Parametric Optimization of Spiked Blunt Bodies in Supersonic Flow Using Surrogate-Assisted Machine Learning and Evolutionary Algorithms

Abstract

This study presents a surrogate-assisted evolutionary optimization framework for parametric design under limited data conditions, integrating computational fluid dynamics (CFD), machine learning, and evolutionary algorithms to optimize spiked blunt body geometries in supersonic flow. A dataset of CFD simulations covering a range of Mach numbers and geometric ratios, including spike length ($L/D$) and diameter ($d/D$), was used to train regression-based surrogate models.Among the evaluated models, the Gradient Boosting Regressor (GBR) achieved the highest predictive accuracy ($R^2=0.8909$, RMSE = 0.00775), effectively capturing the nonlinear relationship between flow conditions, geometry, and drag coefficient ($C_d$). The trained surrogate model was coupled with three evolutionary algorithms—Differential Evolution (DE), Covariance Matrix Adaptation Evolution Strategy (CMA-ES), and Genetic Algorithm (GA)—to identify optimal geometric configurations across different Mach regimes. To validate the proposed framework, the optimal solutions obtained from the surrogate-based optimization were re-evaluated using CFD simulations. A strong agreement between predicted and simulated drag coefficients was observed, confirming the reliability of the surrogate model for guiding optimization within the explored design space. The results reveal consistent geometric trends, with the optimal spike length ratio decreasing as Mach number increases, while the diameter ratio converges to a narrow range around $d/D\approx 0.17$. Additionally, SHapley Additive exPlanations (SHAP) analysis identified $L/D$ as the most influential parameter affecting drag, followed by Mach number and $d/D$, supporting the physical interpretation of the flow behavior. Overall, the proposed framework demonstrates that the integration of CFD, machine learning, and evolutionary algorithms provides an efficient and reliable approach for geometric optimization in supersonic applications, enabling accurate design exploration with a limited number of high-fidelity simulations.

1. Introduction

Once the flow velocity exceeds the speed of sound, compressibility effects become significant and lead to the formation of shock waves, typically appearing as detached bow shocks in front of blunt bodies [1]. These shock waves induce abrupt changes in pressure and temperature, significantly increasing aerodynamic drag and generating intense thermal loads. Such conditions are especially critical in high-speed applications involving reentry vehicles and rockets, where structural integrity and aerodynamic efficiency are strongly coupled [2,3,4]. From an aerodynamic standpoint, streamlined geometries are generally preferred due to their lower drag characteristics. However, blunt-body configurations are widely employed in practical applications owing to their large internal volume and structural benefits [5]. Under supersonic conditions, the flow over a blunt body is characterized by a strong detached shock wave that propagates upstream of the body, producing substantial pressure drag and heat transfer due to gas compression at the shock front [6]. These effects not only reduce aerodynamic performance but also impose severe thermal constraints on the system.

To mitigate these drawbacks, several drag-reduction and flow-control strategies have been proposed, including spikes, opposing jets, and forward-facing cavities, among which spikes have gained considerable attention. A spike is a slender protrusion mounted at the nose of the blunt body and aligned with the flow [7]. Its primary function is to modify the shock wave structure by generating a leading oblique shock that weakens the main bow shock, effectively redistributing pressure gradients and reducing overall drag [8]. The performance of this mechanism depends on multiple parameters, including Mach number, angle of attack, spike length, and geometric proportions of the blunt body [9,10,11]. Additionally, the spike tip geometry plays a critical role in shaping the shock system, influencing both drag reduction and thermal load distribution.

Significant progress in understanding spike-induced drag reduction has been achieved through experimental investigations and high-fidelity computational fluid dynamics (CFD) simulations [12,13,14,15,16,17,18,19]. Nevertheless, these approaches are computationally expensive, particularly when exploring a wide range of geometric configurations or flow conditions. In practical design scenarios, where rapid evaluation of multiple design alternatives is required, the high computational cost of CFD becomes a limiting factor. Therefore, there is a clear need for alternative methodologies that can accelerate the aerodynamic design process while maintaining sufficient predictive accuracy.

To mitigate the computational cost associated with the CFD design process, surrogate models have been used as fast approximations that emulate complex flow solvers [20,21,22,23,24,25,26,27]. Machine Learning (ML) has enabled the possibility of developing surrogate models capable of addressing some of the limitations of traditional trial and error methods for CFD design optimization [21,22,23,24,25,26,27,28]. Surrogate models based on artificial neural networks [28,29], deep learning [28,29,30] or ensemble methods [31] have shown improved generalization capabilities and scalability, making them suitable for complex flow–geometry interactions, for example aerodynamic and Multiphysics optimization problems, although their performance remains sensitive to the quality of the training data and the strategy used to sample the design space.

In parallel with advances in surrogate modelling, the optimization of CFD-driven problems has highlighted the need for robust global optimization strategies [32,33,34,35]. Genetic algorithms (GAs) have emerged as complements to machine learning surrogate models for CFD design optimization [36,37,38].

Evolutionary algorithms, particularly Genetic Algorithms (GA), are well-suited for geometric optimization problems involving nonlinear and potentially non-smooth response surfaces. In the present study, the objective is to identify optimal geometric parameters (spike diameter and length ratios) that minimize the drag coefficient under compressible flow conditions. Although the design space is defined by a limited number of variables, the mapping between geometry and aerodynamic response is highly nonlinear due to shock wave interactions, flow separation, and recirculation phenomena. These effects can introduce local irregularities in the response surface, reducing the effectiveness of gradient-based methods, which typically require smooth and differentiable objective functions and may converge to local optima. In contrast, GA performs a global search without relying on gradient information, using stochastic operators such as selection, crossover, and mutation to efficiently explore the design space. Additionally, deterministic approaches such as grid search become inefficient as resolution increases, while surrogate-assisted optimization combined with GA significantly reduces computational cost. Therefore, Genetic Algorithms provide a robust and flexible strategy for solving the geometric inverse design problem addressed in this work, particularly under limited data availability and when the objective function is evaluated through surrogate models.

By coupling surrogate models with evolutionary algorithms, it becomes possible to evaluate candidate solutions at a reduced computational cost associated with CFD, while offering a hybrid methodology that enables global exploration of complex design spaces.

Motivated by these challenges, the present study proposes a surrogate-assisted evolutionary optimization framework tailored for small datasets, integrating machine learning-based surrogate models with evolutionary algorithms for geometry optimization of spiked blunt bodies in supersonic flow. Unlike conventional approaches that rely either on computationally expensive CFD-based optimization or data-intensive deep learning methods, the proposed framework is specifically designed to operate under limited data availability while preserving predictive accuracy and robustness.

Several machine learning techniques, including Gradient Boosting and LightGBM, are employed as surrogate models and systematically optimized through hyperparameter tuning to enhance their generalization capability. These surrogates are then coupled with multiple evolutionary algorithms—namely Genetic Algorithm (GA), Differential Evolution (DE), and Covariance Matrix Adaptation Evolution Strategy (CMA-ES)—to efficiently solve the inverse design problem of identifying geometric parameters that minimize aerodynamic drag at a fixed Mach number.

The main contributions of this work are threefold: (i) the development of a surrogate-assisted optimization framework tailored for small CFD datasets, integrating machine learning models with evolutionary algorithms; (ii) the validation of surrogate-based optimal configurations using CFD simulations, demonstrating strong agreement between predicted and simulated drag coefficients; and (iii) a comparative assessment of evolutionary algorithms under compressible flow conditions, highlighting their robustness, convergence behavior, and reliability for surrogate-assisted aerodynamic optimization.

2. Materials and Methods

The methodology is divided into three key stages: dataset generation through CFD, training and validation of ML models, and design optimization using surrogate models for a fixed condition. Figure 1 presents a roadmap of methodology used in this work.

In the next sections, CFD dataset generation is described.

2.1. Dataset Generation (CFD Process)

To generate a reliable and comprehensive dataset (Appendix A) for training the machine learning models, the continuity, momentum, and energy equations were solved using the FloWorks package integrated within SolidWorks 2023 (Dassault Systèmes, Vélizy-Villacoublay, France). These governing equations were discretized in a Cartesian framework using the Finite Volume Method (FVM), ensuring numerical stability and accuracy.

The solver employs structured Cartesian meshes, and the geometry of the blunt body was represented using a cut-cell approach [39], which allows the mesh to remain independent of the geometry while accurately capturing complex boundaries. Turbulent effects were modeled using the Reynolds-Averaged Navier–Stokes (RANS) formulation with a modified two-equation k–ε turbulence model. This model incorporates the Lam and Bremhorst damping functions, enabling the simulation of laminar, transitional, and fully turbulent flow regimes in homogeneous fluids. A detailed description of the numerical formulation of the solver can be found in the official white paper by Sobachkin and Dumnov [40].

The computational domain was defined sufficiently large to avoid boundary interference effects, extending several characteristic lengths upstream and downstream of the geometry. Supersonic inlet conditions were prescribed based on the target Mach number, while outlet boundaries allowed flow exit without reflection. The surfaces of the blunt body and spike were modeled as no-slip adiabatic walls.

The mesh resolution was selected to adequately capture shock structures and recirculation regions. Typical simulations involved on the order of 10⁵ to 10⁶ control volumes, ensuring sufficient spatial resolution for resolving the main flow features.

Convergence was assessed based on the stabilization of integral quantities, particularly the drag coefficient, as well as global mass and energy balance criteria provided by the solver. Simulations were continued until these quantities exhibited negligible variation, ensuring steady-state convergence.

The dataset is composed of the simulation outputs corresponding to the geometric configurations, including input variables as axial force, velocity, density, viscosity, speed of sound, spike diameter, and spike length, and the angle of attack, which is considered constant at α = 0°. Those input variables were transformed into dimensionless parameters to reduce the number of features and simplify the dataset. This normalization not only improves the efficiency of the machine learning model but also allows the results to be generalized across different scales. The final dataset includes the drag coefficient (Cd), Reynolds number (Re), Mach number (Ma), as well as the dimensionless geometric ratios of spike diameter (d/D) and spike length (L/D).

$$C_D=f(Ma,Re,d/D,L/D)$$

Simulations were carried out over a broad range of Mach numbers, Reynolds numbers, and geometric aspect ratios to ensure adequate coverage of the design space. The spike diameter ratio (d/D) varied from 0.06 to 0.18 in increments of 0.03, while the spike length ratio (L/D) ranged from 0.15 to 1.95 in increments of 0.3.

Table 1. Dataset range parameters.

Parameters	Description	Range	Size Step
Mach number (Ma)	Free stream Mach number	1.2–3.6	0.2
Reynolds number (Re)	Based on freestream conditions	4.7 × 105–1.2 × 106	Dependent on Ma
d/D	Spike diameter ratio	0.06–0.18	0.03
L/D	Spike length ratio	0.15–1.95	0.3
Angle	Angle of attack	0°	-

presents a summary of range values selected for this study.

The dataset used in this study consists of 239 CFD simulations using a computer (Dell Precision 7530 wkst intel core Xeon e-2176 m/memory 16 GB, One Dell Way, Round Rock, TX, USA).

Although this size may appear limited compared to typical datasets in data-driven studies, it is important to contextualize it within the framework of surrogate modeling for computational fluid dynamics. First, the dimensionality of the problem is relatively low, with only three independent input variables (Mach number, d/D, and L/D). In such cases, the number of samples required to adequately cover the design space is significantly reduced compared to high-dimensional problems. The dataset was generated using a structured parametric sampling strategy, ensuring a uniform and systematic coverage of the design space, which improves the quality of the training data and reduces the need for large sample sizes.

Second, each data point corresponds to a high-fidelity CFD simulation, which is computationally expensive. Therefore, datasets in this context are inherently limited in size. This justifies the use of the term “small dataset,” which is commonly used in surrogate-assisted optimization studies involving CFD.

Additionally, boosting-based machine learning models such as GBR and LightGBM are well-suited for small datasets, as they can capture nonlinear relationships while maintaining good generalization performance with limited data.

To further mitigate the risks associated with small datasets, cross-validation techniques were employed, and model performance was evaluated using independent test data. The consistent performance observed across validation and test sets suggests that the models do not suffer from significant overfitting.

Finally, it is important to note that the structured nature of the dataset reduces redundancy while maintaining representativeness, ensuring that the surrogate models are trained within the bounds of the design space and minimizing extrapolation during optimization.

2.1.1. CFD Validation Test

Flow simulation was employed in this study to analyze the compressible flow around a spiked blunt body. To assess the reliability of the numerical approach across a wide range of Mach numbers, a sphere was selected as a benchmark geometry, as its drag coefficient behavior from subsonic to supersonic regimes is well documented in the literature [41,42]. Both the sphere and the spiked blunt body were simulated under the same thermal conditions, as detailed in

Table 2. Thermal and transport properties.

Pressure (P, atm)	Temperature (T, K)	Viscosity (μ, Pa·s)	Density (ρ, kg/m3)	Gas Constant (Rg, kJ/kg·K)	Speed of Sound (c, m/s)
1.0	290	1.8 × 10−5	0.816	0.287	340

.

A mesh independence study was conducted using different grid configurations. First, structured orthogonal meshes with varying levels of refinement were tested, including a fine mesh (553,776 cells) and a coarse mesh (144,952 cells). In addition, an adaptive mesh with two refinement levels was evaluated. The external flow over the sphere was used as the baseline case for validation purposes.

The validation was carried out using a smooth sphere over a range of Mach numbers from 0.9 to 6.0, allowing comparison with well-established experimental and numerical results for drag coefficient. Mesh sensitivity was evaluated by comparing results obtained with coarse and fine structured grids, showing differences below 5% between them. When compared against reference data from the literature, both meshes exhibited relative errors slightly above 10%. Additionally, an adaptive mesh with two refinement levels was tested, yielding improved accuracy, with relative errors remaining below 9% across the entire Mach range.

Although the adaptive mesh provided better agreement with reference data, it required significantly higher computational time. Therefore, the fine structured mesh was selected as a suitable compromise between accuracy and computational efficiency for both validation and dataset generation (see Figure 2). Overall, the numerical results show good agreement with available experimental data.

2.1.2. Spike Blunt Configuration

Following the approach of Kalimuthu et al. [11], a flat-faced spiked blunt body configuration was analyzed, where the spike diameter (d), spike length (l), and a reference blunt body diameter (D) of 1 inch were considered (see Figure 3). For generality and scalability, these geometric parameters are expressed in dimensionless form, as described in the following section.

All simulations were conducted in a three-dimensional orthogonal Cartesian coordinate system, as shown in Figure 4, with the freestream flow aligned along the positive x-direction.

As in the validation study, a mesh sensitivity analysis was carried out using three different grid configurations: a coarse mesh (approximately 1.5 × 10⁵ cells), a fine mesh (approximately 6 × 10⁵ cells), and an adaptive mesh with two refinement levels. Based on the results of this analysis, the fine mesh was selected for all simulations, providing a suitable balance between accuracy and computational cost. This mesh configuration was therefore used to generate the drag coefficient dataset.

2.2. Machine Learning Models

To predict the drag coefficient (Cd) from geometric and flow parameters, supervised machine learning regression models were developed. In particular, Gradient Boosting Regressor (GBR) and Light Gradient Boosting Machine (LightGBM) were selected as surrogate models.

These models were chosen due to their strong performance in nonlinear regression problems under limited data availability, which is a common constraint in computational fluid dynamics (CFD), where each simulation is computationally expensive. Unlike deep learning approaches, which typically require large datasets, boosting-based ensemble methods can achieve high predictive accuracy with relatively small datasets by iteratively correcting residual errors and capturing complex interactions among input variables.

All machine learning models, and optimization algorithms were implemented in Python 3.10 to ensure reproducibility and computational efficiency. The Scikit-learn 1.3.0 (Scikit-learn) library was used to implement the Gradient Boosting Regressor, while LightGBM 4.0.0 (Microsoft) was employed for gradient boosting with decision trees.

For the optimization stage, evolutionary algorithms were implemented using the DEAP (Distributed Evolutionary Algorithms in Python) framework, while the CMA-ES algorithm was implemented using the pycma library.

2.2.1. Training Protocol

The dataset was randomly shuffled using a fixed random seed to ensure reproducibility, and then split into training (80%) and testing (20%) subsets. To prevent data leakage, all preprocessing steps, including feature scaling and normalization, were performed exclusively on the training data and subsequently applied to the test set. A 5-fold cross-validation strategy was employed during model training, ensuring that each fold preserves the distribution of Mach numbers and geometric parameters.

Given the structured nature of the dataset, special care was taken to avoid the inclusion of near-duplicate configurations across training and validation folds, which could artificially inflate model performance. Additionally, the training process was repeated multiple times with different random splits to verify the stability of the results. The low variance observed across these runs indicates that the surrogate models exhibit robust generalization within the explored design space.

2.2.2. Hyperparameter Tuning

A grid search approach (

Table 3. Hyperparameter search space used for tuning the machine learning models.

Model	Hyperparameter	Values Explored
GBR	n_estimators	50, 100, 200
GBR	learning_rate	0.01, 0.1, 0.2
GBR	max_depth	3, 5, 7
LightGBM	n_estimators	50, 100, 200
LightGBM	num_leaves	20, 31, 50
LightGBM	learning_rate	0.01, 0.1, 0.2
LightGBM	max_depth	3, 5, 7

) was employed to optimize the hyperparameters of the machine learning models. The tuning process was integrated within the cross-validation framework described in Section 2.2.1 to ensure consistent and unbiased model evaluation. For the Gradient Boosting Regressor (GBR), the number of estimators, learning rate, and maximum depth were tuned. For LightGBM, the number of leaves, learning rate, and maximum depth were optimized. The best hyperparameters were selected according to the highest cross-validated R² score.

2.2.3. Model Evaluation Metrics

The performance of the prediction model was evaluated using the coefficient of determination ($R^2$) and the root mean square error (RMSE). The R² and RMSE were calculated as follows:

$$R^2=1-{\displaystyle \frac{{\sum }_1^N{\left(y_p-y_0\right)}^2}{{\sum }_1^N{\left(\overline{y}-y_0\right)}^2}}$$

$$RSME=\sqrt{{\displaystyle \frac{{\sum }_1^N{\left(y_p-y_0\right)}^2}{N}}}$$

where $N$ is the sample size, $y_p$ is the predicted value, $y_0$ is the test observation, and $\overline{y}$ is the average of $y_0$.

In next sections a brief description of machine learning methods selected for this work.

2.2.4. Gradient Boosting Regressor (GBR)

Gradient boosting regressor (GBR) is a sequential ensemble learning method that constructs a predictive model by combining multiple weak learners [43,44,45,46], typically decision trees. In this approach, the prediction at iteration m, denoted as $F_m\left(x\right)$, is obtained by adding a new tree $h_m\left(x\right)$ to the previous ensemble:

$$F_m\left(x\right)=F_{m-1}\left(x\right)+\eta h_m\left(x\right)$$

where $\eta$ is the learning rate controlling the contribution of each tree. The weak learner $h_m\left(x\right)$ is trained to fit negative gradient of the loss function $L(y,F\left(x\right))$ with respect to the current predictions:

$$r_i^{\left(m\right)}={\left[{\displaystyle \frac{\partial L(y_i, F(x_i\left)\right)}{\partial F\left(x_i\right)}}\right]}_{F\left(x\right)=F_{m-1\left(x\right)}}$$

$$h_m(x)\approx {argmin}_h\sum _{i=1}^N{\left(r_i^{\left(m\right)}-h\left(x_i\right)\right)}^2$$

The final ensemble prediction after M iterations is expressed as:

$$\widehat{y}\left(x\right)=F_M\left(x\right)=F_0\left(x\right)+\sum _{m=1}^M\eta h_m\left(x\right)$$

where $F_0\left(x\right)$ is the initial prediction, often the mean of the target variable in regression tasks. GBR is highly effective at modelling complex, non-linear relationships and achieves proper performance when hyperparameters—such as the number of estimators M, learning rate $\eta$, and maximum tree depth—are properly tuned. This makes it particularly suitable for regression tasks involving structured tabular data.

2.2.5. Light Gradient Boosting Regressor (LGBR)

LightGBM is a gradient boosting framework optimized for high efficiency and scalability [47,48], particularly in large-scale machine learning tasks. Like traditional Gradient Boosting, it constructs an additive model of M weak learners (decision trees), where the prediction at iteration m is updated as:

$$F_m\left(x\right)=F_{m-1}\left(x\right)+\eta h_m\left(x\right)$$

with $\eta$ representing the learning rate and $h_m\left(x\right)$ the newly fitted tree. LightGBM employs a histogram-based algorithm that discretizes continuous feature values into $k$ bins, reducing memory usage and accelerating the computation of split gains. The optimal split for a feature $j$ is determined by maximizing the information gain:

$$Gain\left(j,s\right)={\displaystyle \frac{G_L^2}{H_L+\lambda }}+{\displaystyle \frac{G_R^2}{H_R+\lambda }}-{\displaystyle \frac{{(G_L+G_R)}^2}{H_L+H_R+\lambda }}$$

where $G$ and $H$ are the first- and second-order gradients of the loss function with respect to the predictions, $\lambda$ is the $L_2$ regularization parameter, and $\gamma$ controls the minimum split gain.

A key innovation of LightGBM is its leaf-wise growth strategy, where each tree expands by splitting the leaf that yields the highest gain, rather than a level-wise as in conventional gradient boosting. While this strategy enhances training efficiency and model expressiveness, it may increase the risk of overfitting on small datasets. Proper tuning of hyperparameters—such as maximum depth, number of leaves, and learning rate—ensures that LightGBM achieves predictive performance comparable to or exceeding traditional gradient boosting methods while offering significantly faster training for large and high-dimensional datasets.

2.3. Evolutionary Algorithms

Although the optimization problem considered in this study involves a relatively low-dimensional design space (two geometric variables: d/D and L/D), evolutionary algorithms were selected due to their suitability for nonlinear and potentially non-smooth objective functions.

Under compressible flow conditions, the relationship between geometric parameters and drag coefficient may exhibit complex behavior due to shock interactions, flow separation, and recirculation regions. These effects can introduce local irregularities in the response surface, reducing the effectiveness of gradient-based methods, which typically require smooth and differentiable objective functions and may converge to local optima.

Furthermore, this work aims not only to identify optimal configurations within a predefined design space, but also to establish a surrogate-assisted optimization framework that can be extended to higher-dimensional problems. In such scenarios, grid search methods become computationally prohibitive due to the exponential growth of the search space (curse of dimensionality).

Even in the present low-dimensional case, grid search becomes inefficient when a fine resolution is required to accurately capture the nonlinear behavior of the response surface. Increasing the discretization density significantly raises the number of required evaluations, particularly when multiple Mach regimes are considered. Moreover, grid-based approaches are not well suited for handling non-smooth objective functions, which may arise due to shock interactions and flow discontinuities.

In contrast, evolutionary algorithms enable an efficient exploration of the design space and remain robust in the presence of irregularities and surrogate modeling errors. These characteristics make them particularly suitable for optimization based on machine learning predictions rather than direct CFD evaluations.

To support this approach, multiple evolutionary algorithms were considered in this work to evaluate their consistency and performance under different flow conditions.

Once the surrogate model was trained and validated, it was used to solve the inverse design problem, which consists of identifying the optimal pair of geometric parameters that minimizes the predicted drag coefficient for a fixed Mach number. The optimization problem can be formally defined as:

$$mi{n}_{ d/D, L/D} Cd\left(d/D,L/D,Ma\right)$$

Subject to:

$$\begin{gathered} 0.06\le d/D\le 0.18 \\ 0.15\le L/D\le 1.95 \end{gathered}$$

where the Mach number (Ma) is treated as a fixed parameter during each optimization run. The objective function is evaluated using the surrogate model trained on CFD data, ensuring a computationally efficient evaluation of candidate solutions within a bounded design space.

Given the nonlinear nature of the problem, three evolutionary algorithms were implemented: Genetic Algorithm (GA), Differential Evolution (DE), and Covariance Matrix Adaptation Evolution Strategy (CMA-ES). In all cases, the surrogate model was embedded as the fitness function, allowing candidate solutions to be iteratively evaluated and refined across generations.

The performance of each algorithm was assessed in terms of convergence behavior, optimal drag coefficient values, and computational cost.

The evolutionary algorithms were implemented using standard configurations to ensure a balance between exploration capability and computational efficiency. For the Genetic Algorithm (GA), a population size of 50 individuals and 100 generations were used, with a crossover probability of 0.8 and a mutation probability of 0.1. Differential Evolution (DE) was configured with a population size of 50, a scaling factor F = 0.5, and a crossover probability CR = 0.9. The CMA-ES algorithm was initialized with a population size of 50 and default adaptation parameters.

The number of generations was selected based on convergence analysis, ensuring that the optimization process reached stable solutions without significant improvements in subsequent iterations.

2.3.1. Genetic Algorithms (GA)

The Genetic Algorithm (GA) is a population-based stochastic optimization method inspired by the principles of natural selection and biological evolution [49,50]. A GA operates on a population of P candidate solutions (chromosomes) $x_i^{\left(t\right)}\in R^n$ at generation $t_i$, where each candidate represents a potential solution to the optimization problem. Everyone is evaluated using a fitness function $f\left(x\right)$, which quantifies the quality of the solution:

$$f_i^{\left(t\right)}=f\left(x_i^{\left(t\right)}\right), i=\mathrm{1,2},\dots \dots .,P$$

The evolutionary cycle of GA comprises three main operators:

• Selection: Individuals with higher fitness are probabilistically favored for reproduction. A common approach is the roulette-wheel selection, where the probability of selecting an individual is proportional to its fitness:

$$P_i^{\left(t\right)}={\displaystyle \frac{f_i^{\left(t\right)}}{{\sum }_{j=1}^P{f}_j^{\left(t\right)}}}$$

• Crossover (Recombination): Two parent solutions $x_p$ and $x_q$ are combined to produce offspring $x_{child}$. A simple arithmetic crossover for continuous variables is:

$$x_{child}={\alpha x}_p+\left(1-\alpha \right)x_q, \alpha \in \left[0,1\right]$$

• Mutation: A small random perturbation is applied to offspring to maintain diversity and explore new regions of the search space:

$$x_{mutated}=x_{child}+ϵ, ϵ~\mathbb{N}(0,{\sigma }^2)$$

After generating a new population through these operations, the process iterates for generations or until a convergence criterion is met. The best solution at generation t is tracked as:

$$x*=\arg {max}_{x_i^{\left(t\right)}}f\left(x_i^{\left(t\right)}\right)$$

Through successive generations, GA converges towards optimal or near-optimal solutions, making it a flexible and widely adopted approach for solving complex, multi-modal, and non-differentiable optimization problems in engineering and design applications.

2.3.2. Differential Evolution (DE)

Differential Evolution (DE) is a robust and efficient population-based evolutionary algorithm designed for solving continuous optimization problems [51,52]. It operates through the differential mutation of individuals and combines exploration and exploitation effectively.

Each generation g, for each target vector $x_i^{\left(g\right)} \in {\mathbb{R}}^d$ in the population, a donor vector $v_i^{\left(g\right)}$ is generated using the differential mutation rule:

$$v_i^{\left(g\right)}=x_{r1}^{\left(g\right)}+F(x_{r1}^g-x_{r3}^{\left(g\right)})$$

where:

• $x_{r1}^{\left(g\right)}, x_{r2}^{\left(g\right)}, x_{r3}^{\left(g\right)}$ are three distinct vectors randomly selected from the current population, different from $x_i^{\left(g\right)}$.
• $F\in \left[0, 2\right]$ is scaling factor controlling the amplification of the differential variation.

Next, crossover is applied to produce a trial vector $u_o^g$ binomial crossover:

$$u_{ij}^{\left(g\right)}=\left\{\begin{array}{cc}{v}_{ij}^{\left(g\right)}\hfill & \hfill if {rand}_i\le C_r or j=j_{rand}\\ x_{ij}^{\left(g\right)}\hfill & \hfill otherwise\end{array}\right.$$

where:

• $C_r \in \left[0, 1\right]$ is the crossover probability,
• $j_{rand}$ is the randomly chosen index to ensure that at least one component comes from the donor vector.

Finally, selection is applied based on the fitness function $f\left(·\right):$

$$x_i^{(g+1)}=\left\{\begin{array}{c}{u}_i^{\left(g\right)} if f\left(u_i^{\left(g\right)}\right)\le f\left(x_i^{\left(g\right)}\right)\\ x_i^{\left(g\right)} otherwise\end{array}\right.$$

This process repeats until a stopping criterion is met (e.g., a maximum number of generations or convergence threshold). Due to its few control parameters and strong global search capabilities, DE is well-suited for real-valued, non-linear, and non-convex optimization problems with limited computational budgets.

2.3.3. Covariance Matrix Adaptation Evolution Strategy (CMA-ES)

Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a population-based evolutionary strategy that focuses on adapting the search distribution to solve continuous optimization problems [53,54]. The main idea is to sample candidate solutions from a multivariate normal distribution, whose mean and covariance matrix are iteratively updated to reflect the most promising search directions.

• Initialization
  The algorithm starts by defining an initial distribution:

  ○

  Initial mean: $m^{\left(0\right)}\in {\mathbb{R}}^n$

  ○

  Initial standard deviation: ${\sigma }^{\left(0\right)}>0$.

  ○

  Initial covariance matrix: $C^{\left(0\right)}=I_n\left(Identity matrix\right)$.

  ○

  Population size: λ

• Sampling of individuals

At generation $g, \lambda$ candidate solutions are sampled as:

$$x_k^{\left(g\right)}=m^{\left(g\right)}+{\sigma }^{\left(g\right)}·N_k^{\left(g\right)}, where N_k^{\left(g\right)}~\mathbb{N}(0, C^{\left(g\right)})$$

That is, each individual is sampled from a multivariate Gaussian distribution centered at the current mean $m^{\left(g\right)}$.

• Evaluation and selection

Each candidate solution is evaluated using the objective function $f\left(x_k^{\left(g\right)}\right)$, and the top $\mu$ individuals with the best fitness values are selected:

$$x_{1:\lambda }^{\left(g\right)}, x_{2:\lambda }^{\left(g\right)}\dots \dots \dots \dots ..x_{\mu :\lambda }^{\left(g\right)}$$

• Update of the mean

The new mean is computed as a weighted sum of the selected top individual:

$$m^{(g+1)}=\sum _{i=1}^{\mu }{\omega }_i·x_{i:\lambda }^{\left(g\right)}, where \sum {\omega }_i=1$$

The weighs ${\mathsf{\omega }}_i$ typically satisfy ${\omega }_1>{\omega }_2>\cdots >{\omega }_{\mu }$, assigning more importance to better-performing individuals.

• Evolution paths
  Two evolution paths are updated:

  ○

  Step-size control path: $p_{\sigma }$

  ○

  Covariance matrix adaptation path: $p_C$

The paths help accumulate information about the direction and magnitude of the search over generations and influence subsequent adaptation steps.

• Covariance matrix update

The covariance matrix is updated to capture the shape and orientation of search landscape:

$$C^{\left(g+1\right)}=\left(1+c_1+c_{\mu }\right)·C^{\left(g\right)}+c_1·p_c^{\left(g+1\right)}({p_c^{\left(g+1\right)})}^T+C_{\mu }\sum _{i=1}^{\mu }{\omega }_i·y_i^{\left(g\right)}{\left(y_i^{\left(g\right)}\right)}^T$$

where

$$y_i^{\left(g\right)}={\displaystyle \frac{x_{i:\lambda }^{\left(g\right)}-m^{\left(g\right)}}{{\sigma }^{\left(g\right)}}}$$

This update combines the rank-one and rank-$\mu$ updates based on evolution paths and individual contributions.

• Step size adaptation

The global step size $\sigma$ is adapted based on the length of the evolution path $p_{\sigma }$:

$${\sigma }^{(g+1)}={\sigma }^{\left(g\right)}·exp\left({\displaystyle \frac{c_{\sigma }}{d_{\sigma }}}\left({\displaystyle \frac{‖p_{\sigma }^{(g+1)}‖}{E\left[‖N\left(\mathrm{0,1}\right)‖\right]}}-1\right)\right)$$

This mechanism allows dynamic control of the search radius, promoting global exploration or local refinement depending on the search progress.

3. Results and Discussion

To assess the simulation performance and ensure that the relevant flow physics are properly captured, Figure 5 illustrates the shock wave structure using a numerical Schlieren approach for three different Mach numbers and for different $L/D$, $d/D$ geometric configurations: (a) Ma = 1.6 $L/D$ = 1.8733, $d/D$ = 0.1734; (b) Ma = 2.4 $L/D$ = 1.39, $d/D$ = 0.1722; and (c) Ma = 3.2, $L/D$ = 1.3362, $d/D$ = 0.1726. This visualization is based on contours of the magnitude of the density gradient (|∇ρ|) [55]. The results reveal the overall flow behavior, showing how the strong bow shock typically observed in blunt bodies is transformed into a series of weaker oblique shock waves. In Figure 5a–c, characteristic flow features such as recirculation and reattachment regions can be clearly identified, along with the presence of a reattachment shock wave, which is commonly observed in spiked blunt bodies operating at high Mach numbers [56]. In addition, the angle of the oblique shock waves decreases as the Mach number increases. Larger L/D ratios tend to increase the stand-off distance of the shock system and enlarge the recirculation zone, which may contribute to drag reduction. Capturing these shock structures supports the reliability of the CFD tool for predictive purposes.

This section presents the results of simulations for the drag coefficient of different $L/D$, $d/D$ and fixed Mach numbersas well. Figure 6 shows heatmaps of the aerodynamic drag coefficient ($Cd$) at three different Mach numbers (1.6, 2.4, and 3.2) while varying the geometric ratios $L/D$ and $d/D$. Cd values are color-mapped across a matrix of spike length ratios ($L/D$, y-axis) and spike diameter ratios ($d/D$, x-axis).

For a fixed Mach number, there is a clear tendency for the drag coefficient ($C_d$) to decrease as the spike length ratio $L/D$ increases. Longer spikes, corresponding to larger $L/D$ values, consistently show lower drag, particularly for intermediate and high spike diameter ratios $d/D$. The influence of the spike diameter ratio, however, appears to be more subtle. Although its effect is less pronounced than that of $L/D$, very small or very large diameter ratios do not consistently yield the lowest $C_d$ values. Instead, optimal aerodynamic configurations tend to occur within an intermediate range of d/D (approximately 0.09–0.12) values in the discrete CFD dataset. However, the surrogate-assisted optimization results presented later indicate that the true optimum lies at slightly higher values (around d/D ≈ 0.17), suggesting that the heatmap resolution may not fully capture the exact optimal region.

The overall trend indicates a progressive reduction in Cd values as the Mach number increases, which is consistent with the literature. These findings align with prior research [9,10,12,13,18,19] on spike effectiveness in shock wave manipulation and drag reduction for blunt bodies.

This visualization supports that increasing the spike length is more effective than increasing the diameter in reducing drag, and that optimal performance shifts slightly depending on the Mach regime.

3.1. Machine Learning Lodels Performance

Table 4. Performance comparison between optimized machine learning models.

Model	CV R2 (Mean)	Test R2	Test RMSE	Optimized Parameters
Gradient Boosting	0.8553	0.8909	0.00775	n_estimators = 100 max_depth = 3 learning_rate = 0.2
Light GBM	0.7252	0.8459	0.00922	n_estimators = 100 max_depth = 3 learning_rate = 0.1 num_leaves = 20

presents the results of the optimized Gradient Boosting Regressor (GBR), which yielded the highest performance metrics, with a cross-validated R² of 0.8553, a test R² of 0.8909, and a test RMSE of 0.00775. These results were obtained after tuning key hyperparameters using GridSearchCV, with the best configuration identified as: 100 estimators, maximum tree depth of 3, and a learning rate of 0.2. The relatively shallow tree depth helped prevent overfitting, while the moderately high learning rate ensured fast convergence.

In the case of the LGBM model, while also demonstrating good generalization capability, it showed slightly lower performance with a test R² of 0.8459 and RMSE of 0.00922. The optimal hyperparameters included 100 estimators, maximum depth of 3, learning rate of 0.1, and 20 leaves per tree. Although LGBM benefits from its histogram-based training and leaf-wise tree growth—enabling faster training and scalability—these features may become limiting when the dataset is small, and the numerical precision of the features is critical. The discretization process inherent to histogram-based algorithms can potentially reduce the model’s sensitivity to subtle variations in the input space.

Figure 7 presents a comparison between the predicted and CFD-generated values of the drag coefficient ($Cd$) for the evaluated surrogate models, including Gradient Boosting Regressor (GBR), LightGBM (LGBM), and Kriging. The scatter points represent the predictions of each model plotted against the corresponding CFD values, while the dashed line denotes the ideal one-to-one agreement. All models exhibit a strong correlation with the CFD data, indicating their capability to capture the underlying nonlinear relationships of the problem.

Among the evaluated approaches, the Gradient Boosting model shows the highest predictive accuracy, with data points more closely clustered around the ideal line. LightGBM provides comparable performance with slightly larger dispersion, while the Kriging model exhibits a broader spread of predictions, suggesting reduced accuracy for this dataset.

In previous work by the authors using the same CFD dataset, Kriging-based surrogate models were implemented independently for each Mach number, requiring the construction of multiple local models to represent the aerodynamic behavior across different flow conditions [57]. In contrast, machine learning approaches such as Gradient Boosting and k-nearest neighbors were formulated as global models, incorporating Mach number as an input feature and enabling a unified representation of the drag coefficient across all regimes.

This distinction has important implications for scalability and generalization. While Kriging provides accurate local approximations, its need for separate models limits its ability to capture cross-regime interactions and increases computational complexity. In contrast, global machine learning models can learn the coupled effects of geometric parameters and flow conditions within a single framework, making them more suitable for surrogate-assisted optimization involving multiple operating regimes.

Table 5. Prediction error metrics of the surrogate model across different Mach number regimes.

Mach	Mean Relative Error (%)	Max Relative Error (%)	RMSE
1.2	1.021	2.68	0.0040
1.6	2.075	6.40	0.0079
2.0	2.7091	4.70	0.0091
2.4	2.1463	4.54	0.0074
2.8	2.5343	5.70	0.0078
3.2	3.2370	5.53	0.0089
3.6	3.23	5.47	0.0102

presents the prediction error metrics of the surrogate model across different Mach number regimes. The results show that the model maintains consistent accuracy throughout the explored range, with mean relative errors generally between 1% and 3.2% and RMSE values below 0.01.

At lower Mach numbers, the model achieves high accuracy, with a mean relative error close to 1% at Ma = 1.2, although slightly higher maximum errors indicate localized prediction variability. In the intermediate Mach range (2.0–2.8), the model exhibits stable and uniform performance, suggesting that the underlying relationships between geometric parameters and drag are well captured.

At higher Mach numbers (3.2–3.6), a moderate increase in prediction error is observed, which may be attributed to increased flow complexity or reduced data density in this region. However, the overall error levels remain within acceptable bounds, indicating that the surrogate model preserves its generalization capability across all regimes.

These results confirm that the model does not overfit specific regions of the dataset and provides reliable predictions across the entire design space considered in this study.

Figure 8 presents the residual distribution as a function of Mach number. The residuals are distributed around zero across the entire range, with both positive and negative values, indicating the absence of systematic bias in the model predictions. Additionally, no clear trend is observed with respect to Mach number, suggesting that the model maintains consistent accuracy across different flow regimes without degradation at higher Mach numbers.

Although slightly higher errors are observed in intermediate Mach regimes, the overall dispersion remains relatively uniform. This behavior confirms that the model does not overfit specific regions of the dataset and can provide reliable predictions throughout the explored design space.

In summary, the GBR model demonstrated superior accuracy and consistency in comparison with LGBM. LGBM is an excellent alternative for cases where computational efficiency is prioritized, although some predictive accuracy may be sacrificed.

3.2. Optimization Models Performance vs. CFD

Once the best-performing machine learning model was selected, the performance of the evolutionary algorithms was evaluated under fixed Mach number conditions. For each Mach number within the supersonic range (1.2–3.6), the optimization problem was formulated to minimize the drag coefficient $C_d$ by adjusting the geometric parameters $d/D$ and $L/D$, within the bounds $d/D\in [0.06, 0.18]$ and $L/D\in [0.15, 1.95]$.

The Gradient Boosting Regressor was used as the surrogate model, and three evolutionary algorithms—Differential Evolution (DE), Covariance Matrix Adaptation Evolution Strategy (CMA-ES), and Genetic Algorithm (GA)—were employed to identify the optimal geometric configurations for each flow condition.

To validate the reliability of the surrogate-assisted optimization framework, the optimal configurations obtained from each evolutionary algorithm were re-evaluated using high-fidelity CFD simulations. Figure 9 compares the predicted minimum drag coefficient values with those obtained from CFD for each Mach number.

As observed, a strong agreement between the surrogate predictions and CFD results is achieved across all flow regimes. The discrepancies remain relatively small, confirming that the surrogate model provides accurate estimates of the drag coefficient even at the optimized design points. The maximum deviation between surrogate predictions and CFD results remains within an acceptable range (below approximately 5%), further supporting the robustness of the proposed optimization framework.

Although minor deviations are observed at intermediate Mach numbers, the overall trend is consistently captured, demonstrating that the optimization process does not exploit poorly represented regions of the surrogate model. These results validate the capability of the proposed framework to identify physically meaningful and reliable optimal configurations.

The results show a clear decreasing trend in the minimum drag coefficient as the Mach number increases, indicating a higher effectiveness of the spike in supersonic regimes. While CMA-ES tends to predict slightly lower drag values in some cases, the CFD validation reveals that Differential Evolution and Genetic Algorithm provide more consistent and reliable optimal solutions across all Mach numbers. This suggests that although CMA-ES may achieve lower surrogate-based predictions, these do not always translate into improved performance when evaluated using high-fidelity CFD simulations.

To quantitatively assess the effectiveness of the optimization strategy, a comparative analysis was performed between evolutionary algorithms and a grid search baseline using the surrogate model as the objective function. For each Mach number, the optimization problem was defined over the bounded design space of the geometric parameters d/D ∈ [0.06, 0.18] and L/D ∈ [0.15, 1.95]. The grid search approach discretized the design space using a uniform resolution of 50 × 50 points, resulting in 2500 evaluations per Mach condition. For each parameter combination, the drag coefficient was predicted using the trained surrogate model, and the global minimum was identified. In parallel, three evolutionary algorithms—Differential Evolution (DE), Genetic Algorithm (GA), and Covariance Matrix Adaptation Evolution Strategy (CMA-ES)—were applied to the same optimization problem using identical bounds and the surrogate model as the fitness function, as shown in Figure 10.

The results show that, although grid search can identify configurations with competitive drag values, it requires a significantly larger number of evaluations due to the exhaustive sampling of the design space. In contrast, evolutionary algorithms explore the search space adaptively, reducing the number of required evaluations. Among the tested methods, CMA-ES exhibits the fastest convergence in terms of function evaluations; however, it tends to reach slightly suboptimal solutions. Differential Evolution requires a higher number of evaluations than CMA-ES but consistently identifies the global optimum across all Mach regimes, while the Genetic Algorithm achieves similar solutions at a higher computational cost. These findings highlight a trade-off between convergence speed and solution accuracy and demonstrate that evolutionary algorithms provide a more efficient and scalable optimization strategy compared to grid-based approaches, even in low-dimensional problems. This reduction in evaluations is particularly relevant when each function evaluation corresponds to a CFD simulation.

On the other hand, a notable observation from Figure 11 is the strong convergence of the optimal diameter ratio d/D to a narrow range across all Mach numbers. In particular, the solutions obtained with Differential Evolution consistently cluster around approximately 0.17, indicating the presence of a well-defined optimal geometric configuration for spike thickness. The Genetic Algorithm (DEAP) shows a very similar trend, with only minor variations, while CMA-ES systematically predicts slightly lower values of d/D, around 0.14–0.15. Despite these differences, all algorithms converge within a relatively narrow band, reinforcing the existence of a stable and robust optimal region rather than a highly sensitive design parameter.

In contrast, the optimal spike length ratio L/D exhibits a clear decreasing trend as the Mach number increases. This behavior suggests that shorter spikes become more effective at higher flow velocities, likely due to changes in the shock structure and the associated recirculation region, which require less geometric extension to achieve drag reduction. The close agreement between Differential Evolution and Genetic Algorithm further supports the robustness of this trend across different optimization strategies.

Additionally, the results indicate that multiple geometric configurations can yield similar minimum drag coefficients, particularly for Differential Evolution and Genetic Algorithm. Even when slight differences in d/D and L/D are observed, the resulting drag values remain nearly identical. This suggests that the optimization landscape is characterized by a region of near-optimal solutions rather than a single isolated optimum, providing flexibility in geometric design while maintaining aerodynamic performance.

To further analyze the behavior of the surrogate model, a SHAP 0.44.0 (SHapley Additive exPlanations) analysis was performed, as shown in Figure 12 [58,59]. Since the surrogate model was trained using Mach number, d/D, and L/D as input features, the SHAP values represent the global feature importance across the entire dataset rather than the conditional importance within the fixed-Mach optimization problem. The results indicate that the spike length ratio L/D has the strongest overall influence on the predicted drag coefficient, followed by the Mach number, while the diameter ratio d/D exhibits a comparatively smaller contribution. However, it is important to emphasize that Mach is treated as a fixed parameter during the optimization process; therefore, its relative importance in Figure 12 should not be interpreted as part of the decision variables in the optimization stage. Instead, the SHAP analysis provides complementary insight into the global behavior of the surrogate model, whereas the geometric trends observed in the optimization results more accurately reflect the influence of d/D and L/D under fixed flow conditions. Therefore, the optimization results provide a more accurate representation of the relative importance of geometric parameters under fixed Mach conditions.

These findings are consistent with the previously observed trends, where $L/D$ exhibits a clear dependence on Mach number and plays a dominant role in drag reduction, whereas $d/D$ remains within a relatively narrow range and acts as a secondary tuning parameter. This agreement between the SHAP analysis and the optimization results provides additional confidence in the surrogate model and supports the physical consistency of the proposed methodology.

In summary, the dominant contribution of the $L/D$ ratio indicates that spike length plays a critical role in modifying shock structures and flow separation patterns, directly influencing drag reduction. In contrast, the $d/D$ ratio acts as a secondary tuning parameter, refining the aerodynamic response without significantly altering the overall flow behavior.

4. Conclusions

This study presented a surrogate-assisted optimization framework that integrates computational fluid dynamics (CFD), machine learning, and evolutionary algorithms to investigate geometric drag reduction mechanisms in spiked blunt bodies under supersonic flow conditions. The results show a clear decreasing trend in the drag coefficient ($Cd$) as the Mach number increases, indicating an enhanced effectiveness of spike configurations in higher-speed regimes.

The optimization process revealed consistent geometric trends, where the spike length ratio L/D decreases with increasing Mach number, suggesting that shorter spikes are sufficient to effectively modify the shock structure at higher flow velocities. In contrast, the diameter ratio d/D converges to a narrow range around approximately 0.17, indicating the presence of a well-defined optimal geometric configuration that remains relatively stable across different flow conditions. These findings are consistent with the observed flow behavior, where the spike alters the shock system and reduces pressure drag through flow separation and recirculation effects.

The machine learning models demonstrated strong predictive capabilities for approximating the aerodynamic response. More importantly, the optimal configurations obtained through surrogate-based optimization were validated using CFD simulations. The good agreement between predicted and simulated drag coefficients confirms that the surrogate models are not only accurate but also reliable for guiding geometric optimization in compressible flow conditions. This result is particularly relevant given the limited dataset size, demonstrating that properly trained ensemble-based models can achieve robust performance even under data-constrained scenarios.

In addition, previous evaluations using the same dataset showed that Kriging-based surrogate models require the construction of independent models for each Mach number, limiting their ability to generalize across flow regimes. In contrast, the machine learning models employed in this study were formulated as global functions of Mach number and geometric parameters, enabling a unified representation of the aerodynamic response and improving scalability for multi-condition optimization problems.

In terms of optimization performance, Differential Evolution (DE) and Genetic Algorithm (GA) consistently achieved accurate solutions and exhibited robust behaviour across all Mach numbers. Although CMA-ES exhibited faster convergence in terms of function evaluations, it consistently converged to slightly suboptimal solutions when compared to Differential Evolution and Genetic Algorithm, indicating a trade-off between convergence speed and solution accuracy. Additionally, the presence of multiple geometric configurations yielding similar drag values indicates that the optimization landscape contains a region of near-optimal solutions rather than a single global optimum, providing flexibility in design selection.

A quantitative comparison with a grid-search baseline further demonstrated the efficiency of evolutionary algorithms. While grid search was able to identify the same optimal solutions, it required a significantly larger number of evaluations due to exhaustive sampling of the design space. In contrast, evolutionary algorithms achieved comparable results through adaptive exploration, highlighting their advantage in terms of computational efficiency and scalability, even in low-dimensional problems.

Furthermore, the SHAP analysis provided additional insight into the surrogate model by identifying the spike length ratio L/D as the most influential parameter affecting the drag coefficient, followed by the Mach number and the diameter ratio d/D. This result is consistent with the observed optimization trends and reinforces the physical interpretation that spike length plays a dominant role in modifying the shock structure, while the diameter acts as a secondary tuning parameter.

Overall, this work demonstrates that the integration of CFD, machine learning, and evolutionary optimization constitutes a reliable and computationally efficient framework for geometric design under supersonic flow conditions. The proposed approach not only achieves accurate predictions and validated optimal configurations, but also provides a scalable methodology that can be extended to more complex design problems. In particular, the combination of global surrogate models and adaptive optimization strategies enables efficient exploration of nonlinear design spaces, even under limited data conditions.

Future work will focus on extending the methodology to higher-dimensional design spaces, incorporating additional geometric variables, and exploring more complex flow conditions, including transient and three-dimensional effects, to further generalize the proposed framework.