Multi-Objective Optimization of a Transonic Compressor Rotor by Using an Adjoint Method

TO IMPROVE the aerodynamic performance and the economic benefits of an aircraft engine, one strives to increase the total pressure ratio, the adiabatic efficiency et al., and decrease the size and weight of the engine. Multi-objective optimization has been proposed and studied to satisfy the higher-level requirements in recent years. Benini [1] successfully performed the multi-objective optimization of the NASA Rotor 37 to maximize the total pressure ratio and the compressor efficiency by using a multi-objective evolutionary algorithm. Lian and Liou [2] redesigned the blade of NASA Rotor 67 to maximize the total pressure ratio while minimizing the compressor weight, and an approximately 1.8% total pressure ratio gain was achieved. Because of its robustness and excellent compatibility in design optimization, the nongradient-based optimization methods, such as evolutionary algorithm and the surrogate model-based methods [1–4] have been widely applied to the design optimization. The nongradient-based design optimization can support the global optimum in a wide design space. However, for the design optimization of complex aerodynamic shapes, numerous flow calculations are necessary because of the large number of design parameters. The adjoint method proposed by Jameson [5] can support the gradient information fast for the gradient-based design optimization. The design optimization by using the adjoint method can significantly improve the computational efficiency because it requires about only two flow calculations in each design cycle to determine the complete gradient information of each cost function, regardless of the number of design parameters. In the past decades, the adjoint method was widely used in the design optimization of external flow. Jameson and Reuther [6,7] successfully performed aerodynamic design optimization of airfoil, wing, and wing–body configuration by using a continuous adjoint method. In recent years, this method has been introduced to the design optimization of turbomachinery blades by Dreyer and Martinelli [8] and Yang et al. [9]. Recently, by using the adjoint method, the aerodynamic shape design optimization [10], the multistage design optimization [11,12], the aeroelastic design optimization [13], and the multipoint design optimization [14] of turbomachinery blades are successfully performed. Because of its high efficiency and sufficient accuracy on gradient calculation, the adjoint method has already been used in the multiobjective design optimization [15–17]. A simple but widely used approach for the gradient-based multi-objective optimization is the weighted-sum method with a single cost function consisting of a linear combination of multiple cost functions with appropriate weights, which converts the multi-objective optimization problem into a single-objective optimization problem. However, as pointed out by Shankaran and Barr [15], this method is unable to capture the concave portions and the disjoint Pareto fronts. Furthermore, this method cannot always guarantee sufficient design convergence because the improvement of some objectives brings with it unexpected defects not allowed in reality. For example, in the multiobjective optimization of total pressure ratio and adiabatic efficiency of a transonic compressor rotor, the total pressure ratio is not allowed to increase without limit because increasing total pressure ratio induces 1) increased turning to a critical degree, after which the mass flow rate decreases away from the constraint, and 2) compressor stall triggers at a lower backpressure due to the stronger shock and the more intensive shock/tip-leakage interaction. The detrimental performance brings difficulties on obtaining the optimal aerodynamic shape. To overcome the drawbacks of the traditional gradient-based multi-objective optimization mentioned previously, an approach for multi-objective optimization is introduced in the present study. The multi-objective optimization is decomposed into two steps. The first step favors obtaining a series of initialized blades avoiding the compressor stall triggers at the design condition, and the second step, consisting of a series of single-objective optimizations, favors determining the Pareto front. The aerodynamic shape of a transonic compressor rotor NASA Rotor 67 is redesigned at the operating condition near peak efficiency to maximize the total pressure ratio and the adiabatic efficiency with the constraint of mass flow rate by using the adjoint method. The Pareto front of the multi-objective optimization is finally given, and the effects of blade profile modification on the performance improvement are presented.