Real-Time Performance Optimization for a Camber Morphing Wing Based on Domain Incremental Model under Concept Drifting

Abstract

Compared with traditional wings equipped with conventional control surfaces, variable-camber morphing wings have become a hot research topic in the field of aviation due to their ability to maintain a smooth and continuous overall shape while ensuring excellent aerodynamic performance. This study focuses on a high aspect ratio wing with a continuous variable-camber trailing edge. Two precision models were constructed: an aerodynamic model and an aeroelastic model. Based on simulation data obtained from these models, we developed and updated a surrogate model for the wing, with particular emphasis on an incremental modeling approach that takes concept drift into account. Subsequently, using the aforementioned models, we conducted real-time optimization with feedback considerations to reduce drag, lower stress on the main beam, and minimize actuator energy under either steady or slowly varying target lift conditions. Notably, the optimization process resulted in a 4% reduction in drag or a significant decrease of 18.3% in maximum stress. Through computational comparisons, the accuracy of the proposed surrogate model and incremental learning method is demonstrated, along with their efficiency in the context of optimization problems.

1. Introduction

There is a growing global awareness regarding environmental conservation, prompting engineers to focus on improving aircraft efficiency to achieve energy savings and environmental sustainability. The concept of a morphing aircraft pertains to an aircraft whose shape can be dynamically changed in real time throughout the entire flight envelope based on the flight mission, speed, and conditions. By adopting different aerodynamic configurations to meet various flight missions, a morphing aircraft can maintain excellent aerodynamic and flight performance. Among these configurations, variable-camber wings have shown great potential [1,2,3]. Due to the absence of a vortex generated by trailing-edge gaps near the wing control surfaces, the aerodynamic efficiency of a variable-camber wing is generally superior to that of traditional rigid control surface aircraft. Numerous studies have focused on the design aspects of variable-camber aircraft [4,5,6,7] and morphing skins [8,9,10,11,12]. Those concepts can often be extended to long-span wings equipped with multiple control surfaces.

However, the optimal aerodynamic state of an aircraft continuously changes due to systematic variations such as flight conditions and changes in lift requirements resulting from fuel consumption. Traditionally, optimal control tables for aircraft cruising are pre-established through simulation analysis, wind tunnel tests, and flight experiments, with the independent variables being aircraft weight, wind speed, and flight altitude [13]. These tables are then used to achieve real-time deformation during flight to maintain the optimal state. However, tabular methods rely on existing flight test data and cannot be updated with changing flight conditions and aircraft states. Moreover, distributed variable-camber trailing edges often involve a large number of control surfaces, for which the relationship between aerodynamic performance and the trailing-edge shape is complex. This makes it challenging to establish a precise correlation between aerodynamic performance and trailing-edge shape through experimental means, rendering the use of tables increasingly impractical.

Therefore, optimization methods are commonly employed to determine the optimal aircraft shape. Depending on the timing of optimization in experiments, data sources, and optimization strategy, these can be classified into offline and online optimization methods. Offline optimization relies on pre-obtained data for optimization and is suitable when considering static conditions and parameter settings, while online optimization involves real-time monitoring and adjustment, making it suitable for dynamic and real-time changing conditions.

In terms of offline optimization research, Zhang et al. [14] developed a fluid-structure coupling simulation considering static aeroelastic phenomena to achieve shape optimization of a variable-camber trailing-edge wing. A global optimizer based on the radial basis function interpolation was employed to optimize the aerodynamic performance, successfully reducing the induced drag by 2.3%. Lyu et al. [15] also performed shape optimization of a variable-camber trailing-edge wing under multiple operating conditions, thereby reducing the average drag coefficient. A total of 90 design variables for the wing were optimized under each flight condition. They performed CFD simulations for the wing based on the Spalart–Allmaras turbulence model. The optimization approach combined gradient-based methods with the adjoint method for the calculation of the required derivatives and ultimately achieved a 1% reduction in fuel consumption. In comparison, they utilized a low-precision panel method model but considered the static aeroelastic effects of the wing. Later, they employed a high-precision model and accelerated the optimization process using conjugate gradient information, but both methods had relatively slow optimization speeds. However, during the flight of an aircraft, the optimization objective may change in real time. Therefore, if physical field solutions are required for different optimization objectives each time, the time cost would be high, making real-time optimization difficult to achieve.

In the field of online optimization research, V. Popov [16] explored the feasibility of delaying the transition position in real time through the use of a morphing rectangular finite-aspect-ratio wing. Their wing was instrumented with Kulite pressure sensors and two smart memory alloy actuators and could achieve camber morphing. Their basic idea was to implement an optimization algorithm into the controller that would search the optimal airfoil profile with the real system in real time without the need for pre-computed airfoil databases. The optimization method they used was a hybrid of the gradient ascent (or hill-climbing) method and simulated annealing. NASA and Boeing developed the Variable Camber Continuous Trailing Edge Flap (VCCTEF) project [17], in which they investigated the concept of drag reduction by actively controlling the wing aeroelasticity. The aerodynamic modeling in the VCCTEF project was based on aerodynamic derivatives, requiring extensive parameter estimation and identification. The optimization approach used was the Newton–Raphson method. Mkhoyan et al. [13] proposed an online black-box performance optimization method for morphing wings with distributed control. They built a global radial basis function neural network surrogate model and used a derivative-free evolutionary optimization algorithm to verify the effectiveness of the optimization strategy. In comparison, Popov’s work only focused on airfoil shapes rather than the entire wing. The VCCTEF project still relied on lookup tables for flexible multi-control surfaces, which may introduce certain errors. Although the Newton–Raphson method is computationally efficient, it is often only effective in the local region around the trim condition, making it challenging to achieve global optimization throughout the flight envelope. Remarkable contributions have been made by developing a surrogate model for wing optimization, resulting in significantly improved computational efficiency and real-time capability. However, they did not account for the effects of wing aeroelasticity in their simulations; instead, they relied on empirical wind tunnel test data to calibrate the aerodynamic model, leading to increased costs and limited scalability.

The goal of our study was to provide a preliminary experiment before achieving real-time optimization of camber morphing wing shapes in future wind tunnel and flight tests. The main significance of this study lies in proposing the use of surrogate models as a replacement for simulation calculations and real-wing testing, thereby improving efficiency and reducing costs. We introduce a method to maintain the generalization capability through incremental learning when data undergo concept drift due to various circumstances. Furthermore, we demonstrate the applicability of surrogate models in a real-time multi-objective optimization, allowing for real-time calibration of the surrogate model and more accurate optimization results to be obtained. It is important to note that the focus of this study was on the experimental objectives, procedures, and methods rather than specific data. To ensure high computational efficiency, the aerodynamic calculation method used in this study was the panel method, which may not yield precise drag coefficient values. Therefore, the final results obtained should not be considered as the conclusive findings of the experiment.

The remainder of this paper is organized as follows: Section 2 presents an overview of the camber morphing wing used in this study, including its basic profile and the methods utilized for parameterization and physical modeling. Section 3 and Section 4 describe the approach for constructing and updating the surrogate model of the wing, using data obtained from simulations of physical models, as well as introducing the concept of incremental learning to address model concept drift. In Section 5, specific methods for aerodynamic performance optimization are described, including the use of physical models, static surrogate models, and incremental surrogate models. A comparison—in terms of accuracy and efficiency—is conducted to demonstrate the feasibility of the surrogate model construction and optimization methods proposed in this paper. Section 6 introduces multiple objective functions for shape optimization, confirming the applicability and extensibility of the proposed methods in practical applications. Section 7 provides the discussion and conclusions.

2. System Modeling

2.1. Morphing Wing Overview

First, we provide an overview of the basic configuration of the reference wing. The wing under consideration in this study is depicted in Figure 1 and belongs to a high aspect ratio electric unmanned aircraft sourced from a previous study [18]. The aircraft operates at a cruising speed of 30 m/s at sea level with a span of 2300 mm. The camber morphing region is located from approximately 13% to 100% of the spanwise direction and covers the trailing 40% of the chord length.

The trailing-edge morphing wing was designed with a compliant structure, where the trailing-edge section can achieve chordwise bending and differential deformation in the spanwise direction (i.e., twist). The overall wing deformation is smooth and continuous, without any gap between the control surfaces.

2.2. Wing Shape Parameterization

The parameterization of the aerodynamic shape of the wing was the first step in defining the problem.

Initially, the improved Class/Shape Transformation (CST) method was employed to achieve parameterization of the morphing airfoil. The CST method serves as a versatile tool for wing shape parameterization [19], which maps the non-dimensional coordinates $\psi \in \left[0,1\right]$ in Cartesian coordinates to the ordinate $\zeta =\zeta \left(\psi \right)$, formulated using a class function $C\left(\psi \right)$, shape function $S\left(\psi \right)$, leading-edge boundary condition term ${\zeta }_{\mathrm{LE}}$, and trailing-edge boundary condition term ${\zeta }_{\mathrm{TE}}$ [20].

Next, a machine learning approach was used to establish the correlation between the CST parameterization of the morphing airfoil and the deformation characteristics of the designed trailing-edge morphing mechanism. By simulating the deformation of the compliant trailing-edge mechanism under different actuation commands and fitting the shapes using the CST parameterization method, the correspondence between the deflection angle and the CST parameters was obtained. This enabled representation of the morphing wing’s shape through inputting the trailing-edge deflection angle.

Finally, three-dimensional parameterization of the wing was conducted. The three-dimensional wing shape was represented using multiple cross-sectional shapes distributed along the span, with the transition between active control sections achieved through piecewise cubic Hermite interpolating polynomial interpolation, as shown in Figure 2a, where the red sections represent the wing control sections and the black surfaces were obtained through interpolation. In this study, a total of 24 wing control sections are considered, with the root section labeled as No. 1 and the tip section as No. 24.

However, if sections No. 5–24 were considered as deformable control surfaces, in order to fully characterize the wing’s shape, 20 deflection angles would have to be used as independent variables in the shape optimization problem, resulting in a control problem with 20 degrees of freedom. As each parameter took continuous values, an infinite number of trailing-edge shapes resulted, posing a challenge in terms of computational efficiency when conducting optimization.

In order to improve the computational efficiency, we arranged the independent trailing edges in the form of the first five orders of Bernstein polynomials to approximate any deformation along the span. Consequently, the 20 degrees of freedom were transformed into 5 coefficients of virtual shape functions. Additionally, as the polynomials and their linear combinations present multi-order continuity, obtaining the trailing-edge shapes of each section through polynomials ensured smoother continuity between adjacent sections, avoiding abrupt jumps in shape. The trailing-edge shape of the morphing part, as shown in Figure 2a, is represented by the black line obtained by the superposition of the fifth-order polynomials represented by the colored lines in Figure 2b. It is important to note that the shape function was normalized. Each section of the wing allows for an upward and downward deflection of 20°, while the upper and lower bounds are ±1 in the plot. Moreover, a positive deflection angle or a positive virtual shape function represents a downward deflection of the wing.

Therefore, by inputting the weights of the fifth-order shape function, the shape of the morphed wing can be obtained, simplifying parametric modeling of the three-dimensional wing.

2.3. Physical Modeling

To obtain data reflecting the aeroelastic characteristics of the morphing wing, it was essential to establish a model of the wing. Physical models at two levels of fidelity—considering aerodynamic or aeroelastic activity—were developed, which were used to generate a significant amount of random samples for constructing the surrogate models in the subsequent analysis.

The input of the model is defined as $x=\left(\alpha ,u\right)$, where $\alpha$ represents the angle of attack and $u$ denotes the virtual shape functions for the wing’s trailing-edge deformation.

2.3.1. Aerodynamic Model

To achieve a balance between computational accuracy and efficiency, the high-order panel method was employed to calculate the aerodynamic loads on the wing. Specifically, the PANAIR solver program [21] was utilized for the implementation. The high-order panel method utilizes linearly distributed sources and quadratically distributed doublets as singular fundamental solutions, ensuring the continuity of the doublet strength distribution and thereby enhancing computational accuracy and numerical stability. This program allows for the use of freely shaped aerodynamic configurations as input, captures the effects of wing thickness and camber on aerodynamic performance, and exhibits robustness under complex geometries. However, due to the limitations of potential flow theory, the viscous drag cannot be computed [22].

The obtained aerodynamic model is represented by ${\mathit{ℳ}}_{\mathrm{A}}$, where the subscript $\mathrm{A}$ denotes aerodynamics. The evaluation of ${\mathit{ℳ}}_{\mathrm{A}}$ with input $x=\left(\alpha ,u\right)$ yields the result $y_{\mathsf{A}}$, which includes the lift coefficient $C_{\mathrm{L},\mathrm{A}}$, drag coefficient $C_{\mathrm{D},\mathrm{A}}$, etc.

2.3.2. Aeroelastic Model

To accurately capture the aerodynamic and structural characteristics of the morphing wing, it was essential to consider aeroelastic effects. For aerodynamics structure coupling, spatial radial basis function interpolation methods were used for structural displacement interpolation. Aerodynamic force interpolation ensures the equivalence of forces and moments before and after interpolation using the virtual work principle. A two-way static–aeroelastic loosely coupled iterative method was employed, utilizing the Conventional Serial Staggered (CSS) format [23,24]. In this iterative process, the aerodynamic forces and structural deformations were solved alternately.

Employing this approach, the obtained aeroelastic model is represented by ${\mathit{ℳ}}_{\mathrm{AE}}$, where the subscript $\mathrm{AE}$ denotes aeroelastic. The evaluation of ${\mathit{ℳ}}_{\mathrm{AE}}$ with input $x=\left(\alpha ,u\right)$ yields the result $y_{\mathsf{A}\mathrm{E}}$, which includes the lift coefficient $C_{\mathrm{L},\mathrm{AE}}$, drag coefficient $C_{\mathrm{D},\mathrm{AE}}$, etc.

The aeroelastic simulation method employed in this study has been previously validated in wind tunnel experiments. As an example, the results for one particular wing are shown in Figure 3.

3. Surrogate Model Construction Methodology

In this section, we detail the construction and updating of the surrogate model for the wing, emphasizing the incremental learning approach used to account for model concept drift. The necessity to build and update the surrogate model for the wing can be justified according to the following needs:

• The computational cost of simulating the wing model is substantial, and wind tunnel or flight tests incur even higher costs.
• Existing aeroelastic coupling methods for the wing are not capable of meeting the real-time optimization demands for aerodynamic performance, while surrogate models offer significantly faster computation.
• Different wing models, such as aerodynamic models, aeroelastic models, and real-wing prototypes, exhibit varying degrees of fidelity. Each model’s data may be subject to concept drift, making one-time data collection insufficient to fully capture the characteristics of the wing accurately.
• Wing optimization should begin with a pre-trained model and continuous improvement of accuracy during the experimental process is necessary.

The focus of this section is the proposal of the surrogate model, including both the offline learning and incremental learning methods, and the proposed error-based incremental learning strategy’s ability to handle data concept drift. The proposed approach has strong practical applicability and can be extended in the following aspects:

• Data Source Selection: The surrogate model presented in this paper was initially constructed using aerodynamic simulation data and later updated with aeroelastic simulation data. However, the ideal application of this method is to train the surrogate model with aeroelastic simulation data and then update it with wind tunnel or flight test data.
• Input and Output of the Surrogate Model: In this section, the inputs of the surrogate model are the angle of attack and shape, while the outputs are lift and drag coefficients. However, the surrogate model’s outputs can be expanded to include the lift distribution along the span of the wing, stress–strain distribution of the wing, and other parameters. Such an extension would allow for optimization of the lift distribution along the span and a reduction in maximum stress in the wing’s main beam through subsequent performance optimization tasks.

3.1. Model Architectures

Gaussian Kernel regression models and neural network models are commonly used model architectures for training.

Kernel models employ kernel functions to map data into a higher-dimensional feature space, enabling linear operations in this space to deal with non-linear problems. Common examples of kernel methods include Support Vector Machines (SVM) and least-squares regression [25,26].

Neural networks are computational models built on artificial neurons, which can be used to model complex non-linear problems. Neural networks operate by propagating and processing data through layers of weighted connections and non-linear activation functions, allowing them to learn intricate input–output mappings. The neural network used here is a feed-forward shallow neural network [27]. The weight and bias values are updated according to Levenverg–Marquardt optimization.

3.2. Offline Static Learning

Static learning requires providing the entire training data set at once and utilizes an offline approach for training. Once training is completed, the model is deployed in the production environment and learning stops.

A static learning model is built based on the following two assumptions: First, all training samples can be obtained at once. Second, the model operates in a steady environment, where the training data and test data are independent and identically distributed [28].

The advantages of static learning include high model stability, as well as ease of model validation and evaluation. However, when model drift occurs, re-training from scratch is often required, leading to poor scalability for large-scale applications [29].

3.3. Online Incremental Learning

Incremental learning refers to a machine learning approach where new tasks are learned from non-stationary data streams while retaining performance in previous tasks. Depending on the application scenario, incremental learning is commonly categorized into three types: task-, domain-, and class-based incremental learning [30]. According to the different forms of model drift, it can be further classified into co-variate shift, prior probability shift, and concept drift [31].

Considering our problem, there were several physical models with different accuracies, meaning the data were derived from different domains, thus falling into the domain-based incremental learning scenario. The data streams from different domains were non-stationary, manifesting as concept drift, significant noise (especially in wind tunnel test data), and uneven distribution. Therefore, it was necessary to choose an incremental learning method that can handle model concept drift effectively.

To balance the accuracy of the surrogate model at the beginning of its application and after multiple iterations of incremental learning, as well as to prevent the model from forgetting the earlier training data, we propose a novel incremental learning method based on drift error in this paper.

First, we used an initial training set for offline training and obtained a static model. Then, we continuously evaluated the incremental data on the static model and used the difference between the actual and predicted outputs for incremental learning modeling. By focusing on model errors for incremental learning, this approach can address the issue of concept drift in modeling. This modeling method ensures that, after establishing the initial static surrogate model, its accuracy can continuously improve through online learning without compromising the original model, thus adapting to the features of incremental data and dealing with drift issues effectively.

4. Surrogate Model Construction

In the following, we use two model architectures—namely, the kernel model and the neural network—for offline and online training of the wing surrogate model. Taking $x=\left(\alpha ,u\right)$ as input and the lift coefficient ${\widehat{C}}_{\mathrm{L}}$ as the output, we consider a multi-input, single-output non-linear regression problem. The main focus of this section is to propose an incremental learning modeling method that considers model concept drift and compares its performance with other methods used in the control group.

4.1. Preparation of Training Samples

To generate sufficient sample data, random inputs $\left\{\begin{array}{lllll}{x}_1\hfill & \cdots \hfill & x_i\hfill & \cdots \hfill & x_N\hfill \end{array}\right\}\in X$ satisfying the constraints on the angle of attack and the morphing capability of the wing were generated; here, $x_i={\left(\alpha ,u\right)}_i$. By solving the aerodynamic model ${\mathit{ℳ}}_{\mathrm{A}}$ and the aeroelastic model ${\mathit{ℳ}}_{\mathrm{AE}}$, we obtained certain performance parameter outputs, denoted as $\left\{\begin{array}{lllll}{y}_{\mathrm{A},1}\hfill & \cdots \hfill & y_{\mathrm{A},i}\hfill & \cdots \hfill & y_{\mathrm{A},N}\hfill \end{array}\right\}\in Y_{\mathrm{A}}$ and $\left\{\begin{array}{lllll}{y}_{\mathrm{AE},1}\hfill & \cdots \hfill & y_{\mathrm{AE},i}\hfill & \cdots \hfill & y_{\mathrm{AE},N}\hfill \end{array}\right\}\in Y_{\mathrm{AE}}$. The performance parameters include lift, drag coefficient, structural stress, and so on. The inputs and outputs of ${\mathit{ℳ}}_{\mathrm{A}}$ and ${\mathit{ℳ}}_{\mathrm{AE}}$ were taken to form the data sets $\left(x_i,y_{\text{A},i}\right)\in {\mathcal{D}}_{\text{A}}$ and $\left(x_i,y_{\mathrm{AE},i}\right)\in {\mathcal{D}}_{\mathrm{AE}}$, respectively. Finally, the data sets were each split into a training set $\mathcal{T}$ and validation set $\mathcal{V}$, with the training set accounting for 80% of the total data, and the validation set accounting for the remaining 20%.

After training, the model’s quality was evaluated using the validation set. The loss value is defined as the average of the absolute difference between the predicted values $\widehat{Y}$ and the actual values $Y$, divided by the average of actual values; mathematically, this can be expressed as $ℒ=\left|\frac{\overline{{\widehat{y}}_i-y_i}}{{\overline{y}}_i}\right|\times 100\%$.

4.2. Offline Static Learning

First, the static surrogate model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$ was trained using the aerodynamic model data set ${\mathcal{D}}_{\mathrm{A}}$. The estimation results of the static surrogate model are denoted by ${\widehat{Y}}_{\mathrm{A},\mathrm{S}}$. To examine the fitting efficiency and accuracy of the surrogate model, training data sets of differing sizes were tested. To ensure that the model was not over-fitted, it was validated on the validation dataset ${\mathcal{V}}_{\mathrm{A}}$. Additionally, loss evaluation was also performed on the validation data set of the static aeroelastic model ${\mathcal{V}}_{\mathrm{AE}}$, in order to demonstrate the systematic differences between ${\mathcal{D}}_{\mathrm{A}}$ and ${\mathcal{D}}_{\mathrm{AE}}$.

The neural network and kernel models were tested, and the results are provided in

Table 1. Loss of surrogate model with various training data sizes and model architectures.

	$\mathbf{Loss}\mathbf{on}{\mathcal{T}}_{\mathbf{A}}$		$\mathbf{Loss}\mathbf{on}{\mathcal{V}}_{\mathbf{A}}$		$\mathbf{Loss}\mathbf{on}{\mathcal{V}}_{\mathbf{AE}}$
Data Size	NN	K	NN	K	NN	K
50	1.33%	1.85%	4.82%	2.85%	7.20%	5.43%
400	0.18%	2.01%	0.70%	1.83%	4.84%	4.96%
2000	0.13%	2.22%	0.13%	1.95%	4.35%	4.46%
4000	0.13%	1.67%	0.13%	1.62%	4.37%	4.46%

(NN, neural network; K, the kernel model). The number of neurons in the hidden layer of the neural network was 10, which was selected through prior testing. The results for the kernel model shown in the table were achieved through K-fold cross-validation, and all hyperparameters were optimized during the training process, including the linear regression model and kernel scale. This allowed for a more accurate fitted model to be obtained. However, this approach required several tens of seconds for the training duration. The fitting results without hyperparameter optimization were extremely poor and, so, are omitted from the table.

The results indicate that:

• Taking both the accuracy and time consumption into account, the Neural Network (NN) method outperformed the Kernel Model (K) method.
• When the training data size reached 2000, the model achieved good accuracy, and the loss on the validation set was comparable to that on the training set.
• The NN and K models trained with aerodynamic data did not perform well on the aeroelastic data. The difference between the aeroelastic data and the aerodynamic data was approximately 4.23%.

Based on these findings, in subsequent testing, we primarily focused on the neural network and utilized current hyperparameters for further investigation.

The static-aeroelastic surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{S}}$ was trained using the aeroelastic data ${\mathcal{D}}_{\mathrm{AE}}$. After training, the loss value was 0.7974%. This static surrogate model exhibited high accuracy in fitting the aeroelastic data but required a considerable amount of aeroelastic evaluations, making its efficiency relatively low. Hence, it was considered as a baseline for incremental learning.

4.3. Online Incremental Learning

In order to minimize the number of aeroelastic evaluations while ensuring that the model accuracy remained close to the aeroelastic physical model ${\mathcal{D}}_{\mathrm{AE}}$, we adopted an incremental learning approach based on error modeling.

As shown in Figure 4, the process started by initializing the static surrogate model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$ using the aerodynamic data ${\mathcal{D}}_{\mathrm{A}}$. Subsequently, during acquisition of the aeroelastic data ${\mathcal{D}}_{\mathrm{AE}}$, it was continuously used as streaming data to be evaluated by using the surrogate model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$. The evaluation results were then compared with the aeroelastic data, and the difference $y_{\mathrm{AE}}-{\widehat{y}}_{\mathrm{A}}$ was used to train and update the new surrogate model ${\mathsf{\Omega }}_{\mathsf{\epsilon },\mathrm{Inc}}$. This approach models the error $\epsilon$ caused by concept drift.

Therefore, the incremental surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ considering concept drift is composed of the static aerodynamic surrogate model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$ and the incremental model ${\mathsf{\Omega }}_{\mathsf{\epsilon },\mathrm{Inc}}$, which captures the drift error. By comparing the difference in error $y_{\mathrm{AE}}-{\widehat{y}}_{\mathrm{AE}}$ between the incremental surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ with the aeroelastic model ${\mathcal{D}}_{\mathrm{AE}}$ during the incremental learning process, we can assess the accuracy of the incremental surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ and judge its convergence. When the two are very close, the incremental learning can be temporarily stopped. However, if the error has not reached a certain range (or if the experiment is still ongoing and there is a subjective desire to continue online learning), new inputs $x_i={\left(\alpha ,u\right)}_i$ will be continuously fed into the model, and the loop shown in the figure will be repeated to update the incremental surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ of the wing system.

It should be noted that a buffer pool with a window of 100 samples was set up for incremental learning in order to improve the sample utilization efficiency. The samples chosen for the buffer pool are of critical importance for training. An improper buffer can lead to catastrophic forgetting, which is the tendency of an artificial neural network to drastically forget previously learned information upon learning new information [32]. Our buffer is managed based on three key principles: uniformity, timeliness, and adaptability. The underlying concept involves assigning an importance factor to each sample in the pool and updating these factors based on its distance from new samples. Upon the arrival of a new sample, it is incorporated into the pool, while the least important sample is removed. In practical implementation, the sample $x_i$ in the pool has its importance factors as ${\mu }_i$, and the new sample is denoted as $x_{*}$. An adaptive adjustment parameter $\sigma =\frac{1}{3}{\left(\frac{V}{Cn}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$ is introduced. Where $C$ represents the pool’s capacity, $n$ is the dimensionality of $x$, and $V$ denotes the volume of the minimum $n$-dimensional hypercube enclosing all samples in the pool. The updated formula for ${\mu }_i$ is as follows:

$$\begin{array}{l}{\mu }_i={\mu }_i\times G_i\\ G_i=1-\exp \left(-{\Vert x_{*}-x_i\Vert }^2/2{\sigma }^2\right)\end{array}$$

(1)

The effect of this approach is that samples closer to the latest sample experience a greater attenuation, and samples that have been present for a longer period of time are attenuated more times. This ensures that the samples in the pool are evenly distributed within the data domain and are obtained relatively new.

In addition, we introduce another two incremental learning strategies for reference:

• Reference incremental learning strategy 1: First, create an empty neural network structure and incrementally train the neural network from scratch using the aeroelastic data ${\mathcal{D}}_{\mathrm{AE}}$.
• Reference incremental learning strategy 2: Perform incremental learning based on the pre-trained neural network ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$.

In each case, the aeroelastic data ${\mathcal{D}}_{\mathrm{AE}}$ are sequentially input in batches of 10 samples for updating.

The fitting losses obtained when using the above three incremental strategies are shown in Figure 5.

Comparing reference strategies 1 and 2, it can be observed that having a well-initialized base neural network can significantly improve the performance of the incremental learning process. If there is no pre-trained neural network as an initial model, directly performing incremental learning from scratch may not yield satisfactory results.

For reference strategy 2, due to the presence of concept drift, right after the incremental training begins, the incremental model exhibits systematic biases compared to the aerodynamic model, indicating the feature of catastrophic forgetting. Additionally, fitting the aeroelastic model was not robust, and its performance did not show any significant improvement during incremental learning. The average loss value in the last 50 training iterations, compared to the aerodynamic model, was approximately 6.0787%; meanwhile, compared to the aeroelastic model, it averaged approximately 2.3988%.

Our incremental learning approach based on error modeling achieved excellent fitting performance, with a loss value of only 0.1566% on the validation set.

5. Aerodynamic Performance Optimization

Next, we optimized the trailing-edge deflection distribution based on the physical model and surrogate model of the wing while considering the constraints on the deformation capability of the morphing wing. The objective of this process was to reduce the induced drag while keeping the target lift unchanged.

5.1. Optimization Architecture

Cruise drag is significantly correlated with flight performance, and so, reducing drag in the cruise state is a primary goal of aircraft design. For electric aircraft, the aircraft weight remains constant throughout the full envelope. A longer cruise range requires a higher lift-to-drag ratio, which means minimizing the drag coefficient at a constant lift. Therefore, the optimization objective is given as $\underset{x=\{\alpha ,u\}}{\min }{C}_D$, where the design variables $x$ consist of the angle of attack α and virtual shape functions $u$. A constant target lift coefficient of 1.2 was set. Due to the limitations of aerodynamic methods, only induced drag is considered in this study.

Under all flight conditions, the lift–gravity equilibrium must be satisfied. Additionally, considering the limitation of the trailing-edge differential deflection capability (verified through preliminary experiments), constraints were applied to the range of virtual shape functions.

5.2. Basic Optimization Methods

Aerodynamic performance optimization was conducted based on the physical aeroelastic model ${\mathit{ℳ}}_{\mathrm{AE}}$. A simplified flowchart of this process is illustrated in Figure 6.

A global optimization method based on radial basis function interpolation was employed. This method is suitable for optimization problems with time-consuming objective functions, such as the aeroelastic coupling simulation used here [33].

The design variables in the outer loop consist only of the virtual shape functions $u$, while the trim process is in the inner loop to produce the trim angle of attack $\alpha$, thus forming a two-level optimization approach. This method ensures that each iteration of the outer-level optimization is in the trim state.

First, the wing with zero deflections was trimmed with $C_{\mathrm{L}}{}_{{}_t}=1.2$. After performing static aeroelastic coupling iterations, the trimmed angle of attack was found to be 3.9883 degrees, resulting in a lift coefficient of 1.1998 and an induced drag coefficient of $C_{\mathrm{D},\mathrm{A}\mathrm{E},\mathrm{i}\mathrm{n}\mathrm{i}}=0.0235$.

Subsequently, the shape optimization process was carried out with 500 iterations, and the results gradually converged. The final optimized angle of attack reached 1.8206 degrees, with a corresponding lift coefficient of 1.2001 and an induced drag coefficient of $C_{\mathrm{D},\mathrm{A}\mathrm{E},\mathrm{o}\mathrm{p}\mathrm{t}}=0.0227$. The optimized wing trailing-edge shape is shown in Figure 7.

As a basic optimization method, this approach is computationally accurate and can achieve a reduction in induced drag while maintaining constant lift. The pressure coefficient distributions along the span before and after optimization are shown in Figure 8. Due to the wing root lacking the ability to morph, as indicated in Section 2.1, the lift distribution is not discussed. It can be observed that the pressure coefficient distribution without deflection was closer to that of a trapezoidal wing. However, after shape optimization, the camber increases and varies continuously between approximately 30% and 80% of the wingspan, resulting in an increased lift distribution in this section of the wing, which means that the lift distribution became close to that of an elliptical wing. According to the lifting-line theory, when a wing has no twist and sweep, an elliptical wing shape exhibits an elliptical lift distribution with minimal induced drag. Manufacturing challenges and high costs make elliptical wings uncommon in application. However, an elliptical-like lift distribution can be achieved through continuous morphing of the trailing edge.

It should be noted that, due to the relatively long duration of the aeroelastic simulation, the optimization process with this method took 4 h with six-thread parallel computation, making it difficult to achieve real-time optimization. Therefore, in this study, it served as a control group for comparison with the following approach, which introduces a surrogate model to replace the physical model.

5.3. Optimization Methods Based on Surrogate Models

This section introduces the optimization method based on the surrogate model.

For construction of the surrogate model, we adopted the incremental learning approach based on error modeling described in Figure 4. ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ is composed of the static aerodynamic surrogate model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$ and the incremental model ${\mathsf{\Omega }}_{\mathsf{\epsilon },\mathrm{Inc}}$ that captures the drift error.

Then, under the target lift coefficient $C_{{\mathrm{L}}_t}$, the optimizer uses the current surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ to optimize a set of states $x_i={\left(\alpha ,u\right)}_i$. The optimized lift and drag coefficients are represented as ${\widehat{y}}_{\mathrm{AE}}=\left\{{\widehat{C}}_{\mathrm{L},\mathrm{A}\mathrm{E}},{\widehat{C}}_{\mathrm{D},\mathrm{A}\mathrm{E}}\right\}$. These optimized states $x_i={\left(\alpha ,u\right)}_i$ are then input into the aeroelastic physical model ${\mathit{ℳ}}_{\mathrm{AE}}$ for validation, and the resulting lift and drag coefficients are denoted as $y_{\mathrm{A}\mathrm{E}}=\left\{C_{\mathrm{L}}{}_{,\mathrm{A}\mathrm{E}},C_{\mathrm{D}}{}_{,\mathrm{A}\mathrm{E}}\right\}$. The validation results are then used for incremental training of ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$, and the process is iterated for the next shape optimization.

A simplified flowchart of the optimization process is depicted in Figure 9. The model was initialized with 40 sets of random inputs.

Due to the high computational efficiency requirement of our problem, we used the gradient descent algorithm for optimization. As our problem was non-convex and the optimizer’s results were sensitive to the initial values, we took inspiration from ensemble learning and ran multiple optimization processes in parallel with different random initial values. The best solutions obtained from several optimizers were selected as the final solution.

To expedite the computational efficiency, we removed the embedded relationship of trim in the previous method. Instead, we treated the attack angle and virtual shape functions $x=\left(\alpha ,u\right)$ as design variables in the optimization problem. To ensure that the trim was satisfied, it was incorporated as a non-linear constraint in the optimizer. Each optimization iteration costs approximately 2 s on a computer equipped with a sixteen-core processor running at a frequency of 4.3 GHz.

Figure 10 illustrates the variations of parameters during the iterative process of the optimization approach based on incremental learning surrogate models, where the drag coefficients $C_{\mathrm{D},\mathrm{A}\mathrm{E},\mathrm{i}\mathrm{n}\mathrm{i}}$ and $C_{\mathrm{D},\mathrm{AE},\mathrm{o}\mathrm{p}\mathrm{t}}$ are as detailed in Section 5.2.

The optimizer consistently achieved the target lift coefficient in each iteration. In the last 40 iterations of optimization, the average drag value was 0.0228, and the minimum value was 0.0226, demonstrating a consistent drag reduction of about 4%. With the continuous iteration and updating of ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$, the relative fitting errors improved. The relative errors in the lift coefficient were generally within 1%, and the relative errors in the drag coefficient were generally within 7%. Throughout the iteration process, the optimized states $x=\left(\alpha ,u\right)$ remained relatively stable, and the trailing-edge shape resembled the result obtained with the basic optimization method described in Section 5.2.

The optimization result showed an upward deflection of the wingtip. By decreasing the camber of the wingtip, the growth rate and the strength of the wingtip vortices were reduced, and the downwash and induced angle of attack were mitigated, leading to a noticeable decrease in overall induced drag. This phenomenon is consistent with the literature [13]. This effect is similar to the traditional wing design method of twisting the wing to reduce the angle of attack at the wingtip.

We set up two control groups to compare the results of our optimization approach based on incremental learning surrogate models:

• Control group 1: optimization based on the offline static model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$.
• Control group 2: optimization based on the offline static model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{S}}$.

By repeating the process for both control groups multiple times, multiple sets of optimized solutions ${\widehat{y}}_{\mathrm{A},\mathrm{S}},{\widehat{y}}_{\mathrm{A}\mathrm{E},\mathrm{S}}$ were obtained and validated using the aeroelastic model ${\mathit{ℳ}}_{\mathrm{AE}}$. This resulted in $y_{\mathrm{A}\mathrm{E},\mathrm{S}}^{\mathrm{G}1},y_{\mathrm{A}\mathrm{E},\mathrm{S}}^{\mathrm{G}2}$, in which the superscript represents the group of data. We extracted the results from the last eight iterations of our optimization approach based on ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ and compared them, as shown in Figure 11.

From the results, we can observe that:

• The optimization results for Control group 1 may appear good in the optimizer, but they did not perform well in validation. This is due to the static aerodynamic model ${\mathsf{\Omega }}_{\mathrm{A},\mathrm{S}}$ and the aeroelastic model ${\mathit{ℳ}}_{\mathrm{AE}}$ having systematic biases, making the optimization results unreliable.
• The optimization results of Control group 2 were more reliable compared to Control group 1; however, due to the inability to correct model fitting errors during the iterations, it was challenging to search for more reliable data near the target point for model updates, resulting in only moderate accuracy.
• The optimization results of our approach showed more stability and repeatability than those based on the static model. Results yielded from multiple repeated experiments of the static model are circled by the dashed line and the dash-dot line in Figure 11.
• The optimization results of our approach based on ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ gradually improved over time. Notably, in the last few iterations, both the model accuracy and optimization results were excellent.

6. Multi-Objective Optimization

We next optimized the distribution of the trailing-edge deflection based on the surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$, considering the constraints on the deformation capability of the morphing wing. Various objectives were introduced, building upon the method described in Section 5.3, which makes this chapter an extension of the scenarios proposed for the optimization method.

6.1. Real-Time Drag Reduction with Varying Target Lift

In real-world aviation, flight conditions seldom remain static. In order to investigate the tracking capability of the real-time shape optimization method under different operating conditions, the target lift coefficient $C_{{\mathrm{L}}_t}$ was dynamically varied over time. The optimization objective was real-time drag reduction.

The optimization method and process in this section were similar to those described in Figure 9. The main difference was the varying target lift coefficient, which changed over time (as shown by the light-blue dotted line in Figure 12a). Specifically, the target lift coefficient underwent a step decrease of 0.2 every 20 steps, three times. After that, it gradually increased with a linear ramp for 100 steps. Following this, it exhibited a non-linear variation (modeled by a sinusoidal function) over the subsequent 157 steps.

Figure 12 illustrates the variations of parameters observed during the optimization process. Based on Figure 12a,b, it can be seen that under each target lift coefficient $C_{{\mathrm{L}}_t}$, both the optimizer and aeroelastic validation presented excellent tracking performance with respect to $C_{{\mathrm{L}}_t}$. In Figure 12c, the trailing-edge shape of the wing along the span is presented when $C_{{\mathrm{L}}_t}$ was set to 1.2, 1.0, 0.8, or 0.6. With a larger $C_{{\mathrm{L}}_t}$, the deflection angle increased, meaning that the wing experienced more significant camber morphing. Furthermore, the deflection angle at the wingtip was relatively smaller compared to the other locations, which was consistent with the previous findings. To further analyze the results, the 39th and 79th iterations of the optimization process were chosen as typical states for $C_{{\mathrm{L}}_t}=1.0$ and $C_{{\mathrm{L}}_t}=0.6$, respectively. The spanwise pressure distributions under the two states are plotted in Figure 13. Notably, in the spanwise area with morphing trailing edge, the lift distribution uniformly exhibited an elliptical shape.

In addition, it is worth mentioning that we also evaluated the fitting performance of ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ with respect to ${\mathit{ℳ}}_{\mathrm{AE}}$ during the optimization process. For this evaluation, we utilized a validation data set ${\mathcal{V}}_{\mathrm{AE}}$ consisting of 150 randomly selected inputs. With the iterative updates, we observed an improvement in the relative fitting errors. Specifically, the relative error in the lift coefficient remained within 1%, while the relative error in the drag coefficient remained within 3%. In the following cases in this paper, this criterion will be omitted, as the effect was generally similar to that in this case.

The lift-to-drag ratios obtained from random input states, as shown in Figure 14, validated the reduction in the lift-to-drag ratio achieved during the optimization process. For each target lift coefficient, the lift-to-drag ratios of the optimized solutions were significantly higher than those of the random inputs, showing a consistent, substantial improvement. Moreover, the optimized results adhered to the theoretical proportionality between the induced drag coefficient and the square of the lift coefficient.

6.2. Drag and Stress Reduction Simultaneously

When aerodynamic forces act on the wing structure, the main beam inevitably undergoes bending. This is especially prominent in high aspect ratio wings, where aeroelastic effects can lead to significant stress issues in the main beam. To ensure structural integrity and flight safety, we focused on achieving the optimization goals of simultaneously reducing induced drag and minimizing the maximum stress on the main beam through active trailing-edge deflection.

For this test, the target lift coefficient was maintained at $C_{{\mathrm{L}}_t}=1.2$. The main difference was in the cost function of the optimizer, represented as $\mathrm{Cost}=\frac{C_{\mathrm{D}}}{C_{\mathrm{D},\mathrm{A}\mathrm{E},\mathrm{i}\mathrm{n}\mathrm{i}}}+\rho \frac{\sigma }{{\sigma }_{\mathrm{i}\mathrm{n}\mathrm{i}}}$, where $\rho$ denotes the weighting coefficient for the stress term in the cost function.

In addition, the surrogate model ${\mathsf{\Omega }}_{\mathrm{AE},\mathrm{Inc}}$ includes not only the fitting of the lift coefficient and drag coefficient but also the maximum stress in the main beam; thus, ${\widehat{y}}_{\mathrm{AE}}=\left\{{\widehat{C}}_{\mathrm{L},\mathrm{A}\mathrm{E}},{\widehat{C}}_{\mathrm{D},\mathrm{A}\mathrm{E}},{\widehat{\sigma }}_{\mathrm{A}\mathrm{E}}\right\}$. Furthermore, after inputting the optimized state $x=\left(\alpha ,u\right)$ into the aeroelastic physics model ${\mathit{ℳ}}_{\mathrm{AE}}$ for validation, the corresponding stress distribution, denoted as ${\sigma }_{\mathrm{AE}}$, can also be obtained. The surrogate model of $C_{\mathrm{L}}$ and $C_{\mathrm{D}}$ was initialized with 40 sets of random inputs, while the stress did not have an initial static model as a basis. Therefore, to achieve a better initialization state, it was initialized with 180 sets of random inputs.

The Pareto front is depicted in Figure 15. Changing the weight coefficient $\rho$ influences the trade-off between the two objectives and further affects the optimization results.

It can be observed that as long as the objective function includes stress the drag is hardly decreased. This might be attributed to the wider range of stress variations compared to the relatively smaller range of drag changes. As the weight coefficient for drag is progressively reduced, the stress experiences more significant reductions. Two special cases are highlighted by black boxes in Figure 15. Specifically, the lowest optimized stress relative to the reference initial value was approximately $\frac{{\sigma }_{\mathrm{mini},\mathrm{A}\mathrm{E},\mathrm{I}\mathrm{n}\mathrm{c}}}{{\sigma }_{\mathrm{i}\mathrm{n}\mathrm{i}}}\times 100\%=81.7\%$, obtained at $\rho =1.36$, showing an 18.3% stress reduction.

Moreover, the optimized states $x=\left(\alpha ,u\right)$ in the optimization process between $\rho =1$ and $\rho =1.36$ are compared in Figure 16. When the stress weight coefficient was larger, the upward deflection at the wingtip significantly increased, even reaching the maximum of the allowable deflection range. This is because when the wingtip deflects upward, the camber decreases, leading to a significant reduction in the lift provided by the wingtip. With the total lift remaining constant, more lift needs to be generated by the wing root, resulting in a smaller moment arm relative to the wing root and thus reducing the bending moment and stress experienced by the wing root. When $\rho =1.36$, the overall wing camber was smaller, compared to the case of $\rho =1$. To achieve the same total lift, a larger angle of attack was required.

6.3. Drag, Stress, and Actuation Energy Reduction Simultaneously

The process of inducing deformation in the trailing edge of a wing through active deflection undoubtedly consumes energy, which should be considered as a factor in shape optimization. Therefore, we aimed to achieve a simultaneous reduction in induced drag on the wing, a decrease in maximum stress on the main beam, and a lower power consumption of the actuator as the optimization criteria through active trailing-edge deflection. The relevant cost function of the optimizer is represented as $\mathrm{Cost}=\frac{C_{\mathrm{D}}}{C_{\mathrm{D},\mathrm{A}\mathrm{E},\mathrm{i}\mathrm{n}\mathrm{i}}}+\rho \frac{\sigma }{{\sigma }_{\mathrm{i}\mathrm{n}\mathrm{i}}}+\beta \frac{Energy}{Energ{y}_{\mathrm{ave}}}$. In which the energy is proportional to the sum of the absolute value of each deflection angle.

The primary considerations under various flight conditions are different. In takeoff and climb phases, the main goal is to increase lift, which means that the aircraft is more susceptible to structural damage. Therefore, the primary consideration should be to reduce maximum stress. During the cruise phase, which constitutes a significant portion of the entire flight, the primary objective should be to minimize drag. Additionally, it becomes more crucial to consider reducing actuator energy consumption when the energy supply for the actuators is limited.

The Pareto front is illustrated in Figure 17. It is evident that drag reduction was almost not achieved. The stress could be reduced by up to 38.1% when significant trade-offs were made with the other two objectives, as indicated by the marked point ‘B’. At point ‘C’, neither stress nor energy were reduced much, in exchange for the minimal sacrifice of drag, amounting to 0.3%. The most remarkable result was in terms of energy savings, with complete elimination of actuation energy consumption being achieved, as indicated by the marked point ‘A’.

7. Conclusions and Discussion

In this paper, we presented an incremental error-based learning approach for constructing a surrogate model to optimize the real-time shape of continuous camber morphing wings, taking into consideration the concept drift caused by changes in the data domain.

By building a surrogate model, the efficiency was significantly improved over that of conventional shape optimization methods based on physical models. Moreover, the proposed method preserved the original static surrogate model while incrementally learning from errors, leading to enhanced model accuracy. By continuously searching for data points at the optimization target, the local accuracy of the surrogate model was further improved.

Based on the incremental surrogate model, real-time aerodynamic performance optimization of the wing shape was achieved. Through comparisons with other optimization schemes, the effectiveness of the proposed surrogate model’s construction was demonstrated, and its efficiency for optimization applications was proven. Finally, real-time multi-objective optimization was accomplished under slowly varying lift conditions, and drag reduction, stress reduction, and actuation energy reduction were achieved jointly. The optimization yielded stable results, showing potential for achieving a 4% drag reduction in single-objective optimization and an 18.3% maximum stress decrease in multi-objective optimization.

Furthermore, it is important to note that the proposed method is extensible to various applications. Although the data domain considered in this study encompassed aerodynamic and aeroelastic simulation models, the ultimate target domain also includes wind tunnel and flight test data. Regarding the output of the surrogate model, other parameters, such as the lift distribution along the span, may also be included. This flexibility is expected to allow for more comprehensive analysis and optimization of the wing shape based on the specific needs and objectives of the research.