An Optimal Shape Design Problem for Fan Noise Reduction

Noise generated by the rotating blades of an aircraft turbofan engines continues to pose a challenge for many researchers and of course, the reduction of the noise have been received more attention in the recent studies. Williams and Hall studied aerodynamic sound generation by turbulent flow around a scattering half plane [1]. Howe in [2] discussed the trailing edge noise at low Mach numbers. It is well known that the aircraft noise may be minimized through the two signification tasks: acoustic shape optimization of the inlet [3] and impedance optimization of the liner [4]. In much of the pervious work on this topic, the computational methods are gradient-based that bear some drawbacks such as mesh regularization and the requirement of knowledge of derivatives. The aim of this article is to present a method to find an optimal solution of a shape design problem that models reducing the amount of noise radiated from aircraft turbofan engines. Our method is based on measure theory and has some features, for instance, the design process is not iterative and therefore requires no initial design to be suggested; it is computationally efficient and flexible enough to accommodate general design problem. The paper proceeds as follows. In Section 2 we first describe the mathematical formulation of the optimal control problem and then a brief review of the weak formulation of the problem is given. In Section 3, we metamorphose the optimization problem to a linear programming problem by the aid of our method. Finally, we present some numerical results in Section 4. 2. Problem Description and Weak Formulation