To enhance the comprehensive performance of artillery internal ballistics—encompassing power, accuracy, and service life—this study proposed a multi-stage multidisciplinary design optimization (MS-MDO) method. First, the comprehensive artillery internal ballistic dynamics (AIBD) model, based on propellant combustion, rotation band engraving, projectile axial motion, and rifling wear models, was established and validated. This model was systematically decomposed into subsystems from a system engineering perspective. The study then detailed the MS-MDO methodology, which included Stage I (MDO stage) employing an improved collaborative optimization method for consistent design variables, and Stage II (Performance Optimization) focusing on the independent optimization of local design variables and performance metrics. The methodology was applied to the AIBD problem. Results demonstrated that the MS-MDO method in Stage I effectively reduced iteration and evaluation counts, thereby accelerating system-level convergence. Meanwhile, Stage II optimization markedly enhanced overall performance. These comprehensive evaluation results affirmed the effectiveness of the MS-MDO method.

Artillery retains its crucial role in modern warfare, not only due to its battlefield importance but also due to advancements in internal ballistics, which significantly improves firing power and accuracy. The artillery’s internal ballistic system is an intricate assembly involving the propellant, the projectile, and the barrel. The sequential processes of propellant ignition and combustion, the engraving of the rotation band upon the projectile, and the projectile’s subsequent axial progression through the barrel constitute the entirety of the internal ballistic launch sequence. Within this firing process, internal ballistic parameters are intricately interlinked; minute variations in any parameter may precipitate substantial effects on the overall launch performance. Consequently, a holistic optimization of the internal ballistic firing process is essential for enhancing the overall performance of the artillery firing sequence.

There are three methods to optimize the internal ballistic firing performance of an artillery. The first one is to build independent optimization models for each stage. For propellant performance enhancement, Gonzalez [

The second method is to construct a comprehensive dynamic model that simultaneously covers the burning of the firing charge, the engraving of the rotation band, and the movement of the projectile inside the barrel [

The third approach is to consider each stage as a different subsystem and adopt a multidisciplinary design optimization (MDO) method to coordinate the shared design variables among these subsystems, so as to achieve the integrated optimization of the performance of the whole process [

Scholars have provided several distributed MDO strategies such as Collaborative Optimization (CO), the Enhanced Collaborative Optimization (ECO), the Concurrent SubSpace Optimization (CSSO), the Bi-Level Integrated System Synthesis (BLISS), the Analytical target cascading (ATC), the Quantitative Safety Assessments (QSA) and so on [

In summary, this paper proposed a multi-stage multidisciplinary design optimal (MS-MDO) method for the artillery firing dynamics problem, aiming at solving the optimization problem of the integrated performance metrics of the artillery internal ballistics with multiple sequential stages. The methodology consists of two stages: Stage I (MDO Stage) employs a modified CO (M-CO) approach to achieve consistency of subsystem design variables and faster system convergence; Stage II, the performance optimization (PO) stage, focuses on optimizing the local variables of the subsystems to improve the system performance metrics.

The rest of this paper is organized as follows: In

The ballistic firing mechanism in artillery, although extremely short in time, fundamentally comprises a series of successive and interactive processes, ignition, engraving of the rotation band, axial movement of the projectile, and ablation and wear of the rifling as a result of multiple firings, as shown in

The firing process begins with the ignition stage, in which the gun’s propellant is ignited, rapidly generating high-temperature and high-pressure gases. These gases accumulate in the combustion chamber and rapidly build up pressure. This pressure serves as the pivotal force propelling the entire firing process. Although this stage is extremely brief, its efficiency and stability have a decisive influence on the projectile’s movement and firing accuracy.

Engraving of the rotation band is the subsequent stage where the projectile is driven by high-pressure gas into contact with the rifling and rotates. The engraving process not only plays a decisive role in projectile stability but also directly affects the degree of barrel wear. The resistance encountered during engraving impacts both muzzle velocity and barrel lifespan.

Next comes the axial movement stage, where the projectile, guided by the rifling, swiftly progresses along the barrel. The projectile’s velocity gradually increases until ejection from the muzzle. The interaction force between the projectile and the barrel, the axial velocity of the projectile, and the change of gas pressure inside the barrel in this process all affect the muzzle dynamic energy and firing accuracy of the projectile.

Ultimately, with repeated firings, the barrel’s rifling undergoes inevitable ablation and wear, significantly impacting the AIBD’s performance. Rifling wear increases the space in the powder combustion chamber and gradually reduces the base pressure of the projectile, while also reducing the effective contact between the projectile and the rifling, decreasing firing accuracy and barrel life.

As each stage of the internal ballistic process has an impact on the final shot, precise control, and optimization is required to ensure efficient and accurate shooting. Therefore, we developed an artillery internal ballistics dynamics (AIBD) Model in the ABAQUS® software environment, which contains a Finite Element Model (FEM) and a VUAMP subroutine. This FEM is based on the 3D models of the projectile with rotation band and the barrel with rifling, and the VUAMP subroutine includes a propellant combustion model with integrated parameters [

The AIBD model is implemented by coupling the FEM model of the projectile and barrel with the VUAMP subroutine of the propellant combustion model, as shown in

The RAME models were integrated into the AIBD model in the form of the UMAT subroutine of ABAQUS, as shown in

The FEM results for the extrusion of an engraving band in different states are shown in

As shown in

The RAME model used in this paper was developed by Wang et al. [

Multiple FEMs, include the RBE model, the PAM model, and the RAME model, are utilized to depict the entire process of AIBD in this study. Since these models primarily concentrate on their inputs and outputs, they are considered “black-box” models and are referred to as subsystem analyzers in this paper. The output response

The goal of this paper is to address the challenge of achieving consistency between design variables in these “black boxes” or subsystems through the MDO method, an effective optimization strategy. The multidisciplinary solution framework for AIBD is illustrated in

Given that obtaining response

Due to the complexity of the AIBD finite element model used in this paper, each computation is time-consuming. Hence, it is essential to employ an effective Design of Experiments (DOE) method to reduce the number of sample points while ensuring a uniform distribution of samples in the design space. We have chosen the Optimal Latin Hypercube Design (OLHD) method, which further enhances the uniformity of the Latin Hypercube Design (LHD) through additional criteria. The OLHD approach ensures that the experimental sample matrix exhibits good projection uniformity across factor intervals and spatial uniformity within the sample space. Consequently, it allows the surrogate model to more accurately fit the relationship between the design variable space and response values.

In this study, a backpropagation (BP) surrogate model for each subsystem is developed utilizing the OLHD scheme and the BP surrogate model approach. The accuracy and generalization ability of these surrogate models are statistically validated using the coefficient of determination,

where

Model | Rotating band engraving model | Projectile axial motion model | Rifling ablation and mechanical wear model | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Perfor |
Engraving resistance | Projectile |
Projectile |
Maximum chamber pressure | Projectile |
Projectile |
Projectile |
Projectile |
Maximum value of |
Maximum value of |

Response symbol | ||||||||||

0.9587 | 0.9249 | 0.9737 | 0.9795 | 0.9328 | 0.9269 | 0.9347 | 0.9216 | 0.9832 | 0.9832 |

This chapter proposes a multi-stage multidisciplinary design optimization (MS-MDO) approach based on collaborative optimization (CO) [

The CO method is a two-layer optimization structure, which decomposes the problem into two levels: system and subsystem, and is suitable for solving multidisciplinary and highly complex engineering optimization problems. This system decomposition makes it possible to deal with global objectives at the system and local objectives at the subsystem at the same time. CO method can coordinate the design variables among subsystems efficiently by setting consistency constraints in order to achieve coordinated optimization of the system and the subsystem.

At the system-level optimization level, the primary objective is to optimize the overall system function

where

At the level of subsystem optimization, the optimization objective is to minimize the consistency constraint

For the

To solve the system-level convergence problem, this paper adopts the dynamic relaxation (DR) method, and the dynamic factor

where

where

In the performance optimization stage, the optimization objective is to use the performance metrics obtained by the subsystem analyzer as the objective function of the individual subsystems, respectively, and its mathematical model is shown in

For the

The implementation flowchart for the developed MS-MDO method is illustrated in

Step 1: Define the optimization design problem, including the objective function, design variables, and constraint functions.

Step 2: Proceed to Stage I, the MDO Stage, and construct the MDO optimization model for the system and subsystems optimization problem.

Step 3: Initialize or update the design variables, starting points, boundaries, and other parameters at both the system and subsystem levels. And sequentially solve each subsystem to determine the design variables for each.

Step 4: Calculate the inconsistency information

Step 5: Check whether the system level meets the convergence consistency tolerance and the maximum iteration limit. If these criteria are satisfied, the iteration stops, and proceeds to output the optimal shared design variables at the system level and the performance indexes for each subsystem. If not, decide whether to apply the modified collaborative optimization method (M-DRCO) or the original collaborative optimization method (DRCO). For M-DRCO, return to Step 3, updating both system and subsystem level starting points to the shared design variables from the previous cycle, this is a significant advancement in the acceleration of system-level convergence within the MDO framework in this paper. For DRCO, also return to Step 3 but continue optimizing with the original starting points.

Step 6: Proceed to Stage II, the performance optimization stage. Construct the performance optimization model, treating the optimal shared design variables as fixed parameters. Redefine the objective function and design variables for each subsystem. Use the performance index of each subsystem as the new objective function and assign the subsystem’s local design variables as the design variables.

Step 7: Optimizing the solution for each subsystem in turn, output all optimal performance indices and design variables for both Stage I and Stage II.

In evaluating the performance of artillery firing dynamics, it is crucial to consider several key aspects: safety of the bore firing process, muzzle kinetic energy, firing accuracy, and service life. For safety, the focus is on minimizing the engaging resistance of the rotation band. Projectile velocity at the muzzle is used to assess muzzle kinetic energy, with higher velocities being preferable [

For the performance assessment of AIBD, four metrics are introduced to comprehensively reflect the firing performance. These metrics are the normalized engraving resistance coefficient

where

In the context of optimizing AIBD, the prudent selection of constraint functions is paramount. This study delineates a comprehensive suite of constraint functions, meticulously considering a spectrum of factors that encompass safety, power, accuracy, and the extended longevity of the firing mechanism.

(1) Constraints of safety

Three constraints are considered for safety: 1) Engraving resistance (

(2) Constraints on the minimum muzzle velocity of the gun.

Minimum muzzle velocity, essential for firing performance, must exceed 950 m/s [

(3) Constraints on artillery disturbance

The gun disturbance is effectively constraining it can minimize errors during the firing process and enhance accuracy. Consequently, limiting gun disturbance is considered a significant constraint in our study [

The constraints on the projectile’s lateral (

(4) Constraints on the life of artillery firing

Rifling wear has been shown to lead to a reduction in muzzle velocity, generally not exceeding 10% over the lifespan of a gun, and is primarily associated with the number of shots fired, as evidenced by studies of Li et al. [

Scholars have delved into the sensitivity of various response indicators and their corresponding design variables of AIBD [

Next, we detail the system-level optimization model and the subsystem-level models within the relaxation-variable-based collaborative optimization framework for artillery firing dynamics.

The system-level optimization model can be represented by

where

The optimization model for the rotation band engraving subsystem is presented as

where the design variable

The optimization model for the projectile’s axial motion subsystem is presented in

where the design variable

The optimization model for the rifling wear subsystem is presented in

where the design variables

Both the RBE and PAM subsystems include local design variables in their respective optimization models as listed in

The optimization model for the rotating band subsystem is delineated in

where the design variable

The performance optimization model for projectile movement within the barrel subsystem is presented in

where the design variable

The boundary of design variables settings for both system-level and subsystem are detailed in

Design |
Mass eccentricity/ |
Groove width/mm | Rifling depth/mm | Rotating band width/ |
Propellant mass/kg | Projectile mass/kg | Forcing cone angle/° | Propellant thickness/ |
---|---|---|---|---|---|---|---|---|

Symbol | ||||||||

Upper boundary/ |
0.30 | 6.90 | 3.10 | 62.00 | 18.00 | 45.80 | 5.71 | 2.40 |

Lower boundary/ |
0.00 | 5.90 | 2.33 | 54.00 | 16.40 | 45.20 | 1.91 | 2.10 |

Start point 3 | 0.15 | 6.4 | 2.7125 | 58 | 17.2 | 45.5 | 3.81 | 1.275 |

Start point 4 | 0.13 | 6.41 | 2.57 | 54.00 | 17.36 | 45.66 | 3.80 | 2.10 |

Subsystem name | Rotation band engraving subsystem | Projectile axial motion subsystem | ||||
---|---|---|---|---|---|---|

Design variable name | Outer diameter of the rotating band/mm | Bourrelet-to-rifling distance/ |
Rotating band location/mm | Powder length/mm | Powder aperture/mm | Propellant chamber/dm^{3} |

Symbol | ||||||

Upper boundary | 81.00 | 0.60 | 192.00 | 15.00 | 0.80 | 27.00 |

Lower boundary | 79.80 | 0.10 | 172.00 | 13.00 | 0.60 | 25.00 |

Sequential Least Squares Optimization (SLSQP) is a local optimization algorithm based on gradient information that solves nonlinear minimization problems with constraints. Scipy.SLSQP has been utilized as the optimizer at both the system and subsystem levels during the MDO Stage to solve complex constraints in order to ensure the convergence stability of the CO method. The results of the system-level calculations using both the original and modified CO methods Stage I are presented in

Starting point | Solution strategy | Iteration count | Evaluation count | |||
---|---|---|---|---|---|---|

Start point 1 | DRCO | 25 | 320 | 7.44E-14 | 1.98E-16 | 5.46E-09 |

M-DRCO | 23 | 303 | 1.08E-07 | 0 | 1.02E-09 | |

Start point 2 | DRCO | 20 | 269 | 5.85E-09 | 1.49E-08 | 2.75E-07 |

M-DRCO | 16 | 220 | 2.76E-07 | 0 | 9.28E-10 | |

Start point 3 | DRCO | 13 | 174 | 2.23E-10 | 1.90E-09 | 7.22E-05 |

M-DRCO | 8 | 112 | 1.22E-08 | 0 | 4.12E-07 | |

Start point 4 | DRCO | 23 | 332 | 3.07E-16 | 3.20E-16 | 1.54E-08 |

M-DRCO | 19 | 220 | 1.65E-12 | 1.65E-12 | 1.36E-09 |

The RBE and PAM subsystems have been transformed into single-objective optimization problems in the PO Stage, unlike the two-level nested optimization challenges in the MDO Stage. Pygmo2.GPPSO (Gaussian Process and Particle Swarm Optimization) [

Starting point | Stages | Rotation band engraving subsystem | Projectile axial motion subsystem | Rifling ablation and mechanical wear subsystem (Only stages I) | ||
---|---|---|---|---|---|---|

Engraving resistance/kN | Projectile velocity at muzzle/(m/s) | Projectile disturbance coefficient | Mechanical wear/mm | Ablative wear/mm | ||

Start point 1 | Stage I | 913.8 | 990.40 | 4.06 | 0.53 | 1.74 |

Stage II | 899.9 | 1005.00 | 1.767 | |||

Start point 2 | Stage I | 1574.7 | 981.83 | 0.70 | 0.53 | 1.71 |

Stage II | 899.9 | 985.82 | 0.54 | |||

Start point 3 | Stage I | 902.6 | 962.19 | 2.73 | 0.48 | 1.67 |

Stage II | 899.9 | 973.43 | 0.58 | |||

Start point 4 | Stage I | 1286.2 | 990.71 | 2.87 | 0.50 | 1.66 |

Stage II | 900.0 | 997.02 | 0.54 |

From ^{−4}). By comparing the number of iterations and evaluations for the original CO and M-CO, it is clear that the M-CO strategy significantly outperforms the original.

Taking the starting point 4 as an example,

Consider that the number of times the algorithm evaluates the objective function (or loss function) is the number of evaluations, especially when “black-box” models are involved (e.g., hydrodynamics or explicit dynamics finite element models), where each evaluation incurs a high computational cost. Consequently, the M-DRCO strategy aimed at reducing the number of evaluations can significantly lower the overall computational costs. As can be seen from the data in column 4 of

Statistical analysis of the data in

An in-depth analysis of

In summary, the MS-MDO method proposed in this paper effectively reduces the number of iterations and evaluations while successfully coordinating the optimization of the design variables in Stage I, and improves the performance metrics in Stage II, which achieves the original intention of this paper.

It can be found in

These validation curves affirm that the finite element model’s outcomes align with the surrogate model’s performance obtained through the MS-MDO method, verifying the surrogate model’s effectiveness. Notably, performance enhancements in Stage II, such as reduced engraving resistance and increased muzzle velocity, significantly surpass those in Stage I. This underscores that the MS-MDO method successfully enhances the overall performance of artillery internal ballistics by effectively coordinating parameters across various subsystem models.

This study introduces a multi-stage multidisciplinary optimization (MS-MDO) method to address the challenge of enhancing the comprehensive performance of the comprehensive artillery internal ballistic dynamics (AIBD) model. The power, accuracy, and lifetime of artillery firing are improved through the MS-MDO method. These research results provide effective methods and strategies for artillery design and performance enhancement and are expected to have a positive impact on the military field and engineering practice.

1) This study established and validated a comprehensive ballistic dynamics model in artillery, integrating the propellant combustion model, the rotation band engraving model, the projectile axial motion, and the rifling wear model. The model was effectively decomposed into subsystems using a system engineering approach. We then applied a multi-stage multidisciplinary optimization (MS-MDO) method to address the optimization challenges of both the reducer model and the AIBD model, demonstrating the method’s efficacy through validation.

2) The Stage I (MDO Stage), featuring the M-DRCO method proposed in this paper, showcased considerable benefits. It significantly reduced the number of iterations and evaluations, thereby cutting computational costs. The method enhanced the convergence speed of the consistency constraint function and effectively coordinated shared variables across different stages’ optimization models.

3) In Stage II (Performance Optimization), we observed substantial improvements in performance metrics over Stage 1 results. Notably, the average engraving resistance was reduced by 18.7% in Stage 2. The normalized projectile disturbance coefficient decreased by 59.8%, and while the increase in muzzle velocity was modest, it achieved satisfactory levels.

The MS-MDO method proposed in this paper has achieved remarkable results and can be further generalized for use in complex engineering problems. Furthermore, its effectiveness is mainly limited to subsystems containing local design variables, and the traditional CO method may be more suitable for systems that do not contain local design variables. In addition, the sensitivity of this study’s method to the starting design point may limit its ability to directly obtain a globally optimal solution in some cases.

Looking ahead, integrating uncertainty factors with multidisciplinary design optimization (MDO) and exploring the combination of multi-objective optimization with MDO is pivotal for enhancing optimization stability and tackling complex engineering challenges.

Powder thickness

Propellant chamber

Rifling depth

Bourrelet-to-rifling distance

Mass eccentricity of the projectile

Rotating band width

Rotating band location

Forcing cone angle

Anode width

Groove width

Powder mass

Outer diameter of the rotating band

The rotating band location

Powder length

Powder aperture

Propellant chamber

Chamber volume-to-bore area ratio

Projectile base pressure

Maximum chamber pressure

Increment time step

Engraving resistance

Projectile axial displacement

Projectile lateral displacement

Projectile vertical displacement

Projectile axial speed

Projectile axial speed at the muzzle

Projectile lateral speed

Projectile vertical speed

Projectile axial angular displacement

Projectile lateral angular displacement

Projectile vertical angular displacement

Projectile axial angular speed

Projectile lateral angular speed

Projectile vertical angular speed

The projectile lateral angular displacements at the muzzle

The projectile vertical angular displacements at the muzzle

The projectile lateral angular speed at the muzzle

The projectile vertically angular speed at the muzzle

Maximum value of ablative wear

Maximum value of mechanical wear

The normalized engraving resistance coefficient,

The normalized initial muzzle velocity coefficient

The normalized projectile disturbance coefficient at the muzzle

The normalized service life coefficient

Regularization factor

Weight coefficient

Global objective function

The primary objective of the overall system function

Shared variable of the system level

The shared parameters from the system pass to the subsystem

The design variables of the

The parameter passed to the system level by the

Local design variables in subsystem

The coupled state variable in the system

The coupled parameters from the system pass to the subsystem

The coupling variable of the

The parameter of the coupling state passed to the system level by the

Consistency constraints

The consistency constraint of the

System-level constraints

The constraint of the

The consistency constraint function for subsystem

Subspace objective function in the subspace

The number of test samples, the number of subsystems

The dynamic relaxation factor

Control factor

The inter-disciplinary inconsistency information

The three inter-disciplinary inconsistency information of the RBE model

The three inter-disciplinary inconsistency information of the PAM model

The three inter-disciplinary inconsistency information of the RAME model

Parameter, copy of design variables provided by the system-level

Parameters, copy of design variables provided by the subspace a

Parameters, copy of design variables provided by the subspace b

Parameters, copy of design variables provided by the subspace c

Output response of the subsystem analyzers

Approximate response of the subsystem analyzers

The actual response of the finite element model for each test sample

The average of the actual responses

The surrogate model response for each test sample

Determination coefficient

The authors express their appreciation to Bing Wang, Fengjie Xu, and Shuli Li of Nanjing University of Science and Technology for developing a finite element mesh model of the artillery internal ballistics dynamic models based on the parameters specified in the experimental design table. And thanks to Yu Fang from Nanjing University of Science and Technology Zijin College for debugging the code. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.

This research was financially supported by the “National Natural Science Foundation of China” (Grant Nos. 52105106, 52305155), the “Jiangsu Province Natural Science Foundation” (Grant Nos. BK20210342, BK20230904), the “Young Elite Scientists Sponsorship Program by CAST” (Grant No. 2023JCJQQT061).

The authors confirm contribution to the paper as follows: study conception and design: Jipeng Xie, Guolai Yang; data collection: Liqun Wang; analysis and interpretation of results: Jipeng Xie, Lei Li; draft manuscript preparation: Jipeng Xie. All authors reviewed the results and approved the final version of the manuscript.

Due to the ongoing nature of the project on which this research is based, the original data cannot be publicly shared. For those interested in this study, please contact the corresponding author of this paper for further information and access to the relevant data.

The authors declare that they have no conflicts of interest to report regarding the present study.