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.

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

This paper presents a numerical solution to multi-objective shape optimization problems of steady heat-convection fields. In previous study, it has been dealt with a shape optimization problem for total dissipated energy minimization in the domain of a viscous flow field and a shape determination problem of temperature distribution prescribed problem in sub-domains of heat-convection fields. In this study, a multi-objective shape optimization problem using normalized objective functional is formulated for the total dissipated energy minimization problem and the temperature distribution prescribed problem in steady heat-convection fields. In addition, another multi-objective shape optimization problem is formulated for the temperature distribution prescribed problem, while the total dissipated energy is constrained to less than a desired value, in the steady forced heat-convection fields. Shape gradients of these multi-objective shape optimization problems are derived theoretically using the Lagrange multiplier method, adjoint variable method, and the formulae of the material derivative. Reshaping is carried out by the traction method proposed as an approach to solving shape optimization problems. The validity of proposed method is confirmed by results of 2D numerical analysis.

Multi-Objective Shape Optimization in Forced Heat-Convection Fields

This paper presents the application of a viscous adjoint method to the multi-objective design optimization of a transonic compressor rotor blade row. The adjoint method requires about twice the computation effort of flow calculation to obtain the complete gradient information for each cost function, regardless of the number of design parameters. NASA Rotor 67 is redesigned to maximize the total pressure ratio and adiabatic efficiency. The multi-objective design optimization is decomposed into multiple single-objective design optimizations. Firstly a set of blades with different gains of total pressure ratio can be obtained through the design initialization maximizing total pressure ratio with the constraint of mass flow rate. The upper limit of total pressure ratio can be predicted to avoid stall in the design. Then a series of single-objective optimizations maximizing adiabatic efficiency for fixed total pressure ratio are performed starting from the blades determined in the initial step to obtain the Pareto front. The cost function is defined as the entropy production per unit mass flow rate combined with the constraints of mass flow rate and total pressure ratio. The results are presented in detail and the effects of blade profile modification on performance improvement and shock/tip-leakage interaction are examined.

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

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

With smooth objective functions and constraint conditions, gradient-based methods can be used to solve multi-objective optimization problems efficiently. However, when applied to structural sizing optimization problems, using the Finite Element Method (FEM) and a finite difference scheme to calculate sensitivities can be computationally expensive. The adjoint variable method can be used to reduce computational cost. In order to solve multi-objective structural sizing and shape optimization problems efficiently, this paper proposes using the adjoint variable method. The adjoint variable method efficiently calculates multiple sensitivities for objectives that involve structural responses and cuts down computational cost by reducing the number of sensitivity calculations required per design variable.

Adjoint Variable Method for Multi-Objective Sizing and Shape Optimization

Creative Design with Multi-objective Topology Optimization

This paper proposes a systematic scheme of removing void elements to achieve fast and efficient level set based topology optimization. When performing optimization, unless special treatment is applied to the stiffness matrix, the density of these void elements are usually represented numerically by a small positive value. In level set based topology optimization, since the amount of computational resources required for FEM dwarfs those required for level set evolution, the removal of these elements from the global stiffness matrix can drastically reduce total computation time. The proposed scheme removes the void elements, determined by their nodes' level set values, from the optimization process by use of mapping procedures. The results presented here show time reductions of at least 70%. An additional advantage of the presented scheme is that it can be easily used with any black box FEM routine.

Chen Jian Ken Lee

Papers

Adjoint Variable Method for Multi-Objective Sizing and Shape Optimization

Creative Design with Multi-objective Topology Optimization

Removing Void Elements for Structural Level Set Topology Optimizaiton