Airfoil Design and Optimization by the One-Shot Method

An efficient numerical approach for the design of optimal aerodynamic shapes is presented in this paper. The objective of any optimization problem is to find the optimum of a cost function subject to a certain state equation (governing equation of the flow field) and certain side constraints. As in classical optimal control methods, the present approach introduces a costate variable (Lagrange multiplier) to evaluate the gradient of the cost function. High efficiency in reaching the optimum solution is achieved by using a multigrid techniique and updating the shape in a hierarchical manner such that smooth (low frequency) changes are done separately from high-frequency changes. Thus, the design variables are changed on a grid where their changes produce nonsmooth (high frequency) perturbations that can be damped efficiently by the multigrid. The cost of solving the optimization problem is approximately two to three times the cost of the equivalent analysis problem.

[1]  M. G. Salvadori,et al.  Numerical methods in engineering , 1955 .

[2]  Krishnamurty Karamcheti,et al.  Principles of ideal-fluid aerodynamics , 1966 .

[3]  Frances Bauer,et al.  A Theory of Supercritical Wing Sections, with Computer Programs and Examples , 1972 .

[4]  Joseph L. Steger,et al.  A Finite-Difference Method for Transonic Airfoil Design , 1973 .

[5]  R. M. Hicks,et al.  An assessment of airfoil design by numerical optimization , 1974 .

[6]  G. B. Cosentino,et al.  Numerical optimization design of advanced transonic wing configurations , 1985 .

[7]  M. Giles,et al.  Newton solution of direct and inverse transonic Euler equations , 1985 .

[8]  G. Volpe,et al.  The design of transonic aerofoils by a well‐posed inverse method , 1986 .

[9]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[10]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[12]  Gregory A. Wrenn,et al.  An indirect method for numerical optimization using the Kreisselmeir-Steinhauser function , 1989 .

[13]  V. Korivi,et al.  Sensitivity analysis, approximate analysis, and design optimization for internal and external viscous flows , 1991 .

[14]  Sinan Eyi,et al.  Transonic Airfoil Design by Constrained Optimization , 1991 .

[15]  S. Taasan One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control , 1991 .

[16]  M. D. Salas,et al.  Aerodynamic design and optimization in one shot , 1992 .