Indirect Treatment of State Constraints in Aerodynamic Shape Optimization Using Simultaneous Pseudo-Time-Stepping

Introduction

Practical shape design problems involve additional state constraints, e.g., shape optimization of aircraft for drag reduction with constant lift. Effectiveness of the optimization methods for aerodynamic shape design depends crucially on proper choice of the cost function, the constraints and their treatment during the optimization. One can treat the constraints ‘indirectly’, e.g., by making some kind of transformation so that the constraints are added to the objective function (with some weighting) and the constrained problem is reduced to an unconstrained one. This kind of treatment is termed as ‘soft’ constraints in [49]. Indirect treatment of constraints using traditional gradient methods can be found, among others, in [103, 142, 143].

Since problems of practical applications involve additional state constraints, we extend the simultaneous pseudo-time-stepping method to such problems in this chapter. The constraint is treated indirectly, that is by adding it to the objective function. The correspondence to a rigorous treatment is explained below.We present applications to wing and body optimizations of an SCT aircraft for drag reduction with constant lift. The number of iterations required for the full optimization problem is about 8 times that of the analysis problem. This means a drastic reduction of the computational cost compared to traditional ‘black-box’ gradient methods.