Multi-objective optimization of a muzzle brake to enhance overall performance

The use of muzzle brakes in automatic weapons has become widespread due to their ability to counteract the harmful effects of the high pressure and velocity of the propellant gases, thereby enhancing the overall technical performance. These devices typically comprise a series of baffles positioned perpendicular to the gun barrel axis, either attached to or integrated with the muzzle. Due to the impact of the expanding propellant gases during the after-effect period, muzzle brakes can generate specific mechanical effects on the weapons.¹ 

Standard structures have emerged over time based on theory or experience. However, there have always been defects. The reason is that the muzzle flow fields within and outside the brake are 3D unsteady turbulent flows with a complex shock wave system accompanied by chemical changes, and, thus, they cannot be accurately represented by quantitative mathematical models. Therefore, in the traditional design procedure, a muzzle brake is designed mostly based on the experience of the engineers. Moreover, it is difficult for them to take into account some adverse factors.² To some extent, the design process is still limited and subjective. Hence, optimizing a muzzle brake, especially to enhance overall performance, is still a complex problem.

In recent years, advances in computer technology have led to an increase in the use of numerical simulation methods, such as computational fluid dynamics (CFD) and computational aeroacoustics (CAA), as design aids that complement the traditional theoretical, empirical, and experimental approaches.^3–5 Researchers have demonstrated the effectiveness of numerical simulations in capturing and accurately predicting the complex flow characteristics of muzzle brakes.^6–10 Zhang et al.¹¹ numerically investigated the launching of a projectile under three different conditions: a bare muzzle, a three-way muzzle brake, and a multi-hole muzzle brake. They verified the numerical method with corresponding experiments. Zhao et al.¹² analyzed the flow fields around a reaction muzzle brake and studied the contribution of the airflow through a side hole to the efficiency of the brake. Chaturvedi and Dwivedi¹³ designed a tunable muzzle brake, showing that it was better than the existing designs based on a simulation. Zhao et al.¹⁴ investigated the muzzle flow and noise fields simultaneously with a hybrid CFD-CAA method and found that the directivity distribution of the noise was strongly related to the flow structure. These studies demonstrate that numerical simulations can be a useful and efficient way to design muzzle brakes.

In addition, optimization using approximations and genetic algorithms has become common in engineering applications.^15–18 Jiang and Wang¹⁹ optimized the performance of a muzzle brake using a multi-island genetic algorithm with an approximate model. However, their research targeted only the impact force on the muzzle brake. Sherif et al.²⁰ optimized the efficiency, force, and recoil force for a sniper rifle (12.7 × 99 mm²) with a muzzle brake using a multi-objective genetic algorithm (MOGA), but the analytical model for calculating the response parameters was based on the ballistic equations, which do not have good accuracy for a muzzle brake with a complex structure.

This study describes the multi-objective optimization of a muzzle brake, targeting both its efficiency and impulse noise. The aim was to enhance overall performance. The design process includes design of experiments (DoE), the CFD-CAA numerical method, an approximation model, and the MOGA. It was applied to optimize the structure of a standard impact-reaction muzzle brake. The correlation and sensitivity between the overall performance and structural parameters were investigated. A new muzzle brake with the optimum values of the design parameters was modeled and simulated to verify the results, which were compared with the original design.

The methodology for the multi-objective optimization of a muzzle brake to improve overall performance is depicted in Fig. 1. The procedure commences with the definition of the principal structural variables that need to be optimized. The design samples were determined with the DoE technique. Subsequently, each sample was modeled in 3D. A numerical simulation was used to compute the performance indices. Finally, by utilizing the MOGA with an approximation based on kriging, the optimal values of the structural parameters were obtained.

In this study, a typical muzzle brake for a 12.7 mm heavy machine gun was selected as the research object, as shown in Fig. 2(a). The brake has a standard structure that consists of two symmetrical chambers of equal length. The top and bottom of the brake are closed to prevent any expanding gases from posing a threat to the gun crew. The side holes are perpendicular to the barrel axis, enabling the gases to escape after impacting the baffles. Based on prior design experience, the following four design parameters (or factors) were taken into consideration: baffle disk angle (θ), brake standoff distance (L₁), baffle spacing distance (L₂), and baffle bore diameter (D). Figure 2(b) provides a definition for each variable. The fillets in the structure have been simplified.

The DoE method is a well-established approach for maximizing the information gain with minimal resource utilization. The orthogonal experiment method was employed in this study. For each factor, three levels were set, as shown in Table I. For four factors with three levels, nine groups of experiments were performed on the samples according to the L9 (3⁴) orthogonal test table,²¹ with the results listed in Sec. III A.

The objective of this study was to design a new muzzle brake that demonstrates improved overall performance, reflected in both a higher muzzle brake efficiency and a reduction in the muzzle impulse noise. Therefore, the optimization process had two objectives: the muzzle brake efficiency η_T and the overall sound pressure level (OASPL) at the shooter’s left ear.

Given that the increased impulse noise caused by installing a muzzle brake can lead to serious physical and psychological harm for the shooter, in this study, sound receivers were placed at 0.2, 0.5, and 1.0 m to the left of the shooter’s left ear, as shown in Fig. 4. These locations were chosen based on the assumption that the center of the head passes through the axis of the barrel. Using the average dimensions of a male adult head, the distance from the left ear to the axis was taken as 73.5 mm.²² 

F_pt, F_R, and p′, which are variables in Eqs. (2)–(4), were obtained by a 3D unsteady numerical simulation of the muzzle flow and noise fields during the entire after-effect period, which starts at the moment the propellant gas begins to flow outward after the bottom of the projectile leaves the muzzle and ends when it is exhausted. The grid in the model was based on the structure of the muzzle brake. The model was refined in the area around the muzzle in the axial direction. The grid had nearly 2.7 × 10⁶ cells. The grid model with boundary conditions is shown in Fig. 5. The computational domain had a range of 200d in the direction of the jet, and the cylinder radius was 80d, where d is the diameter of the barrel.

The numerical simulation was conducted with the ANSYS Fluent software. It used a large eddy simulation (LES) coupled with the Ffowcs-Williams and Hawkings analogy (FWH). The time-varying force at the bottom of the barrel and the reaction force on the muzzle device were monitored at each time step. The parameters of the unsteady flow field required to calculate the SPL were collected every five time steps, resulting in a total time of 0.025 s. The detailed setup for the transient flow solver is listed in Table III.

In order to determine the accuracy and validity of the computational model, the muzzle brake efficiency and OASPL under the same condition of the experiment in Ref. 23 were calculated using Eqs. (1) and (5). Considering the directionality of muzzle noise, the microphones were set with a radius of 2.0 m at φ = 0°, 30°, 60°, 90°, and 120° counterclockwise from the downstream direction of the jet, as shown in Fig. 6. Thus, the OASPL at the same location was also calculated as listed in Table IV. The deviations between the calculated and measured data were all within 5%, suggesting that the numerical method can be applied to practical scenarios.

The optimal design of a muzzle device has to maximize the muzzle brake efficiency and minimize the SPL, so it is a multi-objective optimization problem in a high-dimensional design space. To address this issue, the MOGA, which is based on the velocity and position of particles, was adopted in this study. The MOGA searches for non-dominated solutions of the proposed multi-objective problem in a feasible search space. Each particle or gene is organized and ranked according to its position.²⁷ The detailed settings for the MOGA are listed in Table V.

The results for the nine samples are presented in Table VI. It gives the average value of the OASPL calculated at the three sound receiver points.

A range analysis was utilized to investigate the impact of the various factors on the objectives, resulting in the calculation of K_j, k_j, and R_j for each factor. K_j is the sum of the test values corresponding to the jth level of the factor, and k_j is the average value corresponding to K_j and is used to determine the optimal combination of this factor. R_j is the difference between the maximum and minimum values of k_j and reflects the significance of the corresponding factors. A higher value of R_j indicates that the factor has a greater impact. The results of the range analysis are displayed in Table VII, where each column represents the impact of changes in the factor on the indicator.

The corresponding maps for k_i for the muzzle brake efficiency are shown in Fig. 7(a). When the muzzle brake efficiency was used as the evaluation index, the range of factor A (θ) reached its maximum, signifying that the angle of the baffle disk is the most sensitive design parameter among the four. This is primarily due to the direct influence of the side hole angle on the distribution of propellant gas within the muzzle and the structure of the muzzle flow field, both of which affect the braking efficiency. As the angle increased, more gas flowed to the rear of the jet, leading to a significant improvement in the muzzle brake efficiency. On the other hand, the impact of the other three factors was relatively insignificant in comparison to that of the baffle disk angle. The brake efficiency decreased with an increase in the brake standoff length, baffle spacing, or baffle bore. Based on the sensitivity analysis, the order for the influence of the various design parameters on the braking efficiency is A (θ) > B (L₁) > D (D) > C (L₂). The optimal design for each factor was determined as the level resulting in the maximum efficiency. Hence, the best combination was A₃B₁C₁D₁, meaning θ = 120°, L₁ = 20 mm, L₂ = 20 mm, and D = 15 mm.

In comparison, there was little difference in the sensitivities of the various structural parameters on the OASPL, as shown in Fig. 7(b). The SPL at the shooter’s position increased with θ and L₁ and did not vary linearly with an increase in L₂, reaching a maximum of 143.05 dB when L₂ = 24 mm. Increasing the outlet diameter from 15 to 17 mm resulted in a decrease of about 0.5 dB in the noise value, whereas there was almost no change in going from 17 to 19 mm. According to the sensitivity analysis for the OASPL, the order of the various parameters is A (θ) > B (L₁) > C (L₂) > D (D). The optimal choice for each parameter was determined as the level resulting in minimum noise. Hence, the best combination selected for the SPL is A₁B₁C₃D₃, meaning θ = 90°, L₁ = 20 mm, L₂ = 28 mm, and D = 19 mm.

The optimal levels for the factors A (θ), C (L₂), and D (D) were different for each metric. To determine the optimal value for these three factors, six more design samples were simulated, with L₁ set to a constant value of 20 mm. The corresponding results are listed in Table VIII.

Cross-validation was conducted after the approximation model was generated. The results of the error analysis for the kriging method are shown in Table IX. For both indices, the average error and root mean square of the error are within the acceptable level of 0.2, which indicates that the approximation model is sufficiently accurate.

The optimization process resulted in the formation of a Pareto front, as depicted in Fig. 8, where each design represents the “best” combination of objectives and where an improvement in one objective comes at the cost of degradation of the other. All the points on the Pareto front have a higher muzzle brake efficiency than the original design, which is depicted as an orange circle in Fig. 8. For some, the impulse noise was higher, but there is better overall performance than the original design. The objectives of the optimization were, first, to increase the braking efficiency to more than 50% while, second, keeping the noise below that of the original design. Three candidates for the optimal solution are shown as red stars. Table X shows a detailed breakdown of these plans. The results of the optimization were verified through a numerical simulation, with the value of each parameter rounded off to take manufacturing accuracy into account. The results of the numerical verification are shown in Fig. 9, which indicates that there is good agreement in terms of trend and accuracy with the Pareto prediction.

The optimized candidate plan 1 was selected for a detailed comparison with the standard muzzle brake. The geometries are compared in Fig. 10, and the corresponding design parameters and calculated results are listed in Table XI.

The change in the reaction force F_R in the first 1 ms and the pressure distribution in the muzzle device are presented in Figs. 11(a) and 12. The two curves in Fig. 11(a) exhibit a similar trend. Initially, the propellant gas flowed into the first part of the muzzle brake and struck the baffle, imparting an additional impulse on the recoiling components, which resulted in a rapid increase in the reaction force. Simultaneously, the expanding gas also exerted a force on the rear wall of the device, which slowed down its rate of expansion. The gas continued to expand and flowed into the second part of the device, and the reaction force reached its maximum negative value. As the gas flowed out of the mouth of the muzzle brake, F_R began to decrease. The comparison reveals that the optimized structure generated a greater braking force in a shorter time with more complex shock wave structures and interactions within the muzzle brake. The optimized design thus leads to a higher muzzle brake efficiency.

The resultant force F, which is used to calculate the reaction coefficients in Eqs. (2) and (3), is plotted in Fig. 11(b). The original and optimized devices are compared with the case without a muzzle brake. The brake efficiency was calculated with Eq. (1), as listed in Table XI, which clearly shows that the muzzle brake efficiency of the optimized plan improved from 29.02% to 53.95%, indicating an improvement in brake performance.

To gain a deeper understanding of the performance in terms of muzzle impulse noise, we plotted noise direction diagrams. The sound receivers were at a distance of r = 2 m from the muzzle, and the angle θ was varied in increments of 10° in a counterclockwise direction relative to the downstream jet, within the range of 0°–150°. The simulated directivity map for the OASPL is presented in Fig. 13. For the original plan with the baffle disk angle θ = 90°, the majority of the noise was concentrated within the azimuth range of θ = 30°–90°. For the optimized design with the baffle disk angle of 120°, except for the azimuth range of θ = 30°–70°, there was an increase in the impulse intensity within the azimuth range of θ = 90°–130°. This correlation between the noise directivity and the structure of the muzzle flow fields is evident when comparing the velocity contours of the flow fields for both the original and optimized designs, as displayed in Fig. 14.

Consequently, it can be deduced that the noise directivity was influenced by the structure of the muzzle flow fields, which differ significantly due to the variation in the deflection for side holes with different disk angles. This conclusion confirms that the baffle disk angle is the most sensitive design variable. Although the optimized design extends the spread of the propellant gas to the rear of the muzzle, the OASPL in the direction to the left of the shooter’s head remained nearly constant.

A multi-objective optimization approach was employed in the present study to obtain a new design of a muzzle brake with enhanced overall performance. The key findings of this research are as follows:

1. The angle of the baffle disk (θ) was found to be the most sensitive design variable affecting both the muzzle brake efficiency and impulse noise.

2. Increasing the muzzle brake efficiency does not always prevent a decrease in the noise. Reducing the baffle spacing distance (L₂) leads to an increase in the braking efficiency but a decrease in the noise.

3. The error between the numerical results and the Pareto prediction was within 0.5%, which demonstrates the accuracy of the optimization method.

4. The optimized muzzle brake was demonstrated to have superior overall performance to the original design as it significantly improved the braking efficiency by nearly 25% with little increase in the SPL.

The optimization method proposed in this paper is highly recommended for the preliminary design and optimization of muzzle brakes and can also be applied to other muzzle devices. Future research could consider more design parameters and more objectives.

This work was supported by the Natural Science Foundation of China (Grant No. 11802138).

The authors have no conflicts to disclose.

Xinyi Zhao: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Methodology (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). Ye Lu: Conceptualization (equal); Data curation (equal); Funding acquisition (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.