Zbl 1041.76550
[32] Juedes D (1991) A taxonomy of automatic differentiation tools. In: Automatic differentiation of algorithms: theory, implementation, and application. SIAM, Philadelphia, pp 315–329 · Zbl 0782.65029
[33] Bischof C, Carle A, Corliss G, Griewank A, Hovland P (1991) ADIFOR Generating derivative codes from Fortran programs. Report CRPC-TR91185-S, Center for Research and Parallel Computation, Rice University
[34] Giering R, Kaminski T (1998) Recipes for adjoint code construction. ACM Trans Math Softw 24:437–474 · Zbl 0934.65027
[35] Berz M (1990) The DA precompiler DAFOR. Technical Report, Lawrence Berkeley National Laboratory, Berkeley, CA
[36] Horwedel J (1991) GRESS a preprocessor for sensitivity studies of Fortran programs. AIAA Paper 91-005 · Zbl 0782.68121
[37] Faure C (2005) An automatic differentiation platform: Odyssée. Future Gener Comput Syst 21(8):1391–1400
[38] Stephens B (1991) Automatic differentiation as a general-purpose numerical tool. PhD thesis, School of Mathematics, University of Bristol, UK
[39] Shiriaev D, Griewank A, Utke J (1996) A user guide to ADOL–F: automatic differentiation of Fortran codes. IOKOMO-04-1995
[40] Rhodin A (1997) IMAS–integrated modeling and analysis system for the solution of optimal control problems. Comput Phys Commun 107:21–38 · Zbl 0938.65091
[41] Christianson B (1992) Automatic Hessians by reverse accumulation. J Numer Anal 12:135–150 · Zbl 0754.65022
[42] Bischof C, Roh L, Mauer-Oats A (1997) ADIC An extensible automatic differentiation tool for ANSI-C. Preprint ANL/MCS-P626-1196, Argonne National Laboratory
[43] Griewank A, Juedes D, Mitev H, Utke J, Vogel O, Walther A (1996) ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. ACM Trans Math Softw 22(2):131–167 · Zbl 0884.65015
[44] Pierce NA, Giles MB (2000) An introduction to the adjoint approach to design. Flow Turbul Combust 65(3–4):393–415 · Zbl 0996.76023
[45] Nadarajah S, Jameson A (2000) A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization. AIAA Paper 2000-0667
[46] Nadarajah S, Jameson A (2001) Studies of the continuous and discrete adjoint approaches to viscous automatic aerodynamic shape optimization. AIAA Paper 2001-2530
[47] Spalart PR, Allmaras SR (1994) A one-equation turbulence model for aerodynamic flows. Rech Aerosp (1):5–21
[48] Roe P (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J Comput Phys 43:357–371 · Zbl 0474.65066
[49] Papadimitriou DI, Giannakoglou KC (2007) A continuous adjoint method with objective function derivatives based on boundary integrals for inviscid and viscous flows. J Comput Fluids 36:325–341 · Zbl 1177.76369
[50] Anderson WK, Venkatakrishnan V (1997) Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation. AIAA Paper 97-0643
[51] Anderson WK, Venkatakrishnan V (1997) Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation. Comput Fluids 28:443–480 · Zbl 0968.76074
[52] Asouti VG, Zymaris AS, Papadimitriou DI, Giannakoglou KC (2008) Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning. Int J Numer Methods Fluids 57:1485–1504 · Zbl 1210.76141
[53] Arian E, Salas MD (1997) Admitting the inadmissible: adjoint formulation for incomplete cost functionals in aerodynamic optimization. NASA/CR-97-206269, ICASE Report No 97-69
[54] Baysal O, Ghayour K (2001) Continuous adjoint sensitivities for optimization with general cost functionals on unstructured meshes. AIAA J 39(1)
[55] Jameson A, Kim S (2003) Reduction of the adjoint gradient formula in the continous limit. AIAA Paper 2003-0040
[56] Denton JD (1993) Loss mechanisms in turbomachines. ASME Paper 93-GT-435
[57] Davies MRD, O’Donnell FK, Niven AJ (2000) Turbine blade entropy generation rate, part I: the boundary layer defined. ASME Paper 2000-GT-265
[58] O’Donnell FK, Davies MRD (2000) Turbine blade entropy generation rate, part II: the measured loss. ASME Paper 2000-GT-266
[59] Papadimitriou DI, Giannakoglou KC (2006) Compressor blade optimization using a continuous adjoint formulation. ASME TURBO EXPO, GT2006/90466, Barcelona
[60] Papadimitriou DI, Giannakoglou KC (2007) Total pressure losses minimization in turbomachinery cascades, using a new continuous adjoint formulation. Proc Inst Mech Eng, Part A: J Power Energy 222(6):865–872
[61] Papadimitriou DI, Zymaris AS, Giannakoglou KC (2005) Discrete and continuous adjoint formulations for turbomachinery applications. In: UROGEN 2005, international conference proceedings, Munich, September 2005
[62] Papadimitriou DI, Giannakoglou KC (2006) A continuous adjoint method for the minimization of losses in cascade viscous flows. AIAA Paper 2006-0049
[63] Kim SK, Alonso JJ, Jameson A (2000) Two-dimensional high-lift aerodynamic optimization using the continuous adjoint method. AIAA Paper 2000-4741
[64] Kim SK, Alonso JJ, Jameson A (2002) Design optimization of high-lift configurations using a viscous continuous adjoint method. AIAA Paper 2002-0844
[65] Leoviriyakit K, Jameson A (2003) Aerodynamic shape optimization of wings including planform variations. AIAA Paper 2003-0210
[66] Leoviriyakit K, Kim S, Jameson A (2003) Viscous aerodynamic shape optimization of wings including planform variations. AIAA Paper 2003-3498
[67] Leoviriyakit K, Kim S, Jameson A (2004) Aero-structural wing planform optimization using the Navier-Stokes equations. AIAA Paper 2004-4479
[68] Leoviriyakit K, Jameson A (2005) Multi-point wing planform optimization via control theory. AIAA Paper 2005-0450
[69] Giles MB, Pierce NA (1997) Adjoint equations in CFD: duality, boundary conditions and solution behaviour. AIAA Paper 97-1850
[70] Giles MB, Pierce NA (1998) On the properties of solutions of the adjoint Euler equations. In: 6th ICFD conference on numerical methods for fluid dynamics, Oxford, UK, 1998
[71] Harbeck M, Jameson A (2005) Exploring the limits of transonic shock-free airfoil design. AIAA Paper 2005-1041
[72] Reuther J, Alonso JJ, Rimlinger MJ, Jameson A (1999) Aerodynamic shape optimization of supersonic aircraft configurations via an adjoint formulation on distributed memory parallel computers. Comput Fluids 28:675–700 · Zbl 0983.76077
[73] Nadarajah S, Kim SK, Jameson A, Alonso JJ (2002) Sonic boom reduction using an adjoint method for supersonic transport aircraft configuration. In: Symposium transsonicum IV, international union of theoretical and applied mechanics, September 2–6, 2002, DLR Gottingen, Germany
[74] Nadarajah S, Jameson A, Alonso JJ (2002) Sonic boom reduction using an adjoint method for wing-body configurations in supersonic flow. AIAA Paper 2002-5547
[75] Alonso JJ, Kroo IM, Jameson A (2002) Advanced algorithms for design and optimization of quiet supersonic platform. AIAA Paper 2002-0144
[76] Nadarajah S, Jameson A, Alonso JJ (2002) An adjoint method for the calculation of remote sensitivities in supersonic flow. AIAA Paper 2002-0261 · Zbl 1184.76750
[77] Taasan S, Kuruvila G, Salas MD (1992) Aerodynamic design and optimization in one-shot. AIAA Paper 91-005
[78] Kuruvila G, Taasan S, Salas MD (1995) Airfoil design and optimization by the one-shot method. AIAA Paper 95-0478
[79] Hazra SB (2004) An efficient method for aerodynamic shape optimization. AIAA Paper 2004-4628
[80] Hazra S, Schulz V, Brezillon J, Gauger N (2005) Aerodynamic shape optimization using simultaneous pseudo-timestepping. J Comput Phys 204(1):46–64 · Zbl 1143.76564
[81] Hazra SB, Schulz V (2006) Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with state constraints. SIAM J Sci Comput 28(3):1078–1099 · Zbl 1130.49032
[82] Held C, Dervieux A (2002) One-shot airfoil optimisation without adjoint. Comput Fluids 31:1015–1049 · Zbl 1014.76077
[83] Dadone A, Grossman B (2000) Progressive optimization of inverse fluid dynamic design problems. Comput Fluids 29:1–32 · Zbl 0955.76081
[84] Dadone A, Grossman B (2003) Fast convergence of inviscid fluid dynamic design problems. Comput Fluids 32:607–627 · Zbl 1084.76548
[85] Soto O, Lohner R (2004) On the computation of flow sensitivities from boundary integrals. AIAA Paper 04-0112
[86] Jameson A, Shankaran S, Martinelli L (2003) A continuous adjoint method for unstructured grids. AIAA Paper 2003-3955 · Zbl 1245.76014
[87] Kim S, Leoviriyakit K, Jameson A (2003) Aerodynamic shape and planform optimization of wings using a viscous reduced adjoint gradient formula. In: 2nd MIT conference on computational fluid and solid mechanics, Cambridge, MA, June 17–20, 2003
[88] Othmer C, de Villiers E, Weller HG (2007) Implementation of a continuous adjoint for topology optimization of ducted flows. AIAA Paper 2007-3947
[89] Mohammadi B, Pironneau O (2001) Applied shape optimization for fluids. Clarendon, Oxford · Zbl 0970.76003
[90] Mohammadi B, Pironneau O (2004) Shape optimization in fluid mechanics. Annu Rev Fluid Mech 36:255–279 · Zbl 1076.76020
[91] Soto O, Lohner R (2001) CFD shape optimization using an incomplete-gradient adjoint formulation. Int J Numer Methods Fluids 51:735–753 · Zbl 1051.76012
[92] Soto O, Lohner R (2000) CFD optimization using an incomplete-gradient adjoint approach. AIAA Paper 00-0666
[93] Soto O, Lohner R (2000) CFD shape optimization using an incomplete-gradient adjoint approach. In: ECCOMAS, Barcelona, September 2000
[94] Soto O, Lohner R (2001) General methodologies for incompressible flow design problems. AIAA Paper 01-1061
[95] Soto O, Lohner R (2002) A mixed adjoint formulation for incompressible rans problems. AIAA Paper 02-0451
[96] Kim HJ, Sasaki D, Obayashi S, Nakahashi K (2001) Aerodynamic optimization of supersonic transport wing using unstructured adjoint method. AIAA J 39(6)
[97] Nielsen EJ, Park MA (2005) Using an adjoint approach to eliminate mesh sensitivities in computational design. AIAA Paper 2005-0491
[98] Mavriplis DJ (2005) Formulation and multigrid solution of the discrete adjoint for optimization problems on unstructured meshes. AIAA Paper
[99] Mavriplis DJ (2006) A discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes. AIAA Paper
[100] Nielsen EJ, Anderson WK (2002) Recent improvements in aerodynamic design optimization on unstructured meshes. AIAA J 40(6):1155–1163
[101] Nielsen EJ, Anderson WK (2001) Recent improvements in aerodynamic design optimization on unstructured meshes. AIAA Paper 2001-0596
[102] Elliot J, Peraire J (1996) Aerodynamic design using unstructured meshes. AIAA Paper 96-1941
[103] Elliot J, Peraire J (1997) Aerodynamic optimization using unstructured meshes with viscous effects. AIAA Paper 97-1849
[104] Elliot J, Peraire J (1997) Practical 3d aerodynamic design and optimization using unstructured meshes. AIAA J 35(9):1479–1485 · Zbl 0900.76420
[105] Pulliam TH, Nemec M, Holst TL, Zingg DW (2003) Comparison of genetic and adjoint methods for multi-objective viscous airfoil optimizations. AIAA Paper 2003-0298
[106] Elliot J, Peraire J (1998) Constrained, multipoint shape optimisation for complex 3d configurations. Aeronaut J 102(1017):365–376
[107] Martins JRRA, Alonso JJ, Reuther JJ (2005) A coupled-adjoint sensitivity analysis method for high-fidelity aero-structural design. Optim Eng 6(1):33–62 · Zbl 1145.76418
[108] Leoviriyakit K, Jameson A (2004) Case studies in aero-structural wing planform and section optimization. AIAA Paper 2004-5372
[109] Leoviriyakit K, Jameson A (2004) Aero-structural wing planform optimization. AIAA Paper 2004-0029
[110] Nadarajah S, Jameson A (2002) Optimal control of unsteady flows using a time accurate method. AIAA Paper 2002-5436
[111] Nadarajah S, McMullen M, Jameson A (2002) Non-linear frequency domain based optimum shape design for unsteady three-dimensional flow. AIAA Paper 2002-2838
[112] Nadarajah S, Jameson A (2002) Optimum shape design for unsteady three-dimensional viscous flows using a non-linear frequency domain method. AIAA Paper 2002-2838
[113] Campobasso MS, Duta MC, Giles MB (2001) Adjoint methods for turbomachinery design. In: ISOABE conference, 2001
[114] Duta MC, Giles MB, Campobasso MS (2002) The harmonic adjoint approach to unsteady turbomachinery design. Int J Numer Meth Fluids 40(3–4):323–332 · Zbl 1017.00047
[115] Campobasso MS, Duta MC, Giles MB (2003) Adjoint calculation of sensitivities of turbomachinery objective functions. AIAA J Propuls Power 19(4)
[116] Giannakoglou KC, Papadimitriou DI (2006) Formulation and application of the continuous adjoint method in aerodynamics and turbomachinery. Von-Karman institute lecture series
[117] Anderson WK, Bonhaus DL (1997) Aerodynamic design on unstructured grids for turbulent flows. NASA Technical Memorandum
[118] Anderson WK, Bonhaus DL (1999) Airfoil design on unstructured grids for turbulent flows. AIAA J 37(2):185–191
[119] Nielsen EJ, Lu J, Park MA, Darmofal DL (2004) An implicit exact dual adjoint solution method for turbulent flows on unstructured grids. Comput Fluids 33:1131–1155 · Zbl 1103.76346
[120] Sherman LL, Taylor III AC, Green LL, Newman PA, Hou GW, Korivi VM (1996) First- and second-order aerodynamic sensitivity derivatives via automatic differentiation with incremental iterative methods. J Comput Phys 129:307–331 · Zbl 0933.76070
[121] Papadimitriou DI, Giannakoglou KC (2007) Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems. Int J Numer Methods Fluids 56:1929–1943 · Zbl 1141.76058
[122] Papadimitriou DI, Giannakoglou KC (2007) Computation of the Hessian matrix in aerodynamic inverse design using continuous adjoint formulations. Comput Fluids 37:1029–1039 · Zbl 1237.76162
[123] Tortorelli D, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Probl Eng 1(1):71–105
[124] Hou GW, Sheen J (1993) Numerical methods for second-order shape sensitivity analysis with applications to heat conduction problems. Int J Numer Methods Eng 36:417–435 · Zbl 0770.73086
[125] Le Dimet FX, Navon IM, Daescu DN (2002) Second-order information in data assimilation. Mon Weather Rev 130(3):629–648
[126] Veerse F, Auroux D, Fisher M (2000) Limited-memory BFGS diagonal preconditioners for a data assimilation problem in meteorology. Optim Eng 1:323–339 · Zbl 0991.65051
[127] Daescu DN, Navon IM (2003) An analysis of a hybrid optimization method for variational data assimilation. Int J Comput Fluid Dyn 17(4):299–306 · Zbl 1161.76449
[128] Arian E, Taasan S (1999) Analysis of the Hessian for aerodynamic optimization: inviscid flow. Comput Fluids 28(7):853–877 · Zbl 0969.76015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.