Burning fossil fuels leads to increased greenhouse gas emissions, which will be one of the major environmental issues for years to come. This results in the development of fuel-efficient vehicles around the world, of which an aircraft is one. It is well known that an aircraft wing with a higher lift-to-drag ratio leads to longer endurance and range. Therefore, airfoil shape design is a critical aspect of aircraft wings and other aerodynamic structure designs, as it affects aerodynamic performance significantly, including lift and drag [1,2]. Another important aspect of airfoil shape optimization is the trade-off between maximum lift and minimum drag. In some cases, a highly cambered airfoil shape may provide increased lift, but at the cost of increased drag. As a result, the design of the airfoil shape needs to be performed by means of optimization, which is usually purposed to explore the trade-off between lift and drag, as well as other objectives, such as stall angle, angle of attack and maximum lift-to-drag ratio [3,4].

An important factor in airfoil shape optimization design is the selection of a tool for aerodynamic analysis. This analysis can either be low-fidelity, e.g., using the vertex lattice method [5], or high-fidelity, using computational fluid dynamics (CFD) [5–7]. Although CFD is more accurate, it also requires complex, computationally expensive models, and design optimization through this approach is nearly impossible. To address this, surrogate models are utilized to approximate the airfoil's aerodynamic performance through the limited number of CFD simulations, enabling a rapid optimization process [8–11]. Despite their advantages, the use of surrogate models presents certain challenges, including the risk of misfit and overfitting. Nevertheless, ongoing development and refinement in this field hold the promise of overcoming these challenges. Enhancement of surrogate-assisted optimization can be attained by implementing multiple surrogate models [12–14], utilizing infill criteria [8,9], or applying multi-fidelity simulation techniques [15,16]. Although infill criteria and multi-fidelity surrogate models have proven effective in improving surrogate models, most research tends to focus on one approach at a time, rather than utilizing both techniques simultaneously. Furthermore, the majority of infill sampling studies focus on single-objective optimization problems, making the exploration of multi-objective optimization with multiple points infill sampling a unique and challenging area of study.

Therefore, this work proposes multi-fidelity multi-objective infill-sampling surrogate-assisted optimization for an airfoil shape design optimization [17,18]. The objective function is set to be maximization of the ratio of lift-to-drag coefficients subjected to airfoil geometry constraints. Computational Fluid Dynamics is used for high-fidelity simulation of airfoils, while the XFoil flow solver is used for low-fidelity simulation of the airfoil. To validate and further assess the proposed method, conventional surrogate assisted optimization and a traditional infill sampling criterion with multi-fidelity simulation are applied. In Section 3, the multi-fidelity analysis tools are described in more details. Section 4 provides details of the multi-surrogate-assisted metaheuristics case. The numerical experiment of each methodology at each test case is illustrated in Section 5. Section 6 demonstrates the results obtained and discusses them, followed by conclusions expressed in Section 7.

Optimization has become a vital part of multi-disciplinary design optimization [19–24], while computational tools based on both high-fidelity simulation [25–27] and low-fidelity simulation [23,28,29] are used practically to simulate the real-world environment and render the outcomes as a function of design variables within the design objectives and constraints framework. Typically, high-fidelity simulation requires extensive computational cost; however, the outcome is more precise than low-fidelity simulation [23,30]. Conversely, the low-fidelity simulations take short times but are less accurate. Therefore, multi-disciplinary design optimization, which requires several function evaluations or simulation runs, is almost impossible based on using high-fidelity simulation alone, while using low-fidelity simulation obtains unacceptable “optimum” solutions. Consequently, the techniques of surrogate-assisted optimization are being used extensively for numerous applications of multi-disciplinary design optimization to reduce [25,31–33] computational competency and analysis times, based on the high-fidelity simulation.

Generally, the surrogate-assisted optimization technique consists of three main steps; generating a set of sampling points, constructing the surrogate model (SuMo), and performing optimization based on the constructed surrogate model [33]. The set of sampling points can be searched for using a design of experiment method, such as the Latin hypercube sampling method (LHS) or the Optimum Latin hypercube sampling method (OLHS) [34,35], while SuMo can be constructed based on several approaches such as Response Surface Methods (RSM) [36], a Gaussian Process or Kriging Model [20,37–41], Radial Basis Functions (RBFs) [42,43], Artificial Neural Networks (ANNs) [44], Support Vector Machines (SVMs) [36], etc. [19,45]. Among these models, Kriging is arguably the most popular one, due to its ability to effectively capture complicated responses and its ability to provide an exact interpolation, objective prediction, and error estimation in the spatial distribution.

Improving the surrogate model method can be achieved by applying infill criteria strategies. The infill sampling technique finds additional points to improve the model by solving an optimization sub-problem, such as maximising an Expected Improvement (EI) indicator [46–49], maximizing the Probability of Improvement function (PI) [50], minimizing the Lower Confidence Bounding (LCB) [51], and Minimizing the Prediction of surrogate models (MP) [52]. Among the various infill criteria, maximizing EI is arguably the most preferred and efficient criterion, while others are also still in use [42,53,54]. Rather than using infill criteria strategies for improving the surrogate model, multi-fidelity surrogate models can also be applied for the reduction of expensive high-fidelity computations with the enhancement of cheaper low-fidelity data [19,53]. Successful use of multi-fidelity surrogate models for several applications of multi-disciplinary design optimization has been reported worldwide [26,27,55,56], e.g., the use in aircraft design [57]. Although the strategy of infill criteria and multi-fidelity surrogate models are used successfully for improving the surrogate model, most of the research usually applies a single strategy; either infill sampling or multi-fidelity modelling in a single work, while the application of using both techniques at the same time is rarely studied. In addition, the infill sampling technique is mostly performed based on a single objective optimization problem, while studying a multi-objective optimization problem for multiple points infill sampling is interesting and challenging.

Numerical simulation and optimization techniques [58–60] are widely used in the aerodynamic shape optimization field. Lift and Drag are counted as aerodynamic forces because they exist due to the movement of the aircraft through the air. For aircraft aerodynamic design, a majority of researchers have investigated the lift to drag ratio as a metric of aircraft range and endurance. The literature depicts that the panel method with low-fidelity simulation and CFD simulation with high-fidelity are the two most widely applied and preferred techniques. However, the present work explored both fidelity tools in an effort to accelerate the design process to predict the lift to drag ratio. The airfoil shape design problem is posted to maximize the ratio of lift to drag coefficients which can be expressed as:

(1)Maximise:f(x)=Cl/Cd

Subject to

xl≤x≤xu,where f(x) is an objective function; x_l and x_u are lower bound and upper bound of the design variables of vector x; C_l = coefficient of lift and C_d = coefficient of drag. Here, x=[x1,x2,x3,…,x10]T illustrates the airfoil design variables that have direct control over the airfoil shape, while the NACA2412 airfoil is used as a baseline geometry. The design vector determines airfoil shape change from the baseline NACA2412 as shown in Fig. 1. The airfoil is constructed using a cubic spline function though the ten design variables points, while the leading and trailing edge points are fixed. The lower and upper bounds are considered as x_l = [0.0177, 0.0646, 0.0629, 0.0427, 0.0098, −0.020, −0.0203, −0.0541, −0.0274, −0.00640]T and x_u =[0.0253, 0.0884, 0.0567, 0.0324, 0.0130, −0.0127, −0.0304, −0.0236, −0.0134, −0.00320 ]T, respectively.

Multi-fidelity tools are applied for the airfoil aerodynamic analysis to balance between accuracy performance and computational time. The XFoil tool and ANSYS 2020 student version are applied to perform numerical simulation for low-fidelity and high-fidelity, respectively.

In real-world problems, optimization is usually time-consuming and computationally expensive in the evaluation of objective functions. Surrogate models can successfully alleviate this problem by reducing the computation time for every numerical simulation code.

In order to examine the performance of the proposed method, three surrogate assisted optimization techniques including the conventional Kriging based surrogate assisted optimization, infill sampling Kriging based surrogate assisted optimization and the proposed multi-objective infill sampling Kriging based surrogate assisted optimization are applied for solving the proposed airfoil shape optimization problem as detailed in Section 2. Multi-fidelity modeling is also applied for objective function calculation. The details of the surrogate model techniques used in this study are as follows:

Conventional-KRG; Conventional Kriging-based surrogate model is used. The Kriging model is constructed based on the high-fidelity simulation for 75 samplings.

For each surrogate model strategy, the initial sampling points are generated using an optimum Latin hypercube sampling technique proposed by Pholdee et al. [34]. A differential evolution (DE) algorithm [70] is used for solving the infill sampling sub-problem, while for the main airfoil shape optimization, the hybrid real-code population-based incremental learning and differential evolution (RPBILDE) [71,72] is used to solve the proposed multi-objective infill sampling sub-problem.

In order to investigate the performance of the above surrogate assisted MH methods, each method is used to tackle the proposed airfoil shape optimization design problem for 20 optimization runs. The number of iterations is set to be 200 while the population size is 50 in the present work. The termination criterion for all optimizers is set as 10,000 function evaluations, which is the multiplication of 50 population size and 200 iterations.

In this investigation, the airfoil shape optimization is executed using the multi-fidelity surrogate assisted MH while the surrogate models are constructed based on the Kriging model, infill sampling, and the MO-infill sampling methods. The performance of the various surrogate-assisted MH techniques is investigated based on the percentage error between surrogate model approximation and the real objective function values by high-fidelity obtained at the optimum points of each techniques. After performing the optimization process for 20 independent runs, Fig. 5 shows the box plot of the percent error of the objective function values at the optimum points for all techniques. For each boxplot, the upper and lower horizontal lines correspond to the maximum and minimum percent error at the optimum points, while the internal line shows the median of the percent error at the optimum points. According to Fig. 5, the best performer based on the median values is the proposed MO-infill-KRG2, while the second best and the third best are MO-infill-KRG1 and infill-KRG2, respectively. The minimum percent of error can be archived from the proposed MO-infill-KRG2.

Table 2 shows the real objective function values at the optimum points for each method after performing high-fidelity simulation. From the results, the best performer according to the mean of the real optimum values is MO-infill-KRG2, while the second and third best are MO-infill-KRG1 and infill-KRG2, respectively. For the consistency performance according to the standard deviation (STD), the best algorithm is Conventional-KRG, while the second and third best methods are Infill-KRG1 and Infill-KRG2, respectively. The best objective function value obtained is from using MO-infill-KRG2. Table 3 shows the real objective function value of the baseline airfoil, and the optimum airfoil obtained from the various surrogate-assisted MH techniques. According to this table, MO-infill-KRG2 obtained the best optimum airfoil while the second and third optimum airfoils are from MO-infill-KRG1 and Infill-KRG2, respectively.

Fig. 6 shows the best airfoil accomplished by MO-infill-KRG2 after the optimization process. The black line represents the baseline geometry, the red line depicts the optimized airfoil shape, and the blue line illustrates the upper and lower bounds of the airfoil nodal points. From Fig. 6, the optimum airfoil geometry shows the lower leading-edge radius, smaller airfoil thickness and different lower and upper surfaces, leading to the aerodynamic efficiency obtained as illustrated in Figs. 7–9. Figs. 7–9 respectively show the comparison of variation of drag coefficient, lift coefficient and ratio of lift and drag coefficient with respect of the angle of attack of the optimized airfoil from MO-infill-KRG2 and the base line airfoil (NACA2412) obtained from CFD analysis. From Figs. 7 and 8, the drag coefficient obtained from the optimum airfoil is better than the base line airfoil for the angle of attack more than 6°, while the lift coefficient obtained from the optimum airfoil is better than the base line airfoil for the angle of attack lower than 8°. For the angle of attack between 8° and 10°, the lift coefficient of both the optimum airfoil and the base line airfoil stall, whereas the optimum airfoil still has higher lift coefficient. Based on Fig. 9, the ratio of lift to drag coefficient obtained from the optimum airfoil is better than the base line airfoil for all angles of attack. Overall, the optimum airfoil obtained from the proposed MO-infill-KRG2 is aerodynamically superior to the baseline airfoil.

As per the aforementioned data, it can be argued that MO-infill-KRG2 is the best while MO-infill-KRG1 is the second-best technique in the vicinity of the Infill-KRG2 approach for airfoil optimization with the investigated surrogate-assisted MHs. Applying EI and space-filling quality functions in the proposed technique can balance model accuracy and function landscape capturing, leading to the best percentage error at the optimum points and the best optimum solution with a smaller number of high-fidelity simulations. In addition, when considering the same surrogate model technique, it was found that using higher fidelity gives better model accuracy at the optimum points and a better optimum solution.

This work successfully proposes multi-objective infill sampling surrogate-assisted metaheuristic based on multi-fidelity simulation for airfoil shape optimization. The optimum LHS technique along with the Kriging surrogate model is applied, while a differential evolution algorithm is used as an optimizer. A special multi-objective optimization sub-problem is proposed for multiple points infill sampling exploration to improve the constructed surrogate model. To validate and further assess the proposed methods, a conventional surrogate-assisted optimization method and infill sampling surrogate-assisted optimization methods are applied with multi-fidelity simulation and their performances are investigated. The quantitative and qualitative comparative analysis demonstrates the dominance of the proposed method (MO-infill-KRG2) over other considered techniques for the presented airfoil shape optimization problem. This work only studied aerodynamics at a single value of AOA with a computational fluid dynamic simulation and various surrogate model method with one optimizer. Future work may try to study various AOAs with various algorithms based on this work. The whole aircraft design based on this simulation process and various algorithms is also a future topic of interest.