Aerodynamic Optimization Problems Research Articles

Data-driven hierarchical collaborative optimization method with multi-fidelity modeling for aerodynamic optimization

The aerodynamic optimization of aircraft based on evolutionary algorithms generally requires thousands or even tens of thousands of iterations. Moreover, it faces challenges such as high-dimensional design variables and high computational costs, leading to low efficiency in algorithm search and poor optimization results. In order to enhance the efficiency of aerodynamic optimization, a data-driven collaborative optimization and multi-fidelity hierarchical modeling method is proposed. Firstly, a Clustering-Enhanced Teaching-Learning-Based Optimization (CETLBO) algorithm is introduced to improve its exploration and exploitation capabilities. By employing a collaborative optimization strategy, a divide-and-conquer paradigm is utilized to efficiently solve high-dimensional, nonlinear, and complex engineering optimization problems. Secondly, a two-stage hierarchical aerodynamic optimization framework is designed to reduce computational costs by integrating data from different fidelity levels. Stage 1 adopts a data-driven approximate optimization method to provide prior knowledge for the high-fidelity optimization of stage 2. Finally, the proposed method is applied to the optimization design of supersonic wing shapes. Compared to traditional optimization design systems, it demonstrates high efficiency in high-dimensional aerodynamic optimization problems, yielding better optimization results.

High-dimensional aerodynamic shape optimization framework using geometric domain decomposition and data-driven support strategy for wing design

Traditional aerodynamic shape optimization methods usually suffer from difficulties in optimization and modeling for high-dimensional aerodynamic optimization problems. To overcome these difficulties, a new high-dimensional aerodynamic shape optimization framework is proposed from the perspectives of geometric domain decomposition for dimensionality reduction and a data-driven model support strategy for changing the role of surrogate model and assisting the optimizer. Geometric domain decomposition uses proper orthogonal decomposition to transform the high-dimensional Class-Shape Function Transformation method into low-dimensional geometric modes to reduce design dimensions for parameterizing the geometric shape, which reduces optimization and modeling difficulties. Also, a data-driven support strategy is proposed to improve the effectiveness and efficiency of the optimizer by learning from historical aerodynamic data; this strategy modifies the role of surrogate model in the optimization framework to assist the optimizer; the surrogate model, serving as an auxiliary role, can reduce the dependence on model accuracy, which overcomes the difficulty of high-dimensional modeling to some extent. We applied the developed framework to transonic drag reduction of wings to validate its performance. The optimization results show that the developed framework converges faster and offers better designs compared to direct optimization and state-of-the-art surrogate-based optimization frameworks for high-dimensional aerodynamic shape optimization.

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

The aerodynamic design of supersonic aircraft necessitates the comprehensive consideration of multiple conflicting design objectives and multi-disciplinary characteristics. However, the efficiency of heuristic algorithms in solving multi-objective aerodynamic problems significantly decreases with the increase in the number of design variables and objectives. Although gradient-based adjoint optimization methods can eliminate the constraints of design variables on aerodynamic optimization and have high optimization efficiency, they are prone to getting stuck in local optima and are difficult to apply in multi-objective optimization. To overcome these shortcomings, a manifold-guided multi-objective gradient-based algorithm (MG-MOGBA) is proposed. This algorithm enhances the optimization efficiency of the algorithm by coupling the multi-objective gradient operator with the predicted Pareto front (PF) manifold structure to guide the population search direction and reduce the search space. A multi-objective aerodynamic optimization framework based on the discrete adjoint method is developed to automate the aerodynamic optimization process. Test functions and airfoil optimization results confirm that the proposed algorithm achieves superior results with fewer iterations and is more efficient than heuristic algorithms. The four-objective supersonic civil aircraft example demonstrates that the algorithm can effectively handle high-dimensional multi-objective aerodynamic optimization problems. A potential equilibrium solution increased the cruise factor by 17.5% and 12.8% under supersonic and transonic conditions respectively, reduced the maximum overpressure of the sonic boom by 25.1%, and decreased the root bending moment by 20.3%. Overall, MG-MOGBA can efficiently solve high-dimensional multi-objective aerodynamic optimization problems and has good scalability for the dimensions of design objectives and variables.

Surrogate-assisted differential evolution using manifold learning-based sampling for high- dimensional expensive constrained optimization problems

To address the challenges of high-dimensional constrained optimization problems with expensive simulation models, a Surrogate-Assisted Differential Evolution using Manifold Learning-based Sampling (SADE-MLS) is proposed. In SADE-MLS, differential evolution operators are executed to generate numerous high-dimensional candidate points. To alleviate the curse of dimensionality, a Manifold Learning-based Sampling (MLS) mechanism is developed to explore the high-dimensional design space effectively. In MLS, the intrinsic dimensionality of the candidate points is determined by a maximum likelihood estimator. Then, the candidate points are mapped into a low-dimensional space using the dimensionality reduction technique, which can avoid significant information loss during dimensionality reduction. Thus, Kriging surrogates are constructed in the low-dimensional space to predict the responses of the mapped candidate points. The candidate points with high constrained expected improvement values are selected for global exploration. Moreover, the local search process assisted by radial basis function and differential evolution is performed to exploit the design space efficiently. Several numerical benchmarks are tested to compare SADE-MLS with other algorithms. Finally, SADE-MLS is successfully applied to a solid rocket motor multidisciplinary optimization problem and a re-entry vehicle aerodynamic optimization problem, with the total impulse and lift to drag ratio being increased by 32.7% and 35.5%, respectively. The optimization results demonstrate the practicality and effectiveness of the proposed method in real engineering practices.

Review of different vertical axis wind turbine modeling methods

Wind energy can be used as an inexhaustible option for human consumption. Wind turbines offer a promising solution for off-grid areas. Darrieus turbine is one of the types of turbines that can be more efficient than other types if it is used by knowing its characteristics. The complex dynamic mechanism of the flow around the machine has led to its aerodynamic optimization problems remaining complex. This article reviews the applied methods in modeling vertical axis turbines and implicitly shows the turbine operation by presenting the results of these methods. Knowing these methods is important because each has its advantages and disadvantages and should be selected depending on the purpose of the research. One of the problems with Darrieus turbines is their poor start-up, which has been little researched so far. Finally, a suitable method that can model the start-up is introduced.

Surrogate and Autoencoder-Assisted Multitask Particle Swarm Optimization for High-Dimensional Expensive Multimodal Problems

In practice, some optimization problems require expensive calculation and exhibit multimodal characteristics simultaneously. These problems are called high-dimensional expensive multimodal optimization problems. When addressing such problems, existing surrogate-assisted evolutionary algorithms (SAEAs) encounter the “curse of dimensionality," which severely affects their capability to search optimal solutions. Therefore, this study proposed a surrogate and autoencoder-assisted multitask particle swarm optimization algorithm. First, an autoencoder-embedded multitask evolutionary framework was established to transform a high-dimensional multimodal optimization problem into multiple low-dimensional subproblems or subtasks. Further, a multi-level surrogate model management mechanism combining mirror learning was proposed. An appropriate local surrogate model can be rapidly generated for each modality of the problem. Moreover, a dual-mode local exploitation strategy was developed to improve the capability of swarm to exploit each subtask. The proposed algorithm was compared with seven existing SAEAs on 33 benchmark functions and the aeroengine aerodynamic design optimization problem. Experimental results revealed that the proposed algorithm can obtain multiple highly competitive optimal solutions, including global optimal solutions.

An Efficient Hybrid Multi-Objective Optimization Method Coupling Global Evolutionary and Local Gradient Searches for Solving Aerodynamic Optimization Problems

Aerodynamic shape optimization is frequently complicated and challenging due to the involvement of multiple objectives, large-scale decision variables, and expensive cost function evaluation. This paper presents a bilayer parallel hybrid algorithm framework coupling multi-objective local search and global evolution mechanism to improve the optimization efficiency and convergence accuracy in high-dimensional design space. Specifically, an efficient multi-objective hybrid algorithm (MOHA) and a gradient-based surrogate-assisted multi-objective hybrid algorithm (GS-MOHA) are developed under this framework. In MOHA, a novel multi-objective gradient operator is proposed to accelerate the exploration of the Pareto front, and it introduces new individuals to enhance the diversity of the population. Afterward, MOHA achieves a trade-off between exploitation and exploration by selecting elite individuals in the local search space during the evolutionary process. Furthermore, a surrogate-assisted hybrid algorithm based on the gradient-enhanced Kriging with the partial least squares(GEKPLS) approach is established to improve the engineering applicability of MOHA. The optimization results of benchmark functions demonstrate that MOHA is less constrained by dimensionality and can solve multi-objective optimization problems (MOPs) with up to 1000 decision variables. Compared to existing MOEAs, MOHA demonstrates notable enhancements in optimization efficiency and convergence accuracy, specifically achieving a remarkable 5–10 times increase in efficiency. In addition, the optimization efficiency of GS-MOHA is approximately five times that of MOEA/D-EGO and twice that of K-RVEA in the 30-dimensional test functions. Finally, the multi-objective optimization results of the airfoil shape design validate the effectiveness of the proposed algorithms and their potential for engineering applications.

Aerodynamic optimization with large shape and topology changes using a differentiable embedded boundary method

Embedded (or immersed) boundary methods (EBMs) for CFD and fluid-structure interaction (FSI) are attractive for aerodynamic optimization problems characterized by large shape deformations and surface topology changes. At each iteration, they eliminate the need for explicitly remeshing a computational fluid domain and avoid the pitfalls of transferring information from one CFD mesh to another. However, they are vulnerable to discrete events that compromise the smoothness of the aerodynamic objective and/or constraint functions they may compute. For this reason, the realization of EBMs for gradient-based aerodynamic shape optimization requires first their endowment with sufficient smoothness guarantees. Drawing on a recently developed EBM for the solution of compressible viscous fluid and FSI problems with such guarantees, this paper demonstrates the potential of shape-differentiable EBMs for discovering optimal aerodynamic designs featuring significant shape deformations and surface topology changes. Specifically, it highlights their advantages in terms of reliability with respect to large shape adjustments and surface topology changes, efficiency, simplicity, and reduced need for user intervention. To this end, the paper showcases the application of an advanced EBM with smoothness guarantees to wing shape, wing section, and nacelle-pylon placement optimization problems formulated for a NASA Common Research Model (CRM).

Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem

The forecasted growth in dynamic global air fleet size in the coming decades, together with the need to introduce disruptive technologies supporting net-zero emission air transport, demands more efficient design and optimization workflows. This research focuses on developing an aerodynamic optimization framework suited for multi-objective studies of small aircraft engine air-intake ducts in multiple flight conditions. In addition to the refinement of the duct’s performance criteria, the work aims to improve the economic efficiency of the process. The optimization scheme combines the advantages of Kriging-based Efficient Global Optimization (EGO) with the Radial Basis Functions (RBF)-based mesh morphing technique and the Chebyshev-type Achievement Scalarizing Function (ASF) for handling multiple objectives and design points. The proposed framework is applied to an aerodynamic optimization study of an I-31T aircraft turboprop engine intake system. The workflow successfully reduces the air-duct pressure losses and mitigates the flow distortion at the engine compressor’s front face in three considered flight phases. The results prove the framework’s potential for solving complex multi-point air-intake duct problems and the capacity of the ASF-based formulation to guide optimization toward the designer’s preferred objective targets.

Kriging-Based Space Exploration Global Optimization Method in Aerodynamic Design

For complicated aerodynamic design problems, the efficient global optimization method suffered from the defect of the incorrect portrayal of the design space, resulting in bad global convergence and efficiency performance. To this end, a Kriging-based global optimization method, named the Kriging-based space exploration method (KSE), was proposed in this paper. It selected multiple promising local minima and classified them into partially and fully explored minima in terms of the fitting quality of the surrogate model. Then, the partially explored minima would be furtherly exploited. During the local search, an enhanced trust-region method was adopted to make deep exploitation. By combining local and global searches, the proposed method could improve the fitting quality of the surrogate model and the optimization efficiency. The KSE was compared to other global surrogate-based optimization methods on 12 bound-constrained testing functions with 2 to 16 design variables and 2 aerodynamic optimization problems with 24 to 77 design variables. The results indicated that the KSE generally took fewer function evaluations to find the global optima or reach the target value in most test problems, holding better efficiency and robustness.

Efficient aerodynamic analysis and optimization under uncertainty using multi-fidelity polynomial chaos-Kriging surrogate model

Surrogate model has been extensively employed in uncertainty-based design optimization (UBDO) for computationally expensive engineering problems. However, it often causes great difficulties to designers due to the unsatisfactory accuracy and the high sensitivity of surrogate prediction in presence of uncertainties. Worse still, some popular metamodeling methods also require a substantially higher computational cost than that in deterministic design to get an acceptable accuracy. To address the challenging problem, an UBDO framework based on the proposed multi-fidelity polynomial chaos-Kriging (MF PC-Kriging) surrogate model is proposed, with particular superiority for complex aerodynamic applications. The construction principle of the MF PC-Kriging model and the rationality of the superiority of it with respect to popular surrogate models are explained in detail. Meantime, it is examined by investigating an analytical function and a transonic aerodynamic application with both geometrical and operational uncertainties. Thus, the MF PC-Kriging with easier understanding and better modeling capabilities is involved in UBDO to resolve the proposed difficulty. Finally, an uncertainty-based aerodynamic design optimization problem is performed using this proposed framework. It is observed that for the considered examples, the developed methodology is more efficient and provides the better performance for aerodynamic uncertainty analysis, and complex aerodynamic analysis and optimization under uncertainty compared with universal Kriging and PC-Kriging methods.

Including the power regulation strategy in aerodynamic optimization of wind turbines for increased design freedom

AbstractIn wind turbine optimization, the standard power regulation strategy follows a constrained trajectory based on the maximum power coefficient. It can be updated automatically during the optimization process by solving a nested maximization problem at each iteration. We argue that this model does not take advantage of the load alleviation potential of the regulation strategy and additionally requires significant computational effort. An alternative approach is proposed, where the rotational speed and pitch angle control points for the entire operation range are set as design variables, changing the problem formulation from nested to one‐level. The nested and one‐level formulations are theoretically and numerically compared on different aerodynamic blade design optimization problems for AEP maximization. The aerodynamics are calculated with a steady‐state blade element momentum method. The one‐level approach increases the design freedom of the problem and allows introducing a secondary objective in the design of the regulation strategy. Numerical results indicate that a standard regulation strategy can still emerge from a one‐level optimization. Second, we illustrate that novel optimal regulation strategies can emerge from the one‐level optimization approach. This is demonstrated by adding a thrust penalty term and a constraint on the maximum thrust. A region of minimal thrust tracking and a peak‐shaving strategy appear automatically in the optimal design.

Dimensionality Reduction, Modelling, and Optimization of Multivariate Problems Based on Machine Learning

Simulation-based optimization design is becoming increasingly important in engineering. However, carrying out multi-point, multi-variable, and multi-objective optimization work is faced with the “Curse of Dimensionality”, which is highly time-consuming and often limited by computational burdens as in aerodynamic optimization problems. In this paper, an active subspace dimensionality reduction method and the adaptive surrogate model were proposed to reduce such computational costs while keeping a high precision. In this method, the active subspace dimensionality reduction technique, three-layer radial basis neural network approach, and polynomial fitting process were presented. For the model evaluation, a NASA standard test function problem and RAE2822 airfoil drag reduction optimization were investigated in the experimental design problem. The efficacy of the method was proved by both the experimental examples in which the adaptive surrogate model in a dominant one-dimensional active subspace is given and the optimization efficiency was improved by two orders. Furthermore, the results show that the constructed surrogate model reduced dimensionality and alleviated the complexity of conventional multivariate surrogate modeling with high precision.

Surrogate-Assisted Differential Evolution Using Knowledge-Transfer-Based Sampling for Expensive Optimization Problems

To sufficiently reuse the knowledge from previous optimization efforts, a surrogate-assisted differential evolution using knowledge-transfer-based sampling (denoted as SADE-KTS) method is proposed for solving expensive black-box optimization problems. In SADE-KTS, a novel knowledge-transfer-based sampling method is integrated with the differential evolution framework to generate promising initial sample points. In this way, a least-squares support vector machine classifier is constructed based on the prior optimization knowledge database to calibrate the initial sample points adaptively, which improves the exploration performance via transferring the existed optimization efforts to the current optimization task. Moreover, the radial basis function and kriging surrogates are employed to replace the expensive simulation models for evolutionary operations, where the tailored differential evolution operators are cooperated with the sequential quadratic programming optimizer to lead the search to the global optimum efficiently. A number of numerical benchmarks are tested to illustrate the optimization capacity of SADE-KTS compared with several competitive optimization algorithms. Finally, SADE-KTS is applied to an airfoil aerodynamic knowledge-based optimization problem considering the existed optimization knowledge, which demonstrates the practicality and effectiveness of the proposed SADE-KTS in engineering practices.

Research of Airfoil Optimization Based on CST Method and Genetic Algorithm

Aiming at the aerodynamic optimization problem of guided ammunition airfoil, genetic algorithm is used to optimize the lift-drag ratio of the airfoil based CST geometric description method with given inflow velocity and fixed angle of attack. Firstly, the parameter space generated by CST parameterized airfoil description method is performed with binary code and the basic operations of genetic algorithm such as selection, crossover and mutation. Then, the external CFD solver is called to automatically divide the grid and calculate the lift-drag ratio through the control script. Finally, the target solution is obtained by searching. The calculation results show that this method can obtain a optimal target solution in relative less generation for a group of airfoils generated randomly in a certain search space.

Hypersonic Vehicle Aerodynamic Optimization Using Field Metamodel-Enhanced Sequential Approximate Optimization

Hypersonic vehicle has gained increasing attention due to its high cruise speed and long voyage. In this paper, an enhanced Sequential Approximate Optimization method is proposed for aerodynamic optimization of a hypersonic vehicle. In this method, a constrained design of experiment method is adopted to generate the initial sampling set with adequate number of feasible points. A field metamodel is proposed to surrogate the aerodynamic parameters distributions in the flow field obtained by the aerodynamic simulation. A hybrid metamodel combing radial basis functions and polynomial chaos expansion is applied to construct the field metamodel with good approximate performance. A robust mesh morphing scheme based on radial basis functions is developed to generate high-quality meshes for the sequential optimization scheme. The hypersonic vehicle aerodynamic optimization problem is performed using the proposed optimization framework and satisfactory results are obtained with limited computational budgets. Results show that the proposed field metamodel-enhanced Sequential Approximate Optimization method possesses powerful optimization performance and promising prospects in the field of hypersonic vehicle optimization design.

Two-stage aerodynamic optimization method based on early termination of CFD convergence and variable-fidelity model

Efficient aerodynamic design optimization method is of great value for improving the aerodynamic performance of little UAV's airfoil. Using engineering or semi-engineering estimation method to analyze aerodynamic forces in solving aerodynamic optimization problems costs little computational time, but the accuracy cannot be guaranteed. However, CFD method ensuring high accuracy needs much more computational cost, which is unfordable for optimization. Surrogate-based optimization can reduce the number of high-fidelity analyses to increase the optimization efficiency. However, the cost of CFD analyses is still huge for aerodynamic optimization due to multiple design variables, multi-optimal and strong nonlinearities. To solve this problem, a two-stage aerodynamic optimization method based on early termination of CFD convergence and variable-fidelity model is proposed. In the first optimization stage, the solutions by early termination CFD convergence and the convergenced CFD solutions are regarded as low-and high-fidelity data respectively for building variable-fidelity model. Then, the multi-island genetic algorithm is used in the global optimization based on the built variable-fidelity model. The modeling efficiency can be greatly improved due to many cheap low-fidelity data. In the second stage optimization, the global optimum from the first optimization stage is treated as the start of the Hooke-Jeeves algorithm to search locally based on convergenced CFD computations in order to acquire better-optimum. The proposed method is utilized in optimizing the aerodynamic performance of the airfoil of little UAV, and is compared with the EGO method based on single-fidelity Kriging surrogate model. The results show that the present two-level aerodynamic optimization method consumes less time.

Design sensitivity analysis with polynomial chaos for robust optimization

Polynomial chaos (PC) methods with Gauss-type quadrature formulae have been widely applied for robust design optimization. During the robust optimization, gradient-based optimization algorithms are commonly employed, where the sensitivities of the mean and variance of the output response with respect to design variables are calculated. For robust optimization with computationally expensive response functions, although the PC method can significantly reduce the computational cost, the direct application of the classical finite difference method for the analysis of the design sensitivity is impractical with a limited computational budget. Therefore, in this paper, a semi-analytical design sensitivity analysis method based on the PC method is proposed, in which the sensitivity is directly derived based on the Gauss-type quadrature formula without additional function evaluations. Comparative studies conducted on several mathematical examples and an aerodynamic robust optimization problem revealed that the proposed method can reduce the computational cost of robust optimization to a certain extent with comparable accuracy compared with the finite difference-based PC method.

Data-Driven Evolutionary Algorithm With Perturbation-Based Ensemble Surrogates.

Data-driven evolutionary algorithms (DDEAs) aim to utilize data and surrogates to drive optimization, which is useful and efficient when the objective function of the optimization problem is expensive or difficult to access. However, the performance of DDEAs relies on their surrogate quality and often deteriorates if the amount of available data decreases. To solve these problems, this article proposes a new DDEA framework with perturbation-based ensemble surrogates (DDEA-PES), which contain two efficient mechanisms. The first is a diverse surrogate generation method that can generate diverse surrogates through performing data perturbations on the available data. The second is a selective ensemble method that selects some of the prebuilt surrogates to form a final ensemble surrogate model. By combining these two mechanisms, the proposed DDEA-PES framework has three advantages, including larger data quantity, better data utilization, and higher surrogate accuracy. To validate the effectiveness of the proposed framework, this article provides both theoretical and experimental analyses. For the experimental comparisons, a specific DDEA-PES algorithm is developed as an instance by adopting a genetic algorithm as the optimizer and radial basis function neural networks as the base models. The experimental results on widely used benchmarks and an aerodynamic airfoil design real-world optimization problem show that the proposed DDEA-PES algorithm outperforms some state-of-the-art DDEAs. Moreover, when compared with traditional nondata-driven methods, the proposed DDEA-PES algorithm only requires about 2% computational budgets to produce competitive results.

Sequential Radial Basis Function-Based Optimization Method Using Virtual Sample Generation

Abstract To further reduce the computational expense of metamodel-based design optimization (MBDO), a novel sequential radial basis function (RBF)-based optimization method using virtual sample generation (SRBF-VSG) is proposed. Different from the conventional MBDO methods with pure expensive samples, SRBF-VSG employs the virtual sample generation mechanism to improve the optimization efficiency. In the proposed method, a least squares support vector machine (LS-SVM) classifier is trained based on expensive real samples considering the objective and constraint violation. The classifier is used to determine virtual points without evaluating any expensive simulations. The virtual samples are then generated by combining these virtual points and their Kriging responses. Expensive real samples and cheap virtual samples are used to refine the objective RBF metamodel for efficient space exploration. Several numerical benchmarks are tested to demonstrate the optimization capability of SRBF-VSG. The comparison results indicate that SRBF-VSG generally outperforms the competitive MBDO methods in terms of global convergence, efficiency, and robustness, which illustrates the effectiveness of virtual sample generation. Finally, SRBF-VSG is applied to an airfoil aerodynamic optimization problem and a small Earth observation satellite multidisciplinary design optimization problem to demonstrate its practicality for solving real-world optimization problems.