Multiobjective optimization of UAV winglet performance using surrogate models and CFD

Surrogate-based method can dramatically reduce the number of expensive function evaluations in real-world multi- objective optimization problems (MOP). When the number of objectives is small, using surrogate models combined with expected hypervolume improvement (EHVI) infill sampling criterion (ISC) has been proved to be efficient to provide a set of solutions with good diversity and good proximity to the Pareto front (PF) in aerodynamic shape optimization. However, traditional hypervolume-based infilling strategies use only one kind of ISC to generate one or multiple sample points, the advantages of various kinds of ISC cannot be comprehensively utilized and the parallelization is not easy to implement. This paper proposes a combined multi-point infilling strategy based on Kriging models and develops an efficient global multi-objective constrained optimization method (EGMOCO) to solve multi- objective aerodynamic shape optimization with complex constraints. Multiple sample points are generated by using four ISC considering hypervolume at each iteration and then evaluated in parallel. Firstly, the performance of EGMOCO is compared with that of single criterion EHVI strategy on six benchmarks within the same computational budget to prove its effectiveness, and then EGMOCO is implemented in an aerodynamic shape optimization problem with complex constraints. The result shows that EGMOCO has good performance in balancing local exploitation and global exploration with faster convergence rate and high robustness, the whole PF can be fully explored in limited evaluations and the constraint handling is effective especially for real-world problems with complex and nonlinear constraints, the comprehensive aerodynamic performance of the airfoil is greatly improved. It can be confirmed that Kriging-based multi-objective optimization method combined with multi-point infilling strategy performs better than single infilling criterion EHVI, since different sample infilling criteria can complement with each other, both local exploitation and global exploration can be considered and well balanced.

In this work, a volume and longitudinal stability-constrained multiobjective aerodynamic shape optimization is conducted. The aerodynamic shapes of lifting bodies are parameterized by using class function/shape function transformation parameterization method for maximum design flexibility. Hypersonic aerodynamic objectives and constraints are analyzed by solving the Reynolds-averaged Navier–Stokes equations in conjunction with a two-equation turbulence model. The Kriging technique is adopted to construct surrogate models aiming at reducing the computational cost. A two-stage method of infill sampling combined with multiobjective optimization is proposed to improve the performance of the surrogate models. Multiobjective evolutionary algorithm based on decomposition (MOEA/D) combined with penalty function method is employed to handle the multiobjective optimization problem with nonlinear constraints. The optimization results reveal that the two-stage method based on surrogates can reduce the computational cost significantly, and the accuracy of the surrogates around the Pareto front is sufficient. The two objectives are competing such that a set of Pareto optimal solutions are obtained, which are the best tradeoffs among the objectives. The unconstrained multiobjective optimization problem is also investigated with MOEA/D and nondominated sorting genetic algorithm II (NSGA-II) respectively to make a further comparison. The results show that the Pareto sets based on MOEA/D are more excellent and distribute more evenly than that obtained by NSGA-II. The computational efficiency of MOEA/D is about six times faster than that of NSGA-II. Lastly, aerodynamic characters of typical shape of Pareto front are analyzed under different flight conditions, and the results reveal favorable robustness of this shape.

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.

The Class-Shape function Transformation (CST) method is used to describe the parameterized airfoil geometry. The parameterized models for aerodynamic and stealthy performance of airfoil are constructed. The aerodynamic analysis model of airfoil is constructed by Computational Fluid Dynamics (CFD) method based on N-S equations. And the stealthy performance analysis model of airfoil is constructed by Computational Electromagnetic Method (CEM) based Method of Moments (MoM). The multi-objective aerodynamic and stealthy performance optimization method for airfoil using Kriging surrogate model is presented in this paper. The Latin hypercube method is employed to get a set of sample points. The aerodynamic and stealthy performance Kriging models are built. The multi-objective aerodynamic and stealthy performance optimization of airfoil is optimized by combining Pareto genetic algorithm with Kriging surrogate model. The presented method is validated by two applications. The results of the investigation show that the constructed analysis models are reasonable and the presented multi-objective optimization design method is feasible, which can improve the performance of airfoil and the efficiency of optimization effectively.

For models with large numerical simulation costs, such as high-speed trains, using as few samples as possible to construct a high-precision surrogate model during aerodynamic multi-objective optimization is critical to improving optimization efficiency. This study proposes a sequential infill criterion (SIC) appropriate for the Kriging surrogate model to address this issue. Three multi-objective functions are employed to test the feasibility of constructing a surrogate model based on SIC, and the SIC surrogate model then performs multi-objective aerodynamic optimizations on the high-speed train. The findings indicate that the expected improvement infill criterion (EIC) in the first stage can enhance the global prediction accuracy of the SIC. An infill criterion based on EIC that fuses gradient information (PGEIC) in the second stage is proposed to seek samples in the Pareto front. The PGEIC surrogate model achieves the lowest generational distance and prediction error. The performance of EIC for global search, EIC for Pareto front search, and infill criterion for Pareto front search using only gradient information is poor. The final PGEIC–SIC surrogate model of train aerodynamics has less than 1% prediction error for the three optimization objectives. The optimal solution reduces the aerodynamic drag force of the head car and the aerodynamic drag and lift force of the tail car by 4.15%, 3.21%, and 3.56%, respectively, compared with the original model. Furthermore, sensitivity analysis of key parameters revealed that the nose height v1, cab window height v3, and lower contour line have a greater impact on aerodynamic forces. Moreover, the nose and cab window heights of the optimal model have been reduced, and the lower contour line is concave. Correspondingly, the streamlined shape appears more rounded and slender.

Aspects concerning resonance and global stability of a wind turbine blade must be carefully considered in its optimal design. In this paper, a composite wind turbine blade with an external geometry based on the NREL 5 MW model was subjected to multi-objective structural optimization considering these aspects. Four multi-objective structural optimization problems are formulated considering the blade mass, the maximum blade tip displacement, the natural frequencies of vibration, and the critical load factor as objective functions. The design variables are the number of plies, material, and fiber orientation. The design constraints are the materials’ margin of safety, the blade’s allowable tip displacement, and the minimum load factor. The blade model is submitted to the loads determined by the actuator lines theory and discretized in a finite element parameterized model using the Femap software according to geometric design variables. Among many multi-objective evolutionary algorithms available in the literature concerning evolutionary computation, the NSGA-II is the adopted evolutionary algorithm to solve the multi-objective optimization problems. Pareto fronts are obtained and performance indicators are used to evaluate the distribution of the non-dominated solutions. Multi-criteria decision-making is used to extract the solutions from the Pareto fronts according to the decision-maker’s preferences. The values of the objective functions, design variables, and constraints are presented for each extracted solution. The proposed study is expected to contribute to the multi-objective optimization and the structural design of wind turbine blades.

By transforming a multi-objective optimization problem into a number of single-objective optimization problems and optimizing them simultaneously, decomposition-based evolutionary multi-objective optimization algorithms have attracted much attention in the field of multi-objective optimization. In decomposition-based algorithms, the population diversity is maintained using a set of predefined weight vectors, which are often evenly sampled on a unit simplex. However, when the Pareto front of the problem is not a hyperplane but more complex, the distribution of the final solution set will not be that uniform. In this paper, we propose an adaptive method to periodically regenerate the weight vectors for decomposition-based multi-objective algorithms according to the geometry of the estimated Pareto front. In particular, the Pareto front is estimated via Gaussian process regression. Thereafter, the weight vectors are reconstructed by sampling a set of points evenly distributed on the estimated Pareto front. Experimental studies on a set of multi-objective optimization problems with different Pareto front geometries verify the effectiveness of the proposed adaptive weights generation method.

Adaptive surrogate assisted multi-objective optimization approach for highly nonlinear and complex engineering design problems

Manifold-guided multi-objective gradient algorithm combined with adjoint method for supersonic aircraft shape design

Study of a hull form optimization system based on a Gaussian process regression algorithm and an adaptive sampling strategy, Part II: Multi-objective optimization

In this study, efficient global optimization (EGO) with a multi-fidelity hybrid surrogate model for multi-objective optimization is proposed to solve multi-objective real-world design problems. In the proposed approach, a design exploration is carried out assisted by surrogate models, which are constructed by adding a local deviation estimated by the kriging method and a global model approximated by a radial basis function. An expected hypervolume improvement is then computed on the basis of the model uncertainty to determine additional samples that could improve the model accuracy. In the investigation, the proposed approach is applied to two-objective and three-objective optimization test functions. Then, it is applied to aerodynamic airfoil design optimization with two objective functions, namely minimization of aerodynamic drag and maximization of airfoil thickness at the trailing edge. Finally, the proposed method is applied to aerodynamic airfoil design optimization with three objective functions, namely minimization of aerodynamic drag at cruising speed, maximization of airfoil thickness at the trialing edge and maximization of lift at low speed assuming a landing attitude. XFOILis used to investigate the low-fidelity aerodynamic force, and a Reynolds-averaged Navier–Stokes simulation is applied for high-fidelity aerodynamics in conjunction with a high-cost approach. For comparison, multi-objective optimization is carried out using a kriging model only with a high-fidelity solver (single fidelity). The design results indicate that the non-dominated solutions of the proposed method achieve greater data diversity than the optimal solutions of the kriging method. Moreover, the proposed method gives a smaller error than the kriging method.

The optimization of grinding is a multi-objective problem characterized by high dimensionality, non-linearity, and complexity. Solving this multi-objective optimization (MOO) problem is one of the most challenging tasks in the field of mechanical engineering. In-depth research on multi-objective parameter optimization technology for grinding is of great significance for improving processing efficiency, optimizing product quality, and reducing energy consumption. This paper takes the multi-objective optimization problem of grinding as its starting point. First, it introduces the basic theory of multi-objective optimization and two primary methods for solving such problems: optimization target dimension reduction and multi-objective optimization. Second, the key technologies of the two methods are reviewed, including the modeling method of the optimization problem, the multi-objective optimization algorithm for solving the optimization model, and the prior and posterior trade-off methods used to obtain the compromised optimal solutions. Finally, the existing problems of the multi-objective optimization methods in grinding processing are summarized and the future development trends are predicted. This paper aims to provide researchers with a comprehensive understanding of the multi-objective optimization technology in grinding processing, enabling them to make more reasonable decisions when dealing with actual multi-objective optimization problems.

Optimization problems in many industrial applications are very hard to solve. Many examples of them can be found in the design of aeronautical systems. In this field, the designer is frequently faced with the problem of considering not only a single design objective, but several of them, i.e., the designer needs to solve a Multi-Objective Optimization Problem (MOP). In aeronautical systems design, aerodynamics plays a key role in aircraft design, as well as in the design of propulsion system components, such as turbine engines. Thus, aerodynamic shape optimization is a crucial task, and has been extensively studied and developed. Multi-Objective Evolutionary Algorithms (MOEAs) have gained popularity in recent years as optimization methods in this area, mainly because of their simplicity, their ease of use and their suitability to be coupled to specialized numerical simulation tools. In this chapter, we will review some of the most relevant research on the use of MOEAs to solve multi-objective and/or multi-disciplinary aerodynamic shape optimization problems. In this review, we will highlight some of the benefits and drawbacks of the use of MOEAs, as compared to traditional design optimization methods. In the second part of the chapter, we will present a case study on the application of MOEAs for the solution of a multi-objective aerodynamic shape optimization problem.

Groundwater management involves a complex decision‐making process, often with the need to balance the trade‐off between meeting society's demand for water and environmental protection. Therefore effective management of groundwater resources often involves some form of multi‐objective optimization (MOO). Many existing software tools offer simulation model‐enabled optimization, including evolutionary algorithms, for solving MOO problems. However, such analyses involve a huge amount of numerical process‐based model runs, which require significant computational effort, depending on the nonlinearity and dimensionality of the problem, in order to seek the optimal trade‐off function known as the Pareto front. Surrogate modeling, through techniques such as Gaussian Process Regression (GPR), is an emerging approach to significantly reduce the number of these model evaluations thereby speeding up the optimization process. Yet, surrogate model predictive uncertainty remains a profound challenge for MOO, as it could mislead surrogate‐assisted optimization, which may result in either little computational savings from excessive retraining, or lead to suboptimal and/or infeasible solutions. In this work, we present probabilistic Pareto dominance criteria that considers the uncertainty of GPR emulation during MOO, producing a “cloudy” Pareto front which provides an efficient decision space sampling mechanism for retraining the GPR. We then developed a novel acquisition strategy to manage the solution repository from this cloud and generate an ensemble of infill points for retraining. We demonstrate the capabilities of the algorithm through benchmark test functions and a typical density‐dependent coastal groundwater management problem.

This dissertation presents an efficient optimization methodology to solve the CFD-based shape optimization problems. This methodology is based on evolutionary algorithms (EAs) for their well-known derivative-free property as well as the advantages in dealing with multiobjective optimization problems (MOOPs) and providing the global optimal solutions. Meanwhile, the approximation models and the deterministic optimization methods are combined with EA to improve the optimization efficiency and the local convergence. The optimization process consists of two parts: the design space exploration using EA (global search) and the convergence acceleration using deterministic methods (local search). When solving a shape optimization problem, the optimizer controls the whole optimization process. The shape variation and flow simulation are incorporated to perform the objective function evaluations and construct the database for the training of the approximation models. Free form deformation (FFD) is employed for the shape variation because it directly modifies the computational grids required by the flow solver and provides a flexible deformation by only moving a small number of the control points. The flow simulation is performed using the in-house developed finite-volume flow solver FASTEST. A modified, elitist evolutionary method NSGA-II is employed as the global explorer. During the evolutionary optimization process, in some generations the online and locally trained RBFN models are utilized to substitute the expensive function evaluations conducted by the high-fidelity flow solver. The adaptive exchange between the exactly and approximately evaluated generations is accomplished through an approximation control procedure. Afterwards, using the achieved results as the starting points, two derivative-free trust-region algorithms DFO and CONDOR are chosen to perform the local search. The proposed optimization methodology is first applied to several analytical and numerical optimization problems, and the optimization results show that it works well for both convex and non-convex Pareto front. The incorporation of approximation models overcomes the requirement of large number of computationally expensive function evaluations. Compared to conventional EA, this hybrid optimization method is able to achieve a set of optimal solutions with good diversity and better convergence with much less computational cost. Furthermore, the influence of RBFN construction methods and the number of solutions in the initial database on the approximation accuracy, as well as the performance of two local search methods, DFO and CONDOR, are studied in this work. Another contribution of the present work is to provide a methodology to construct the approximation model by combining the interpolation methods (spline interpolation or radial basis function interpolation) with the proper orthogonal decomposition (POD) technique in order to approximate the complete flow region in an efficient manner. Applied in the optimization process, this kind of surrogate model has the ability not only to predict the objective functions but also to provide a detailed estimation of the underlying flow region. The efficiency and accuracy of the POD-based approximation models as well as the quality of the optimization results are investigated by two shape optimization test cases.