Axisymmetric Adaptive ES-FEM-SPH Coupling Algorithm for Simulating Impact Problems

Abstract

Impact dynamics problems are ubiquitous in various engineering applications, often involving nonlinear phenomena such as material fracture, damage, and fragmentation. It poses significant challenges to numerical simulation methods. To deal with these challenges, this paper develops an adaptive axisymmetric coupling method that combines the edge-based smoothed finite element method (ES-FEM) with smoothed particle hydrodynamics (SPH), referred to as the ES-FEM-SPH method. Initially, the entire computation employs ES-FEM, which effectively alleviates the excessive stiffness inherent in conventional FEM while maintaining high accuracy, particularly when using linear triangular elements. During the simulation, if any element undergoes severe distortion, the algorithm converts it into an SPH particle and continues the computation with SPH automatically. Thus, it can effectively address issues such as large deformation. To validate the efficacy and reliability of the proposed method, this study performs numerical simulations on several representative cases, including Taylor bar impact, projectile penetration into aluminum plates, and flat-nosed projectile impact on metal target plates. The results demonstrate that the adaptive axisymmetric ES-FEM-SPH coupling method exhibits good performance in both computational accuracy and efficiency, making it well suited for numerical simulations of impact-related problems and holding substantial promise for engineering applications.

1. Introduction

Issues pertaining to impact dynamics represent a critical bottleneck in achieving core technological breakthroughs in national defense, military industry, and advanced manufacturing. These issues are ubiquitously encountered in pivotal engineering scenarios, including tank airdrops, missile penetration, and explosive damage, directly influencing the damage efficacy, protective capability, and operational reliability of weaponry and high-end equipment. They provide essential support for the iterative advancement of equipment.

Impact dynamic problems are usually related to nonlinear and multi-field coupling. Therefore, many complex physical phenomena will occur in the impact process, such as phase change induced by high pressure and high speed, multiphase medium interaction, large deformation of materials and structural failure. Thus, numerical simulation has been widely used and has gradually become the main technical means of relevant research. In the numerical solution of impact dynamics, researchers mainly focus on the coupled nonlinearities of geometry, material and boundary conditions. The traditional Eulerian and Lagrangian mesh algorithms have obvious defects in the actual calculation: it is difficult for Eulerian mesh to track the material interface and collect effective dynamic data, while Lagrangian meshes are prone to severe distortion under extreme deformation, which seriously affects computational accuracy and numerical stability. As a result, limited by these inherent defects, traditional methods are not adequate for complex impact simulation. Therefore, it is necessary to develop advanced numerical techniques.

The smoothed particle hydrodynamics (SPH) method has demonstrated significant advantages in simulating impact dynamics problems. The SPH method was originally proposed by Lucy and Gingold and Monaghan [1,2,3], and it has unique advantages in simulating large-scale material deformation, especially in impact conditions. The SPH method combines the interface tracking capability of the Lagrangian method with the large deformation handling capability of the Eulerian method, and uses particles to discretize the medium and solve the control equations. In the field of impact dynamics, Zang et al. [4] explored the penetration characteristics of curved linear shaped charges. They analyzed the formation of jets and their penetration efficiency, clarified the influence of structural parameters on penetration capability, and verified the effectiveness of the algorithm for simulating the penetration of such charges. This provides theoretical and simulation basis for the engineering application of curved linear shaped charges. Zhang et al. [5] used the SPH method to conduct numerical simulation of the dynamic penetration process of shaped charges. This analysis examines the impact of charge structure and operating conditions on jet formation and penetration effectiveness, aiming to elucidate the laws governing dynamic penetration. The results provide a simulation reference for the design and performance optimization of shaped charges. Guan et al. [6] proposed an improved single-layer particle boundary algorithm that avoids boundary penetration issues. By utilizing the SPH method, it completes complex three-dimensional fluid–structure interaction numerical simulations, providing reliable technical support for fluid–structure interaction simulations related to ocean engineering. Chen et al. [7] employed SPH modeling to simulate the penetration of a shaped charge jet into a ceramic–metal dual-layer medium, aiming to elucidate its penetration mechanism and damage patterns. Kim et al. [8] developed a GPU-based parallel SPH solver to achieve precise simulation of ultra-high-speed impact of cohesive jets on concrete structures, enhancing the efficiency and accuracy of numerical calculations for high-speed penetration. Deng et al. [9] relied on the SPH method to build a three-dimensional parallel simulation model for high-speed penetration, completed relevant numerical calculation research, and explored the deformation characteristics, and impact response. This provides a feasible approach and technical reference for efficient simulation analysis of similar high-speed impact problems. Villodi et al. [10] optimized the solid boundary treatment method for the compressible SPH method, and developed a stable and reliable boundary treatment scheme. They improved the boundary calculation defects in numerical simulation. Wang et al. [11] conducted research on the penetration of 12.7 mm projectiles into steel target plates of different strengths using numerical simulation methods. They analyzed the projectile’s motion state, penetration depth, and the damage and failure modes of the target plate. They clarified the specific impact of target material strength differences on the overall penetration effect, providing effective data support for the research on the penetration performance of firearms ammunition and the design of target protection. Li et al. [12] conducted numerical analysis using the SPH method to investigate the two major parameters of shaped charge explosives: burst height and jet inclination. They found that these two parameters significantly alter the jet formation state and motion posture. They clarified the mechanism by which parameter variations affect overall penetration performance, providing a theoretical basis for optimizing the practical application parameters of shaped charge explosives. Wang et al. [13] improved the SPH algorithm and applied it to the simulation study of projectile penetration into targets. They found that the optimized method could effectively mitigate the numerical defects of traditional algorithms, and enhance the simulation accuracy and computational stability of the penetration process. It provides a superior technical approach for numerical simulation of projectile-target impacts. SPH and the finite element method (FEM) [14] show notable complementarity in the applicability of computational methods. The SPH algorithm is good at managing large deformations, addressing the limitations of Lagrangian grid methods. However, its efficiency and accuracy fall short of those of FEM in small deformation analyses, and its boundary handling is comparatively less convenient.

Combining the finite element method with the smoothed particle hydrodynamics (FEM-SPH) method has advantages over the single SPH method. Furthermore, the combined method has also seen rapid development in the simulation of impact dynamics problems in recent years.

The key to the FEM-SPH coupled algorithm lies in the allocation method based on the deformation characteristics of different regions. It applies SPH simulation in large deformation regions and uses FEM calculation in small deformation regions. This strategy combines the advantages of SPH in handling large deformations and FEM in solving small deformations effectively. It also enhances the simulation capability and computational efficiency of impact dynamics problems. Wu et al. [15] constructed an FEM-SPH adaptive coupling model for studying the penetration depth of sapphire. They found that this coupling model exhibited stronger adaptability, and reproduced the impact damage characteristics of sapphire material. They also improved the computational accuracy of simulations for the penetration of hard transparent materials. Liu et al. [16] conducted research using the FEM-SPH coupling method to analyze the variation patterns of perforation penetration under different geological engineering conditions. They explored the characteristics of how formation environments and working conditions affect penetration effects, pore formation morphology, and operational efficiency, providing a reference for the simulation and scheme design of perforation operations in complex formations. Wang et al. [17] applied the three-dimensional FEM-SPH hybrid algorithm to the study of API projectile penetration into target plates. They found that this combined algorithm can balance computational efficiency and simulation accuracy, reproducing the mechanical behavior and failure characteristics throughout the entire penetration process. This provides a practical technical approach for similar ammunition penetration simulation analysis. Fang et al. [18] conducted a study on the penetration characteristics of lanthanum-modified 93 tungsten alloy by combining experiments with adaptive FEM-SPH numerical simulations. They aimed to elucidate the impact penetration mechanism of this alloy material and determine the enhancement effect of modification treatment on the material’s anti-penetration performance, providing a research basis for the development of high-performance penetration-resistant alloy materials. Yang et al. [19] conducted research using the FEM-SPH coupled numerical method to analyze the anti-penetration capability of multi-layer protective structures, and investigate the impact resistance and failure mechanisms under different structural arrangements and material combinations. They summarized effective approaches to enhance the anti-penetration performance of multi-layer protective systems. Su et al. [20] conducted numerical simulations of the penetration process of depleted uranium alloy relying on the FEM-SPH coupled algorithm. They clarified the deformation patterns and failure characteristics of the alloy material during high-speed penetration. They identified its mechanical behavior characteristics during penetration, providing simulation support for related research on alloy penetration applications. Shi et al. [21] developed a three-dimensional multi-resolution FEM-SPH coupled model incorporating boundary penetration prevention technology, designed for studying strong fluid–structure interaction problems. They discovered that this model effectively mitigates particle boundary penetration issues, enhancing the stability and computational accuracy of fluid–structure interaction simulations under extreme operating conditions. Wu et al. [22] conducted numerical research on the penetration of ceramic target plates by long-rod projectiles, aiming to elucidate the impact process and damage evolution laws of long-rod projectiles penetrating ceramic media. They clarified the influence of material properties and working conditions on penetration effects, providing theoretical references for the anti-penetration design of ceramic protective structures. Jiang et al. [23] conducted a study on the ballistic penetration characteristics of columnar ceramic fiber laminated composite armor by combining experimental testing and numerical simulation. They aimed to understand the failure mode and ballistic resistance mechanism of this composite armor under impact, and to clarify the influence of structural composition on overall protective performance. This research provides a basis for the optimal design of new composite protective armor. He et al. [24] utilized the FEM-SPH adaptive algorithm to investigate the ballistic resistance of array-perforated ceramic armor, exploring the influence of perforated structure layout on the armor’s penetration resistance, impact deformation, and damage evolution. This research provides valuable insights for the structural optimization and engineering application of porous ceramic protective armor.

The FEM-SPH coupled algorithm has been applied in numerical simulation of impact problems widely. However, it also faces some challenges, such as the “over-stiffness” of the traditional finite element stiffness matrix, which leads to poor computational accuracy in the case of linear triangular elements. To address this, Liu Guirong et al. with Chen et al. [25] combined the finite element method with the smoothing gradient technique, and introduced the smoothed finite element method (SFEM). Because the smoothing domains’ construction are different, the SFEM has developed into various forms, including those based on elements [25], edges [25], nodes [26], and faces [27]. Ma Yu’e et al. [28] applied the SFEM to the study of thermo-elastic materials under thermo-mechanical coupling effects. Cui et al. [29] proposed the high-order edge-based smoothed finite element method (ES-FEM) based on four-node triangular elements to solve solid mechanics problems. They found that this method effectively improves computational accuracy and convergence performance, and addresses the operational defects of traditional elements. They provided an efficient and reliable new approach for numerical solution of solid mechanics. Chen et al. [30] utilized a hybrid ES-FEM/FEM phase field model to investigate the fracture issues of flexible materials with interfaces. They explored the evolution patterns of crack initiation and propagation at the interfaces of such materials, replicated the entire process of flexible material fracture failure, and refined the numerical analysis method for the fracture behavior of soft material interfaces. Zhou et al. [31] developed a hybrid selective edge-based smoothed PFEM method combined with second-order cone programming for the analysis of large deformations in geotechnical engineering. They explored the computational advantages of this algorithm in scenarios involving large deformations in geotechnical engineering, effectively enhancing the stability and accuracy of complex soil deformation simulations. This provides a new approach for numerical analysis of large deformations in geotechnical engineering. He et al. [32] employed the edge-based smoothed finite element method to establish a fluid–structure interaction numerical model, exploring the application effectiveness of this method in fluid–structure interaction calculations. They found that it could stably simulate the interaction process between fluid and structure, effectively enhancing the reliability and applicability of fluid–structure interaction simulations. Ho et al. [33] proposed the adaptive quadtree edge-based smoothed finite element method for structural limit state analysis, finding that this method can flexibly adjust the computational grid. They determined the critical instability and ultimate bearing state of the structure, and improved the efficiency and accuracy of structural limit mechanical analysis.

The SFEM can achieve higher accuracy results in both linear triangular and tetrahedral elements. It offers more advantages than traditional FEMs. This paper integrates the advantages of SFEM and SPH fully, developing the ES-FEM-SPH coupling algorithm for simulating impact dynamics problems.

High-speed projectiles penetrate into thick metal target plates; it is an important engineering challenge in the field of weapon damage and armor protection. This process exhibits axisymmetric characteristics and is accompanied by strong nonlinear features such as local extreme large deformation, perforation and fragmentation. The traditional FEM is prone to mesh distortion in large deformation areas, leading to computation interruptions, while SPH also lacks efficiency and accuracy in small deformation areas. The FEM-SPH coupling relies on manual partitioning and demonstrates poor accuracy when using linear triangular and tetrahedral elements, thereby struggling to meet the combined accuracy and efficiency demands of engineering design. Hence, this paper proposes an adaptive axisymmetric ES-FEM-SPH coupling algorithm.

First, the entire computational domain is discretized into elements at the initial time step and computed using the ES-FEM. During the numerical simulation, once the deformation of an element meets a specific criterion, the system converts it from an element into a particle and subsequently employs the SPH algorithm to update and compute the particle domain. Ultimately, the element–particle coupling algorithm enables collaborative computation within a unified framework. This paper proposes the adaptive axisymmetric ES-FEM-SPH coupled algorithm. This algorithm can capture the large deformation and fragmentation process in the impact area of the target plate accurately. It can provide more precise numerical support for weapon damage and armor protection design.

2. The Discrete Formulation of the Axisymmetric SPH Method

This study employs the governing equations in the following form when addressing impact problems [34].

Mass conservation equation:

$${\displaystyle \frac{\mathrm{d}{\rho }_i}{\mathrm{d}t}}=-{\displaystyle \frac{{\rho }_{\mathrm{i}}{v}_i^r}{r_i}}+{\displaystyle \frac{1}{2\pi }}{\sum }_{j=1}^N{\displaystyle \frac{m_j}{r_j}}\left(\left(v_i^r-v_j^r\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial r_i}}+\left(v_i^z-v_j^z\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial z_i}}\right)$$

(1)

Momentum conservation equation:

$${\displaystyle \frac{\mathrm{d}{v}_i^r}{\mathrm{d}t}}=-{\displaystyle \frac{{\sigma }_i^{\theta \theta }}{{\rho }_i{r}_i}}+2\pi {\sum }_{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rr}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rr}}}{{\eta }_i^2}}\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial r_i}}+2\pi {\sum }_{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rz}}}{{\eta }_i^2}}\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial z_i}}$$

(2)

$${\displaystyle \frac{\mathrm{d}{v}_i^z}{ \mathrm{d}t}}=2\pi {\sum }_{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rz}}}{{\eta }_i^2}}\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial r_i}}+2\pi {\sum }_{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{zz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{zz}}}{{\eta }_i^2}}\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial z_i}}$$

(3)

Energy conservation equation:

$$\begin{gathered} {\displaystyle \frac{\mathrm{d}{u}_i}{\mathrm{d}t}}={\displaystyle \frac{{\sigma }_i^{\theta \theta }{v}_i^r}{{\rho }_i{r}_i}}+\pi \sum _{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rr}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rr}}}{{\eta }_i^2}}\right)\left(v_j^r-v_i^r\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial r_i}}+\pi \sum _{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rz}}}{{\eta }_i^2}}\right)\left(v_j^r \\ -v_i^r\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial z_i}}+\pi \sum _{j=1}^{ N}{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{rz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{rz}}}{{\eta }_i^2}}\right)\left(v_j^z-v_i^z\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial r_i}} \\ +\pi \sum _{j=1}^N{m}_j\left({\displaystyle \frac{r_j{\sigma }_j^{\mathit{zz}}}{{\eta }_j^2}}+{\displaystyle \frac{r_i{\sigma }_i^{\mathit{zz}}}{{\eta }_i^2}}\right)\left(v_j^z-v_i^z\right){\displaystyle \frac{\partial W_{\mathit{ij}}}{\partial z_i}} \end{gathered}$$

(4)

where $\rho$ represents the three-dimensional density formula, $\eta =2\pi r\rho$ denotes the planar density, $m_j$ is defined as the full circumferential ring mass of particle j, v is for velocity, σ for stress, and e for internal mass energy; r, z and $\theta$ denote the radial, axial, and circumferential coordinates, respectively. W represents a smoothing function, specifically chosen as a B-spline function [35]. The initial smoothing length:

$$h_{i0} = {\alpha }_h\Delta$$

(5)

where ${\alpha }_h$ represents the smoothing length amplification factor, and $\mathbf{\Delta }$ denotes the particle spacing. The smoothing length remains constant throughout the calculation. To stabilize the computation and eliminate numerical oscillations, the SPH method employs the combined artificial viscosity proposed in the literature [36]; the expression is as follows:

$$Q_{\mathit{ij}}=\left\{\begin{array}{cc}{C}_L{\rho }_i{c}_i\left|{\mu }_{\mathit{ij}}\right|+C_Q{\rho }_i{\mu }_{\mathit{ij}}^2,& {\mu }_{\mathit{ij}}<0\\ 0,& {\mu }_{\mathit{ij}}\ge 0\end{array}\right.$$

(6)

where $C_L$ and $C_Q$ are dimensionless coefficients; the values taken in this article are 0.5 and 1. ${\mu }_{\mathit{ij}}={\overline{h}}_{\mathit{ij}}{x}_{\mathit{ij}}^{\alpha }{v}_{\mathit{ij}}^{\alpha }/\left(d_{\mathit{ij}}^{ 2}+{\phi }^2\right)$, ${\overline{h}}_{\mathit{ij}}=0.5\left(h_i+h_j\right), \phi =0.1{\overline{h}}_{\mathit{ij}}, d_{\mathit{ij}}=\sqrt{x_{\mathit{ij}}^{\alpha }{x}_{\mathit{ij}}^{\alpha }},x_{\mathit{ij}}^{\alpha }=x_i^{\alpha }-x_j^{\alpha }$; $h_i$ represents the smoothing length of particle i; ${\rho }_i$ and $c_i$ respectively denote the density and sound velocity of particle i.

In order to incorporate the aforementioned artificial viscosity into the smoothed particle method, the stress tensors in the momentum Equations (2) and (3) and the energy Equation (4) are replaced.

$${\sigma }_i^{\alpha \beta }\to {\sigma }_i^{\alpha \beta }-Q_{\mathit{ij}}{\delta }^{\alpha \beta }, {\sigma }_j^{\alpha \beta }\to {\sigma }_j^{\alpha \beta }-Q_{\mathit{ij}}{\delta }^{\alpha \beta }$$

(7)

where ${\delta }^{\alpha \beta }$ represents the Dirac function.

3. Axisymmetric Smoothed Finite Element Method

3.1. Fundamental Theory of the Smoothed Finite Element Method

This paper introduces the application of the edge-based smoothed finite element method (ES-FEM) [37] for solving structural motion deformation, wherein the entire solid structure is discretized through a linear triangular mesh.

Under the updated Lagrangian formulation, the current configuration is chosen as the reference configuration. In the ES-FEM approach, the velocity $v_c^s$ and the shape function interpolation approximation for the spatial position coordinates $x_c^s$ are

$$v_c^s={\sum }_{I=1}^{n_{\mathrm{node}}}{N}_I{v}_{\mathit{Ic}}^s$$

(8)

$$x_c^s={\sum }_{I=1}^{n_{\mathrm{node}}}{N}_I{x}_{\mathit{Ic}}^s$$

(9)

where $N_I$ represents the shape function at node I, $n_{\mathrm{node}}$ denotes the total number of nodes, $v_{\mathit{Ic}}^s$ denotes the velocity of node I, and $x_{\mathit{Ic}}^s$ signifies the spatial coordinate of node I at the current instant. In the ES-FEM, calculating the smoothed gradient of the field variable initially requires partitioning the smoothing domain, with the condition that these domains must not overlap. In this paper, the configuration of the smoothing domain in ES-FEM is depicted in Figure 1 [37].

On the smoothing domain ${\Omega }_{\mathit{ke}}^s$, the expression for the smoothed velocity gradient is given by

$${\overline{v}}_{c,d}^{\mathrm{s}}={\displaystyle \frac{1}{A_{\mathit{ke}}^s}}{\int }_{{\Omega }_{\mathit{ke}}^s}{\displaystyle \frac{\partial v_c^s}{\partial x_d}}\mathrm{d}\Omega$$

(10)

where $A_{\mathit{ke}}^s$ denotes the area of the smoothing domain ${\Omega }_{\mathit{ke}}^s$. Upon applying the divergence theorem, we obtain the expression for the smoothed velocity gradient as

$${\overline{v}}_{c,d}^s={\displaystyle \frac{1}{A_{\mathit{ke}}^s}}{\int }_{{\Gamma }_{\mathit{ke}}^s}{v}_c^s{n}_d\mathrm{d}\Gamma ={\displaystyle \frac{1}{A_{\mathit{ke}}^s}}{\int }_{{\Gamma }_{\mathit{ke}}^s}{N}_I{v}_{\mathit{Ic}}^s{n}_d\mathrm{d}\Gamma ={\overline{N}}_{I,d}{v}_{\mathit{Ic}}^s$$

(11)

where $n_d$ denotes the unit normal vector on the smoothing domain boundary ${\Gamma }_{\mathit{ke}}^{ s}$, while ${\overline{N}}_{I,d}$ represents the derivative of the smoothed shape function, expressed as

$${\overline{N}}_{I,d}={\displaystyle \frac{1}{A_{\mathit{ke}}^s}}{\int }_{{\Gamma }_{\mathit{ke}}^s}{N}_I{n}_d\mathrm{d}\Gamma$$

(12)

By utilizing the derivatives of smoothed shape functions, it is straightforward to derive quantities such as smoothed strain rates, smoothed Jaumann stress rates, and smoothed Cauchy stresses.

3.2. Axisymmetric Discretized Equations for the ES-FEM

The governing equations for axisymmetric problems can be represented in cylindrical coordinate system as follows [38].

Mass conservation equation:

$${\displaystyle \frac{\mathrm{d}\rho }{\mathrm{d}t}}=-\rho \left({\displaystyle \frac{\partial v^r}{\partial r}}+{\displaystyle \frac{v^r}{r}}+{\displaystyle \frac{\partial v^z}{\partial z}}\right)$$

(13)

Momentum conservation equation:

$$\rho {\displaystyle \frac{\mathrm{d}{v}^r}{\mathrm{d}t}}={\displaystyle \frac{\partial {\sigma }^{\mathit{rr}}}{\partial r}}+{\displaystyle \frac{1}{r}}\left({\sigma }^{\mathit{rr}}-{\sigma }^{\theta \theta }\right)+{\displaystyle \frac{\partial {\sigma }^{\mathit{rz}}}{\partial z}}+{\rho b}^r$$

(14)

$$\rho {\displaystyle \frac{\mathrm{d}{v}^z}{\mathrm{d}t}}={\displaystyle \frac{\partial {\sigma }^{\mathit{rz}}}{\partial r}}+{\displaystyle \frac{{\sigma }^{\mathit{rz}}}{r}}+{\displaystyle \frac{\partial {\sigma }^{\mathit{zz}}}{\partial z}}+{\rho b}^z$$

(15)

Energy conservation equation:

$${\displaystyle \frac{\mathrm{d}u}{\mathrm{d}t}}={\displaystyle \frac{{\sigma }^{\mathit{rr}}}{\rho }}{\displaystyle \frac{\partial v^r}{\partial r}}+{\displaystyle \frac{{\sigma }^{\mathit{rz}}}{\rho }}{\displaystyle \frac{\partial v^r}{\partial z}}+{\displaystyle \frac{{\sigma }^{\theta \theta }}{\rho }}{\displaystyle \frac{v^r}{r}}+{\displaystyle \frac{{\sigma }^{\mathit{rz}}}{\rho }}{\displaystyle \frac{\partial v^z}{\partial r}}+{\displaystyle \frac{{\sigma }^{\mathit{zz}}}{\rho }}{\displaystyle \frac{\partial v^z}{\partial z}}$$

(16)

where r, z and $\theta$ denote the radial, axial, and circumferential coordinates, respectively, $b^r$ and $b^z$ denote the body force per unit mass.

By taking into account the aforementioned momentum conservation equation and employing the weak-form Galerkin variational principle within the framework of the smoothed finite element method, along with the shape function derived from linear polynomial interpolation, we can derive the equations for the r direction and the z direction, respectively.

$$\begin{gathered} {\int }_{\Omega }{\overline{N}}_{I,r}{\overline{\sigma }}^{\mathit{rr}}\mathrm{d}\Omega +{\int }_{\Omega }{\overline{N}}_{I,z}{\overline{\sigma }}^{rz}\mathrm{d}\Omega +{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\theta \theta }}{r}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\mathit{rr}}}{r}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I\rho b^r\mathrm{d}\Omega \\ -{\int }_{{\Gamma }_t}{N}_I{\overline{t}}^r\mathrm{d}\Gamma +{\int }_{\Omega }{N}_I\rho N_J{\ddot{u}}_J^r\mathrm{d}\Omega =0 \forall I\notin {\Gamma }_v \end{gathered}$$

(17)

$$\begin{gathered} {\int }_{\Omega }{\overline{N}}_{I,r}{\overline{\sigma }}^{\mathit{zr}}\mathrm{d}\Omega +{\int }_{\Omega }{\overline{N}}_{I,z}{\overline{\sigma }}^{\mathit{zz}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\mathit{rz}}}{r}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I\rho b^z\mathrm{d}\Omega \\ -{\int }_{{\Gamma }_t}{N}_I{\overline{t}}^z\mathrm{d}\Gamma +{\int }_{\Omega }{N}_I\rho N_J{\ddot{u}}_J^z\mathrm{d}\Omega =0 \forall I\notin {\Gamma }_v \end{gathered}$$

(18)

where ${\overline{\sigma }}^{\mathit{rz}}$, $\overline{t}$, ${\Gamma }_t$, and ${\Gamma }_v$ denote the smoothed Cauchy stress, surface traction force, surface traction boundary, and surface velocity boundary, respectively. ${\overline{N}}_{I,r}$ and ${\overline{N}}_{I,z}$ represent the derivatives of the smoothed shape functions. This work adopts an axisymmetric volume/boundary integration measure for the r-z meridional plane: $\mathrm{d}\Omega =2\pi \mathit{rdrdz}$ and $\mathrm{d}\Gamma =2\mathrm{\pi }r\mathrm{d}{\Gamma }_{\mathit{rz}}$.

The discrete form of the ES-FEM for the momentum equation addressed in this paper can be expressed as follows:

$$M_I{\ddot{u}}_I^s=f_{\mathrm{ext}}^{ \alpha }-f_{\mathrm{int}}^{ \alpha }$$

(19)

$$f_{\mathrm{int}}^{ r}={\int }_{\Omega }{\stackrel{\mathrm{-}}{N}}_{I,r}{\overline{\sigma }}^{\mathit{rr}}\mathrm{d}\Omega +{\int }_{\Omega }{\stackrel{\mathrm{-}}{N}}_{I,z}{\overline{\sigma }}^{\mathit{rz}}\mathrm{d}\Omega +{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\theta \theta }}{r}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\mathit{rr}}}{r}}\mathrm{d}\Omega$$

(20)

$$f_{\mathrm{int}}^{ z}={\int }_{\Omega }{\stackrel{\mathrm{-}}{N}}_{I,r}{\overline{\sigma }}^{\mathit{rz}}\mathrm{d}\Omega +{\int }_{\Omega }{\stackrel{\mathrm{-}}{N}}_{I,z}{\overline{\sigma }}^{\mathit{zz}}\mathrm{d}\Omega -{\int }_{\Omega }{N}_I{\displaystyle \frac{{\overline{\sigma }}^{\mathit{rz}}}{r}}\mathrm{d}\Omega$$

(21)

$$f_{\mathrm{ext}}^{ r}={\int }_{\Omega }{N}_I{\rho b}^r\mathrm{d}\Omega +{\int }_{{\Gamma }_t}{N}_I{\overline{t}}^{ r}\mathrm{d}\Gamma$$

(22)

$$f_{\mathrm{ext}}^{ z}={\int }_{\Omega }{N}_I\rho b^z\mathrm{d}\Omega +{\int }_{{\Gamma }_t}{N}_I{\overline{t}}^{ z}\mathrm{d}\Gamma$$

(23)

where ${\ddot{u}}_I^s$ represents the acceleration component at node I; $M_I$ denotes the lumped mass at node I; $f_{\mathrm{int}}^{ \alpha }$ and $f_{\mathrm{ext}}^{ \alpha }$ represent the equivalent nodal internal force and external force, respectively; ${\overline{t}}^z$ denotes surface traction, respectively; $\rho$ is the density; and ${\Gamma }_t$ denotes the traction boundary. Finally, the central difference method is utilized for time integration to explicitly solve for the structural motion and deformation. Compared to the discrete form of the ES-FEM momentum equation presented in the literature [38], the formula in this paper incorporates the smoothed gradient technique, thereby improving the computational accuracy of linear triangular elements.

4. Adaptive Coupling Algorithm of ES-FEM and SPH

Based on the FE-SPH adaptive coupling algorithm proposed by Xiao Yihua et al. [38], this paper develops an adaptive ES-FEM-SPH coupling algorithm for simulating impact problems. When solving impact problems with the adaptive ES-FEM-SPH coupling algorithm, the edge-based smoothed finite element method is used for the entire computational domain at the initial stage. During the simulation, distorted elements are automatically converted into particles, and the computation is subsequently continued with the SPH method. The adaptive ES-FEM-SPH coupling method involves two key techniques. One is the conversion algorithm that is used to convert FEM elements to SPH particles, and the other is the coupling algorithm of ES-FEM with SPH to handle the interface of the elements region and particles region.

4.1. Adaptive Conversion Algorithm of Converting Elements into Particles

The initial computational model is completely composed of elements, and the elements of large deformation will be converted into particles automatically during the computational process. With the adaptive conversion algorithm of converting elements to particles, the particles will be utilized as little as possible and the efficiency can thus be improved. Xiao et al. developed an algorithm to convert FEM elements to SPH particles when modeling high velocity impact. In this work, we develop a similar adaptive element–particle conversion algorithm, in which FEM elements are automatically converted into SPH particles. The adaptive element–particle conversion algorithm includes four main steps: (1) identifying the elements of large deformation that should be converted into particles, (2) determining the elements treated as imaginary particles for interface treatment, (3) converting elements to real particles, and (4) updating the boundary segment and smoothing domain of the element region. Steps (1), (2), and (3) are the same as those in Xiao et al. [39]. The only difference lies in step (4). After the conversion of elements to particles, the boundary segment and the smoothing domain of the element region should be updated. The method for updating the boundary segment is the same as that developed by Xiao et al. [39]. Additionally, the smoothing domain should be updated using the approach developed by Zhang and Long et al. [40].

For triangular elements, the minimum interior angle can well reflect the degree of element distortion. When the minimum interior angle becomes too small, the element becomes severely distorted, leading to significant deterioration of interpolation accuracy and finite element solution accuracy. Therefore, in the calculation process, the shape of each element should be kept as regular as possible to avoid such distortion. This paper uses the minimum interior angle (MIA) principle as the criterion for element-to-particle conversion [39]. If the minimum interior angle of an element is less than a specified critical value, the element is considered severely distorted and is converted into SPH particles (and removed from the finite element mesh). The critical value is taken as 30° in this work, which is a commonly used threshold ensuring that the remaining elements maintain acceptable aspect ratios [39].

In order to quickly search for coupling interface elements and make the geometric shape of the coupling interface more continuous and regular, this paper adopts a group-based conversion method combined with the principle of MIA developed by Xiao et al. [39]. As shown in Figure 2 [39], monitoring is carried out group by group. When any element in a group of elements meets the minimum internal angle conversion criterion, the entire group of elements will be triggered for synchronous conversion.

The size of the sub-regions for unit grouping is determined as [39]

$${\Delta }_g=\phi \left(2h_{\max }\right)$$

(24)

where $\phi$ typically takes a value greater than 1, which enables the influence range of particles generated due to element distortion to be well confined within the area covered by the neighboring element group. In impact problems, $\phi$ is generally set to 1.5. h_max represents the maximum smoothing length of the element corresponding to the transformed particles [39]. By using the aforementioned formula, the computational domain is divided into several rectangular subdomains, thus assigning a clear “group label” to each element and achieving element grouping based on geometric location.

After determining the element to be converted, the algorithm updates the element list and generates particles: it removes the original element and creates new particles. The newly added particles directly inherit their mass, density, and stress from the original element, while their coordinates and velocity are taken as the arithmetic mean of the node values. The radius and smoothing length of the particles are determined as

$$r_I=\sqrt{A_E}/2, h_I={\delta }_h\sqrt{A_E}$$

(25)

where $r_I$ and $h_I$ are the radius and smoothing length of the newly added particles, $A_E$ is the area of the triangular element, and ${\delta }_h$ is the smoothing length amplification factor.

4.2. Coupling Schemes of ES-FEM with SPH

In the present adaptive ES-FEM-SPH coupling algorithm, for areas with large deformation, FEM elements will be automatically converted to SPH particles using the aforementioned adaptive element–particle conversion algorithm. Then the domain consists of two regions: the FEM region with small deformation, and the SPH region with large deformation. Hence, the treatment of the interface between the FEM region and SPH region is a key technique to maintain the consistency of stress and velocity across the interface. The interface includes the element–particle coupling interface within the same object, as well as contact interfaces between different objects. The contact interfaces include element–element contact, element–particle contact, and particle–particle contact. For the element–particle coupling interface, the imaginary particles scheme proposed by Xiao et al. [39] is adopted. For element–element contact and element–particle contact, the contact algorithm proposed by Johnson et al. [41,42] is referenced. Particle–particle contact is handled by referring to the particle–particle penalty contact algorithm proposed by Campbell et al. [43].

4.3. Update Cell Boundaries and Smoothing Domains

After converting the elements into particles, it is necessary to update the boundaries and smoothing domains of the element region. The boundary update adopts the method developed by Xiao Yihua [39]. Additionally, the smoothing domain should be updated using the approach developed by Zhang and Long et al. [40,44]. As shown in Figure 3 [44], for example, the triangle element N3-N4-N5 is converted into a particle. Then smoothing domains based on edges of N3-N4, N4-N5 and N5-N3 should be updated. There are two situations for updating smoothing domains: (1) If the smoothing domain is based on a boundary edge, such as smoothing domain C1-N3-N4 based on the boundary edge N3-N4 and smoothing domain C1-N4-N5 based on the boundary edge N4-N5, it should be removed. (2) If the smoothing domain is based on the inner edge, such as smoothing domain C1-N5-C2-N3 based on the inner edge N3-N5, it should remove the sub-region (N5-N3-C1) of the element converted to a particle and become the smoothing domain (N5-C2-N3) based on boundary edge (N5-N3) [44].

5. Numerical Examples

In the axisymmetric penetration examples, the domain is discretized using three-node linear triangular elements solved by ES-FEM. When the minimum interior angle of an element falls below 30°, the element is marked for conversion. A group-based strategy is adopted: if any element in a group satisfies the MIA criterion [39], all elements in that group are converted to SPH particles and removed.

5.1. Taylor Rod Impact Problem

In this example, the adaptive ES-FEM-SPH coupling algorithm is used to numerically simulate the classical Taylor bar impact problem, in which a metal round bar strikes a rigid plane at high velocity. By comparing the simulated deformation contour dimensions of the rod with experimental measurement results, the material constitutive parameters are calibrated, and the accuracy and reliability of the coupled algorithm in simulating impact problems are verified.

At the initial time step, the circular rod is discretized into 948 finite elements. The numerical simulation settings followed the experimental configuration described in reference [45]. The Taylor rod is an Armco iron circular bar with a diameter of 6.5 mm and a length of 25.4 mm. The constitutive relationship adopts an ideal elastic–plastic model, and the material pressure is determined by the Mie–Grüneisen equation of state. The specific material property parameters are detailed in

Table 1. Material parameters of Amco iron.

$\mathit{\rho }$ (kg/m3)	G/GPa	${\mathit{Y}}_{\mathbf{0}}$/GPa	${\mathit{C}}_{\mathit{s}}$ (m/s)	${\mathit{S}}_{\mathit{s}}$	Γ
7890	80	0.5	3600	1.80	1.81

[46].

Fixed displacement constraints are applied at the bottom edge of the target, where both radial and axial displacements are completely restricted ($u_r=0,{ u}_z=0$). The nodes on the symmetry axis (r = 0) are constrained in the radial direction to satisfy the axisymmetric hypothesis, allowing only axial movement. On the symmetry axis, axisymmetric boundary conditions are also imposed on the SPH particles.

Figure 4 presents the deformation process of the circular rod calculated by the algorithm at four time points of 10 μs, 20 μs, 40 μs, and 60 μs. From the image, it can be clearly observed that as the plastic deformation accumulates, the elements at the bottom of the circular rod transform into SPH particles according to the criteria. The coupling interface between the particle zone and the finite element zone has a regular shape and the particle region exhibits good continuity. The simulation is terminated at 60 μs, when the shape of the bar remains unchanged.

The data that can be obtained from Figure 4d are as follows: the characteristic dimensions of the deformed rod include a length $L_c$ of 19.3, the impact-end diameter $D_c$ of 13.6, and a width $W_c$ of 8.5 at a distance of 0.2 $L_0$ from the impact end. Finally, the result is approximately 0.01472 calculated by Equation (26), which is close to the errors of the smoothed particle method in

Table 2. Simulation results of characteristic dimensions after deformation of Taylor rod.

Methodology	Deformed Length L/mm	Diameter at the Impact End D/mm	Width at a Distance of 0.2 L0 from the Impact End W/mm
Experiment [45]	19.8	13.7	8.6
SPH [38]	19.4	13.8	8.5
FEM-SPH [38]	19.1	13.9	8.6
ES-FEM-SPH	19.3	13.6	8.5

and the FEM-SPH adaptive coupling algorithm (0.013 and 0.017) [38].

$$\mathit{Re}=\left(\left|L_c-L_e\right|/L_e+\left|D_c-D_e\right|/D_e+\left|W_c-W_e\right|/W_e\right)/3$$

(26)

where the subscripts “c” and “e” represent the calculated value and the experimental value, respectively.

The results indicate that the ES-FEM-SPH adaptive coupling algorithm developed in this study demonstrates excellent performance in terms of computational accuracy and stability, thereby validating the correctness and feasibility of the proposed method.

5.2. Normal Penetration into Aluminum Plate

Piekutowski et al. [47] carried out a normal penetration test on aluminum plates. The objective of this experiment was to simulate the process of an ogival-nosed projectile normally penetrating a metallic target. The specific configuration was as follows: the projectile was made of 4340 steel, with a diameter of 12.9 mm and a length of 88.9 mm; the ratio of the head curvature radius to the diameter was 3.0. The target plate was a 6061-T651 aluminum plate with dimensions of 304 mm × 26.3 mm (length × thickness). Figure 5 illustrates the computational model for the normal penetration of the aluminum plate, where the initial computational model consists of 11,444 triangular elements.

The material constitutive model in our study is configured as follows: the projectile adopts an ideal elastoplastic model, with material parameters specified as: density ρ = 7.83 g/cm³, Young’s modulus E = 206 GPa, Poisson’s ratio ν = 0.3, and initial yield strength Y₀ = 1.43 GPa. The target is regarded as a circular target with a diameter of 304 mm. Its model follows the Johnson–Cook viscoplastic constitutive equation and the Mie–Grüneisen state equation. The material parameters of the target used in this paper, including density (ρ), Young’s modulus (E), Poisson’s ratio (ν), and the parameters A, B and n (obtained by fitting experimental data in the Johnson–Cook constitutive model), are all taken from reference [47] and can be seen from

Table 3. Material properties of target.

ρ (kg/m3)	E/GPa	υ	CS (m/s)	SS	Γ	CV (J/(kg·K))
2710	69	0.33	5380	1.337	2.1	880
A/GPa	B/GPa	C	n	Tm/K	m	Pmin/GPa	$\dot{{\mathit{\epsilon }}_{\mathbf{0}}}$	T0/K
0.262	0.161	0	0.3156	920	1.0	−0.5	1.0	300

. For other parameters in the model, their values are set in accordance with those in reference [48].

In terms of initial conditions, the entire projectile has a constant initial penetration velocity consistent with experimental tests in reference [47], while all the nodes of the metal target are initialized at zero velocity. For boundary conditions, the right boundary of the target plate is fully fixed with $u_r=0 \mathrm{and} u_z=0$. The nodes on the symmetry axis (r = 0) are constrained in the radial direction to satisfy the axisymmetric hypothesis, allowing only axial movement. On the symmetry axis, axisymmetric boundary conditions are also imposed on the SPH particles.

The group-based conversion method integrated with the MIA principle [39] to determine the elements that should be converted to particles. If one or more elements in an element group satisfy the MIA principle [39] during the computational process, then all elements in the group are converted to particles. In addition, the leapfrog method is used for time integration.

Figure 6 shows deformation images, local enlarged images, and velocity contour plots at four different moments, with an initial velocity of 508 m/s. On the left side of Figure 6 are the deformation images, the middle are the local enlarged images, and the right are the velocity contour plots. From the deformation images and local enlarged images in Figure 6, it can be observed that the material at the contact interface between the projectile and the target undergoes severe distortion due to the huge compressive and shear stresses after the impact. This mechanical state directly triggers the transformation of the elements in this region into particles. The continuous penetration of the projectile causes the contact interface to shift in the direction of the projectile, and the new contact front region repeats the above transformation process.

From the velocity contour plots, it can be observed that at t = 20 μs, the velocity of the projectile’s head is the highest (close to the initial velocity), and the tail maintains its original velocity due to inertia, forming a distinct velocity gradient (decreasing velocity from head to tail). A high-velocity deformation zone appears near the impact point on the target plate, and the surrounding material experiences low-velocity disturbance due to the propagation of stress waves. At t = 40 μs and t = 60 μs, the overall velocity of the projectile tends to be uniform (velocity gradient decreases), penetrating into the target plate at an approximately constant velocity. A conical penetration channel forms inside the target plate, and the velocity of the material surrounding the channel decreases from the center to the edge. At t = 80 μs, it can be seen that the projectile has penetrated through, and the material on the back of the target plate has collapsed due to tensile failure (some target plate fragments are transformed into particles by the bullet’s head in the diagram). The velocity of the projectile’s tail is slightly lower than that of the head (with a minor velocity gradient), and the residual target plate fragments near the bullet’s head still have high velocity. It indicates that the bullet’s residual velocity is high at this time. This suggests that the energy loss of the projectile is small and its penetration capability is strong.

The ES-FEM-SPH adaptive coupling algorithm exhibits excellent region identification accuracy. Specifically, this coupling algorithm can identify important regions where large deformations occur under penetration, and convert only a small number of elements within this local range into particles to participate in subsequent SPH calculations selectively. At the same time, other parts of the computational domain continue to use the finite element method, thus ensuring both computational accuracy and efficiency. The ES-FEM-SPH adaptive coupling algorithm is not only highly accurate, efficient, and continuous, but also successfully adopts the minimum interior angle as the criterion for element transformation.

Figure 7 presents a comparative analysis of three numerical methods (ES-FEM-SPH, FEM-SPH, and SPH) with the experimental results [47]. This figure compares the remaining velocities of the projectile under five initial velocities. The research results indicate that the numerical methods ES-FEM-SPH and SPH are highly consistent with the experimental results [47]. From an error analysis perspective, the average relative errors for all three methods compared to the experimental results are maintained within 3.5% (ES-FEM-SPH: 3.2%; FEM-SPH: 3.4% [38]; SPH: 2.9% [38]), indicating comparable computational accuracy.

5.3. The Plugging Failure in Metal Plates Impacted by Flat-Nosed Projectiles

The numerical example employs the experiment conducted by Børvik et al. [49], involving a flat-nosed projectile impacting a metal plate, as a basis for simulation. As depicted in Figure 8, the flat-nosed projectile labeled No. 1 above, with a diameter of 20 mm and a length of 80 mm, is shown penetrating the metal target plate labeled No. 2 below, which has a thickness of 8 mm. The projectile is made of Arne tool steel (with a mass of 0.197 kg), while the target plate is composed of Weldox460E steel.

A two-dimensional axisymmetric model is employed for the calculations. Both the flat-nosed projectile and the metal target plate are discretized using uniform triangular elements, with the initial computational model comprising 22,400 such elements.

The bilinear hardening model is utilized for the projectile material, with the material parameters detailed in

Table 4. Material parameters of projectile body.

ρ (kg/m3)	E/GPa	${\mathit{Y}}_{\mathbf{0}}$/MPa	υ	Et/MPa
7850	204	1900	0.33	15,000

. For the target material, the Johnson–Cook viscoplastic model combined with the Mie–Grüneisen state equation is applied, and the corresponding material parameters are presented in

Table 5. Material properties of target.

ρ (kg/m3)	E/GPa	υ	CS (m/s)	SS	Γ	CV (J/(kg·K))	ρ (kg/m3)
7850	200	0.33	5380	1.337	2.1	452	7850
A/GPa	B/GPa	C	n	Tm/K	m	Pmin/GPa	$\dot{{\mathit{\epsilon }}_{\mathbf{0}}}$	T0/K
0.49	0.807	0.114	0.3156	1800	0.94	−0.54	1.0	293

. The target material parameters align with those reported in the literature [50].

Figure 9 presents a comparative analysis between the adaptive ES-FEM-SPH coupling algorithm and experimental results [49], considering eight distinct impact velocities and their corresponding residual velocities. It is evident from the figure that the curves exhibit a good fit. The two fitting curves depicted in the figure are derived from the analytical models proposed by Recht and Ipson [51], obtained through regression analysis of the residual velocity data obtained from both the numerical simulations and experiments, employing the least-squares method.

The expression for the analytical model is given as

$$v_r=a{\left(v_i^p-v_{\mathrm{bl}}^p\right)}^{1/p}$$

(27)

where a, p and $v_{\mathrm{bl}}$ are constants, while $v_i$ and $v_r$ represent the incident velocity and residual velocity, respectively. The physical meaning of the constant $v_{\mathit{bl}}$ in the analytical model is the ballistic limiting velocity.

The fitted values of the analytical model constants (which characterize the ballistic limit velocities) are presented in

Table 6. Fitting results of analytical model constants.

Methodology	${\mathit{V}}_{\mathbf{b}\mathbf{l}}^{\mathit{p}}$ (m/s)	p	a
Experiment [49]	154.3	3.59	0.83
ES-FEM-SPH	155.7	3.28	0.85

. As evident from the table, the ballistic limit velocities derived from fitting the coupled computational and experimental results [49] are 155.7 m/s and 154.3 m/s, respectively. The close agreement between these two sets of values substantiates the reliability of the computational findings. The residual velocity of the flat-nosed projectile exhibits greater sensitivity to the impact velocity. The reason for the reduced residual velocity at lower impact speeds is attributed to the abrupt velocity decrease caused by “slug formation” wherein the slug and the projectile move in unison, thereby increasing the penetration resistance.

Figure 10 presents a comparison between the adaptive ES-FEM-SPH coupling algorithm and experimental data [49]. Two initial launch velocities of the bullet, namely 173.7 m/s and 298.0 m/s, were selected for analysis. The figure clearly shows the calculated curve depicting the variation in the bullet’s velocity over time. Both the calculated and experimental results indicate that the trend of the bullet’s velocity variation is consistent.

Figure 11 presents the dynamic deformation process at four key time points, simulated by the adaptive ES-FEM-SPH coupling algorithm when the incident speed is 173.7 m per second. It was compared with the corresponding experimental results reported in reference [50]. The research results show that the deformation processes of the projectile and the target plate obtained through the simulation are basically consistent with the experimental observations. The projectile undergoes minimal deformation, while the target plate experiences a “punching” effect, resulting in the formation of a nearly cylindrical plug in the impact zone due to high stress. Near the shear zone where the plug connects to the target, the conversion of some elements into particles due to stress concentration suggests that this area represents a concentrated region of impact damage.

A comparison is made between the experimental plug in Figure 12 [49] and the calculated plug in Figure 13. It is evident from the figure that the calculated deformation of the projectile is minimal. The calculated maximum diameter $d_{\mathrm{plm}}$ of the plug is 21.6 mm, and its height $h_{\mathrm{pl}}$ is 7.6 mm, both of which are in good agreement with the experimental results [49].

6. Conclusions

In this paper, an adaptive axisymmetric coupling algorithm is developed by integrating the edge-based smoothed finite element method (ES-FEM) with smoothed particle hydrodynamics (SPH), tailored specifically for simulating impact dynamics problems. In this approach, ES-FEM is used to discretize the entire computational domain at the initial stage. During the simulation, distorted elements are automatically converted into SPH particles, enabling adaptive dynamic partitioning and seamless coupling computation. Three-node triangular elements are adopted to discretize the domain, making the ES-FEM-SPH model well suited for impact problems with complex geometries. Three typical examples are investigated using the proposed algorithm, and the results are compared with those from other sources. Based on the analyses and numerical results, we conclude that the adaptive axisymmetric ES-FEM-SPH coupling algorithm has good applicability and can effectively address impact dynamics problems, providing a feasible numerical simulation approach for engineering challenges such as armor protection and ammunition penetration. In future work, we will conduct further numerical investigations on different target materials and non-ideal penetration conditions to further validate the proposed ES-FEM-SPH coupling algorithm.