zbMATH — the first resource for mathematics

Aerodynamic shape optimization using simultaneous pseudo-timestepping. (English) Zbl 1143.76564
Summary: The paper deals with a numerical method for aerodynamic shape optimization. It is based on simultaneous pseudo-timestepping in which stationary states are obtained by solving the non-stationary system of equations representing the state, costate and design equations. The main advantages of this method are that it requires no additional globalization techniques and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP method. A design example for drag reduction for an RAE2822 airfoil, keeping its thickness fixed, is included. The overall cost of computation is less than four times that of the forward simulation run.

76N25 Flow control and optimization for compressible fluids and gas dynamics
49Q10 Optimization of shapes other than minimal surfaces
65K10 Numerical optimization and variational techniques
76G25 General aerodynamics and subsonic flows
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI
[1] W.K. Anderson, V. Venkatakrishnan, Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, AIAA 97-0643, 1997
[2] Bock, H.G.; Egartner, W.; Kappis, W.; Schulz, V., Practical shape optimization for turbine and compressor blades, Optimiz. eng., 3, 395-414, (2002) · Zbl 1079.90623
[3] Battermann, A.; Heinkenschloss, M., Preconditioners for karush-Kuhn-Tucker systems arising in the optimal control of distributed systems, (), 15-32 · Zbl 0909.49015
[4] Battermann, A.; Sachs, E.W., Block preconditioners for KKT systems in PDE-governed optimal control problems, (), 1-18 · Zbl 0992.49022
[5] G. Biros, O. Ghattas, Parallel Lagrange-Newton-krylov-schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver, Tech. Rep., Laboratory for Mechanics, Algorithms and Computing, Carnegie Mellon University, 2000 · Zbl 1091.65061
[6] O. Fromman, SynapsPointerPro v2.50, Synaps Ingenieure Gesellschaft mbH, Bremen, Germany, 2002
[7] N.R. Gauger, Aerodynamic shape optimization using the adjoint Euler equations. in: Proceedings of the GAMM Workshop on Discrete Modelling and Discrete Algorithms in Continuoum Mechanics (ISBN 3-89722-683-9) Berlin: Logos Verlag, 2001, pp. 87-96
[8] N.R. Gauger, Das Adjungiertenverfahren in der aerodynamischen Formoptimierung, DLR-Report No. DLR-FB-2003-05 (ISSN 1434-8454), 2003
[9] Gauger, N.R.; Brezillon, J., Aerodynamic shape optimization using adjoint method, J. aero. soc. India, 54, 3, (2002) · Zbl 1273.76312
[10] Gill, P.E.; Murry, W.; Wright, M.H., Practical optimization, (1981), Academic Press London
[11] S.B. Hazra, V. Schulz, Simultaneous pseudo-timestepping for PDE-model based optimization problems. To appear in BIT 2004 · Zbl 1066.65071
[12] S.B. Hazra, V. Schulz, Simultaneous pseudo-timestepping for Aerodynamic shape optimization problems with state constraints, Forschugsbericht Nr.04-6, Department of Mathematics/Computer Science, University of Trier, Germany, ISSN 0944-0488, SIAM J. Opt., 2004, submitted
[13] Hicks, R.M.; Henne, P.A., Wing design by numerical optimization, J. aircraft, 15, 407-412, (1978)
[14] Hörmander, L., Pseudo-differential operators, Comm. pure appl. math., 18, 501-517, (1965) · Zbl 0125.33401
[15] Jameson, A., Aerodynamic design via control theory, J. scientific comput., 3, 23-260, (1988)
[16] A. Jameson, Automatic design of transonic airfoils to reduce shock induced pressure drag, MAE Report 1881, presented in at the 31st Israel Annual Conference on Aviation and Aeronautics, February, 1990
[17] A. Jameson, Optimum aerodynamic design using CFD and control theory, in: AIAA 12th Computational Fluid Dynamics Conference, AIAA 95-1729-CP, June, 1995 · Zbl 0875.76497
[18] A. Jameson, J. Reuther, Control theory based airfoil design using the Euler equations, AIAA in: Proceedings of 94-4272-CP, 1994
[19] A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the Euler equation by finite volume methods using Runge-Kutta time-stepping schemes, AIAA 81-1259, 1981
[20] Kohn, J.J.; Nirenberg, L., On the algebra of pseudo-differential operators, Comm. pure appl. math., 18, 269-305, (1965) · Zbl 0171.35101
[21] N. Kroll, R.K. Jain, Solution of two-dimensional Euler equations - Experience with a finite volume code, DFVLR-IB-129-84/19, 1984
[22] N. Kroll, C.C. Rossow, K. Becker, F. Thiele, The MEGAFLOW - a numerical flow simulation system, in: 21st ICAS Symposium, Paper 98-2.7.4, Melbourne, Australia, 1998
[23] Kroll, N.; Rossow, C.C.; Becker, K.; Thiele, F., The MEGAFLOW project, Aerosp., sci. technol., 4, 223-237, (2000) · Zbl 0955.76079
[24] G. Kuruvila, S. Ta’asan, M. Salas, Airfoil optimization by the one-shot method, AGARD-FDP-VKI, Special Course on Optimum Design Methods in Aerodynamics, April 1994
[25] Lions, J.L., Optimal control of systems governed by partial differential equations, (1971), Springer-Verlag New York · Zbl 0203.09001
[26] Logashenko, D.; Maar, B.; Schulz, V.; Wittum, G., Optimal geometrical design of Bingham parameter measurement devices, International series of numerical mathematics (ISNM), 138, 167-183, (2001) · Zbl 1172.76317
[27] M. Nemec, D.W. Zingg, From analysis to design of high-lift configurations using a Newton-Krylov algorithm, ICAS Congress 2002
[28] Pironneau, O., On optimum design in fluid mechanics, J. fluid mech., 64, 97-110, (1974) · Zbl 0281.76020
[29] Pironneau, O., On optimum profiles in Stokes flow, J. fluid mech., 59, 117-128, (1973) · Zbl 0274.76022
[30] Pironneau, O., Optimal shape design for elliptic systems, (1982), Springer-Verlag New York · Zbl 0496.93029
[31] A. Iollo, G. Kuruvila, S. Ta’asan, pseudo-time method for optimal shape design using Euler equations, ICASE Report No. 95-59, 1995
[32] J. Reuther, A. Jameson, Aerodynamic shape optimization of wing and wing-body configurations using control theory, AIAA 95-0123, January, 1995
[33] J. Reuther, A. Jameson, J. Farmer, L. Martinelli, and D. Saunders, Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation, AIAA 96-0094, January, 1996.
[34] V.H. Schulz, Reduced SQP methods for large-scale optimal control problems in DAE with application to path planning problems for satellite mounted robots, Ph.D. Thesis, Universität Heidelberg, 1996 · Zbl 0848.49021
[35] Schulz, V.H., Solving discretized optimization problem by partially reduced SQP methods, Comput. vis. sci., 1, 83-96, (1998) · Zbl 0970.65066
[36] V. Schulz, SQP-based direct discretization methods for practical optimal control problems, (guest editor) Special issue of the Journal of Computational and Applied Mathematics 2000, 120, 1-2
[37] von Schwerin, M.; Deutschmann, O.; Schulz, V., Process optimization of reactive systems by partially reduced SQP methods, Computers chem. eng., 24, 89-97, (2000)
[38] S. Ta’asan, Pseudo-time methods for constrained optimization problems governed by PDE, ICASE Report No. 95-32, 1995
[39] S. Ta’asan, G. Kuruvila, Aerodynamic design and optimization in one shot. AIAA 92-0025, January 1992
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.