Antipenetration Performance of Multilayer Protective Structure by the Coupled SPH-FEM Numerical Method

. Multilayer composite structures have signifcant advantages in antipenetration protection. Problems such as element distortion are more likely to occur when the FEM is used to simulate the heterogeneous composite structure against penetration with large deformation. A coupled smoothed particle hydrodynamic (SPH)-FEM is proposed to simulate the antipenetration performance of multilayer protective structures under the penetration of high-speed hemispherical-nosed projectiles. Te large deformation and broken areas are calculated by the SPH method, which overcomes the problem of element distortion in FEM. In other areas, the FEM is used to improve the calculation efciency. Te results indicated that simulating multilayer plates using the coupled SPH-FEM can achieve the ballistic limit and deformation that agree with the experiment. Moreover, the deformation of single-layer, in-contact double-layer, spaced double-layer, and sandwich target plates with a core layer of water was studied and discussed. Te infuence of the faceplate and core layer on the penetration resistance performance was discussed in this paper by applying LSDYNA to establish the model of 3D SPH-FEM and calculate the dynamic process. In addition, the relationship between initial-residual velocity, deformation, and damage failure behavior was obtained.


Introduction
In recent years, multilayer protective constructions have been studied by many scholars, and these investigations have included the material, the structure, numerical simulation methods, and so on. Tese factors have a signifcant infuence on the ballistic limit, which can refect the performance of the antipenetration of the multilayer structure. Diferent materials and structures have been studied by scholars, and their properties can help us analyze possible ways of enhancing the ballistic performance of the composite structures. Most of these were acquired by comparing the results of the numerical simulation with those of the experiment [1,2].
For high-speed impact and penetration, the main research methods are experiments and the fnite element method. Wei et al. [3][4][5] investigated how the blunt-, ogvial-, and hemispherical-nosed projectiles penetrated the steel and determined the infuence of the air gap between layers and the number, order, and thickness of layers on the ballistic performance through experiments. Zhao et al. [6] studied the ballistic performance of Sandwich plates with a steel face-layer and aluminum foam core with the FEM; they obtained the relationship between the ballistic limit and face-layer/core thickness and core density. Ali et al. [7] studied the crashworthiness of the Sandwich plate based on the FEM and an experimental method. Ahmadi et al. [8] investigated the high-velocity impact behavior of the Sandwich plate with syntactic foam cores and FML skins through experiments and the FEM. Abbasi and Alavi Nia [9] studied the high-velocity impact behavior of Sandwich plates with AL skins and foam cores through experiments and the FEM and analyzed the infuence of the foam core layer on ballistic resistance. Ren et al. [10,11] conducted a series of experiments, FEM simulation, and theory to study the free vibration dynamic/damage behavior and ballistic resistance of Sandwich plates with metallic skins and soft cores.
However, high-speed penetration has the characteristics of large deformation and fragmentation, and with the improvement of protection design, composite structures, especially heterogeneous composite structures, have gradually become the main structural form of ballistic protection. When the traditional FEM is used to simulate the penetration resistance performance, it is more prone to the problem of large grid distortion, resulting in reduced calculation accuracy and even interruptions. SPH adopts Lagrange particles to describe the deformation, and it is an ideal numerical method in impact mechanics without relying on elements [12][13][14]. It was frst applied in the fuid computation feld and gradually applied to low-and highvelocity impact and penetration. Kwon and Monaghan [15] investigated the sedimentation of dust in a turbulent fuid by using the SPH method, which had a good agreement with experimental results. Deng et al. [16] successfully determined the way to decay and drive turbulence within a two-dimensional rectangular tank with rigid no-slip boundaries using the SPH method. Tis demonstrated that fuid modeled by particles can have a strong contact with a solid boundary.
Tere are many methods to use SPH coupling, which can be used to handle the problem in many felds where many researchers have made a great progress. Khayyer et al. [17] presented the corresponding ISPH-HSPH FSI solver for 3D composite structures and their interactions with incompressible fuids, which can be capable of handling large material anisotropies and discontinuities at material interfaces without the use of any artifcial stabilizers/ smoothing schemes. Feng and Imin [18] presented the KDF-SPH method, which has higher accuracy and better convergence than the traditional SPH method. Zhao et al. [19] put forward the PB-SPH method to simulate the fracture of the rock with preexisting faws, and the results show that this method can be well applied to rock mechanics engineering.
However, since the SPH needs to search for neighboring particles and calculate the physical quantities of each particle, it is time-consuming, so its computational efciency is lower than that of FEM. Terefore, the coupled SPH-FEM technology came into being. Tis review aims to use the coupled SPH-FEM to investigate the multilayer plates of metal steel and the Sandwich plate. Studying the multilayered plate with the coupled SPH-FEM not only needs to address the problem of contact but also needs to determine whether plates using diferent materials are suited to the SPH method. Xue et al. [20] investigated the infuence of the panel thickness on the underwater lowspeed impacting resistance performance of the pyramid lattice structure based on the coupled SPH-FEM method, and the results were in good agreement with those of the experiment. Wu et al. [21] studied rock-breaking behavior under water jet impact with the SPH-FEM, where SPH was used to simulate the water jet and the FEM was adopted to simulate the rock-breaking response, and the results illustrated the rock impact on the rock of the water jet and breaking. Zhong et al. [22] analyzed the dynamic impact response characteristics and inertia efect based on the coupled SPH-FEM and found that the UHPC concrete slab had good impact resistance; the results of the numerical simulation agreed with those of the experiment. Zhou et al. [23] investigated composite laminates impacted by birdstrike using the SPH-FEM, and the damaging modes of laminates and the energy variations of the bird impact on the laminated plate were analyzed. Zhang et al. [24] investigated how the shaped-charge jets impact the double shell, which is steel and SPS, and found that the polyurethane layer protected the second shell based on SPH-FEM. Wang et al. [25] investigated the intersection of the aircraft and the fragment feld under dynamic conditions based on the FEM-SPH method. Scazzosi et al. [26] used the coupled SPH-FEM to investigate the high-velocity impact of a metal ball against the ceramic-based plate. Yongjun et al. [27] investigated the antipenetration performance of ceramic/fber composite targets which were impacted by high-speed alloy spherical projectiles based on the method of combining FEM-SPH and a range of results were obtained. Cheng et al. [28] studied how the tungsten heavy alloy projectile with diferent velocities impacted the B 4 C ceramic target by using the FEM-SPH method. Islam and Shaw [29] used the SPH method to investigate the monolithic plate penetrated by three diferent projectiles which were hemispherical, conical, and ogival, and analyzed the infuence of the six damage models.
Te coupled SPH-FEM was used in many aspects, but more investigations are needed into the impact and penetration of multilayered plates and Sandwich plates [30,31], especially when using diferent materials. Tis paper aimed to investigate the antipenetration performance of single-and multilayer Sandwich plates impacted by a hemisphericalnosed projectile with the coupled SPH-FEM.
In this study, a coupled SPH-FEM simulated the antipenetration performance of single-, double-layer, and Sandwich target plates subjected to high-speed hemispherical-nosed projectiles. Tis paper includes three main contents: (1) coupled SPH-FEM theory, contact algorithm, and constitutive models; (2) validation of the coupled SPH-FEM used on high-velocity penetration for single-and multilayer target plates; (3) and the ballistic behavior and deformation of single-layer, in-contact doublelayer, spaced double-layer, and Sandwich target plates.

Coupled SPH-FEM Theoretical
2.1. SPH Teory. SPH is a kind of meshless method of particles, a partial diferential equation of the complete Lagrange method, and its essence lies in a discrete domain to disperse matter that needs to be calculated into particles with Lagrange properties. Tere is an interaction among particles through the smooth function to realize their mechanical analysis and calculation, and the discrete domain is continuous equation (1). 〈f(r)〉 is the kernel of the function approximation equation [32,33]. 2 Shock and Vibration where r is the space position vector, which represents the position of the particle in the Ω domain, h denotes the smooth length, dr′ indicates the volume element of partial derivatives, W(r − r ′ , h) is the smooth function, and the smooth function is infnite if the h infnitely approaches 0, and this condition meets the Dirac function that δ is approaching infnity. Te size of r c determines the size of the infuence domain of the smooth function. Figure 1 shows an approximate diagram of continuous variables [16], the infuence of each factor in the infuence domain of continuous smooth on the smooth function, and in this diagram r c � κh, where κ denotes the smooth factor, and both κ and h determine the size of the infuence of the domain of the smooth function.
In the infuence domain, equation (3) is the particle approximation, which is the summation of particle estimates for the spatial derivatives of feld functions; the equation indicates the summation of adjacent particles j of all particles i in the infuence domain.
where m j and ρ i are the mass and density of particle j, respectively, and the N represents the number of the particle j in the domain of the particle i. Especially, the smoothing kernel function W should meet the following conditions: Equation (4) is the normalization condition which means that the integral of a smooth function needs to be equal to 1. Equation (5) indicates that smoothing kernel function will have Dirac delta function properties when the smoothing length goes to 0, as the Dirac delta function is difcult to be satisfed. Equation (6) is the condition of compact support, where κ determines the scope of the smooth function.
In this paper, the most common smoothing kernel function W(r i − r j , h) which used by the SPH community, the cubic B-spline was adopted in LSDYNA and is given as follows: in which α D is 1/h, 15/(7πh 2 ), and 3/(2πh 3 ) in one-, two-, and three-dimensional space, respectively. D is the relative distance between two points r i and r j [34]. Te SPH formalism implies a derivative operator; a particle approximation for the derivative operator must be defned. Before giving the defnition of this approximation, we defne the gradient of a function as: where 1 denotes the unit function. Te SPH formulation for hydrodynamics with material strength is expressed as follows: where ω and ζ are space indices, v ω is the velocity component, ρ is the density, σ ωζ is the total stress tensor which consists of pressure and viscous stress, r ζ is the spatial coordinates, and e is the internal energy. Te abovementioned conservation equations can be solved using weak discrete form, which of the conservation equations can be expressed as follows. Te particle approximation of the weak form of the momentum conservation equation is as follows:  Figure 1: Approximation of the continuum variables (Ω: the support domain, and W is the smoothing kernel for particle i). In two-dimensional space, the support domain is a circle with a cutof radius of r c and r ij is the distance between particle i and j [16].
Shock and Vibration 3 2.2. SPH-FEM Coupling Algorithm. Te coupled SPH-FEM is realized using contact algorithms, and there are two coupling parts between FE and SPH: the attachment and the contact couple [34]. In this paper, the contact of particles and fnite elements used tied contact, which adopted the kinematic constraint algorithm, and the penalty algorithm was used for the contact coupling. Te abovementioned interface algorithms are both based on the master-slave scheme [35,36]. In this scheme, the interface includes two sides, which are the slave and master sides, in three dimensions. As shown in Figure 2, the illustration of the interaction of slave point and master segment, which are the particles and fnite elements, respectively, shows n s is the master surface, and the slave point contacts the nearest master segment in the surface.

SPH-FEM Attachment
Algorithm. Te kinematic constraint algorithm plays an important role in the contact between the SPH and FE; Figure 3 shows the process of the kinematic constraint algorithm in the constraint interface [34]. Te algorithm eliminates the normal degree of freedom of nodes by converting between nodes and imposing global constraints, which can reduce the explicit integration time and improve efciency. When the particles were impacted in calculation, the interface of the contact of the slave nodes and master segment (fnite elements) transformed the force to the master surface which were the fnite elements, and caused the interaction of particles and fnite elements. In addition, it will detect and update independently after each time step, and the interpolated contact point (ξ c , η c ) on the master segment will keep the state at each time step. Te following equations show the procedures for the increments of force and mass for the master nodes.
where ϕ i (ξ, η) is the interpolation function, and i is the number of the master node. After the summation of all slave nodes is completed, the acceleration of the k th node of the master segment a k i can be obtained from the following equation: where a k i is the master segment interface and k is the sequence of the nodes.

SPH-FEM Contact Algorithm.
Te contact between the SPH plate and the FE projectile is realized by the penalty method. In the process of the computation, the contact occurs only when the penetration becomes positive, so every single slave node ns will be checked for penetration l through the master surface [35].
where n is a vector normal to the master segment at the contact point (ξ c , η c ). g and r are position vectors drawn to the slave node and master node, respectively. If penetration l occurs (l < 0), an interface force vector f s which is proportional to the magnitude of the penetration is applied between the slave node and its corresponding contact point.
Ten, the interface force applied to the four nodes (i � 1, 2, 3, 4) of the master segment can be expressed as follows: is the interpolation function. k i is the stifness factor for the master segment, f si is the factor, K i is the bulk modulus, A i is the face area of the element, and V i is the volume [35].

Validation of the SPH-FEM Method
In this section, a series of SPH-FEM numerical simulation validations were carried out on multilayer target plate material, and the projectiles were Q235 and 38CrSi steel. Te numerical simulation process adopted the modifed John-son_Cook and Plastic_Kinematic constitutive models; the former was used to simulate the plate, and the latter was used to simulate the projectile. At the same time, water was added between the two layers of target plates to investigate its hindrance efect on projectile impact, which simulated the impact on a liquid tank of the projectile. Terefore, we needed to verify the feasibility and accuracy of the coupled SPH-FEM by comparing it with the experimental results.

Material Parameters of the Target Plate and Projectile.
Te material of the multilayer target plates was Q235 steel. Te constitutive model of the Q235 steel adopted MAT_-MODIFIED_JOHNSON_COOK, and this constitutive model is a well-known version in LSDYNA (material model 107) [37][38][39]. It is expressed as follows [38]: where σ eq and ε eq represent the equivalent stress and equivalent plastic strain, respectively. A, B, n, C, and m are the material constants; ε * eq indicates the dimensionless strain rate; and the homologous temperature is expressed as where T, T r , T m denote absolute temperature, room temperature, and melting temperature, respectively. Te temperature increment from adiabatic heating is calculated as follows: where ρ and C p denote the density and specifc heat, and χ denotes the Taylor-Quinney coefcient, which gives the proportion of work converted into heat [40]. Te modifed Johnson_Cook material model simulates the steel target impacted by the projectile [41], and the parameters of the material model are shown in Table 1 [42]. Te constitutive model of the projectile is Plastic_Kinematic, as shown in Table 2 [43]. Te constitutive model of water adopted the MAT_NULL in LSDYNA (material model 09) in Table 3 [ 44], and this constitutive model has no yield strength and is generally used in fuid material calculation.

Types and Geometry Parameters of Target Plate.
Te coupled SPH-FEM was used to simulate the hemisphericalnosed projectile penetrating a multilayer target plate in this paper; the particle was adopted in the center region of the target plate penetrated by the projectile, which is called the SPH domain, while the FEM was adopted in the remaining plates. Te hemispherical-nosed projectile also adopted the FEM. Figure 4 shows the model states of the diferent target plates; the larger area in the middle is particle distribution, which is called the SPH domain, and the impervious part surrounding it remains in the form of a fnite element mesh, which is called the FEM domain. Te aim of this arrangement was to reduce the amount of calculation while ensuring its accuracy. Te target plate confguration and codes are listed, in Table 4; the symbol "T" represents the thickness of the plate, and "T2" indicates that the thickness of the plate is 1 mm. Te numerical simulation model of the T4 target plate and projectile is shown in Figure 4(a), where the single-layer target plate thickness was 2 mm. According to the character of SPH, the smaller the space between the particles, the larger the interaction among them [45], so the space between two particles of the plate was 0.9 mm. As is shown in Table 5 the relative error was the smallest, and the rest of the plate, which was the FEM domain, had a mesh size of 3 mm. A total of 16,200 particles were used to simulate the vulnerable part, and the sum of the number of remaining mesh nodes of the target plate and the elements of the projectile was 45,936. Te numerical simulation models of the frst and the second  Te numerical simulational model of the T2_6_T2 (1 mm + 6 mm +1 mm) target plate and projectile is shown in Figure 4(c); two 1 mm steel plates separated by 6 mm air space, in which the number of particles was 4418 and the sum of the number of remaining mesh nodes of the target plate and projectile was 57727. Te numerical simulational Table 1: Material constitutive and damage parameters for Q235 steel [42]. Table 2: Material constitutive and damage parameters for projectile [43].
0.33 1900 15000 Table 3: Material constitutive and damage parameters for water [44].   Figure 4(d), which is represented as three identical steel target plates contacted together; each plate thickness was 2 mm, the number of particles was 48,600, and the remaining mesh nodes of the target plate and projectile element numbers were 129,136. Te distance between the projectile and the target plate was 0.1 mm to reduce the movement distance of the projectile before contact. Te projectile was hemispherical-nosed with a diameter of 12.62 mm and a total length of 39.91 mm.

Boundary Conditions.
Te boundary condition for the coupled SPH-FEM was CONTACT_TIED_NODES_SUR-FACE_ CONST-RAINED_OFFSET, which can be used to transfer the force received by the particle to the fnite element nodes. Te contact method adopted by the projectile when penetrating the target plate was CON-TACT_AUTOMATIC_NODES_TO_SURFACE, which ensures a good contact efect between particles and the projectile surface. Te projectile made an active contact with the plate and made contact with every fnite element per node, and the nodes applied forces to the fnite element mesh in the contact process.

Velocity Validation of the SPH-FEM Method.
In this study, the coupled SPH-FEM was used to investigate the antipenetration performance of the target plate. Te feasibility and accuracy of the coupled SPH-FEM were validated by comparing them with experimental results [5]. Te relationship between initial and residual velocities was obtained based on the least squares ftting method, and the analytical model of ballistic limit velocity frst proposed by Recht and Ipson [46] is expressed as follows: where v i is the initial impact velocity of the projectile, v r and v bl are the residual velocity of the projectile and ballistic limit velocity respectively. In this paper, the formula was used to get the ftting data for the residual velocity. Tables 6 and 7 show the comparison of the ballistic limit velocities of the experiment and the numerical simulation for diferent plates, and the results of the ballistic limit of the experiment and the simulation respectively were very close. a � m p /(m p + m pl ) and m p and m pl are the masses of projectiles and plugs, respectively. Both a and p are ftted to the numerical simulation result using the method of least squares [4].
Comparisons of the SPH-FEM numerical and experimental results for T4, T2T2, T2_6_T2, and T4T4T4 target plates are shown in Figures 5(a)-5(d). It can be concluded that the initial-residual velocity of the coupled SPH-FEM method showed good agreement with the experimental initial-residual velocity.

Comparison of the Deformation Curve and Process of Diferent Target
Plates. In order to validate the single-layer target plate deformation accuracy of the coupled SPH-FEM, the deformation results of the SPH-FEM and the experiments were analyzed and compared. Te comparison of the deformation profles of the experiment and the SPH-FEM simulation for the T4 target plate is shown in Figure 6, and the deformation history is shown in Figure 7. It can be seen in Figure 6 that the simulational deformational profles of SPH-FEM had a good agreement with the experimental results, but there were diferences in the maximum deformation and small deformation regions. Tese occurred because the experimental models were rigidly fxed by bolts, and the SPH-FEM numerical simulations were fxed in the four boundary surfaces. In addition, the SPH particle numbers were fnite, and the particle numbers and particle distance had some efect on fragmentation and maximum deformation. It can be seen in Figure 7 that the target plate deformation process was nearly the same between the SPH-FEM simulation and experiment; the only diference was the location of the plug, because the projectile could not   remain completely horizontal during fight in the experiment, so the experimental plug did not move away from the target plate. In order to validate the contact algorithm and deformation process accuracy of the in-contact double-layer target plate, the T2T2 and T2_6_T2 target plates were simulated by the coupled SPH-FEM, and comparisons of the deformation profles of the numerical simulation and the experiment are shown in Figure 8 and Figure9, and the deformation history is shown in Figure 10. We see in Figure 8 that the simulational deformational profles of SPH-FEM had a very good agreement with the experimental deformations; there were diferences in the small deformational region, which was the coupling place of particles and the fnite mesh. It can be seen in Figure 10 that the target plate deformation process was almost the same between the SPH-FEM simulations and the experiment; the only difference was the size of the plug, owing to the infuence of the particle numbers and particle distance. Tus, we can prove that the coupling SPH-FEM accurately simulated the contact and deformational processes of the multilayer contact target plate.

Comparison of Deformation of Numerical Simulation and
Experiment. Te relationship between the hemisphericalnosed projectile's initial-residual velocity, deformation, and damage to target plates was determined. Terefore, we need to compare the deformation results of diferent target plates in the numerical simulation and the experiment. [5] Figure 11 shows the deformation states of the in-contact double-layer target plate T2T2. Figures 11(a)   From the diagrams, we know that they experienced dishing and bending deformation and all produced the plug, which had nearly been broken down upon impact with the relatively higherspeed projectile. Te plugs produced by the numerical simulation were broken because the velocity of the projectile was so high that the force applied to the nodes was great, and after comparing them, the results of the numerical simulation were a good agreement with experimental results.

Antipenetration Analysis of Single-and Double-Layer Sandwich Target Plate
In this section, the results for the single target plate T8, the in-contact double-layer target plate T4T4, the spaced double-layer target T4_12_T4 and the spaced double-layer target T4_water_T4 are investigated based on the validated coupled SPH-FEM. First, the penetration processes of the four types of target plates are presented and analyzed, and the infuences of target plate thickness, air gap, and water gap on the dynamic process are obtained. Second, the initialresidual velocity relationships of the four diferent types of target plate are studied, and the infuences of target plate thickness, air gap and water gap on residual velocity are obtained. Finally, the deformation is investigated, and the infuences of target plate thickness, air gap, and water gap on deformation and damage are obtained.

Models of Single-and Double-Layer and Sandwich Target
Plate. Te coupled SPH-FEM was validated in Section 3. On this basis, the antipenetration performances of the singlelayer, in-contact double-layer, spaced double-layer, and Sandwich target plates are investigated. Te numerical models of the target plate T8 (4 mm), target plate T4T4 (2 mm + 2 mm), and target plate T4_12_T4 (2 mm + 12 mm +2 mm) are shown in Figures 13(a)-13(d), respectively. Te models separately represent the single target plate T8, meaning the thickness is 4 mm, and the aim was to maintain the thickness consistency of the diferent target plates: the in-contact target plates with a thickness of 2 mm and the spaced double-layered target plate with the same 2 mm thickness separated by 12 mm air fuid. Te sum of the number of particles was 32,400, and the number of fnite elements in the target plate and hemispherical-nosed projectile was 87,536.
Te numerical model of the T4_water_T4 (2 mm + 12 mm +2 mm) target plate and projectile is shown in Figure 13(e). Te thickness of the front and back steel target plates of this model was 2 mm, and the thickness of the water in the middle was 12 mm. And between them, the number of particles was 189,399, and the remaining mesh part of the target plate and the projectile body element was 324,336. It should be pointed out that the number of particles in the water was 156,999, and the number of grid nodes of the water tank was 216,000.
Te overall size of the target plate was 170 mm × 170 mm, and the thickness of the target plate difered according to the diferent target plate types. T4 denoted that the thickness of the target plate was 2 mm, and the thickness of the later target plate was the same, while the size of the middle particle part was 90 mm * 90 mm. Te particle spacing of 0.9 mm corresponded to the grid cell size, while the mesh element size of the residual target plate was 3 mm, and the mesh size of the projectile was set to 1.3 mm. Te distance between the projectile and the target plate was 0.1 mm to reduce the movement distance of the projectile before contact. Te hemispherical-nosed projectile had a diameter of 12.62 mm and a total length of 39.91 mm.

Process Analysis of Projectile Penetration Target Plate.
Te penetration process of the hemispherical-nosed projectile on the single-layer target plate T8, the in-contact double-layer target plate T4T4, the spaced double-layer target plate T4_12_T4 with 12 mm air fuid and    T4_water_T4 with 12 mm water fuid are investigated in this section. Figure 14 illustrates the deformation process of the hemispherical-nosed projectile penetrating the single-layer target plate T8 and the in-contact double-layer target plate T4T4. Te 0∼140 μs deformation process of the hemispherical-nosed projectile penetration on the singlelayer target plate T8 is illustrated in Figures 14(a)-14(c). When the velocity of the projectile was great enough, the target plate produced plunger, expansion, bending, and tensile deformation at the same time, and the model incurred shear and tensile damage. Figures 14(d)-14(f ) shows the 0∼140 μs deformation process of the in-contact doublelayer target plate T4T4 penetrated by the hemisphericalnosed projectile. Te target plate also appears to be punched, but compared with the single-layer target plate T8, the overall deformation and fragmentation of the in-contact double-layer target T4T4 were smaller, and as for the ballistic limit velocity, that of the in-contact double-layer target T4T4 was smaller than that of the single-layer target plate T8. We found that the coupled SPH-FEM can well handle the large deformations and ruptures of target plates. Te penetration processes of the hemispherical-nosed projectile on the double-layer target plate T4_12_T4 with 12 mm air fuid and the in-contact double-layer target plate T4T4 are shown in Figure 15. Te 0∼140 μs deformation process of the spaced double-layer target plate T4_12_T4 with 12 mm air fuid is shown in Figures 15(a)-15(c), and the comparison group 0∼140 μs deformation process of incontact double-layer target plate T4T4 are illustrated Figures 15(d)-15(f ). It can be concluded that the T4_12_T4 target plate produced a plunger under the penetration of the hemispherical-nosed projectile, owing to the large enough spacing between the frst and the second layers of target plate; the plunger of the frst-layer plate infuenced   the second target plate with the projectile, and the second plate also produced a plunger. It can be concluded from the illustration that the plunger of the second-layer target plate was bigger than that of the frst-layer target plate. Both of them produced dishing, bending, and tensile deformation. Te fragment sizes of T4_12_T4 and T4T4 were determined by the particle numbers of the disconnected target plate, and it was obvious that the fragment size of T4T4 was bigger than T4_12_T4, thus proving that the antipenetration performance of the spaced double-layer target plate was better than the in-contact double-layer target plate.
Te penetration processes of the hemispherical-nosed projectile on the T4_12_T4 and target plate T4_water_T4 is described in Figure 16. Te 0∼140 μs deformation process of target plate T4_12_T4 spaced double-layer target plate is illustrated in Figures 16(a)-16(c), and the T4_water_T4 Sandwich plate is illustrated in Figures 16(d) and 16(e). Te fragment size of the target plate T4_12_T4 was bigger than the T4_water_T4 plate, but the deformation of T4_water_T4 second layer bigger than T4_water_T4, proving that the antipenetration performance of the spaced double-layer plate with the water gap better than spaced double-layer with air gap.   Figure 17 shows the change process of the water layer in the middle when the projectile penetrated the spaced double-layer plate with the water gap during over 30∼150 μs. Te water layer was dispersed in all directions when the projectile contacted the target plate, owing to the projectile's large diameter, and the target plate was expanded and deformed because of the interaction and extrusion between the upper and lower steel target plates. Figure 18 shows the velocity curve of the projectile, and the initial penetration velocity v i � 295 m/s. Te tilted part of the frst section is the velocity metabolic process of the projectile striking the frst layer of the steel target plate of the frst layer; the slightly fat curve of the second section is the velocity changing process of the projectile striking the water layer, and the larger amplitude curve of the third section represents the velocity changing process of the projectile penetrating the second layer of the steel target plate. Te reason for the large range variation in the third stage was that the velocity of the projectile decreased sharply after it impacted the frst layer and the   water layer of the steel target, and it took a longer time to impact the second layer of the steel target than it did the frst layer.

Penetration Velocity Analysis of Projectile.
Te initialresidual velocity curves of the single-layer target plate T8, incontact double-layer target plate T4T4, spaced double-layer target plate T4_12_T4, and Sandwich plate T4_water_T4 under hemispherical-nosed projectiles are shown in Figure 19. It can be concluded from the diagram that the singlelayer target plate T8 had the maximum ballistic limit velocity. Te ballistic limit velocity of the spaced double-layer target T4_12_T4 with 12 mm air was the smallest, and that of the in-contact double-layer target plate T4T4 was larger than   that of the Sandwich plate T4_water_T4. Because the thickness of the water layer was not high enough and the size of the projectile was too large, the water layer ofered little resistance to it. Trough calculation, the ballistic limit velocity of the Sandwich plate T4_water_T4 was 1.035 times that of the in-contact double-layer target plate T4_12_T4 with the 12 mm air interval. Te ballistic limit velocity and parameters obtained by numerical simulation corresponding to each target plate's working condition are shown in Table 8.

Deformation Analysis of One-Layer, Double-Layer, and
Sandwich Target Plates. Compared with the experiment results, the infuence of the number of layers of the target plate and the air space on the antipenetration performance of the target plate was determined. In this section, the coupled SPH-FEM is used to study the antipenetration performance of the single-layer target plate T8, the incontact double-layer target plate T4T4, and the spaced double-layer target plate T4_12_T4 with the 12 mm air gap. Tus, the antipenetration of the Sandwich plate T4_water_T4 is further studied. Te deformational curve of the single-layer target plate T8 and the upper target plate T4T4 in contact is shown in Figure 20. It can be seen that the deformation of the single-layer target plate T8 was greater than that of the double-layer in-contact target plate T4T4 when the initial velocity of the projectile was close. Te reason is an interaction between the frst and the second layers of the in-contact double-layer target plate T4T4 that ofset part of the deformation, making its deformation smaller than that of the single-layer target plate. Moreover, the ballistic limit velocity of the single-layer target plate T8 was larger, and the impact time of the projectile body on the target plate was longer.
Te deformations of the in-contact double-layer target plate T4T4 and the spaced double-layer target plate T4_12_T4 with a 12 mm air gap are shown in Figure 21. It can be seen that the deformation of the in-contact doublelayer target plate T4T4 was larger than the spaced doublelayer target plate T4_12_T4 with a 12 mm air gap. Te reason is that the ballistic limit velocity of the in-contact doublelayer target plate T4T4 was much larger than the doublelayer target plate T4_12_T4 with the 12 mm air gap. Overall, the projectile penetrated the in-contact double-layer target plate T4T4 for a long time, resulting in a larger deformation. Moreover, the deformation of the second-layer target plate in the spaced double-layer target plate T4_12_T4 was larger than that of the frst-layer target plate because the projectile penetrated the frst-layer target plate and produced a plug that acted on the second-layer target plate, thus resulting in a larger deformation than the frst-layer target plate. As for the deformation curve of the T4_water_T4 Sandwich target plate after impact by the projectile, we see in Figure 22 that the curve of the T4_water_T4 Sandwich target plate and the double-layer air target plate T4_12_T4 with the 12 mm air gap are shown. It can be concluded that the deformation of the steel target plate at the frst layer of the Sandwich target plate T4_water_T4 was much smaller than that at the second layer. Te deformation of the frst and the second layers of the air double-layer target plate T4_12_T4 was larger than that of the steel target plate T4_water_T4 but smaller than that of the second layer. Te reason is that the water layer in the Sandwich target plate T4_water_T4 was squeezed after being impacted by the projectile, which caused the steel target plates on the upper and lower layers to expand and deform. Moreover, the action time of the target plate on the second layer was longer in the direction of projectile body penetration, resulting in larger deformation.

Energy
Analysis of One-Layer, Double-Layer, and Sandwich Target Plates. Te comparison between kinetic energy variables of T8, T4T4, T4_12_T4, and T4_water_T4 plates are shown in Figure 23, it can be seen that the result accords    with the law of kinetic energy. According to the curve changes in Figure 23 can be analyzed that the increase in the impact velocity of projectiles results in an increase in the energy dissipation. Te single-layer (T8) has the best efect on energy absorption, while the energy absorption efect of T4_12_T4 is the worst when the impact velocity is the same. With the increase in initial velocity, the kinetic energy of the projectile changes more.

Conclusions
In this paper, the coupled SPH-FEM was used to study the antipenetration performance of single-and multilayer plates subjected to high-speed hemispherical-nosed projectiles. Te feasibility and accuracy were validated by comparing them with the experimental results, and on this basis, the single-layer and multilayer Sandwich target plates are studied. Te main conclusions of this study were as follows: (1) Te initial-residual velocity of the hemispherical-nosed projectile, the deformational process and damage to single-and multilayer target plates based on the coupled SPH-FEM were compared with the experimental results. Te comparison indicated that the coupled SPH-FEM had high reliability and accuracy, and was suitable for the antipenetration analysis of high-velocity, large deformation, and multilayer hetero structures. (2) Te gap medium greatly infuences the ballistic limit velocity of double-layer target plates. Te single-layer (T8) had the highest ballistic limit velocity, followed by the in-contact double-layer target plate (T4T4), the water gap double-layer target plate (T4_water_T4), and the air gap double-layer (T4_12_T4) had the lowest ballistic limit velocity. (3) Water had the greatest infuence on the deformation of the double-layer target plate, the diference between T4_water_T4 top and bottom plate deformations was the biggest. Te deformation of the T4T4 and T4_12_T4 top and bottom plates was basically equal. (4) Te single-layer (T8) has the best ability for energy absorption, while the energy absorption efect of T4_12_T4 is the worst. With the increase in initial velocity, the kinetic energy of the projectile changes more.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.