American Institute of Mathematical Sciences

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January  2005, 1(1): 133-148. doi: 10.3934/jimo.2005.1.133

Airfoil design via optimal control theory

K.F.C. Yiu 1, , K.L. Mak 1, and  K. L. Teo 2,

1. 

Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China, China
2. 

Department of Mathematics and Statistics, Curtin University of Technology, Perth, Australia

Received  April 2004 Revised  October 2004 Published  January 2005

In airfoil design, one problem of great interest is to find the target airfoil profile to achieve a given target velocity distribution. It can be formulated as an optimal control problem, with the control being the airfoil profile and the governing equation being the full potential equation in the transonic regime. To discretize the problem, one approach is to employ the finite element method. In the discretized space, a direct relationship between the objective function and the unknown profile co-ordinates can be defined via the finite element basis functions. Moreover, it is advantageous to derive the gradient in the discretized space rather than the continuous space to avoid contamination by discretization errors. In this paper, this approach is studied. In particular, a new formulation is proposed. A novel decomposition of the discrete space for the potential function, the gradient is derived and an efficient algorithm using the quasi-Newton method is described. In generating and adjusting the mesh during iterations, the elliptic mesh generation technique is used.
Keywords: finite elements, full potential equation., optimal control, Airfoil design.
Mathematics Subject Classification: 93C20, 76H05, 49Q1.
Citation: K.F.C. Yiu, K.L. Mak, K. L. Teo. Airfoil design via optimal control theory. Journal of Industrial and Management Optimization, 2005, 1 (1) : 133-148. doi: 10.3934/jimo.2005.1.133
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