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Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with state constraints. (English) Zbl 1130.49032
The paper is started by a very brief review on two difficulties in the numerical solution methods for aerodynamic shape optimization methods which are based on gradient optimization. Then by a nice literature review and reminding some useful points about practical methods, the path for explaining the pseudo-timestepping method is prepared. By defining the problem as a simple general optimal control one, the necessary optimality conditions are formulated via a Lagrangian functional. This is caused to look for the steady-state solutions of a pseudo-time embedded evolution system. In order to accelerate convergence, the preconditioner for the system is used by inexact RSQP method. (The method is explained algorithmic in section 2.2 based on the projected Hessian of the Lagrangian. The inverse of the matrix in the basic formulation of the method (formula (6)) is used as a preconditioner for the process.) Then, theoretically scalar state constraints are added. Using the Gaussian elimination and approximating Newton step for necessary conditions of the new problem corresponding to RSQP method, are reduced the problem into a quadratic one. It is shown that this reduction can be interpreted as a projection of the design velocity toward the linearized lift constraint. In this manner the simultaneous pseudo-timestepping algorithm for the preconditioned system is given in section 3.1. Afterwards, the detailed equations of state, costate, design with discretization, surface parameterization, gradient computation and grid perturbation strategy for the shape optimization of the airfoil problem are explained. In the numerical results, which is an interesting part, the optimization method is applied to test to the RAE2822 airfoil for drag reduction with constant lift and drag reduction with constant pitching moment (in 4 cases with 21 and 40 design parameters). The mathematical computational methods and the best way for choosing the parameters to have the better shape in each cases are also discussed. But there is no comparison with the other methods.
MSC:
49Q10Optimization of shapes other than minimal surfaces
49M30Other numerical methods in calculus of variations