Dynamic Response Analysis of Projectile Target Penetration Based on an FE-SPH Adaptive Coupling Method

: The penetration of projectiles into targets has a broad background in engineering. In this work, numerical simulations of the projectile-target penetration problem are conducted using the Finite Element Method (FEM), the Smoothed Particle Hydrodynamics (SPH) and the Finite Element–Smoothed Particle Hydrodynamics Adaptive Coupling Method (FE-SPH ACM) based on the LS-DYNA software package. First, the penetration experiments using aluminum targets and ceramic targets are simulated. The experimental and simulation results show that the FE-SPH ACM has the better accuracy in calculating the debris cloud head velocity and interface velocity, with an error of no more than 4%. Furthermore, we use the FE-SPH ACM to investigate the anti-penetration performance of aluminum/ceramic composite targets in different combinations. We ﬁnd that the reasonable layout can improve the protective performance of multi-layered target, especially composite target plates with ceramic as the front layer. In addition, the ballistic limit velocities for ceramic-aluminum ratios of 3/7, 5/5 and 7/3 are approximately 1300 m/s, 1400 m/s and 1500 m/s, respectively. Obviously, increasing the proportion of ceramic materials can enhance the anti-penetration performance.


Introduction
Penetration usually refers to the process with which a projectile penetrates or perforates an object.The research on penetration issues has involved multiple fields, such as aerospace, material science, weapon design, etc.The study of material response under penetration is of great significance.Common methods for studying penetration problems include theoretical analysis, experimental exploration, and numerical simulation methods.Because penetration problems involve large deformations, high strain rates, transient mechanics, short duration, and physical processes such as high temperature and pressure, penetration involves the destruction, deformation, and failure of projectiles and target plate materials.The fragmentation distribution law of the debris cloud after penetration and the instantaneous changes in parameters such as velocity and energy make it difficult to obtain results using theoretical and experimental methods.Numerical simulation methods are able to describe phenomena such as material damage, deformation, and fracture during penetration, and have become an important method for exploring penetration problems.
In the numerical simulation process of penetration problems, the finite element method (FEM) and smoothed particle hydrodynamics (SPH) method are commonly used methods.FEM is a numerical method based on grids, and has facilitated significant progress in many fields in recent years.Lu [1] simulated the penetration of a vibrating hammer into a large-diameter thin-walled cylinder using the FEM method, and determined the dynamic response and failure mode of the cylinder under different conditions.Liu [2] Metals 2023, 13, 1074 2 of 18 used the FEM method to simulate the inclined penetration of reinforced concrete and obtained the penetration velocity under different inclination angles.The results are in good agreement with the experimental results.Liu [3] studied the anti-penetration performance of polyurethane/steel plate by FEM method and analyzed the residual velocity of the projectile and the energy absorption characteristics of the composite target plate.However, the FEM method also has many problems and limitations in fields such as high-velocity impact and fluid-structure interaction.When phenomena such as large deformations or crack propagation occur, the finite element method often cannot accurately reflect the problem, and the mesh elements are prone to distortion, leading to calculation termination [4,5].When the material undergoes significant deformation, the element can fail, and after deleting the failed element, the contact interface becomes abnormal [6].Furthermore, in numerical calculations, deleting elements can result in loss of mass, energy and momentum, and the impact results cannot be fully described.Therefore, pure finite element method is more suitable for low velocity and small deformation impact problems [7].Improvements have been made in this area through previous research.R.M. Faergestad [8] used a coupling of FEM and DEM to study ultra-high-velocity impacts.The large deformation behavior in the local impact area was simulated using DEM, while other areas were simulated using FEM.They conducted a parametric study and investigated the sensitivity of key model parameters.In FEM, the stiffness matrix is too stiff, and the volume locking of incompressible or nearly incompressible problems leads to poor adaptability of the deformed mesh.Cui [9] proposed a smoothed finite element method based on high-order edges (ES-FEM), which provides good solutions to the aforementioned problems.In addition, to better address the shortcomings of the FEM method, meshless methods have been proposed and developed.Compared with the finite element method, the notable feature of the meshless method is that it does not require grid division.The basic idea is to remove the grid structure in the finite element method and completely replace it with a series of node arrays, which eliminates the constraints of grid initialization and grid reconstruction on the problem, ensures the accuracy of the solution, overcomes the dependence of the finite element on the grid, and shows significant advantages in problems involving mesh distortion and mesh movement [10][11][12][13].The SPH method is a meshless method.Wang [14] proposed an analysis model to predict motion of debris clouds generated by projectiles with different shapes on multi-target plate structures and verified the accuracy of the model using SPH method, which can successfully describe the motion of debris clouds generated by hypervelocity collisions.Ju [15] studied the fragmentation characteristics of concrete based on the SPH method and proposed a predictive model.Although the SPH method does not require meshes and is good at dealing with large deformations, high pressure, high strain rates, and other issues, due to the calculation principle of the SPH method, the smooth length of particles generally has a small value, which leads to low computational efficiency.The FEM and SPH method have their own advantages and disadvantages.In order to effectively combine the advantages of the two, adaptive coupling methods of FEM and SPH (FE-SPH ACM) have been proposed [16][17][18][19] and developed.He [20] reproduced the process of metal projectile penetrating metal target plate by FE-SPH ACM and proposed a statistical analysis method for fragments.Based on the FE-SPH ACM, Chen [21] established a general analysis method for dangerous fragments in debris cloud.Wen [22] analyzed fragmentation patterns under different impact conditions through the FE-SPH ACM based on the propagation and evolution of shock waves.Scazzosi [23] simulated hypervelocity impacts of bullets on ceramics using the FE-SPH ACM and compared it with commonly used methods in the finite element method.Kala [24] simulated penetration problems of concrete-like materials by FE-SPH ACM and demonstrated that more accurate results can be obtained compared with the FEM method in terms of penetration depth and failure area.Cottrell [25] compared three techniques for simulating hypervelocity impact on ceramics: numerical material erosion, mass-conserving adaptive re-meshing, and finite element conversion to meshless particles.
However, current research lacks a systematic understanding of the ability of using different methods to simulate hypervelocity penetration and target plate fragmentation issues.In this paper, numerical simulations of the projectile-target penetration problem are conducted using Finite Element Method (FEM), Smoothed Particle Hydrodynamics (SPH) and Finite Element-Smoothed Particle Hydrodynamics Adaptive Coupling Method (FE-SPH ACM) using the LS-DYNA software package.The applicability of the three methods is compared by combining the experimental and simulation results, and the process of fragment evolution is analyzed.

FE-SPH ACM Principle
The principle of the FE-SPH ACM [26] is to use the FEM for calculation at the initial moment, automatically converting elements in the large deformation area into particles during the calculation process, and using the SPH method for calculation.When the distortion element reaches the criterion set value, it will automatically be converted to an SPH particle.At the same time, the method also incorporates the SPH algorithm, the finite element-particle coupling algorithm and the finite element-particle contact algorithm.The variables of the converted element, such as stress, strain, Internal energy, and damage are transferred to the particle points.The mass, velocity, and center of mass of the particles are the same as those of the original elements, and the particle velocity is calculated from the momentum of the original element.

Exact Solution of Debris Cloud
In response to the debris cloud formed by target penetration, Bless [27] proposed a debris cloud model through theoretical derivation, and obtained three axial characteristic parameters describing the motion of the debris cloud using one-dimensional shock wave theory.These parameters are the head velocity v 1 , interface velocity v 2 , and tail velocity v 3 .According to the shock wave theory in the solid of the hypervelocity impact field, qualitative analysis results show that a strong shock wave will form after the projectile collides with the target.Based on the one-dimensional shock wave theory, the velocity difference of particles before and after the shock wavefront can be obtained, as shown in Formula (1).
where v 0 is the initial impact velocity, ρ 0b and ρ 1b are the densities of the target before and after the impact, respectively.In addition, ρ 0p and ρ 1p are the densities of the projectile before and after the impact, respectively.The particle velocity in the target is u, and that in the projectile is (v 0 − u) after the shock wave impact.Under the influence of sparse waves, the pressure in the material is unloaded to zero and the material further accelerates.Using one-dimensional shock wave theory, the expressions for v 1 , v 2 , and v 3 can be obtained as follows: The above is the Bless model based on one-dimensional shock wave theory to describe the motion of the debris cloud.The velocity calculated by this model matches well with that of a spherical projectile and has moderate agreement with that of a cylindrical projectile.Therefore, the Bless model can be combined with experimental results as an accurate solution for the expansion velocity of the debris cloud to verify the accuracy of the debris cloud velocity calculated using different numerical simulation methods.

Numerical Simulation of Aluminum Projectile-Target Penetration Experiment
As a ductile material, the use of metals has resulted in systematic progress in the research of different projectiles [28,29], cross-sectional shapes [30,31], and impact angles [32,33] during penetration.Therefore, numerical simulation methods have been widely used in ductile target plates.In this section, with reference to Piekutowski's experiment [34], which involved the hypervelocity impact of a spherical projectile on a target plate, two polar aurora air guns were used to launch aluminum bullets impacting an aluminum alloy target plate.By controlling the impact velocity of the aluminum bullets and changing the ratio of the projectile diameter to the target plate thickness, the factors affecting the characteristics of the debris cloud were explored.

Computational Model
In this paper, LS-DYNA finite element software was used to simulate and establish FEM, SPH, and FE-SPH calculation models for the impact of aluminum alloy projectiles on aluminum alloy target plates.The specific geometric parameters are shown in Figure 1.In the figure, the projectile (in red) and the target plate (in blue) are hexahedral Lagrangian elements.The projectile has a diameter of 9.53 mm and is made of 2017-T4 aluminum alloy, with a velocity of v = 6680 m/s.The target plate is a square plate with a width of 50 mm and a thickness of 0.8 mm, made of 6061-T6 aluminum alloy.In the FE-SPH ACM calculation, the keyword DEFINE_ADAPTIVE_SOLID_TO_SPH is used to convert both the projectile and the target plate from FEM elements to SPH particles, and the failure criterion is set.In this paper, the failure criterion is set as the maximum principal strain failure, and the FEM elements are converted to SPH particles after the principal strain reached 1.2.
The Johnson-Cook (JC) strength model and Mie-Gruneisen equation of state were selected to characterize the materials.The Johnson-Cook constitutive equation [35] describes the strengthening of material yield stress under the combined action of material softening and strength reduction caused by strain, strain rate, and temperature rise.The Mie-Gruneisen equation of state is mainly used to describe the thermodynamic behavior of most metals in the solid state.These two models are the most commonly used models in numerical simulations, and are suitable for high-strain-rate situations such as hypervelocity impact.The maximum tensile stress failure criterion and the JC failure model are used to simulate tensile failure and compressive failure, respectively.In previous studies, aluminum alloy has been a relatively common and widely used material, so the material model parameters can be taken from [20].The specific parameters are shown in Table 1.

Comparison of Results Using Different Simulation Methods
Figure 2 shows the evolution process at different times obtained by simulation and calculation using three different methods under an impact velocity of 6680 m/s.From the figure, it can be seen that both the projectile and target plate were shattered after being subjected to the impact of the projectile.Initially, the edge of the impact area was damaged, and then the convexity of the target plate gradually increased, causing element failure and significant deformation of the projectile.Obviously, the simulation results under the FEM method showed that both the target plate and the projectile had deleted elements, which could observe the movement information of large debris but could not simulate the effect of particle splashing.The SPH method was able to obtain a perfect debris cloud, but could not obtain the situation of large debris.The simulation results of FE-SPH ACM combine the characteristics of the above two methods, making it possible to observe the situation of the debris cloud and directly obtain information on large debris that is more accurate than SPH.Based on the obtained parameter information, such as the shape, size, energy, and distribution of debris, statistical analysis can be performed later.In addition, under the FE-SPH ACM, FEM elements need to fail and be converted into SPH particles to form the debris cloud, so it can be clearly seen from the figure that the size of the debris cloud simulated by FE-SPH ACM is smaller than that with SPH, especially for the ricochet debris cloud located in the front end of the target and the debris cloud at the tail of the projectile.
Figure 3 presents the motion curves of the projectile calculated by different simulation methods at an impact velocity of 6680 m/s.From Figure 3a, it can be seen that the velocity curve drops sharply at first, because the projectile has penetrated the target plate, and then the curve becomes stable.At the same time, the remaining velocity of the projectile is highest under the FEM method (approximately 6340 m/s), while it is lowest under the SPH method (approximately 6100 m/s), and the remaining velocity calculated by FE-SPH ACM is about 6250 m/s.This is because under the FEM method, elements of the target plate are deleted after the damage threshold is reached, and the subsequent calculation reduced the loss of the projectile's velocity.However, some elements of the projectile also reach the failure criterion, so under the FE-SPH ACM, some elements of the projectile are converted into SPH particles, leading to a higher remaining velocity of the projectile than under the SPH method.Figure 3b shows the energy curve of the projectile over time.Under the FEM method, the energy curve of the projectile always decreases.Under the SPH method, the energy curve of the projectile stabilizes after a period of decline.Under the FE-SPH ACM, the energy curve of the unconverted FEM elements always shows a decreasing trend until they have all been converted into SPH particles.The energy curve of the converted SPH particles always shows an increasing trend until it has stabilized.Adding the energy curve of the two together, it can be found that the energy curve of the FE-SPH ACM shows the same trend as that of the SPH method, verifying the rationality of this method.
Metals 2023, 13, x FOR PEER REVIEW 6 of 19 energy, and distribution of debris, statistical analysis can be performed later.In addition, under the FE-SPH ACM, FEM elements need to fail and be converted into SPH particles to form the debris cloud, so it can be clearly seen from the figure that the size of the debris cloud simulated by FE-SPH ACM is smaller than that with SPH, especially for the ricochet debris cloud located in the front end of the target and the debris cloud at the tail of the projectile.Figure 3 presents the motion curves of the projectile calculated by different simulation methods at an impact velocity of 6680 m/s.From Figure 3a, it can be seen that the velocity curve drops sharply at first, because the projectile has penetrated the target plate, and then the curve becomes stable.At the same time, the remaining velocity of the projectile is highest under the FEM method (approximately 6340 m/s), while it is lowest under the SPH method (approximately 6100 m/s), and the remaining velocity calculated by FE-SPH ACM is about 6250 m/s.This is because under the FEM method, elements of the target plate are deleted after the damage threshold is reached, and the subsequent calculation reduced the loss of the projectile's velocity.However, some elements of the projectile also reach the failure criterion, so under the FE-SPH ACM, some elements of the projectile are converted into SPH particles, leading to a higher remaining velocity of the projectile than under the SPH method.Figure 3b shows the energy curve of the projectile over time.Under the FEM method, the energy curve of the projectile always decreases.Under the SPH method, the energy curve of the projectile stabilizes after a period of decline.Under the FE-SPH ACM, the energy curve of the unconverted FEM elements always shows a decreasing trend until they have all been converted into SPH particles.The energy curve of the converted SPH particles always shows an increasing trend until it has stabilized.Adding the energy curve of the two together, it can be found that the energy curve of the FE-SPH ACM shows the same trend as that of the SPH method, verifying the rationality of this method.To further validate the accuracy of the simulation results, the numerical simulation results at 10 µs were obtained with reference to the experimental results of Piekutowski [34], as shown in Figure 4.It can be seen from Figure 4a-c that the FEM method can only obtain the final shape of the projectile and target plate, and cannot obtain information about the debris cloud and particles.Both the SPH method and the FE-SPH ACM can obtain the shape of the debris cloud.The distance from the farthest fragment to the target plate is defined as the maximum distance of the debris cloud, denoted as h.The charac- To further validate the accuracy of the simulation results, the numerical simulation results at 10 µs were obtained with reference to the experimental results of Piekutowski [34], as shown in Figure 4.It can be seen from Figure 4a-c that the FEM method can only obtain Metals 2023, 13, 1074 7 of 18 the final shape of the projectile and target plate, and cannot obtain information about the debris cloud and particles.Both the SPH method and the FE-SPH ACM can obtain the shape of the debris cloud.The distance from the farthest fragment to the target plate is defined as the maximum distance of the debris cloud, denoted as h.The characteristic parameters of the front end of the debris cloud, denoted as a 1 , a 2 , and a 3 , were defined, and the errors of the three methods and the experimental method were compared.Meanwhile, according to the debris cloud model derived from theoretical derivation by Bless [27], three characteristic velocities v 1 , v 2 , v 3 of the debris cloud are calculated, and the errors between the computational results of the three methods and the exact solution are compared.As shown in Table 2.It can be seen from the table that the maximum distances calculated by the three methods are similar (45.447 mm for FEM, 44.497 mm for SPH, 44.682 mm for FE-SPH ACM), and the errors are relatively small and within an acceptable range.However, in terms of the characteristic parameters of the debris cloud, the error of the FE-SPH ACM is smaller than that of the SPH.In terms of the characteristic velocities of the debris cloud, the FE-SPH ACM method calculates a larger result than the SPH method, with smaller v 2 error in FE-SPH ACM calculation and smaller v 3 error in SPH calculation.Overall, it can be concluded that the FE-SPH ACM method is both accurate and effective.
To further validate the accuracy of the simulation results, the numerical simulation results at 10 µs were obtained with reference to the experimental results of Piekutowski [34], as shown in Figure 4.It can be seen from Figure 4a-c that the FEM method can only obtain the final shape of the projectile and target plate, and cannot obtain information about the debris cloud and particles.Both the SPH method and the FE-SPH ACM can obtain the shape of the debris cloud.The distance from the farthest fragment to the target plate is defined as the maximum distance of the debris cloud, denoted as h.The characteristic parameters of the front end of the debris cloud, denoted as a1, a2, and a3, were defined, and the errors of the three methods and the experimental method were compared.Meanwhile, according to the debris cloud model derived from theoretical derivation by Bless [27], three characteristic velocities v1, v2, v3 of the debris cloud are calculated, and the errors between the computational results of the three methods and the exact solution are compared.As shown in Table 2.It can be seen from the table that the maximum distances calculated by the three methods are similar (45.447 mm for FEM, 44.497 mm for SPH, 44.682 mm for FE-SPH ACM), and the errors are relatively small and within an acceptable range.However, in terms of the characteristic parameters of the debris cloud, the error of the FE-SPH ACM is smaller than that of the SPH.In terms of the characteristic velocities of the debris cloud, the FE-SPH ACM method calculates a larger result than the SPH method, with smaller v2 error in FE-SPH ACM calculation and smaller v3 error in SPH calculation.Overall, it can be concluded that the FE-SPH ACM method is both accurate and effective.Based on the numerical simulation results of the FE-SPH ACM, information such as the distribution of fragments during the impact process can be obtained.Figure 5 shows the evolution process of large fragments calculated by the FE-SPH ACM.It can be seen from Figure 5 that large fragments mainly exist in the form of FEM elements, and when the maximum principal strain of the elements reaches the failure criterion of 1.2, the FEM elements in the large fragments are converted into SPH particles, resulting in a reduction in the size of the large fragments.Figure 6 shows the fragment distribution maps at different times and the curve of the number of large fragments over time under the FE-SPH ACM.It can be seen from Figure 6 that large fragments mainly distribute in the central area of the bullet hole, i.e., the head of the projectile.In the initial stage of fragment formation, the fragments gather together, and the number of elements is relatively large.Then, some of the elements fail and are converted into SPH particles, and the fragments begin to break into smaller pieces.With increasing time, the number of debris elements tends to exponentially decrease.
the maximum principal strain of the elements reaches the failure criterion of 1.2, the FEM elements in the large fragments are converted into SPH particles, resulting in a reduction in the size of the large fragments.Figure 6 shows the fragment distribution maps at different times and the curve of the number of large fragments over time under the FE-SPH ACM.It can be seen from Figure 6 that large fragments mainly distribute in the central area of the bullet hole, i.e., the head of the projectile.In the initial stage of fragment formation, the fragments gather together, and the number of elements is relatively large.Then, some of the elements fail and are converted into SPH particles, and the fragments begin to break into smaller pieces.With increasing time, the number of debris elements tends to exponentially decrease.In summary, the accuracy and superiority of FE-SPH ACM in simulating the penetration of aluminum alloy projectiles into aluminum alloy target are verified by comparing the numerical simulation results of three methods.Obviously, FE-SPH ACM can simulate the debris cloud and large debris information during the penetration process, making it applicable to high-velocity penetration problems involving aluminum alloy target plates.

Numerical Simulation of Tungsten-Ceramic Projectile-Target Penetration Experiment
Brittle materials are also commonly used as target plates in penetration problems.Ceramic materials, as a type of brittle material, have excellent mechanical properties and are widely used in protective applications [36].In this section, we refer to the experimental work of S. Nemat-Nasser [37], who considered a tungsten alloy flat-headed projectile impacting a pure ceramic target plate at a velocity of 903.9 m/s.

Computational Model
The model is established using LS-DYNA software, where the projectile is modeled as a cylindrical object with a diameter of 6.14 mm and a length of 20.86 mm, and the ma- In summary, the accuracy and superiority of FE-SPH ACM in simulating the penetration of aluminum alloy projectiles into aluminum alloy target are verified by comparing the numerical simulation results of three methods.Obviously, FE-SPH ACM can simulate the debris cloud and large debris information during the penetration process, making it applicable to high-velocity penetration problems involving aluminum alloy target plates.

Numerical Simulation of Tungsten-Ceramic Projectile-Target Penetration Experiment
Brittle materials are also commonly used as target plates in penetration problems.Ceramic materials, as a type of brittle material, have excellent mechanical properties and are widely used in protective applications [36].In this section, we refer to the experimental work of S. Nemat-Nasser [37], who considered a tungsten alloy flat-headed projectile impacting a pure ceramic target plate at a velocity of 903.9 m/s.

Computational Model
The model is established using LS-DYNA software, where the projectile is modeled as a cylindrical object with a diameter of 6.14 mm and a length of 20.86 mm, and the material is tungsten alloy.The target is modeled as a square with dimensions of 101.6 mm × 101.6 mm and a thickness of 12.7 mm, and the material is aluminum oxide ceramic.All elements are hexahedral Lagrangian elements.When using the FE-SPH ACM for calculations, the target plate is set as a keyword (DEFINE_ADAPTIVE_SOLID_TO_SPH) to convert the FEM element into SPH element.
The Johnson-Cook (JC) strength model and Mie-Gruneisen equation of state are chosen to characterize the tungsten alloy material, with parameters taken from references [38,39], as shown in Table 3.The MAT_JOHNSON_HOLMQUIST_CERAMICS model (referred to as the JH2 model), which is commonly used to simulate the mechanical behavior of brittle materials such as ceramics under impact and penetration [40,41], is chosen to characterize the ceramic material.Material model parameters are taken from reference [23] and are shown in Table 3.

Comparison of Results Using Different Simulation Methods
Figure 7 shows the evolution images at different times obtained through different numerical simulation methods at an impact velocity of 903.9 m/s.From the figure, it can be seen that when using the FEM method for calculation, due to the removal of failed ceramic target plate elements, the tungsten alloy projectile only undergoes a small deformation at its head, without significant damage, making it impossible to simulate the effect of particle splashing.When using the SPH method for calculation, since the elements exist in the form of particles, they will not be removed after failure, resulting in the formation of a debris cloud and more obvious deformation of the projectile.However, when using the FE-SPH ACM for calculation, as the elements that fail will transform into SPH particles and continue to participate in the calculation, the projectile will undergo significant deformation, and a debris cloud will be formed.From the figure, it can be seen that both the SPH method and the FE-SPH ACM can obtain the shape of the debris cloud, and the latter can also obtain information about large fragments contained in the debris cloud.In addition, under FE-SPH ACM, the FEM elements need to fail and transform into SPH particles to form the debris cloud, so the size of the debris cloud simulated by the FE-SPH ACM is smaller than that of the SPH method, especially for the ricochet debris cloud located at the front end of the target.Figure 9 presents the motion curves of the projectile calculated using different simulation methods at an impact velocity of 903.9 m/s. Figure 9a is the curve of the velocity change with time, and Figure 9b is the curve of the energy change with time.As can be seen from the figure, the velocity curve of the projectile first decreases, and then the projectile penetrates the target plate, and the velocity curve becomes stable.In the FEM method, some elements will be removed due to failure during the penetration process, so there is not much change in the projectile's velocity.However, in the SPH method, failed elements will not be removed during calculation, and the failed elements in the FE-SPH ACM are transformed into SPH particles.Therefore, the velocity curve under these two methods will experience a significant decline, and the velocity curve under the SPH method will decrease faster.At the same time, due to the high failure criterion of the tungsten alloy projectile, there are not many failed elements in the projectile, so there is little Figure 8 shows the shape of the projectile and the main strain cloud map at different times obtained by simulating through three methods.It can be seen from the figure that the deformation of the projectile mainly occurs in the head, and the degree of deformation of the projectile simulated by the three methods will be different.Under the FEM method, the projectile undergoes an upset, the range of deformation is the greatest, and the reduction in projectile length is the greatest.Under the SPH method, particles form into splinters.Under the FE-SPH ACM method, the projectile peels off in a petal shape and presents a mushroom-head shape, with a small range of deformation and the least reduction in projectile length.Figure 9 presents the motion curves of the projectile calculated using different simulation methods at an impact velocity of 903.9 m/s. Figure 9a is the curve of the velocity change with time, and Figure 9b is the curve of the energy change with time.As can be seen from the figure, the velocity curve of the projectile first decreases, and then the projectile penetrates the target plate, and the velocity curve becomes stable.In the FEM method, some elements will be removed due to failure during the penetration process, so Figure 9 presents the motion curves of the projectile calculated using different simulation methods at an impact velocity of 903.9 m/s. Figure 9a is the curve of the velocity change with time, and Figure 9b is the curve of the energy change with time.As can be seen from the figure, the velocity curve of the projectile first decreases, and then the projectile penetrates the target and the velocity curve becomes stable.In the FEM method, some elements will be removed due to failure during the penetration process, so there is not much change in the projectile's velocity.However, in the SPH method, failed elements will not be removed during calculation, and the failed elements in the FE-SPH ACM are transformed into SPH particles.Therefore, the velocity curve under these two methods will experience a significant decline, and the velocity curve under the SPH method will decrease faster.At the same time, due to the high failure criterion of the tungsten alloy projectile, there are not many failed elements in the projectile, so there is little difference in the final velocity under these two methods.The energy curve of the projectile shows a similar trend to the velocity curve.Furthermore, we compared the residual velocities calculated using three simulation methods with the experimental results [37], as shown in Table 4. From Table 4, it can be seen that the simulation results using the SPH method and FE-SPH ACM are in good agreement with the experimental results (1.03% error for SPH and 1.76% error for FE-SPH), while the calculation results using the FEM method differ significantly from the experimental results (30.5% error for FEM).This verifies the accuracy of the SPH method and FE-SPH ACM.Additionally, based on the numerical simulation results of the FE-SPH ACM, information about the fragments during the penetration process can be obtained.However, since ceramic materials belong to the brittle materials, the number of pieces of large debris is not easy to count, unlike in the case of ductile materials.Therefore, the number of failed FEM elements was counted here.Figure 10 shows the penetration process of the projectile at different times and the curve of the number of failed elements changing with time.As can be seen from Figure 10, during the penetration process of the projectile, the failed elements are mainly concentrated in the head of the projectile, and some of them form large fragments, which eventually transform into SPH particles, forming a debris cloud.
From the curve, it can be seen that the failed elements mainly form at the beginning of the penetration process, and the number of failed elements becomes stable after the projectile passes through the target board.Furthermore, we compared the residual velocities calculated using three simulation methods with the experimental results [37], as shown in Table 4. From Table 4, it can be seen that the simulation results using the SPH method and FE-SPH ACM are in good agreement with the experimental results (1.03% error for SPH and 1.76% error for FE-SPH), while the calculation results using the FEM method differ significantly from the experimental results (30.5% error for FEM).This verifies the accuracy of the SPH method and FE-SPH ACM.Additionally, based on the numerical simulation results of the FE-SPH ACM, information about the fragments during the penetration process can be obtained.However, since ceramic materials belong to the brittle materials, the number of pieces of large debris is not easy to count, unlike in the case of ductile materials.Therefore, the number of failed FEM elements was counted here.Figure 10 shows the penetration process of the projectile at different times and the curve of the number of failed elements changing with time.As can be seen from Figure 10, during the penetration process of the projectile, the failed elements are mainly concentrated in the head of the projectile, and some of them form large fragments, which eventually transform into SPH particles, forming a debris cloud.
From the curve, it can be seen that the failed elements mainly form at the beginning of the penetration process, and the number of failed elements becomes stable after the projectile passes through the target board.In summary, the accuracy and superiority of FE-SPH ACM in simulating the penetration of tungsten alloy projectiles into ceramic target are verified by comparing the numerical simulation results of three methods.This method not only ensures high accuracy and efficiency, but also provides information about the evolution of fragments.Therefore, the FE-SPH ACM can be applied to the numerical simulation of high-velocity penetration of ceramic targets.

Numerical Simulation of the Projectile Composite Target Penetration Experiment Based on the FE-SPH ACM
As mentioned above, the FE-SPH ACM is suitable for the penetration problem of aluminum and ceramic targets.Therefore, in this section we use FE-SPH ACM to investigate the anti-penetration problem of aluminum/ceramic composite targets with different combinations.

Computational Model
The model of a flat-headed projectile impacting a ceramic-aluminum alloy composite target plate was established using LS-DYNA.The projectile was modeled as a cylindrical body with a diameter of 6 mm and a length of 24 mm, and the material was aluminum alloy.The composite target plate was modeled as a square with dimensions of 100 mm × 100 mm, and the thickness and material could be set according to different model.All elements in the model were hexahedral Lagrangian elements.When using the FE-SPH ACM for calculations, the target plate was set as a keyword (DEFINE_ADAP-TIVE_SOLID_TO_SPH) to convert the FEM element into SPH element.

Simulation Analysis of Penetration of Composite Targets in Different Combinations
Firstly, FE-SPH ACM is used to numerically simulate and analyze composite target plates with different combinations.As shown in Figure 11, ceramic-aluminum alloy composite target plates with different combinations are set up.C/A represents ceramic material for the surface panel and aluminum alloy for the back panel, A/C represents aluminum alloy material for the surface panel and ceramic material for the back panel, and A/C/A represents a three-layer composite target plate with two layers of aluminum alloy sandwiching one layer of ceramic material.d1 and d2 represent the thickness of the composite target plate (red denotes the projectile, blue denotes the ceramic material, and green denotes the aluminum alloy material), and the different combinations are set as shown in Table 5.For composite target plates in different combinations, the total thickness of the In summary, the accuracy and superiority of FE-SPH ACM in simulating the penetration of tungsten alloy projectiles into ceramic target are verified by comparing the numerical simulation results of three methods.This method not only ensures high accuracy and efficiency, but also provides information about the evolution of fragments.Therefore, the FE-SPH ACM can be applied to the numerical simulation of high-velocity penetration of ceramic targets.

Numerical Simulation of the Projectile Composite Target Penetration Experiment Based on the FE-SPH ACM
As mentioned above, the FE-SPH ACM is suitable for the penetration problem of aluminum and ceramic targets.Therefore, in this section we use FE-SPH ACM to investigate the anti-penetration problem of aluminum/ceramic composite targets with different combinations.

Computational Model
The model of a flat-headed projectile impacting a ceramic-aluminum alloy composite target plate was established using LS-DYNA.The projectile was modeled as a cylindrical body with a diameter of 6 mm and a length of 24 mm, and the material was aluminum alloy.The composite target plate was modeled as a square with dimensions of 100 mm × 100 mm, and the thickness and material could be set according to different model.All elements in the model were hexahedral Lagrangian elements.When using the FE-SPH ACM for calculations, the target plate was set as a keyword (DEFINE_ADAPTIVE_SOLID_TO_SPH) to convert the FEM element into SPH element.

Simulation Analysis of Penetration of Composite Targets in Different Combinations
Firstly, FE-SPH ACM is used to numerically simulate and analyze composite target plates with different combinations.As shown in Figure 11, ceramic-aluminum alloy composite target plates with different combinations are set up.C/A represents ceramic material for the surface panel and aluminum alloy for the back panel, A/C represents aluminum alloy material for the surface panel and ceramic material for the back panel, and A/C/A represents a three-layer composite target plate with two layers of aluminum alloy sandwiching one layer of ceramic material.d 1 and d 2 represent the thickness of the composite target plate (red denotes the projectile, blue denotes the ceramic material, and green denotes the aluminum alloy material), and the different combinations are set as shown in Table 5.For composite target plates in different combinations, the total thickness of the composite target plate is kept constant, fixed at 10 mm, and the proportion of the surface and back sections is changed.The table shows that there are three different proportions: 3/7, 5/5, and 7/3, which are used to the deformation characteristics and anti-penetration performance of ceramic and aluminum alloy composite target plates in different proportions.
, 13, x FOR PEER REVIEW 14 of 19 and back sections is changed.The table shows that there are three different proportions: 3/7, 5/5, and 7/3, which are used to explore the deformation characteristics and anti-penetration performance of ceramic and aluminum alloy composite target plates in different proportions.Then, the composite target material with a ceramic-to-aluminum alloy ratio of 5/5 is selected as the research object, and the propagation process of stress waves in different structural target plates is analyzed, as shown in Figure 12.Here, the velocity of the projectile is 1500 m/s.From the figure, it can be observed that stress waves propagate from the impact point to the surroundings.Due to differences in impedance, the size of stress waves in different materials can vary significantly, with stress values being higher in ceramic materials than in aluminum alloy materials.There is a sudden decrease in stress waves when they propagate from ceramic materials to aluminum alloy materials, and a sudden increase in stress waves when propagating from aluminum alloy materials to ceramic materials.At the boundary between materials, there is reflection of interface waves.
Figure 13 presents the images of each layer of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.From the figure, it can be seen that when the combination is C/A, at 10 µs, the ceramic elements of the first layer begin to fail and convert into SPH particles, resulting in the generation of cracks in the ceramic section, while the aluminum alloy changes are not significant.As time increases, at 30 µs, the cracks in the ceramic continue to expand, and the elements of the aluminum alloy part begin to fail, creating pits and some of them converting into SPH particles.When the combination is A/C, the aluminum alloy part is first in contact with the projectile.At 10 µs, the elements of the first layer of aluminum alloy fail and convert into SPH particles, creating pits, while some of the elements of the second layer of ceramics fail and convert into SPH particles, resulting in the formation of cracks.Subsequently, at 30 µs, the aluminum alloy part is penetrated, and a large number of ceramic elements fail, leading to the formation of numerous cracks.The evolution process of the composite target plate in the A/C/A combination combines the characteristics of C/A and A/C.At 10 µs, the elements of the first layer of aluminum alloy fail and convert into SPH particles, creating pits, while some of the elements of the second   Then, the composite target material with a ceramic-to-aluminum alloy ratio of 5/5 is selected as the research object, and the propagation process of stress waves in different structural target plates is analyzed, as shown in Figure 12.Here, the velocity of the projectile is 1500 m/s.From the figure, it can be observed that stress waves propagate from the impact point to the surroundings.Due to differences in impedance, the size of stress waves in different materials can vary significantly, with stress values being higher in ceramic materials than in aluminum alloy materials.There is a sudden decrease in stress waves when they propagate from ceramic materials to aluminum alloy materials, and a sudden increase in stress waves when propagating from aluminum alloy materials to ceramic materials.At the boundary between materials, there is reflection of interface waves.
Figure 13 presents the images of each layer of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.From the figure, it can be seen that when the combination is C/A, at 10 µs, the ceramic elements of the first layer begin to fail and convert into SPH particles, resulting in the generation of cracks in the ceramic section, while the aluminum alloy changes are not significant.As time increases, at 30 µs, the cracks in the ceramic continue to expand, and the elements of the aluminum alloy part begin to fail, creating pits and some of them converting into SPH particles.When the combination is A/C, the aluminum alloy part is first in contact with the projectile.At 10 µs, the elements of the first layer of aluminum alloy fail and convert into SPH particles, creating pits, while some of the elements of the second layer of ceramics fail and convert into SPH particles, resulting in the formation of cracks.Subsequently, at 30 µs, the aluminum alloy part is penetrated, and a large number of ceramic elements fail, leading to the formation of numerous cracks.The evolution process of the composite target plate in the A/C/A combination combines the characteristics of C/A and A/C.At 10 µs, the elements of the first layer of aluminum alloy fail and convert into SPH particles, creating pits, while some of the elements of the second layer of ceramics fail and convert into SPH particles, leading to the formation of cracks, while the changes in the third layer of aluminum alloy are not significant.Later, at 30 µs, the aluminum alloy of the first layer is penetrated, a large number of ceramic elements of the second layer fail and form a large number of cracks, and the elements of the third layer of alloy fail and convert into SPH particles, forming pits.etals 2023, 13, x FOR PEER REVIEW 15 of 1 layer of ceramics fail and convert into SPH particles, leading to the formation of cracks while the changes in the third layer of aluminum alloy are not significant.Later, at 30 µs the aluminum alloy of the first layer is penetrated, a large number of ceramic elements o the second layer fail and form a large number of cracks, and the elements of the third laye of aluminum alloy fail and convert into SPH particles, forming pits.Furthermore, we compare the residual velocity of the projectile penetrating the composite target plate with the variation of penetration velocity under different combinations, as shown in Figure 14.From Figure 14a, it can be seen that when the combination is C/A, the projectile cannot penetrate the composite target plate at lower velocities.The ballistic limit velocity of a composite target plate with a ceramic-to-aluminum alloy proportion of 3/7 is between 1200 and 1300 m/s, while that of a composite target plate with a ceramic-toaluminum alloy proportion of 5/5 is between 1300 and 1400 m/s, and that of a composite target plate with a ceramic-to-aluminum alloy proportion of 7/3 is between 1400 and 1500 m/s.When the combination is A/C, as shown in Figure 14b, the projectile penetrates all the composite target plates, and the residual velocity of the projectile after penetration decreases with the increase in the ceramic proportion.When the combination is A/C/A, as shown in Figure 14c, the ballistic limit velocity of the composite target plate with a ceramic-to-aluminum alloy proportion of 3/7 is between 1200 and 1300 m/s, the ballistic limit velocity of the composite target plate with a ceramic-to-aluminum alloy proportion of 5/5 is between 1000 and 1100 m/s, and the ballistic limit velocity of the composite target plate with a ceramic-to-aluminum alloy proportion of 7/3 is between 1300 and 1400 m/s.In addition, we use the Lambert-Jonas approach [42] to describe the ballistic limit analyses.This method is widely used in the analysis of the critical velocity of projectiles.The main equation is expressed as: where v p is the initial velocity of the projectile, v r is the remaining velocity of the projectile, v 50 is the ballistic limit velocity, and α and n are parameters.Typically, n = 2, α = m m+M , m represents the mass of the projectile, and M represents the mass ejected during the impact process.Based on the Lambert-Jonas approach, the calculation results show that the simulation results are in good agreement with the results obtained from the calculation formula.As the ceramic proportion increases, the remaining velocity of the projectile decreases, which reflects the enhancement of the anti-penetration performance of the composite target.Furthermore, we compare the residual velocity of the projectile penetrating the composite target plate with the variation of penetration velocity under different combinations as shown in Figure 14.From Figure 14a, it can be seen that when the combination is C/A the projectile cannot penetrate the composite target plate at lower velocities.The ballistic limit velocity of a composite target plate with a ceramic-to-aluminum alloy proportion of 3/7 is between 1200 and 1300 m/s, while that of a composite target plate with a ceramicto-aluminum alloy proportion of 5/5 is between 1300 and 1400 m/s, and that of a composite target plate with a ceramic-to-aluminum alloy proportion of 7/3 is between 1400 and 1500 m/s.When the combination is A/C, as shown in Figure 14b, the projectile penetrates al the composite target plates, and the residual velocity of the projectile after penetration In summary, the composite target plate under C/A has the strongest resistance to penetration, followed by the composite target plate under the A/C/A.The composite target plate under the A/C has the weakest resistance to penetration.Additionally, the higher the ceramic proportion, the higher the ballistic limit velocity of the composite target plate.Obviously, increasing the ceramic proportion can enhance the anti-penetration performance of the composite target plate.

Figure 2 .
Figure 2. Evolution images obtained at different times using different numerical simulation methods at an impact velocity of 6680 m/s.(a) FEM; (b) SPH; (c) FE-SPH ACM.

Figure 2 .
Figure 2. Evolution images obtained at different times using different numerical simulation methods at an impact velocity of 6680 m/s.(a) FEM; (b) SPH; (c) FE-SPH ACM.Metals 2023, 13, x FOR PEER REVIEW 7 of 19

Figure 3 .
Figure 3.The motion curves of the projectile calculated using different simulation methods at an impact velocity of 6680 m/s.(a) The curve of velocity change with time.(b) The curve of energy change with time, where E1 is the unconverted FEM elements energy under the FE-SPH ACM, and E2 is the converted SPH particles energy under the FE-SPH ACM.

Figure 3 .
Figure 3.The motion curves of the projectile calculated using different simulation methods at an impact velocity of 6680 m/s.(a) The curve of velocity change with time.(b) The curve of energy change with time, where E1 is the unconverted FEM elements energy under the FE-SPH ACM, and E2 is the converted SPH particles energy under the FE-SPH ACM.

Figure 4 .
Figure 4.The results of different numerical simulation methods at 10 µs and the definition of characteristic parameters.(a) FEM, (b) SPH, (c) FE-SPH ACM.h represents the maximum distance of the debris cloud, and a1, a2, and a3 represent the characteristic parameters of the debris cloud.

Figure 4 .
Figure 4.The results of different numerical simulation methods at 10 µs and the definition of characteristic parameters.(a) FEM, (b) SPH, (c) FE-SPH ACM.h represents the maximum distance of the debris cloud, and a 1 , a 2 , and a 3 represent the characteristic parameters of the debris cloud.

Figure 5 .
Figure 5. Schematic diagram of the process of converting FEM elements into SPH particles when the maximum principal strain reaches 1.2 under the FE-SPH ACM.

Figure 5 . 19 Figure 6 .
Figure 5. Schematic diagram of the process of converting FEM elements into SPH particles when the maximum principal strain reaches 1.2 under the FE-SPH ACM. , 13, x FOR PEER REVIEW 9 of 19

Figure 6 .
Figure 6.Fragment distribution maps at different times and the curve of the number of large fragments over time under the FE-SPH ACM.

Metals 2023 , 19 Figure 7 .
Figure 7. Evolution images different times obtained through different numerical simulation methods at an impact velocity of 903.9 m/s.(a) FEM; (b) SPH; (c) FE-SPH ACM.

Figure 8 .
Figure 8. Principal strain cloud maps calculated by different simulation methods at different times.

Figure 7 .
Figure 7. Evolution images at different times obtained through different numerical simulation methods at an impact velocity of 903.9 m/s.(a) FEM; (b) SPH; (c) FE-SPH ACM.

Figure 8 .
Figure 8. Principal strain cloud maps calculated by different simulation methods at different times.

Figure 8 .
Figure 8. Principal strain cloud maps calculated by different simulation methods at different times.

Metals 2023 ,
13, x FOR PEER REVIEW 12 of 19difference in the final velocity under these two methods.The energy curve of the projectile shows a similar trend to the velocity curve.

Figure 9 .
Figure 9.The motion curves of the projectile calculated by different simulation methods at an impact velocity of 903.9 m/s.(a) The curve of velocity change with time.(b) The curve of energy change with time.

Figure 9 .
Figure 9.The motion curves of the projectile calculated by different simulation methods at an impact velocity of 903.9 m/s.(a) The curve of velocity change with time.(b) The curve of energy change with time.

23, 13 , 19 Figure 10 .
Figure 10.Fragment evolution images at different times time-dependent curves of the number of failed elements.

Figure 10 .
Figure 10.Fragment evolution images at different times and time-dependent curves of the number of failed elements.

Figure 12 .
Figure 12.Von Mises stress evolution history of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.(a C/A; (b) A/C; (c) A/C/A.

Figure 12 .
Figure 12.Von Mises stress evolution history of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.(a) C/A; (b) A/C; (c) A/C/A.

Figure 13 .
Figure 13.Images of each layer of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.(a) C/A; (b) A/C (c) A/C/A.

Figure 13 .
Figure 13.Images of each layer of composite target plates with a ceramic aluminum alloy ratio of 5/5 under different combinations at different times under the velocity of 1500 m/s.(a) C/A; (b) A/C; (c) A/C/A.

Table 2 .
The error between the characteristic parameters obtained by different numerical simulation methods and the experiment, and the error between the characteristic velocity and the exact solution.

Table 4 .
Calculation results and errors of residual velocity.

Table 4 .
Calculation results and errors of residual velocity.

Table 5 .
The thickness settings of composite target plates under different combinations.

Table 5 .
The thickness settings of composite target plates under different combinations.