Aerodynamic Shape Optimization Research Articles

Surrogate-based aerodynamic shape optimization of high-speed train heads: A review of four key technologies

With the increase in running speed, the aerodynamic characteristics of high-speed trains have a significant impact on running stability, energy consumption and passenger comfort. Since the shape of the high-speed train head can directly influence the surrounding airflow, optimizing the head shape is the primary way to improve the aerodynamic performance of the train. This paper reviews current research studies on the surrogate-based aerodynamic shape optimization of high-speed train heads, aiming to provide a comprehensive reference for designers to enhance design efficiency and optimization performance. The entire optimization process is divided into four essential steps, and the key optimization technologies in each step are discussed, including parametric modeling, computational fluid dynamics (CFD) simulation, surrogate model and optimization algorithm. By introducing the practical applications of these technologies, we summarize their advantages and disadvantages and suggest four potential research directions for the future.

A mid-range approximation method assisted by trust region strategy for aerodynamic shape optimization

This paper presents an efficient solution for high-fidelity large-scale aerodynamic shape optimization problems based on several developments in the mid-range approximation method within a trust region optimization framework. The mid-range approximation method is an iterative optimization technique that utilizes mid-range approximations to replace the physical experiments or simulations during the optimization based on the selected trust region strategy. It could transform the original optimization problem into a sequence of approximate sub-optimization problems. In this work, an improved trust region strategy is proposed to contain more optimization states with a flexible and controllable performance to suit different types of problems. A metamodel assembly technique and its gradient-enhanced version are developed to further relax the requirements of computational costs in the mid-range approximation method. Its performance is discussed through a detailed comparison of metamodel performance using a mathematical benchmark case named Vanderplaats scalable beam. The wing only of the Common Research Model is offered to the proposed method for aerodynamic shape optimization. The optimization has 1 design objective, 232 design variables, and 135 design constraints. With all constraints satisfied, the optimized configuration has a 4.85% improvement in wing drag performance. The shock region is greatly reduced and the wing pressure distribution is smooth and nearly parallel. These results show that the proposed method could achieve the design goal successfully within a reasonable computational cost for large-scale problems.

Fast Buffet-Onset Prediction and Optimization Method Based on Pretrained Flowfield Prediction Model

A transonic buffet is a detrimental phenomenon that occurs on supercritical airfoils and limits the aircraft operating envelope. Traditional methods for predicting buffet onset rely on multiple computational fluid dynamics simulations to assess a series of airfoil flowfields and then apply criteria to them, which is slow and hinders optimization efforts. This study introduces an innovative approach for rapid buffet-onset prediction. A machine-learning flowfield prediction model was pretrained on a large database and then deployed offline to replace the simulations in the buffet prediction process for new airfoil designs. Unlike using a model to directly predict buffet onset, the proposed technique offers better visualization capabilities by providing users with intuitive flowfield outputs. It also demonstrates superior generalization ability, as evidenced by a 32.5% reduction in the average buffet-onset prediction error on the testing dataset. This method was used to optimize the buffet performance of 11 distinct airfoils within and outside the training dataset. The optimization results were verified with simulations and proved to yield improved samples across all cases. It was affirmed that the pretrained flowfield prediction model can be applied to accelerate aerodynamic shape optimization, but further work is still needed to raise its reliability for this safety-critical task.

Multi-fidelity deep learning for aerodynamic shape optimization using convolutional neural network

Multi-fidelity deep learning for aerodynamic shape optimization using convolutional neural network

Dynamic-mode-decomposition-based gradient prediction for adjoint-based aerodynamic shape optimization

Accurate and efficient gradient computation is the key to aerodynamic shape optimization. In this paper, dynamic mode decomposition (DMD) is employed to analyze the dynamic characteristics of the early pseudo-time marching of adjoint equations and to predict the gradient. Besides the first-order zero-frequency mode, other zero-frequency modes also contribute to the pseudo iterations of the adjoint equations in the early iterations. Hence, different from existing methods, all zero-frequency modes are retained to reconstruct adjoint fields for gradient prediction. Moreover, to further improve the modeling accuracy, an improved DMD (IDMD) is proposed by omitting the initial snapshots in early iterations. The effect of pseudo-time step on modeling accuracy is also studied. By solving the adjoint equations of the transonic and subsonic flows, the accuracy of the proposed method is verified. Results indicate that the proposed method still works despite the conventional solution process diverges. Through aerodynamic shape optimization examples of transonic flow over an airfoil, the number of adjoint pseudo-time steps is remarkably reduced by 83%, which indicates the proposed IDMD-based gradient prediction method has great potential for improving the efficiency of aerodynamic shape optimization.

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.

A stacking-based ensemble prediction method for multiobjective aerodynamic optimization of high-speed train nose shape

A stacking-based ensemble prediction method is proposed in order to improve the efficiency and accuracy of aerodynamic shape optimization. This can be divided into a two-level model. The first-level model uses various base learners to predict the training dataset and obtain various combinations of predictions. In the second-level model, the meta-learner is trained using various combinations of predictions as inputs and the true response values as outputs. The RMSE and R-Square metrics results show that the performance metrics of the stacked models are significantly better than those of the original surrogate models, except for the stacked RBFNN model. The stacked Kriging model and LSSVR model achieved the best results with RMSE and R-Square index values of 0.0039 and 0.8418 for aerodynamic drag coefficient, respectively. The stacked Kriging model, RBFNN model, and LSSVR model achieved the best results with RMSE and R-Square index values of 0.0024 and 0.76 for aerodynamic lift coefficient, respectively. Four state-of-the-art multiobjective optimization algorithms are used to perform the optimization process. The results of hypervolume and Pareto fronts demonstrate the effectiveness of the selected FDV-NSGAII algorithm. After optimization, the drag and lift coefficients before and after optimization are reduced by 1.9% and 12.7%, respectively.

Aerodynamic shape optimization based on proper orthogonal decomposition reparameterization under small training sets

Complex shape aerodynamic optimization frequently encounters the problem of dimensionality curse. Extracting geometric features from the original design space proves effective in this issue by reconstructing shapes with lower-dimension. While deep learning methods possess strong capabilities in dimensionality reduction, they suffer from longer training times and higher computational resource requirements. However, Proper Orthogonal Decomposition (POD) presents a more practical algorithm that can be effectively utilized for complex high-dimensional shapes. In this study, we propose a combination of the POD-based method and Gaussian Process Regression with an automatic kernel construction algorithm (AKC-GPR) to optimize complex shapes in a lower-dimensional reparameterization space using small training sets. This approach aims to explore the potential for reducing optimization cost. By reducing the design parameters from 56 to 18 with an error rate of 1.28 %, we establish surrogate models for optimization using AKC-GPR with progressively decreasing training sets. For samples below 100, the Mean Absolute Percentage Errors (MAPEs) remain below 2 %, while the Mean Relative Percentage Errors (MRPEs) approach 10 %. It emphasizes the proficiency of POD in geometric dimensionality reduction. The optimized shapes exhibit smoother contour, resulting in a 10 % improvement in lift and a relatively modest reduction in drag of less than 3 %. The lift-to-drag ratio experiences an increase of over 14 %. The shapes optimized by various training sets possess minimal differences in geometric shapes and aerodynamic characteristics. This confirms the AKC-GPR algorithm's high-precision fitting capability in effectively handling complex aerodynamic coefficients of multi-parameter shapes during small-sample shape optimization.

Design of morphing wing for aerodynamic performance considering the wing flexibility effects

This paper provides an insight into ongoing research aimed at designing a morphing wing with the ability to continuously adapt its aerodynamic shape. The wing is targeted at a general purpose unmanned aerial vehicle. The morphing wing concept outlined in the paper is based on continuous camber changes of the wing leading and trailing edges, allowing optimal performance in different flight regimes. The aeroelastic tailoring method is used to design the load carrying structure of the wing in order to define the optimal stiffness and strength of the structure, which are considered as fixed in subsequent design steps. The research proposes a novel modular design approach that combines aerodynamic shape optimisation and aeroelastic considerations for designing morphing wing surfaces.

A novel accelerated convergence method for solving adjoint equations based on modal reduction

The efficiency of adjoint-based aerodynamic shape optimization depends critically on the solution efficiency of adjoint equations. In this letter, we employ the Proper Orthogonal Decomposition (POD) method to analyze the adjoint field samples and project them from the physical space into a low-order modal space. Subsequently, the full-order adjoint equations are reduced to low-order equations using the POD modes. Thus, we can efficiently predict the initial values for pseudo-time marching, thereby accelerating the solution of adjoint equations. Results indicate that the high-order POD modes are crucial for constructing the low-dimensional system. Moreover, this method can be seamlessly integrated with our previously established Dynamic Mode Decomposition (DMD) acceleration method to form a POD+DMD acceleration approach. Application of this approach to the flow past a National Advisory Committee for Aeronautics 0012 airfoil demonstrates a noteworthy 80.9% reduction in iteration numbers when solving the adjoint equations. Even for the airfoil located on the upper boundary of sampling space, the number of iterations is still reduced by 72.6%. Therefore, we believe that the proposed method holds significant promise for improving the efficiency of adjoint-based aerodynamic shape optimization in future research.

Knowledge transfer accelerated turbine blade optimization via an sample-weighted variational autoencoder

The aerodynamic shape optimization (ASO) of turbine blades is a typical expensive black-box problem. With the ever increasingly strict design requirements, high-fidelity simulations are demanded, and the related simulation cost increases sharply, which means fewer simulation calls are allowed within a design optimization cycle. Then, the problem as “how to achieve better solution with fewer simulations” becomes rather crucial. Inspired by the concept of transfer learning, we propose to leverage the samples of related completed tasks (also known as source samples) to address the above problem. Specifically, an sample-weighted variational autoencoder (VAE) is proposed, which includes the source samples as part of the training dataset to boost the proportion of high-scoring solution candidates in the learnt design space of target problem, and thus increasing the probability of attaining substantially better solutions. The including of source samples in VAE training also helps to increase the accuracy of reparameterizing the source samples in target design space. Furthermore, a multi-fidelity surrogate is used to guide the optimization search, which extracts information from the source samples to help the algorithm to more quickly arrive at the small neighborhood of the true optimal of the target problem. With the above, a knowledge transfer accelerated aerodynamic shape optimization procedure is proposed for the turbine blade design, labeled as KT-ASO. Through test on a highly-loaded low-pressure turbine blade and compared against the ASO procedures without knowledge transfer, the proposed KT-ASO is shown to reduce at least 50% function calls when achieving similar optimal solutions. In the meantime, under the same sample budget, our proposed KT-ASO is shown to attain much better solutions than the compared ASO procedures.

A duality-preserving adjoint method for segregated Navier–Stokes solvers

Adjoint methods efficiently compute gradients for systems with many inputs and have been widely used for large-scale gradient-based optimization in fluid mechanics. To ensure optimization's numerical robustness, we need to develop an adjoint solution algorithm that has a similar, if not the same, convergence rate as the primal flow solver at each optimization iteration. This consistent primal-adjoint convergence behavior is also called duality-preserving (DP). Existing DP adjoint methods are mostly for fully coupled Navier–Stokes (NS) solvers with compressible flows. However, few DP adjoints have been proposed for segregated NS solvers, e.g., the semi-implicit method for pressure-linked equations (SIMPLE) algorithm solvers widely used for incompressible flow simulations. This study will fill this gap by deriving a fixed-point segregated adjoint formulation that fully preserves the convergence behavior of primal solvers. We first rewrite the steady-state segregated NS primal solution process into a fully coupled left-preconditioned Richardson format. Then, we transpose the preconditioner matrix to obtain a DP iterative adjoint formulation. In addition to algebraic derivation, we create a new graph representation that significantly streamlines the DP adjoint development for new segregated solvers. We prove that the proposed graph representation is equivalent to the above preconditioner transpose approach and guarantees duality. We evaluate our proposed algorithm using three cases with increasing complexity: a miniature-sized problem, an airfoil, and a wing. To quantify duality, we compare the eigenvalues between the primal and adjoint solvers and observe excellent agreements. We also evaluate to what extent various numerical settings (e.g., inner iteration tolerances and non-dual preconditioner) impact the eigenvalue distributions. Our proposed adjoint achieves machine-precision accurate gradient computation with competitive speed. Finally, we incorporate our adjoint solver into a large-scale gradient-based optimization framework and demonstrate its capability for wing aerodynamic shape optimization. The proposed fixed-point adjoint approach has the potential to make the adjoint solution more robust and efficient for any segregated NS solvers.

Aerodynamic Shape Optimization of Subsonic/Supersonic Flows Integrating Variable-Fidelity Longitudinal Trim Analysis

Most existing studies on aerodynamic shape optimization have not considered longitudinal trim under control surface deflection, typically achieving self-trim through a constraint of zero pitching moment or adjusting the optimized configuration for longitudinal trim. However, adjustments to the optimized configuration might introduce additional drag, reducing overall optimization benefits. In this paper, a novel approach of incorporating control surface deflection for longitudinal trim in aerodynamic optimization is proposed. Firstly, an aerodynamic computation program based on the high-order panel method was developed, introducing velocity perturbations on specific mesh surfaces to simulate actual control surface deflections. Subsequently, a comprehensive optimization framework was established, encompassing parametric modeling, aerodynamic computation, and variable-fidelity control surface deflection analysis. Finally, aerodynamic optimization analysis was conducted under both subsonic and supersonic conditions. Thirty-one design variables were selected with the trimmed lift-to-drag ratio in cruising condition as the objective function and the control surface deflection angle as the constraint. The results indicated an 8.52% increase in the trimmed lift-to-drag ratio compared to the baseline model under subsonic conditions and an 8.1% increase under supersonic conditions.

Aerodynamic shape optimization using design-variables-screening method

Aerodynamic shape optimization involving a complex geometric model or problem may have tens or hundreds of design variables, necessitating multiple accurate but time-consuming computational fluid dynamics simulations to produce optimal designs, which greatly affects the efficiency of optimization and. To address this challenge, this article proposes an efficient optimization method based on design-variables-screening. Within the framework of the method, a complicated input–output relationship is broken down into quantitative effects. The influence of design variables on the objective function is calculated by the Kriging regression model and functional analysis of variance. In the meantime, a screening strategy is proposed to facilitate the selection of design variables for optimization. The less important design variables in the problems of interest are fixed so that the dimensionality of the problems is reduced to save computational cost. Experimental results on the National Advisory Committee for Aeronautics airfoil (NACA0012) demonstrate that the simplified model with the screening strategy achieves nearly the same reduction in drag coefficient as the conventional method that optimizes all design variables. Moreover, it significantly enhances the efficiency of optimization and contributes to the enhancement of flow stability.

An Improved Approach for Reducing the Dimensionality of Wing Aerodynamic Optimization Considering Longitudinal Stability

The wing aerodynamic shape optimization is a typical high-dimensional problem with numerous independent design variables. Researching methods to reduce the dimensionality of optimization from the perspective of aerodynamic characteristics is necessary. One traditional aerodynamic-based approach decouples the wing’s camber and thickness according to the thin airfoil theory, but it has limitations due to unclear application scope and effectiveness. This paper proposes an improved approach that determines the values of certain thickness variables based on a data-driven aerodynamic characteristics model before optimization, which considers longitudinal stability. By reducing the number of design variables, the dimensionality of optimization is decreased effectively. The derivation of the improved approach is accomplished through the design of experiments, parametric modeling, computational fluid dynamics, and sensitivity analysis. The effectiveness of the improved approach is validated by applying it to the aerodynamic optimization of an ONERA-M6 wing in subsonic flow based on the surrogate-based optimization algorithm. The results demonstrate that the improved approach significantly accelerates the optimization process while maintaining global effectiveness.

Adjoint based aerodynamic shape optimization of a semi-submerged inlet duct and upstream inlet surface

Adjoint based aerodynamic shape optimization of a semi-submerged inlet duct and upstream inlet surface

Disturbance observer-based fixed-time control for hypersonic morphing vehicles with uncertainties

Abstract The attitude-tracking problem of hypersonic morphing vehicles (HMVs) is investigated in this research. After introducing variable-span wings, the optimal aerodynamic shape is available throughout the entire flight mission. However, the morphing wings cause significant changes in aerodynamic coefficients and mass distribution, challenging the attitude control. Therefore, a complete design procedure for the flight control system is proposed to address the issue. Firstly, the original model and the control-oriented model of HMVs are built. Secondly, in order to eliminate the influence caused by the multisource uncertainties, an adaptive fixed-time disturbance observer combined with fuzzy control theory is established. Thirdly, the fixed-time control method is developed to stabilise hypersonic morphing vehicles based on a multivariable sliding mode manifold. The control input can be obtained directly. Finally, the effectiveness of the proposed method is proved with the help of the Lyapunov theory and simulation results.

Natural laminar flow airfoil design via adjoint-based transition onset delay

This work is focused on the aerodynamic shape optimization (ASO) of airfoils with the aim of attaining natural laminar flow (NLF) designs. The two-equation Amplification Factor Transport (AFT) model is utilized for an accurate prediction of the transitional boundary layer flows. The NLF designs are achieved by utilizing two different objective functions. In the first approach, the conventional drag minimization problem is solved with a target lift coefficient. In addition, a second approach is proposed in this work that directly aims at delaying the transition onset on both the pressure and suction sides, thus expanding the natural laminar flow region around the airfoil. It is shown that by utilizing a sigmoid fitting of the turbulence index profile, the transition onset locations can be accurately predicted in a differentiable and smooth fashion, which is essential to the adjoint-based sensitivity analysis of the RANS solver. The efficacy of the proposed technique for NLF design has been tested for NACA 0012 and RAE 2822 airfoils at moderate lift coefficients. Results have shown that a significantly lower drag count can be achieved by delaying the transition onset locations compared to the case where only a drag minimization is sought. Additionally, the transition onset locations on both sides of the airfoil are delayed using our proposed technique which results in a significant expansion of the natural laminar flow region.

An approach for formation design and flight performance prediction based on aerodynamic formation unit: Energy-saving considerations

The performance improvement of swarm drones through aerodynamic shape optimization may be challenging due to folded size constraints imposed by the specific launch approach. However, fixed-wing aircraft swarms can benefit from formation flight in terms of energy consumption. This study introduces the concept of the “aerodynamic formation unit”, which consists of two or three aircraft that form an inseparable unit of the formation. Considering the Unmanned Aerial Vehicle (UAV) distribution and wingtip vortex interference in the formation, two typical aerodynamic formation units are optimized by the variable-fidelity aerodynamic optimization method based on space mapping. The aerodynamic characteristics of the formation UAVs that affect flight performance, such as lift-to-drag ratio (L/D ratio) and static stability are analyzed by Computational Fluid Dynamics (CFD) simulations. The L/D ratio (cruising condition) of the following aircraft can be increased by 22.8% and 57.5% in the optimal units that involve two and three aircraft respectively. Moreover, this study conducts several CFD simulations for multi-aircraft formations formed by the units, which show that the average L/D ratio of the formation can be improved by more than 19%. These results verify the feasibility and effectiveness of the “aerodynamic formation unit” concept and the optimization framework for formation parameters.

Simultaneous surrogate modeling and dimension reduction using unsupervised learning. Application to parametric wing shape optimization

This paper presents a machine-learning-based approach that enables simultaneous surrogate modeling and dimension reduction and applies it to aerodynamic parametric shape optimization. Aerodynamic shape optimization is a crucial process in various industries, including aerospace, automotive, and renewable energy. It involves iteratively improving the properties of a system by evaluating an objective function and driving its minimization or maximization using an optimization algorithm. However, the evaluation of aerodynamic objective functions requires computationally expensive operations, such as solving complex fluid dynamics equations and calculating performance metrics like lift and drag coefficients. This computational cost becomes particularly burdensome when derivative-free optimization algorithms need to evaluate numerous samples per iteration. Additionally, when the design space dimension is high, the efficiency and effectiveness of the optimization process decrease. To address these challenges, the paper proposes combining surrogate modeling and dimension reduction. Surrogate modeling constructs a reduced order model that approximates the coefficients of interest in a cost-effective manner, while dimension reduction identifies the most relevant design space dimensions using techniques like Proper Orthogonal Decomposition. The paper suggests an integrative approach that employs Artificial Neural Networks (ANN) and Unsupervised Learning, specifically AutoEncoder networks, to simultaneously build a surrogate model and reduce the problem dimension. This technique is applied to optimize the shape of an airplane wing aerofoil under trans-sonic flight conditions. The wing shape is parameterized using Free Form Deformation (FFD). The paper demonstrates that the suggested approach enables rapid and effective shape optimization.