Robust Optimization Design of the Aerodynamic Shape and External Ballistics of a Pulse Trajectory Correction Projectile

Abstract

To improve the tactical and technical performance of pulse correction projectiles while maintaining stability in uncertain conditions and considering practical engineering constraints, this study performs a multi-objective robust optimization design of the aerodynamic shape and external ballistics of a projectile. The study utilizes an aerodynamic force engineering algorithm and numerical trajectory calculations to obtain the projectile’s performance responses within the Latin hypercube design space. To enhance optimization efficiency, a stochastic Kriging surrogate model is established to capture the inherent uncertainty of limited input data. Ultimately, a Pareto optimal solution for the projectile is obtained using a non-dominated sorting multi-objective sparrow search algorithm. The results of this study demonstrate that the consideration of design uncertainty in the robust optimization of pulse correction projectiles leads to significant enhancements in both lateral correction ability and range while satisfying flight stability requirements. Moreover, when compared to deterministic optimization, the performance variability of the design is markedly improved. This research methodology provides valuable insights for optimizing the performance of pulse correction projectiles.

1. Introduction

Trajectory correction projectiles have gained significant attention both domestically and abroad as a crucial approach for achieving low-cost precision upgrades to traditional standard artillery shells [1]. The main types of trajectory correction projectiles include drag increasing correction projectiles, pulse correction projectiles [2,3], canard correction projectiles [4], and metamorphic center correction projectiles. These projectiles offer several advantages, such as achieving long-range suppression, enhancing shooting accuracy, efficiently destroying targets, and reducing collateral damage [5]. Modular, low-cost trajectory correction mechanisms can help in maximizing the utilization of the existing inventory of standard ammunition while satisfying the necessary requirements for precision targeting. By utilizing simple detection and control units, these mechanisms can considerably reduce costs while still achieving the desired precision. Currently, both domestic and international scholars primarily focus on the research of dynamic modeling [6], aerodynamic shape, ballistics optimization design [7,8,9,10], and control strategies [11] pertaining to two-dimensional trajectory correction projectiles. Pulse trajectory correction projectiles, in comparison to other correction projectile types, offer several advantages, including shorter response time, streamlined and efficient control, and reduced cost. However, due to the limited design space and various engineering constraints involved in the modification of standardized ammunition, the corrective mechanisms have limited capacity for improvement. As a result, the current trajectory correction projectile with lateral pulses has limited corrective ability and efficiency. Furthermore, given that pulse correction projectiles generally lack attitude stability control systems, it becomes increasingly crucial to enhance the tactical performance of the projectile while ensuring optimal flight stability. Hence, selecting appropriate modeling and design methods is of paramount importance to establish the mapping relationship between design variables and to target performance efficiently and accurately. This will help in reducing the design cycle, enhancing the robustness of the design solution, and ensuring that the pulse-controlled correction projectile achieves the desired correction capability while maintaining stable flight.

The comprehensive optimization of aerodynamic layout and external ballistics design has been primarily focused on unguided projectiles in both domestic and international research. Only a few studies have been conducted on guided projectiles, with most of them focusing on canard-controlled trajectory correction projectiles. Chang SJ [12] and Fowler [13,14] are among the researchers who have studied the comprehensive optimization of aerodynamic layout and controlled trajectory for canard-controlled correction projectiles. They have utilized engineering algorithms to efficiently obtain the aerodynamic parameters of the projectile, addressing the issue of high computational costs associated with aerodynamic calculations during the design phase. However, neither of them has considered the issue of maintaining stable performance of the design results when there are small fluctuations in aerodynamic coefficients due to the lack of precision in engineering algorithms during the design process. Previous research on pulse-controlled correction projectiles has predominantly concentrated on optimizing the local parameters of the pulse correction mechanism. However, optimizing solely the local parameters of the correction mechanism cannot guarantee optimal overall performance of the ballistic [15,16]. In references [4,5], the parameters of the pulse correction mechanism were used as design variables, and particle swarm optimization and ant colony optimization algorithms were employed, respectively, to minimize energy consumption and maximize the lateral correction capability for the controlled projectile trajectory design. Nevertheless, the design process relied on the assumption of ideal conditions and did not consider the engineering constraints arising from the limited design space for the correction mechanism parameters.

In recent years, various approximate modeling methods, such as radial basis function neural networks, back propagation neural networks, and support vector machines, have been widely applied to reduce computational costs in the field of aerospace [17,18] and intelligent munitions design [19,20,21], becoming important technical means for multidisciplinary analysis and optimization. Among them, the Kriging surrogate model has received more attention because of its outstanding non-linear approximation modeling ability in medium- to low-dimensional problems when compared to other methods [22]. For uncertain design problems, a stochastic Kriging (SK) surrogate model based on the prototype of deterministic Kriging (DK) can be established. By effectively predicting the mean and variance of the performance, the SK model can significantly reduce computational costs while ensuring a certain level of analysis accuracy. The robustness advantage is also more apparent [23]. However, there have been no reported studies on the integrated optimal design of the aerodynamic shape and exterior ballistics of the pulse-controlled correction projectile using robustness approximation-based modeling methods in the field of pulse-controlled correction projectile research.

In summary, this paper focuses on trajectory correction projectiles with lateral pulses as the research object. It considers the inherent uncertain factors in engineering calculations and carries out a robust optimization design for the aerodynamic shape and exterior ballistics. To solve the mathematical problem, a stochastic Kriging surrogate model is constructed to obtain a rapid response of the mean and variance of the projectile’s performance with respect to design variables. Furthermore, based on the non-dominated sorting multi-objective sparrow search algorithm (MOSSA) [24], simulation and experimental results are compared and analyzed for deterministic Pareto optimal solutions and robust Pareto optimal solutions. This analysis aims to verify the practical engineering significance of robust design for pulse-controlled correction projectiles.

2. Theoretical Foundation

2.1. Optimization Design Process for Pulse-Controlled Correction Projectiles

Figure 1 shows the optimized design process diagram, where the first step is the establishment of the sample design space using Latin hypercube design (LHD), a well-balanced sampling design that uniformly covers the entire experimental range and evenly divides the design space of each variable. The design variables in this study include aerodynamic shape parameters (such as fin count and fin cant angle) and correction mechanism parameters (such as axial eccentricity and number of pulse thrusters), which directly affect the projectile’s flight trajectory performance and tactical indicators. The goal of the comprehensive optimization design is to achieve a well-performing controlled trajectory design by properly matching the aerodynamic shape parameters and correction mechanism parameters. The next step is to calculate the projectile’s aerodynamic force and moment coefficients using engineering algorithms; Projectile Design Analysis System (PRODAS) software V3 is used in this study for solving the aerodynamic coefficients [25,26]. Based on the obtained parameters, a six-degree-of-freedom (6 DOF) ballistic simulation is conducted, and a sample library is created with information on the design variables and the target performance.

The Latin hypercube sampling and 6 DOF model will be used to generate a sample library, which will be used to construct the DK model and SK model. These models will allow for a quick evaluation of target performance at various design points. Verification of the model prediction accuracy will enable selection of several performance indicators as objective functions for the design scheme, along with corresponding constraint conditions. The design process will then be transformed into a constrained nonlinear programming problem with multiple design variables, objectives, and constraints. To obtain the optimal design scheme that meets the performance criteria, the MOSSA, a non-dominated sorting multi-objective optimization algorithm, will be utilized based on the optimization objective functions designed in this paper. For further details, please see Section 2.4. Given the existence of an unknown Pareto frontier in the design problem addressed in this article, the convergence criterion is established such that the degree of algorithm optimization will not exhibit further improvement with increasing iterations.

2.2. Pulse Ignition Logic

The ballistic performance of the pulse-controlled correction projectile will be simulated based on sample points obtained from the LHD experiment. These simulations will provide a sample library for establishing a surrogate model. The 6 DOF ballistic model used in this paper is consistent with the reference literature [27] and will not be elaborated on in here. To help readers better understand the simulation results, the following explanation is provided regarding the pulse control characteristics used in the 6 DOF ballistic simulation:

(1)

The pulse thrusters are arranged uniformly in a circumferential ring along the projectile’s surface. Under ideal conditions, the thrusters generate thrust in a plane located in the projectile’s axis coordinate system, which is perpendicular to the projectile’s axis;

(2)

Unlike traditional aerodynamic control, the pulse force only exists during the action time τ and is of a discrete nature;

(3)

Each correction can only utilize a single pulse thruster;

(4)

Considering the effect of the rotation rate ${\dot{\gamma }}_0$, the direction of the control force ${\gamma }_m$ constantly changes due to the rotation of the projectile when the pulse thruster is active;

(5)

Consider the delay error of pulse ignition;

(6)

Due to the projectile’s right-handed rotation, the pulse thruster number is searched for in the counterclockwise direction;

(7)

The pulse control force $F_{\mathrm{imp}}$ is processed based on the equivalent average control force, as depicted in Figure 2 (a transverse section of the projectile axis), and the final average effect ${\overline{F}}_{\mathrm{imp}}$ is considered to be along the bisector of the angle of rotation during the pulse’s activation.

2.3. Stochastic Kriging Modeling with Limited Samples

The Kriging model is an unbiased estimation model that predicts responses at unknown test points based on known experimental point information, assuming that all data follow an n-dimensional normal distribution [28]. Essentially, the Kriging model is an interpolation model, and its interpolation result is a linear weighting of the known sample response values. To calculate the weighting coefficients, the Kriging model introduces a statistical assumption: the unknown function is regarded as a specific realization of a Gaussian stationary random process. Since the stochastic response surface model based on polynomial chaos expansion (PCE) is modeled against the background of the response surface model, its fitting ability for high-order nonlinear problems is slightly insufficient. Thus, deterministic Kriging surrogate models are often considered more attractive than response surface models. Therefore, reference [29] proposes developing the deterministic Kriging surrogate model into the stochastic space, forming the stochastic Kriging surrogate model. Uncertainty research often requires the numerical calculation of a large number of repeated samples. This method effectively reduces the amount of calculation in uncertainty design by introducing a covariance matrix representing the inherent uncertainty of samples into the mathematical model.

In cases where there are repeated sample points with uncertainty, the DK surrogate model is transformed from the deterministic space to the stochastic space, and random samples in the space are represented in the following form:

$$\begin{array}{l}{Y}_j\left(x\right)=\beta +M\left(x\right)+{\epsilon }_j\left(x\right)\\ \eta \left(x\right)=\mathrm{var}\left(\epsilon \left(x\right)\right)\end{array},$$

(1)

where $Y_j\left(x\right)$ represents the $j$ repeated sample point at position x ($j=1,2\cdots ,p$, $p$ representing the response value dimension), $\beta$ is the global mean, $M\left(x\right)$ represents non-intrinsic uncertainty, and n is a random variable used to represent the difference between the mathematical expectation and trend terms of data with uncertainty at each position. The term ${\epsilon }_j\left(x\right)$ is used to represent the perturbation of the repeated sample at x, which is assumed to be a random variable and follows a normal distribution. It is independent of the perturbations of all repeated samples and independent of the non-intrinsic uncertainty. The term $\eta \left(x\right)$ is the variance of the uncertainty at x. In the SK surrogate model constructed in this paper, the prediction of the mean value at unknown locations ${\overline{Y}}^{\ast }\left(x\right)$ is estimated using the following form:

$${\overline{Y}}^{\ast }\left(x\right)={\omega }_0\left(x\right)+{\displaystyle \sum _{i=1}^n{\omega }_i\left(x\right)}\overline{Y}\left(x\right),$$

(2)

where $\overline{Y}\left(x\right)$ represents the mean vector at known locations. The term ${\omega }_i\left(x\right)$ represents the weight assigned to the mean value at known locations, which is used to estimate the mean value at unsampled locations by weighting the known mean values. The construction form of the mean squared error (MSE) for the SK model with uncertainty can be expressed as the following equation:

$$MSE=E\left[{\left(Y_j\left(x\right)-{\overline{Y}}^{\ast }\left(x\right)\right)}^2\right]=E\left[\beta +M\left(x_0\right)+{\epsilon }_j\left(x_0\right)-{\omega }_0-{\omega }^{\mathrm{T}}\overline{Y}\right].$$

(3)

The construction of the mean square error in the SK model considers the inherent uncertainty information of repeated samples during the model construction process. This allows for the representation of the assumed truth and estimation form. Consequently, the estimate of the mean and variance of the SK model for unknown points with known uncertainty can be obtained. By accurately predicting the mean and variance, the SK model effectively addresses the issue of high computational costs associated with traditional methods of uncertainty estimation.

2.4. Multi-Objective Sparrow Optimization Algorithm with Non-Dominated Sorting

The search performance of the optimization algorithm directly determines the extent to which the optimization results can approach the global optimal solution. The sparrow search algorithm (SSA) is a metaheuristic algorithm that was proposed in 2020. It is known for its simple structure, ease of implementation, and strong local search capabilities. SSA has shown significant advantages in solving single-objective optimization problems, as demonstrated by benchmark function tests [30,31,32]. However, it was originally designed for single-objective optimization problems. To extend its use to multi-objective optimization, non-dominated sorting is required to compare fitness values. In summary, this article builds upon the improvement strategy of a single optimal point proposed in reference [32]. It introduces a variable spiral factor and combines it with a per dimension lens imaging learning strategy to enhance the early algorithm’s exploration ability and escape local extreme points. Additionally, to solve the multi-objective optimization problem more effectively, the article introduces the non-dominated sorting method, which enables the original search algorithm to search for the Pareto optimal frontier instead of a single optimal point. The pseudocode for the algorithm is presented in Figure 3. The MOSSA algorithm utilized in this article has been implemented using MATLAB scripts [33]. Using the DK and SK surrogate models, the article seeks the Pareto optimal solution to the multi-objective optimization problem of the pulse correction projectile through MOSSA.

3. Process of Design Optimization

3.1. Definition of Design Problems

The installation of aft fins on low-spin fin-stabilized projectiles can provide weathervane stability, and a specific cant angle is utilized to achieve low spin and minimize dispersion. This section is dedicated to the robust integrated optimization design of the aerodynamic shape and exterior trajectory of pulse correction projectiles. To achieve this, a standard low-spin fin-stabilized projectile (as depicted in Figure 4) is utilized as the initial configuration. The static stability and spin design of a fin-stabilized correction projectile have a direct impact on its flight performance and correction efficiency. In engineering applications, the choice of aft fin area, shape, number of fins, and their position relative to the projectile’s center of mass determine its static stability. Meanwhile, the cant angle magnitude determines the spin rate of the projectile. In addition, the size of a single pulse thruster, the number of pulse thrusters, and their axial and radial distances from the thrust center, as well as other correction mechanism parameters, are interconnected with the aerodynamic layout and have a direct impact on the controlled trajectory performance of the projectile. Consequently, this paper considers six design variables to optimize the aerodynamic shape and trajectory of the pulse correction projectile. These variables include the number of fins, the lead edge cant angle of the fins, the axial mounting position of the fins (relative to the projectile nose), the fin deployment area, the size of a single pulse thruster, and the axial eccentricity of the pulse thrusters.

Table 1. Example optimization problem design variables.

Symbol	Description	Original Configuration	Minimum Value	Maximum Value
$N$	Fin count	6	Limited to 4, 5, 6
$\epsilon$	Lead edge cant angle	6°	4°	15°
$R$	Distance of the leading edge of the fin from the nose of the projectile	650 mm	620 mm	657 mm
$S$	Area of a single fin	20 cm2	7 cm2	25 cm2
$I_{\mathrm{imp}}$	The impulse size of a single pulse thruster	30 N·s	30 N·s	150 N·s
$L_{\mathrm{imp}}$	Axial eccentricity	100 mm	−200 mm	200 mm

provides details of the relevant parameters of the initial design configuration, specific design variables, and their value ranges. During the design optimization process, the number of fins is discretized and rounded off.

The primary concern for the tactical and technical performance of the pulse correction projectile is its correction capability, followed by its maximum range of controlled trajectory and flight stability. Instead, there are constraints on the angle of attack δ of the projectile, with ${\delta }_{\max }<{15}^{°}$ based on engineering experience. The specific mathematical model and constraints for optimization are discussed in Section 3.2 and Section 3.3. In Section 3.2, a DK surrogate model is utilized to obtain deterministic Pareto optimal solutions, while Section 3.3 applies an SK surrogate model to obtain robust Pareto optimal solutions for the pulse correction projectile. Section 3.4 conducts a comparative analysis between the deterministic and robust optimization design results of the projectile, and the Pareto design frontier of the corrected missile is obtained based on MOSSA. Additionally, Section 3.4 compares and analyzes the deterministic and robust optimization design results of the pulse correction projectile and presents the Pareto design frontier of the projectile under performance trade-offs.

During the design phase, engineering algorithms offer advantages in performing repeated calculations of aerodynamic forces. However, these algorithms are not as accurate as computational fluid dynamic (CFD) simulations and wind tunnel experiments and may have computational errors of up to 10% [6,22]. As the range of the projectile directly affects its impact on the target, the drag coefficient is a crucial aerodynamic force coefficient in projectile design. This paper considers the engineering calculation error of the drag coefficient during the design process and conducts robust optimization on the pulse correction projectile.

Using a combination of CFD simulations and wind tunnel experiments, the error range of the drag coefficient for a projectile was determined based on its original design as shown in Figure 4. The simulation software used is FLUENT V16.2 [34]. To ensure computational accuracy, the time step for solving was initially set at 1 × 10⁻⁶ s and eventually determined to be 1 × 10⁻⁵ s. Unstructured polyhedral grids were chosen for numerical calculations, and a density-based solver was employed. The far-field inflow was specified using the pressure far-field condition. In order to improve simulation accuracy, grid refinement was applied to regions such as the projectile head and tail where shock waves might occur. The size of the refined grids ranged from 0.5 mm to 4 mm, while the remaining parts of the projectile had a uniform grid size between 5 mm and 50 mm, with larger grid sizes in regions farther away from the projectile. The maximum value of Y+ was set to 0.96. As deterministic CFD simulations do not account for the random variations in the operating environment that can affect the performance of the aircraft, uncertainty quantification methods were employed to provide stochastic solutions for the variables of interest, thereby supporting the fine-tuning of the aircraft’s shape, structure, and flight control system [35,36,37]. Hence, the uncertainty quantification is used for both the choice of the time step and the residual settings in the CFD simulations.

To ensure the reliability of the CFD numerical calculation method, the study first examines grid independence (as shown in Figure 5 and

Table 2. Comparison of the drag coefficient under different grid densities.

	Cx	Ma = 0.4			Ma = 2
Grid Scheme		$\mathit{\delta }=-2^{°}$	$\mathit{\delta }=0^{°}$	$\mathit{\delta }=2^{°}$	$\mathit{\delta }=-2^{°}$	$\mathit{\delta }=0^{°}$	$\mathit{\delta }=2^{°}$
1		0.346	0.330	0.357	0.462	0.459	0.463
2		0.349	0.335	0.361	0.465	0.448	0.467
3		0.345	0.331	0.356	0.470	0.452	0.459

). The results in Table 2 demonstrate that the simulated values of the drag coefficient C_x under different grid density schemes are highly consistent, confirming the reliability of the CFD numerical method. Additionally, the accuracy of the CFD calculation results is evaluated using wind tunnel experimental data. The experimental model involves a 1/4 scale model of the projectile. The experimental wind tunnel is a downdraft temporary impact closed high-speed wind tunnel, with an experimental section size of 0.3 m × 0.3 m and a length of 0.6 m. The available Mach number range is 0.5–4.5. The model is supported and the angle of attack changes through the wind tunnel’s angle of attack mechanism. A 6-component strain balance is used to measure the force and torque of the model along the three coordinate axes. The experimental Mach number Ma is 2, and the angle of attack varies from −4° to 8°. The experimental model installation diagram is shown in Figure 6. By comparing the CFD simulation values with the wind tunnel experimental values under the grid scheme 1 (Ma = 2), the study evaluates the accuracy of the CFD calculation. As shown in Figure 7, the variation trend of the simulated and experimental values of the drag coefficient is generally consistent. Both increase nonlinearly with an increasing angle of attack, and the error between the two increases with an increasing angle of attack. The maximum relative error is approximately 0.02, indicating that the CFD numerical method can accurately solve the projectile’s aerodynamic parameters. Furthermore, it can be used to evaluate the error between accurately calculated aerodynamic parameter values and those calculated using the PRODAS engineering algorithm. Figure 8 presents a comparison between the drag coefficient calculated using PRODAS and CFD at δ = 0° for Ma ∊ 0.4~3. The results indicate that the CFD-calculated drag coefficient values are slightly lower than the corresponding PRODAS engineering calculations within this range, with a maximum difference of approximately 0.912 and a minimum difference of approximately 0.943. Considering both the numerical results obtained from CFD and the structural changes in the fins during the design process, this paper proposes an optimization design to ensure the robust performance of pulse correction projectiles within the variation range of drag correction coefficient $aero\_factor\in \left[0.91,0.92,0.93,0.94,0.95\right]$.

Furthermore, the following explanation is provided for the aerodynamic calculations and ballistics simulations in the design process:

(1)

The engineering experience system software PRODAS V3 used for aerodynamic force calculations and reference [26] have different definitions for the pitch damping moment coefficient and roll damping moment coefficient, resulting in numerical discrepancies of a factor of two;

(2)

To assess the maximum lateral correction capability, simulations are conducted considering the effects of all pulse thrusters;

(3)

The actual engineering constraints of the pulse correction mechanism design are considered. Due to limited design space, under the condition of the same propellant material, the magnitude of single pulse impulse is negatively correlated with the number of pulse thrusters, while the magnitude of single pulse impulse is positively correlated with the pulse duration.

3.2. Deterministic Optimization Design Based on DK Model

Based on the performance goals of pulse correction projectiles, deterministic optimization design is conducted using the DK surrogate model. At the design point where aero_factor = 0.93, flight dynamic stability during the optimization process must be ensured The mathematical optimization model can be expressed as follows:

$$\begin{array}{c}ObjectiveFunction:\max \{\hat{c}\left(X\right),\hat{r}\left(X\right)\}\min \{\hat{a}\left(X\right)\}\\ \hfill S.T.{\delta }_{max}<15°\hfill \\ Variable:X=[N,\epsilon ,R,S,I_{imp},L_{imp}]\end{array}.$$

(4)

In this equation, c, r, and a correspond to the representative lateral correction distance, maximum range, and maximum angle of attack of the target, respectively. The symbol (^) denotes that this value is predicted by the DK surrogate model. In the design process, it is not required for the three performance criteria to be strictly mutually exclusive. In the design process, 600 sample points were generated using Latin hypercube design (LHD) as training samples and an additional 30 test samples were generated. The number of samples to select is based on the expected prediction accuracy and computation. The distribution of performance indicators, namely lateral correction distance, maximum range, and maximum angle of attack, categorized according to the number of fins, is shown in Figure 9 and Figure 10. The red dots in the figures indicate unstable points in the design (${\delta }_{\max }>{15}^{°}$). It can be observed that reducing the number of fins can increase the range effectively. However, an improper design of the number and location of fins may cause the center of pressure and center of gravity to be too close, resulting in the static stability reaching the critical value. Moreover, when the projectile is subject to pulse forces during its controlled trajectory, it becomes more challenging to meet the dynamic stability requirements. Additionally, larger correction capabilities often require more significant pulse energy consumption, which puts higher demands on the projectile’s flight stability.

Figure 11 shows that the DK surrogate model constructed has good agreement with the 6 DOF simulation results for the predicted values of most sample points; the prediction of maximum range and lateral correction distance are more accurate than the maximum angle of attack. This indicates that the surrogate model can predict the performance variation trend of the test samples, and its accuracy meets the needs of the subsequent optimization processes. However, it should be noted that the selection of hyperparameters and regression functions in the DK model can affect the model’s accuracy, and the selection of model parameters is somewhat empirical. Therefore, different target performances may require appropriate model parameters. Additionally, the flight stability constraint of the projectile is handled in the MOSSA optimization process by imposing a penalty term on individuals who violate the constraint on the amplitude of the attack angle. Finally, to eliminate accidental errors, each algorithm runs independently for 30 times. The optimization results will be compared and analyzed in Section 3.4.

3.3. Robust Optimization Design Based on SK Model

Based on the constructed SK surrogate model, a robust optimization design is performed for the pulse-corrected projectile. The design state is $aero\_factor\in \left[0.91,0.92,0.93,0.94,0.95\right]$, with the main objective of making the target performance of the pulse-corrected projectile insensitive to small fluctuations in the drag coefficient, i.e., considering the stability of the system performance under uncertain factors. The optimization model can be expressed as follows:

$$\begin{array}{cc}ObjectiveFunction:& \max \{\hat{c}\left(X\right),\hat{r}\left(X\right)\}\min \{\hat{a}\left(X\right),{\sigma }_c^2,{\sigma }_r^2,{\sigma }_a^2\}\\ S.T.& {\delta }_{max}<15°\\ Variable:& X=[N,\epsilon ,R,S,I_{imp},L_{imp}]\end{array}.$$

(5)

The symbol ${\hat{\sigma }}_i^2$ (i = c, r, a) represents the variance prediction value of the target performance by the SK model. The definitions of other related symbols are consistent with those in Formula (4). To ensure consistency with the optimization design results obtained from the DK model, we kept the LHD sample space and MOSSA parameter settings unchanged from Section 3.2. Unlike constructing a deterministic model, a random surrogate model was built by uniformly repeating the sampling process five times within the aero_factor change interval at each sample point. The performance mean and variance at different design variable points were used as input variables for the SK model to obtain the response mean and variance of the design variables at unknown locations. Similarly, we selected 30 test samples outside of the 600 design samples to verify the rationality of the SK model. Figure 12a–c depict the comparison curves between the means and variances predicted by the SK model and the actual computational results. The results demonstrate a close match, indicating that the accuracy of the constructed SK model is adequate for robust optimization design in the future.

3.4. Results and Analysis of Optimization Based on MOSSA

The design of pulse correction projectiles requires an optimal solution that changes with variations in target performance. Improvements in one performance indicator often come at the expense of others. Due to the complexity of the influence of design variables on results, the MOSSA algorithm based on non-dominated sorting is used to seek the Pareto optimal solution using both the DK and SK surrogate models. The flight stability of the projectile is handled using the penalty function method, which applies a penalty term on individuals that violate the constraint of the angle of attack amplitude. The optimization process uses a population size of 80 and a maximum number of iterations of 500 to eliminate accidental errors. Each algorithm is independently run 30 times. The Pareto design sets based on the DK and SK models are shown in Figure 13a,b, respectively. The obtained Pareto solutions all fall on the edge of the LHD experimental samples, indicating that all target performance indicators have been improved. It is observed that there is a conflicting relationship between the maximum angle of attack and the lateral correction distance. Specifically, the larger the lateral correction distance, the larger the maximum trajectory angle of attack. In the decision-making process of the scheme, both the demands for correction capability and flight stability should be comprehensively considered. However, the maximum range and lateral correction capability, as discussed at the beginning of the design, are not strictly mutually exclusive or unrelated. Nonetheless, when improving the maximum vertical correction capability as the target function, the situation is different. Therefore, a better solution can be selected from the Pareto optimal solutions under the premise of meeting dynamic flight stability in the decision-making process of pulse correction projectiles. To conduct a specific performance evaluation of the deterministic optimization design based on the DK surrogate model (Deter_DKM) and the robust optimization design based on the SK surrogate model (Robust_SKM), design points with similar performance indicators from the Pareto solution set are selected for comparative analysis.

Table 3. Comparison of optimized configurations based on DK model and SK model.

Configuration	N	ε (°)	R (mm)	S (cm2)	Iimp (N·s)	Limp (mm)
Original design	6	6	657	20	70	100
Deter_DKM	4	6.43	620.05	9.86	62.54	178.54
Robust_SKM	4	6.44	620.56	11.92	35.87	176.26

presents the values of the Pareto solutions for deterministic optimization and robust optimization.

Table 4. Comparison of optimized performance based on DK model and SK model.

Configuration	${\mathit{\mu }}_{\mathbf{c}}$	${\mathit{\sigma }}_{\mathbf{c}}^2$	${\mathit{\mu }}_{\mathbf{r}}$	${\mathit{\sigma }}_{\mathbf{r}}^2$	${\mathit{\mu }}_{\mathbf{a}}$	${\mathit{\sigma }}_{\mathbf{a}}^2$
Original design	88.85	7.11	11,202.82	25,196.80	4.69	1.13
Deter_DKM	158.38	10.43	12,144.53	26,097.03	7.43	1.27
Robust_SKM	154.84	3.26	12,086.61	23,097.15	6.82	0.63

provides a detailed comparison of the corresponding Pareto design solutions in terms of performance results obtained from 6 DOF calculations. The mean and variance of the lateral correction distance, maximum range, and maximum angle of attack within the aero_factor change interval are represented by ${\mu }_i,{\sigma }_i\left(i=c,r,a\right)$, respectively. Both optimization results have shown a significant improvement in performance at the design points compared to the original configuration. According to Table 4, the deterministic optimization solution of Deter_DKM has increased the mean value of the lateral correction distance by approximately 78.25% and the mean value of maximum range by approximately 8.40% but has increased the mean value of maximum angle of attack by approximately 2.74°; this is within the aero_factor change interval. Similarly, the robust optimization solution based on SK has increased the mean value of the lateral correction distance by approximately 74.27%, the mean value of the maximum range by approximately 7.89%, and the mean value of the maximum angle of attack by approximately 2.13°. The performance of the two optimization solutions is relatively close, and the deterministic optimization solution has a slight advantage in performance indicators. Figure 14a–c present a comparison of the divergence curves for the target performance, and it is evident that the performance divergence characteristics of the robust optimization scheme based on the SK model are superior. As the uncertain design in this study mainly focuses on small-scale fluctuations in the drag coefficient, Figure 14 shows that compared to the lateral correction distance and maximum angle of attack, the robust optimization scheme improves the divergence characteristics of the maximum range more significantly.

4. Validation of Optimization Results

To validate and provide reference data for the optimized projectile aerodynamic layout performance, wind tunnel experiments and CFD simulations were conducted on the deterministic and robust optimization schemes. The wind tunnel experimental model was designed in sections, and different aft fin components could be obtained by replacement. The test model parts and the experimental schlieren image are shown in Figure 15 and Figure 16, respectively. To facilitate comparative analysis of the design schemes, the CFD simulation conditions were consistent with scheme 1 in Figure 5a. The comparison between the wind tunnel experimental drag coefficient $C_x{}_{\left(W\right)}$ and the CFD drag coefficient $C_x{}_{\left(CFD\right)}$ under different operating conditions for the design scheme is shown in

Table 5. Comparison of drag coefficient based on DK model and SK model.

Configuration	$\mathit{Ma}=2,\mathit{\delta }=-4^{°}$		$\mathit{Ma}=2,\mathit{\delta }=0^{°}$		$\mathit{Ma}=2,\mathit{\delta }=2^{°}$
	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{W}\right)}$	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{CFD}\right)}$	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{W}\right)}$	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{CFD}\right)}$	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{W}\right)}$	${\mathit{C}}_{\mathit{x}}{}_{\left(\mathit{CFD}\right)}$
Original design	0.463	0.467	0.457	0.459	0.465	0.463
Deter_DKM	0.412	0.415	0.391	0.394	0.434	0.430
Robust_SKM	0.427	0.430	0.418	0.416	0.447	0.4421

. The pressure distribution on the fin surfaces of the original configuration and the optimized design at δ = 0° and a Mach number of 0.6 is depicted in Figure 17a–c. The optimized design shows a decrease in the pressure values on the fin surface and a weakening of the surface shock wave intensity compared to the original design, as seen from Figure 17. This reduction is attributed to the decreased number of fins and the windward surface area, which effectively reduce the drag on the projectile. Furthermore, Figure 18 compares the curves of the fin surface roll moment coefficient ($m_{xw}$) and the roll damping moment coefficient ($m_{xz}$) obtained from CFD calculations. The optimized design exhibits a larger resultant moment vector in the roll direction due to an increase in the lead edge cant angle. The fin cant angle is an important factor that affects the balance spin rate of the projectile, and therefore its flight characteristics and the correction efficiency of the trajectory. However, it is not the only determining factor. Figure 19 presents curve diagrams of the static moment coefficient ($m_z$) and the center of pressure ($X_p$) as a function of the Mach number, obtained from CFD simulations. The optimized design adjusts the fin installation position to move closer to the projectile’s nose, which reduces the distance between the center of mass and the center of pressure, thereby decreasing the static stability and enhancing the trajectory correction capability. It should be noted that the static stability of a pulse correction projectile should not be excessively high, as it may adversely affect its trajectory correction performance, nor too low, as it may cause flight instability. Therefore, the fin installation position should be rationally arranged based on the overall design requirements during the design process.

Based on the aerodynamic parameters obtained from CFD, Figure 20 and Figure 21 present the corresponding flight trajectories and angle of attack curves for the original configuration and the two optimized designs. In these figures, “Uncor” denotes an uncontrolled trajectory, while “Cor” denotes a controlled trajectory. It is evident that both sets of optimized designs fulfill the requirements for stable flight. This study’s simulation results have demonstrated the practical significance of robust design in improving the performance of pulse correction projectiles.

5. Conclusions

This article delves into the inherent uncertainties that exist in engineering algorithms, as well as the practical engineering constraints that arise during the design process. To address these challenges, the article proposes using the stochastic Kriging surrogate model. By applying this model, the basic mathematical problem can be formulated to perform a robust optimization design of the aerodynamic shape and trajectory of a pulse correction trajectory projectile. Based on the analysis conducted, the main conclusions drawn from this study are as follows:

(1)

The results of this study demonstrate that both the DK and SK surrogate models accurately predict the performance of the pulse correction projectile during optimization. However, the SK model’s ability to predict both the mean and variance of performance provides a significant advantage in addressing uncertainty design issues, compared to the DK surrogate model;

(2)

In comparison to the original configuration, both the deterministic optimization results based on the DK model and the robust optimization results based on the SK model significantly improve the mean performance indicators of the pulse correction projectile’s lateral correction ability and maximum range, while also meeting the flight stability requirements. However, when considering the range of the variable aero_factor, the robust optimization design proves effective in improving the projectile’s performance divergence characteristics. As a result, it achieves a stable improvement in performance for the pulse correction projectile.

(3)

Using the non-dominated sorting MOSSA algorithm, a Pareto optimization scheme for the aerodynamic shape and exterior trajectory of pulse correction projectiles has been developed. Aerodynamic parameters were obtained through wind tunnel tests and CFD simulations, and the correctness of the optimization results was verified through trajectory numerical calculations. The optimization design method presented in this article exhibits strong optimization capability and practical engineering significance, effectively improving the projectile’s performance and shortening the design cycle compared to deterministic optimization methods.