Numerical Study on Interaction Between the Water-Exiting Vehicle and Ice Based on FEM-SPH-SALE Coupling Algorithm

Abstract

The icebreaking process of water-exiting vehicles involves complex nonlinear interactions as well as multi-physical field coupling effects among ice, solids, and fluids, which poses enormous challenges for numerical calculations. Addressing the low solution accuracy of traditional grid methods in simulating large deformation and destruction of ice layers, a numerical model was established based on the FEM-SPH-SALE coupling algorithm to study the dynamic characteristics of the water-exiting vehicle on the icebreaking process. The FEM-SPH adaptive algorithm was used to simulate the damage performance of ice, and its feasibility was verified through the four-point bending test and vehicle breaking ice experiment. The S-ALE algorithm was used to simulate the process of fluid/structure interaction, and its accuracy was verified through the wedge-body water-entry test and simulation. On this basis, numerical simulations were performed for different ice thicknesses and initial velocities of vehicles. The results show that the motion characteristics of the vehicle undergoes a sudden change during the ice-breaking. The head and middle section of the vehicle are subject to greater stress, which is related to the transmission of stress waves and inertial effect. The velocity loss rate of the vehicle and the maximum stress increase with the thickness of ice. The higher the initial velocity of the vehicle, the larger the acceleration and maximum stress in the process of the vehicle breaking ice. The acceleration peak is sensitive to the variation in the vehicle’s initial velocity but insensitive to the thickness of the ice.

1. Introduction

The polar region is rich in natural resources and of extremely high strategic value [1,2,3]. The use of water-exiting vehicles for underwater resource exploration in polar regions has broad application prospects. However, the presence of ice in the polar region poses a huge challenge to the floating up and recycling of water-exiting vehicles. For this reason, the vehicle must be capable of breaking ice while undertaking such missions as resource detection and terrain exploration. The water-exiting vehicle breaking ice is a forcible nonlinear dynamic response process, which involves structural failure of the vehicle and ice crack propagation. For this reason, it is characterized by material nonlinearity, geometrical nonlinearity, and contact nonlinearity. To meet the practical needs of vehicles in the polar region, it is of great significance to explore the coupling effect on the water-exiting vehicle breaking ice.

As a typical solid substance, the mechanical properties of ice are affected by multiple factors such as temperature, salinity, and crystal structure [4,5,6]. Building an accurate and reliable constitutive model for ice material lays a foundation for delving into the interaction of the vehicle and ice. Hence, scholars have explored extensively in this field. The elastic/plastic constitutive model is a one-dimensional model based on plasticity, which has been widely applied in the simulation of ice. Xue et al. [7] combined peridynamic and adaptive dynamic relaxation to establish a peridynamic elastic/plastic constitutive model for ice material. After comparing the three-point bending experimental results with the numerical simulation results, they proved that the model delivered an accurate simulation of the ice material damage process without adding any extra rules on damage. Based on the principle of empirical failure, Gao et al. [8] put forward an elastic/plastic ice material model and introduced a cutting-plane algorithm to describe the plastic stress/strain relationship. Furthermore, they adopted a user-defined procedure to embed the model into the LS-DYNA software. The elastic/plastic constitutive model provides a satisfying simulation of ice plasticity damage, but it ignores such characteristics of ice material as strain rate effect, tension, and compression. For this reason, Carney et al. [9] presented a new sea ice model that takes into account strain sensitivity to flow stress, independent failure stress in compression, and ice’s capability of bearing hydrostatic pressure after failure. Yu et al. [10] proposed a phenomenological model based on strain rate correlation to capture the ductile/brittle transition behavior of ice under loading, which was verified by a rigid indentation test.

In order to better describe the crack expansion of ice when it is damaged, some scholars have developed a variety of constitutive models based on the theories related to fracture mechanics. Tong et al. [11] adopted a Johnson–Holmquist-2 constitutive model to illustrate the characteristics of ice material. While taking into account the damage mode of ice material, they determined the relevant parameters of the model through theoretical analysis and experiment and performed the uniaxial compression numerical simulation to prove that the Johnson-Holmquist-2 constitutive model could accurately simulate the damage to sea ice. Polach et al. [12] proposed an ice model based on the Lemaitre damage pattern, which could reflect the microstructure of ice and physical effects under load. Jang et al. [13] employed the explicit finite element method and introduced the Crushable Foam and Drucker–Prager yield criteria to develop a model for ice material with yield strength, damage mode, and dynamic strain rate. This model could accurately predict the contact force. Pandey et al. [14] obtained the variation curve of ice load through a separated Hopkinson pressure bar (SHPB) experiment and established an ice constitutive model containing fracture strength and damage evolution parameters using ABAQUS 2022 software.

Presently, the studies on structures breaking ice could be categorized into three focuses: structures breaking ice in a waterless environment, structures breaking ice and entering water, and structures breaking ice and exiting water. Without the effect of fluid, the first focus can be easily addressed in experiments and simulations. Zhang et al. [15] built a numerical computation model for a rigid sphere pounding the ice and obtained ice damage results with different sphere speeds, ice thicknesses, and boundary conditions. As revealed in the results, when the speed of the sphere increases, the damage to the ice changes from several non-penetrating radial cracks to a mesh of radial and circumferential cracks. Zhou et al. [16] studied the interaction between a six-degree-of-freedom ship and ice ridges using bending and crushing failure modes to simulate ice loads. Cui et al. [17] carried out an experiment to study the crack pattern and damage mechanism of an ice sample subject to vertical impact and found that when the damage caused by the impact head to the ice sample is more severe, the impact load applied to the ice sample is lower if the size of the ice sample and the kinetic energy of the impact are identical.

As for the problem of structures breaking ice and entering water, Ren et al. [18] constructed a numerical computation model for a rigid sphere breaking ice and entering water based on the Boundary Data Immersion Method (BDIM) and Volume of Fluid (VOF). The results show that the motion characteristics of the sphere is affected by the impact speed, the density of the rigid sphere, and the elastic modulus of ice. Wang et al. [19] delved into the influence of head shape and length-to-diameter ratio on ice damage when a vehicle breaks ice and enters water. They noticed that the water environment has little effect on the pattern of ice damage but greater effect on the velocity and acceleration of the vehicle in motion. Cui et al. [20] established a numerical model for the high-speed water entry projectiles based on the coupled Euler–Lagrange (CEL) method. They compared and analyzed the evolution of cavitation in the presence and absence of ice and obtained the effect of ice sheet on the load characteristics of structures.

At present, the main focus of scholars is the structures breaking ice and exiting water by virtue of test and simulation. Zhang et al. [21] explored the influence of broken ice on the hydrodynamic properties of a water-exiting vehicle. As shown in the results, the deflection angle of the vehicle at the time of exiting water with ice floating around is greater than that without ice around, and the pressure around the vehicle fluctuates more violently when the vehicle approaches the ice/water interface. Dong et al. [22] put forward a new algorithm combining the BDIM and VOF and used it in the numerical calculation of a semispherical vehicle breaking ice and exiting water. They pointed out that the shear effect of fluid must be the main cause for the cavitation of a vehicle. Yue et al. [23] carried out a numerical simulation for a vehicle breaking ice at different initial velocities and concluded that the load characteristics of the vehicle relate to its velocity and the mechanical properties of ice.

Some mesh-based numerical simulation methods have been widely applied in various fields such as the Finite Element Method (FEM) and Finite Volume Method (FVM), but they have certain limitations in simulating the interaction between structures and ice. These limitations are mainly reflected in two aspects: (1) Ice has a large deformation structurally under the impact of a high-speed vehicle, which leads to mesh distortion and undermines the accuracy of solution. (2) Ice cracks extensively under impact, but cracks exist only along the mesh border in the mesh-based methods, which does not match with the practical conditions. Mesh-free methods have become popular in recent years. At the time of computation, these methods need only nodes but do not rely on the link between nodes, so they are greatly adaptive and applicable to addressing the mutual effect of structures and ice [24,25,26]. Smoothed particle hydrodynamics (SPH) is currently the fastest spreading mesh-free method. After being first put forth by Gingold et al. [27], the SPH method has been widely applied in such domains as hydrodynamics and impact dynamics. In the method, particles carry the properties of material, including energy and density, so that it is differentiated from other mesh-free methods that take nodes only for interpolation in computation. For this reason, the SPH method can deliver more accurate tracking of material interface and deformation boundary and has been applied to a certain extent in the field of structural-ice interaction. Han et al. [28] introduced fracture criteria into the SPH method and performed numerical simulations of three-point bending tests on ice, proving the feasibility of the SPH method in solving fracture problems. Zheng et al. [29] built a vehicle/water/ice numerical model based on the SPH method to explore the ice resistance to an icebreaker in various circumstances. Yang et al. [30] adopted the SPH method to simulate the collision of a vertical cylindrical structure with broken ice and studied the effect of such parameters as ice thickness and density on ice resistance.

The SPH method may better simulate the crack spread of sea ice under impact and ensure higher accuracy of computation, but it is less efficient in computation and occupies more computing resources than the mesh-based methods. Therefore, the SPH and FEM coupling algorithm will be practically more effective since it takes into account both accuracy and efficiency of computation. Chen et al. [31] utilized the SPH-FEM adaptive coupling algorithm to simulate the mutual effect of horizontal ice and inclined structure. As revealed in the results, the algorithm, which keeps the strengths of both SPH and FEM methods, can greatly simulate the large deformation of ice and guarantee the accuracy of computation. Feng et al. [32] proposed an SPH-FEM coupling algorithm to simulate the collision of structure and floating ice at sea, and adopted the Von Mises failure criterion to reflect the damage to sea ice with the SPH method for the outer layer and the FEM method for the inner layer of floating ice.

To sum up, scholars have currently conducted extensive research on the lateral ice breaking of structures along the sea level (e.g., ships and vertical structures breaking ice) [33] and the hydrodynamics of structures passing through the mixture of ice and water. However, they have rarely studied the mechanical response and motion characteristics of structures surfacing and breaking ice. In the meantime, most of their studies adopt the mesh-based methods, leading to poor computational accuracy. The experiment provides the significant basis for verifying the accuracy of the simulation model, but there is not much experimental data available for the vehicles breaking ice at high velocity.

In response to the shortcomings of current research, the main contributions of this paper are as follows: (1) In response to the low accuracy of traditional grid methods, this paper combined the FEM-SPH adaptive algorithm with the S-ALE algorithm to propose an FEM-SPH-SALE coupling algorithm suitable for multi-physics field calculations; based on the algorithm, a numerical computation model was constructed for a water-exiting vehicle breaking ice, and the Johnson-Holmquist-2 constitutive model was used to simulate the mechanical properties of ice. (2) To examine the accuracy of FEM-SPH-SALE coupling algorithm, this paper carried out the four-point bending numerical simulation of the ice and the wedge body water entry simulation. (3) An experiment for the high-speed vehicle breaking ice was conducted, providing a basis for verifying numerical calculation models and expanding the test data on high-speed icebreaking by structures. (4) The dynamic characteristics of the vehicle and the damage mechanism of ice with different initial velocities of the vehicle and different thicknesses of ice were analyzed.

2. Simulation Methodology

2.1. Basic Theory of the Smoothed Particle Hydrodynamics Method

2.1.1. Control Equations

Smoothed particle hydrodynamics (SPH) is a mesh-free particle method which discretizes a computational domain into a number of particles that can interact with each other. The particles include mass, density, energy, and other physical quantities [34]. For solid media, the control equations of the SPH method include the conservation of mass equation, conservation of momentum equation, conservation of energy equation, and equation of motion as follows:

$${\displaystyle \frac{d\rho }{dt}}=-\rho {\displaystyle \frac{\partial v^{\alpha }}{\partial s^{\alpha }}}$$

(1)

$${\displaystyle \frac{d{v}^{\alpha }}{dt}}={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\sigma }^{\alpha \beta }}{\partial s^{\beta }}}+g^{\alpha }$$

(2)

$${\displaystyle \frac{dU}{dt}}={\displaystyle \frac{{\sigma }^{\alpha \beta }}{\rho }}{\displaystyle \frac{\partial v^{\alpha }}{\partial s^{\beta }}}$$

(3)

$${\displaystyle \frac{d{s}^{\alpha }}{dt}}=v^{\alpha }$$

(4)

where $\alpha$ and $\beta$ are the Cartesian components in the directions x and y, respectively; $\rho$, $v$, and $s$ stand for the density, velocity, and spatial coordinates of particles, respectively; ${\sigma }^{\alpha \beta }$ is the total stress tensor component; $g$ is the gravitational acceleration; $U$ is the internal energy per unit of mass; and t represents the time. The total stress tensor component ${\sigma }^{\alpha \beta }$ can also be the sum of the hydrostatic pressure $T$ and the stress deviation $S^{\alpha \beta }$:

$${\sigma }^{\alpha \beta }=-T{\delta }^{\alpha \beta }+S^{\alpha \beta }$$

(5)

where ${\delta }^{\alpha \beta }$ is the Dirichlet function, satisfying

$${\delta }^{\alpha \beta }=\left\{\begin{array}{ll}0& \alpha \ne \beta \\ 1& \alpha =\beta \end{array}\right.$$

(6)

2.1.2. Integral Interpolation Theory

The SPH particles interact with each other by a kernel function, so two steps are crucial to the construction of the SPH equation, that is, kernel function interpolation and particle approximation.

The function at a point in the domain Ω is defined as the field function, which is expressed as follows:

$$f(r)={\displaystyle \int f(r^{\prime })}W(r-r^{\prime },h)d{r}^{\prime }$$

(7)

where $r$ is the spatial position vector; $W(r-r^{\prime },h)$ is the kernel function, which depends on the distance between two particles $\left|r-r^{\prime }\right|$ and the smoothing length; and the smoothing factor $\kappa$ and $h$ decide the affected domain of the kernel function (as shown in Figure 1).

The spatial derivative of the field function is

$$\nabla \cdot f(r)=-{\displaystyle \int f(r^{\prime })}\cdot \nabla W(r-r^{\prime },h)d{r}^{\prime }$$

(8)

By virtue of integration, the kernel function interpolation delivers the field function and its spatial derivative, and then the particle approximation is conducted to discretize the kernel function interpolation. In a certain domain of solution, the integral of the kernel function interpolation can be transformed through a set of particles into the discrete form. The particle mass $m_j$ can be expressed as

$$m_j={\rho }_j\Delta V_j\hspace{1em}(j=1,2,\cdots ,N)$$

(9)

where $m_j$ ${\rho }_j$ and $\Delta V_j$ are the mass, density, and volume of the particle $j$, respectively; N is the total number of particles in the affected domain.

After discretizing Equations (7) and (8), we obtain

$$f(r_i)={\displaystyle \sum _{j=1}^N\frac{m_j}{{\rho }_j}f(r_j)}{W}_{ij}$$

(10)

$$\nabla \cdot f(r_i)={\displaystyle \sum _{j=1}^N\frac{m_j}{{\rho }_j}f(r_j)}{\nabla }_i{W}_{ij}$$

(11)

$${\nabla }_i{W}_{ij}={\displaystyle \frac{r_{ij}}{|r_i-r_j|}}{\displaystyle \frac{\partial W_{ij}}{\partial |r_i-r_j|}}$$

(12)

where $W_{ij}$ is the abbreviation for kernel function $W(r_i-r_j,h)$. Equation (11) indicates that the field function value at the position of particle $i$ can be obtained by weighting the kernel function values of all particles in the domain. Equation (12) reveals that the spatial derivative value of the field function depends on the kernel function gradient and the function values of particles in the affected domain. With Equations (10) and (11), the integral form of a function and its spatial derivative is transformed into the discrete summation of particles so that the SPH method can achieve numerical computation without mesh. In order to improve interpolation accuracy and computational stability, this paper uses the Moving Least Squares (MLS) method to correct the integration scheme [35,36].

The kernel function chosen in this paper is the cubic spline kernel function.

$$W_{ij}=A^{\prime }\times \left\{\begin{array}{ll}1-{\displaystyle \frac{3}{2}}{q}^2+{\displaystyle \frac{3}{4}}{q}^3& 0\le q<1\\ {\displaystyle \frac{1}{4}}{(2-q)}^3& 1\le q<\kappa \\ 0& q\ge \kappa \end{array}\right.$$

(13)

where $q=|r_i-r_j|/h$, $A^{\prime }$ is taken ${\displaystyle \frac{1}{\pi h^3}}$.

The traditional SPH method suffers from tensile instability; this paper adopts the artificial stress method to solve this problem [37].

$${\displaystyle \frac{d{v}_i}{dt}}=-{\displaystyle \sum _{j=1}^N{m}_j(\frac{P_i+P_j}{{\rho }_i{\rho }_j}+f_{ij}^{\psi }{R}_{ij}){\nabla }_i{W}_{ij}+{\displaystyle \sum _j^N{m}_j\frac{{\eta }_i+{\eta }_j}{{\rho }_i{\rho }_j}}}+{\displaystyle \sum _j^N{m}_j\frac{{\eta }_i+{\eta }_j}{{\rho }_i{\rho }_j}}{v}_{ij}({\displaystyle \frac{1}{r_{ij}}}{\displaystyle \frac{\partial W_{ij}}{\partial r_{ij}}})+g\hspace{1em}\hspace{1em}\hspace{1em}$$

(14)

where $P_i$, $P_j$ are the pressures of particle i, j. ${\eta }_i$, ${\eta }_j$ are dynamic viscosity of particle i, j. $f_{ij}=W_{ij}/W(\Delta p,h)$, $\Delta p$ is the initial distance between particles. $\psi$ is the variable exponent based on the kernel function. $v_{ij}=v_i-v_j$ denotes relative velocity. $R_{ij}$ is an artificial stress term.

$$R_{ij}=R_i+R_j$$

(15)

$R_i$ and $R_j$ are artificial parameters determined by pressure and density.

$$\begin{array}{ll}{R}_i=-0.2{\displaystyle \frac{P_i}{{\rho }_i^2}}& P_i<0\\ R_i=0.2{\displaystyle \frac{P_i}{{\rho }_i^2}}& P_i\ge 0\end{array}$$

(16)

Similarly, the expression for $R_j$ is the same as for $R_i$. The artificial stress method introduces an artificial stress term $R_{ij}$ among particles to counteract the false attraction caused by negative pressure gradients in a tensile state, thereby preventing particle clumping.

2.2. FEM-SPH Adaptive Algorithm

2.2.1. Conversion of Finite Elements into Smoothed Particles

The FEM and SPH are crucial to numerical computation, but they have some limitations. The FEM delivers poor accuracy of solutions for the problem of large deformations such as damage to sea ice. The SPH is less efficient in computation, and difficult to apply the boundary conditions. In order to resolve these problems, an FEM-SPH adaptive algorithm is adopted in this paper to simulate sea ice so as to effectively integrate the strengths of both methods: high computational efficiency and stable boundary conditions of the FEM and the applicability to solve deformation problems of the SPH [38].

The basic idea of the FEM-SPH adaptive algorithm is to adopt the FEM for the sea ice model at the initial stage and take strain as the basis for judging element conversion. In the computation, if the strain of any finite element exceeds its critical value, the finite element is converted into a smoothed particle. The smoothed particle, which inherits such physical information as pressure, density, and velocity from the finite element, will still play a part in the subsequent computation [39]. In this paper, the critical strain value is 0.35 [32].

The conversion of finite elements into smoothed particles is illustrated in Figure 2. In the beginning, the FEM is employed to discretize the sea ice model. With the collision of the vehicle with sea ice, some finite elements experience mesh distortion. When the distortion of finite elements reaches a certain level (i.e., the critical strain of elements), the finite elements are converted into smoothed particles.

After conversion, the smoothed particles will take the physical quantities of failed finite element nodes as their initial physical quantities.

$$\left\{\begin{array}{l}{m}_p=m_n={\displaystyle \sum _{i=1}^{N_e}\frac{{\rho }_{ei}{V}_{ei}}{N_n}}\\ v_p=v_n\\ s_p=s_n\\ c_p=c_n\\ {\sigma }_p={\sigma }_n={\displaystyle \frac{1}{N_e}}{\displaystyle \sum _{i=1}^{N_e}{\displaystyle \sum _{j=1}^{N_{\chi }}{\omega }_{\chi j}{\sigma }_{\chi j}}}\end{array}\right.$$

(17)

where $c$ and $\sigma$ stand for the sound velocity and stress tensor, respectively; the subscript $n$ represents the node of a finite element; the subscript $p$ indicates the smoothed particle; the subscript $\chi$ stands for the Gaussian integral point; the subscript $e$ denotes the finite element; $N_n$, $N_e$, and $N_{\chi }$ are the number of nodes in each finite element, the number of finite elements linked to the node, and the number of Gaussian integral points in the finite element; $w_{\chi j}$ and ${\sigma }_{\chi j}$ are the coefficient and stress tensor at the Gaussian integral point, respectively.

After obtaining the above physical quantities, we can calculate the strain rate tensor ${\stackrel{\cdot }{\epsilon }}^{\alpha \beta }$ of the SPH particles as follows:

$${\stackrel{\cdot }{\epsilon }}^{\alpha \beta }={\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^N\frac{m_j}{{\rho }_j}[({\nu }_j^{\alpha }-{\nu }_i^{\alpha })\frac{\partial W_{ij}}{\partial s_i^{\beta }}+({\nu }_j^{\beta }-{\nu }_i^{\beta })\frac{\partial W_{ij}}{\partial s_i^{\beta }}]}$$

(18)

Based on the conservation of mass, the smoothing length $h_p$ of SPH particles can be calculated as follows:

$$h_p={\displaystyle \frac{\zeta }{N_e}}{{\displaystyle \sum _i^{N_e}{d}_{0i}(\frac{{\rho }_{0i}}{{\rho }_i})}}^{1/3}$$

(19)

where ${\rho }_0$ is initial density; $\zeta$ indicates the scale factor of the smoothing length and the finite element size; and $d_0$ stands for the initial finite element size.

2.2.2. Time Steps

The time step of the FEM is

$$\Delta t_{FEM}={\displaystyle \frac{d_{\min }}{\sqrt{\frac{E(1-\upsilon )}{\rho (1+\upsilon )(1-2\upsilon )}}}}$$

(20)

$d_{\min }$ is the minimum element size. $E$, $\rho$, and $\upsilon$ are the elastic modulus, density, and Poisson’s ratio of the material, respectively.

The time step of SPH is bounded by the following condition [40]:

$$\Delta t_{SPH}\le \min ({\displaystyle \frac{0.3h}{\sqrt{K/{\rho }_i}+\left|v_i^{\max }\right|}})$$

(21)

$K$ is bulk modulus of the material, $v_i^{\max }$ is the maximum velocity of all SPH particles.

This paper takes the smaller value of $\Delta t_{FEM}$ and $\Delta t_{SPH}$ as the time step of FEM-SPH adaptive algorithm.

$$\Delta t_{FEM-SPH}=\min (\Delta t_{FEM},\Delta t_{SPH})$$

(22)

2.2.3. Explicit Integration Solution

In this paper, the leap-frog method is used to solve the discrete equations of SPH by explicit integration.

$$\begin{array}{l}{\rho }_i^{l+{\displaystyle \frac{1}{2}}}={\rho }_i^{l-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{\Delta t_{SPH}^l+\Delta t_{SPH}^{l+1}}{2}}{\displaystyle \frac{d{\rho }_i^l}{dt}}\\ v_i^{l+{\displaystyle \frac{1}{2}}}=v_i^{l-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{\Delta t_{SPH}^l+\Delta t_{SPH}^{l+1}}{2}}{\displaystyle \frac{d{v}_i^l}{dt}}\\ s_i^{l+{\displaystyle \frac{1}{2}}}=s_i^{l-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{\Delta t_{SPH}^l+\Delta t_{SPH}^{l+1}}{2}}{\displaystyle \frac{d{s}_i^l}{dt}}\end{array}$$

(23)

where l represents the l-th time step.

The central difference method is used to solve the dynamic equations of the FEM.

$$\begin{array}{l}{a}_e^l={\displaystyle \frac{1}{\Delta t_{FEM}^2}}(s_e^{l-\Delta t_{FEM}}-2s_e^l+s_e^{l+\Delta t_{FEM}})\\ v_e^l={\displaystyle \frac{1}{2\Delta t_{FEM}}}(s_e^{l+\Delta t_{FEM}}-s_e^{l-\Delta t_{FEM}})\end{array}$$

(24)

where $a$ represents the acceleration.

2.2.4. Calculation Process

The computation process of the FEM-SPH adaptive coupling algorithm, as presented in Figure 3, is divided into the following steps: (1) calculate the minimum time step of finite elements based on the minimum element size of the FEM model; (2) compare the time steps of the FEM and the SPH and take the smaller one as the time step of the FEM-SPH coupling algorithm; (3) update the displacement of finite element nodes, and recalculate the time step of finite elements based on the current minimum size of the FEM model; (4) determine whether the finite elements fail according to the strain failure criteria and convert them into SPH particles if they fail; (5) apply the relevant physical quantities to the SPH particles with Equations (17)–(19) and update the time step of the SPH; (6) judge whether the finite element nodes are in contact with the SPH particles and calculate the contact force; (7) substitute the contact force into the kinetic equation of the FEM method and the momentum equation of the SPH method; (8) solve and display the kinetic equation with the FEM method and solve the Navier–Stokes equation with the SPH method; (9) update the physical quantities of finite elements and SPH particles, and the time step is finally determined.

2.3. S-ALE Fluid/Structure Coupling Algorithm

The fluid/structure coupling effect occurs among the vehicle, ice, and fluid. In the conventional Lagrangian method, a finite element mesh can be adopted to describe the geometry of an object, but it deforms with the motion of the object. Therefore, mesh distortion is easily caused in the analysis of large deformation, which lowers the computational accuracy and even terminates the procedure. For this reason, it is not applicable to the simulation of water and air flow characteristics. Differently, the arbitrary Lagrangian–Eulerian (ALE) algorithm allows the mesh to move arbitrarily and vary flexibly in space so that it has been widely employed in the calculation of fluid/structure interaction [41].

The structured arbitrary Lagrangian–Eulerian (S-ALE) algorithm is an improved algorithm based on the ALE algorithm, which keeps the same interface reconstruction method as the ALE algorithm. However, it is superior in the following aspects: (1) the S-ALE algorithm can generate orthogonal meshes internally, and it is and easy to maintain and occupies less memory; (2) it is more efficient because of parallel computation; (3) the solution is stable and can effectively solve the fluid leakage problem that often occurs in fluid/structure interaction algorithms. During fluid/structure interaction calculations, the fluid mesh deforms as the fluid moves and the S-ALE algorithm maintains the boundary conditions of the deformed fluid unchanged, performs the remeshing of the internal elements, keeps the topological relationships of the mesh unchanged, and transfers physical quantities such as density and energy from the pre-deformed mesh to the new mesh. By remeshing at the fluid/structure interface, the impact of mesh deformation on computational accuracy can be reduced.

The S-ALE algorithm adopts a reference coordinate system, and the time derivative of the material in connection with the reference coordinate system can be written as

$${\displaystyle \frac{\partial f(X,t)}{\partial t}}={\displaystyle \frac{\partial f(x^{\alpha },t)}{\partial t}}+w^{\alpha }{\displaystyle \frac{\partial f(x^{\alpha },t)}{\partial x^{\alpha }}}$$

(25)

where X is the displacement in the Lagrangian coordinate system; $x$ is the displacement in the Eulerian coordinate system; and $w=v-u$, where $v$ is the velocity of the material and $u$ is the moving speed of the reference coordinate system. The control equations of the S-ALE algorithm are as follows:

$$\left\{\begin{array}{l}\rho {\displaystyle \frac{\partial v^{\alpha }}{\partial t}}={\sigma }^{\alpha \beta }+\rho b^{\alpha }-\rho w^{\alpha }{\displaystyle \frac{\partial v^{\beta }}{\partial x^{\beta }}}\\ {\displaystyle \frac{\partial \rho }{\partial t}}=-\rho {\displaystyle \frac{\partial v^{\beta }}{\partial x^{\alpha }}}-w^{\alpha }{\displaystyle \frac{\partial \rho }{\partial x^{\alpha }}}\\ \rho {\displaystyle \frac{\partial U}{\partial t}}={\sigma }^{\alpha \beta }{\tau }^{\alpha \beta }+\rho b^{\alpha }{v}^{\alpha }-\rho w^{\beta }{\displaystyle \frac{\partial U}{\partial x^{\beta }}}\end{array}\right.$$

(26)

where $b$ is the unit volume force, and ${\tau }^{\alpha \beta }$ is the viscous shear force. The S-ALE algorithm uses overlapping grid technology to handle interpolation issues between fluid grids and solid grids.

In order to solve the energy dissipation and pressure oscillation problems caused by artificial numerical damping in the fluid/structure interaction process, this paper adds an artificial viscosity formula to the S-ALE algorithm.

$$P_q=\left\{\begin{array}{ll}\rho (C_{q}c{d}_a\left|{\dot{\epsilon }}_V\right|+C_l{l}^2{\dot{\epsilon }}_V^2)& {\dot{\epsilon }}_V<0\\ 0& {\dot{\epsilon }}_V\ge 0\end{array}\right.$$

(27)

where $P_q$ is the artificial viscosity pressure added to the actual pressure, $d_a$ is the feature length, $C_q$ is linear artificial viscosity coefficient, $C_l$ is secondary artificial viscosity coefficient, ${\dot{\epsilon }}_V^2$ is volume strain rate. In this paper, $C_q$ is 1.5, and $C_l$ is 0.06.

2.4. Constitutive Model of Ice

In this paper, the Johnson-Holmquist-2 constitutive model (abbreviated as JH-2 model) is employed to simulate sea ice. It contains the state equation, strength model, and damage model so that it can realize a satisfying simulation of the mechanical behaviors of ice material.

2.4.1. State Equation

The state equation of the JH-2 model is a polynomial that can define the relationship between hydrostatic pressure and volumetric strain. The hydrostatic pressure is low when the flow rate is small, so a first-order polynomial is adopted. However, the flow rate will be large when ice is impacted at a high velocity. In this paper, a third-order polynomial is employed as shown in Equation (28).

$$P=J_1\mu +J_2{\mu }^2+J_3{\mu }^3$$

(28)

where $\mu$ is the volumetric strain; $J_1$, $J_2$, and $J_3$ are the strain constants.

2.4.2. Strength Model

The JH-2 strength model takes into account damage factor and strain rate effect. The strength of material can be divided into three states: no damage, damage started, and completely damaged.

$${\sigma }^{\ast }={{\sigma }_y}^{\ast }-D({{\sigma }_y}^{\ast }-{{\sigma }_f}^{\ast })$$

(29)

where ${\sigma }_y$ is the yield stress when the material has no damage; ${\sigma }_f$ is the yield stress when the material is completely damaged; the superscript “*” implies the normalization of the parameter; and D is the cumulative loss index. Equation (29) shows the softening effect on ice material after being damaged, which becomes stronger with the increasing degree of damage.

$${\sigma }_y^{\ast }=A{(T^{\ast }+T_{\max }^{\ast })}^N(1+C\ln {\dot{\epsilon }}^{\ast })$$

(30)

$${\sigma }_f^{\ast }=B{(T^{\ast })}^M(1+C\ln {\dot{\epsilon }}^{\ast })$$

(31)

where $T_{\max }$ is the maximum hydrostatic pressure; A, B, C, M, and N are the constants of material; and $\dot{\epsilon }$ is strain rate. Moreover, $T^{\ast }=T/T_{HEL}$, $T_{\max }^{\ast }=T_{\max }/T_{HEL}$, and $T_{HEL}$ are the hydrostatic pressure at the elastic limit. The Hugoniot elastic limit of ice can be divided into hydrostatic pressure and deviatoric stress:

$$HEL=T_{HEL}+{\displaystyle \frac{2}{3}}{\sigma }_{HEL}$$

(32)

$${\sigma }_{HEL}=2G{\displaystyle \frac{{\mu }_{HEL}}{1+{\mu }_{HEL}}}$$

(33)

where HEL stands for Hugoniot elastic limit; ${\sigma }_{HEL}$ is the yield stress at the elastic limit; ${\mu }_{HEL}$ is the volumetric strain at the elastic limit; G is the shear modulus.

2.4.3. Damage Model

The JH-2 damage model may reflect the accumulation trend of plastic strain when damage happens to sea ice.

$${\epsilon }_p=D_1{(T^{\ast }+T_{\max }^{\ast })}^{D_2}$$

(34)

where ${\epsilon }_p$ is the plastic strain; D₁ and D₂ are the damage factors of the material.

The main parameters of the JH-2 model are shown in

Table 1. Main parameters for JH-2 model.

Parameters	Value
${\sigma }_{HEL}$ (MPa)	5200
$T_{HEL}$ (MPa)	3438.1
G (MPa)	4444.4
A	1.4
B	0.09
C	0.2287
M	1.2
N	0.8918

[42].

It is worth noting that the JH-2 constitutive model assumes ice to be an isotropic material, which cannot reflect the anisotropy of ice. Furthermore, it does not consider the influence of thermodynamic effects on mechanical properties, resulting in some discrepancies from actual conditions. However, it still meets the accuracy requirements of this study.

2.5. Numerical Model Based on FEM-SPH-SALE Coupling Algorithm

In this paper, the FEM-SPH adaptive algorithm and S-ALE fluid/structure coupling algorithm are combined to put forward a new FEM-SPH-SALE numerical computation method. The proposed method is adopted to build a model for a water-exiting vehicle breaking ice as shown in Figure 4. We employ the FEM for the vehicle model, the S-ALE for the fluid model, and the FEM-SPH adaptive algorithm for the ice model. In the initial stage, fluid/structure interaction between the vehicle, ice, and fluid is generated through the S-ALE algorithm. When the vehicle comes into contact with the ice, the vehicle based on the FEM and the ice based on the FEM-SPH adaptive algorithm interact through the penalty function contact algorithm, and the ice begins to deform. When the deformation of the ice reaches the critical strain, the ice elements is converted into SPH particles.

The penalty function contact algorithm defines the vehicle and ice as the primary surface and secondary nodes, respectively. At each time step, it detects whether the secondary nodes penetrate the primary surface. If penetration occurs, a normal spring is placed between the primary surface and the secondary nodes to limit the penetration. The force generated by the spring is called the contact force, which is proportional to the stiffness of the contact $k$ and the penetration depth $\delta$.

$$F=k\delta$$

(35)

The vehicle consists of a head and a cylindrical section. The head is streamlined and made of aluminum alloy. The ice is a cuboid, which is simulated by the JH-2 constitutive model. The state equation of air is as follows [43]:

$$P=C_0+C_1\varphi +C_2{\varphi }^2+C_3{\varphi }^3+(C_4+C_5\varphi +C_6{\varphi }^2)U$$

(36)

where $\varphi =\rho /{\rho }_0$ with $\rho$ for the current density and ${\rho }_0$ for the initial density; $C_0$~$C_6$ are constants.

The state equation of water is as follows:

$$P={\displaystyle \frac{{\rho }_0{c}^2\varphi [1+(1-\frac{{\gamma }_0}{2})\varphi -\frac{\theta }{2}{\varphi }^2]}{1-(S_1-1)\varphi -S_2\frac{{\varphi }^2}{1+\varphi }-S_3\frac{{\varphi }^3}{{(1+\varphi )}^2}}}+(\theta \varphi +{\gamma }_0)U$$

(37)

where $\theta$ is the first-order correction factor of ${\gamma }_0$; ${\gamma }_0$ and $S_1$~$S_3$ are constants [44].

The velocity of the vehicle is along the normal direction of the sea ice and points toward the surface of the ice. The boundary of the ice layer uses the no-reflection boundary condition.

The parameters of the computation model are as given in

Table 2. Parameters of the numerical model.

Properties	Value
Diameter of vehicle/(mm)	10
Length of vehicle/(mm)	100
Length and width of ice/(mm)	200
Thickness of ice/(mm)	10
Length and width of water domain/(mm)	200
Height of water domain/(mm)	120
Length and width of air domain/(mm)	200
Height of air domain/(mm)	30

:

3. Verification of Numerical Simulation Method

3.1. Four-Point Bending Simulation of the Ice Model Based on FEM-SPH Adaptive Algorithm

The damage to ice under vertical load is mainly in the form of bending. Prior to the numerical computation of a water-exiting vehicle breaking ice, it should be verified whether the sea ice model based on the FEM-SPH adaptive algorithm can accurately simulate the process of damage to ice.

A four-point bending numerical computation model was established as shown in Figure 5. The mesh size of the ice model was set to 20 mm × 20 mm × 20 mm, 15 mm × 15 mm × 15 mm, and 10 mm × 10 mm × 10 mm so as to obtain three different sets of meshes, which were denominated as schemes 1, 2, and 3, respectively. The three sets of meshes were used in the numerical computation.

Figure 6 shows the ice load curves over time for different schemes; it can be seen that the computation results of schemes 1 and 2 are much different, but the computation results of scheme 2 are close to those of scheme 3. The average load error for scheme 1 and scheme 2 is 13.85%, and the average load error for scheme 2 and scheme 3 is only 2.18%. In other words, the computation result converges when the mesh size of the ice model is 15 mm × 15 mm × 15 mm. In this paper, the simulation result of scheme 2 is compared with the disclosed four-point bending experimental results [45]. The curve for the relationship between ice load and time in the four-point bending simulation and experiment is presented in Figure 7. Evidently, the numerical simulation result is basically consistent with the experimental result. Hence, the proposed ice model based on the FEM-SPH adaptive algorithm can greatly simulate the characteristics of bending damage to ice.

3.2. Wedge-Body Water-Entry Simulation Based on S-ALE Algorithm

In order to verify the accuracy of the S-ALE algorithm in the analysis of fluid/structure coupling, a water-entry wedge-body model was constructed as described in Reference [46]. It is presented in Figure 8.

The mesh size of the water and air models is set to 4 mm × 4 mm × 4 mm, while the mesh size of the wedge body is 2 mm × 2 mm × 2 mm. A convergence test is performed for the models. The mesh size of the models is reduced by 0.5 times and 0.25 times, respectively, to obtain three different sets of meshes, which are denominated as schemes 1, 2, and 3.

As shown in Figure 9, the resistance and time curves of schemes 2 and 3 highly overlap each other, indicating that the computation results of scheme 2 have converged. The results of scheme 2 are compared with the experimental results in references as shown in Figure 10 and Figure 11. It is evident that the numerical simulation result of the water-entry wedge body is highly consistent with the experimental result. This could prove that the S-ALE algorithm adopted in this paper can deliver a great simulation of fluid/structure coupling.

3.3. Vehicle Breaking Ice Experiment

3.3.1. Experimental Device

We designed a scale experiment of a vehicle breaking through ice vertically. In the designed system, the experimental device is composed of five parts, including gas supply module, launch module, ice fixture module, control module, and data acquisition module, as shown in Figure 12.

The gas supply module is intended to supply high-pressure nitrogen to drive the ejection. It contains a high-pressure cylinder, a regulator, a storage tank, pipes, a high-speed solenoid valve, a hand valve, and a one-way check valve. Among them, the high-pressure cylinder in Figure 13 used high-purity nitrogen as the input gas with the highest pressure of 16 MPa.

The launch module consists of a launch tube and a vehicle as shown in Figure 14. The vehicle has a length of 1000 mm, a diameter of 100 mm, and a mass of 5 kg. It is made of aluminum alloy, and the head shape is hemispherical. The diameter of the launch tube is 114 mm, and its length is 1130 mm.

The data acquisition module mainly includes a high-speed camera and sensors. The frame rate of the high-speed camera is 1000 frames/s. At the same time, in order to meet the brightness requirements of the high-speed camera, two 200 W LED lights are used as supplementary light sources in the experiment.

3.3.2. Experiment Analysis

The length, width, and thickness of the ice in the experiment were 600 mm, 400 mm, and 200 mm, respectively. The ejection pressure was 3 MPa, 4 Mpa, and 5 Mpa for the vehicle to break ice vertically. The phenomenon in the experiment is illustrated in Figure 15, Figure 16 and Figure 17. Evidently, some small pieces of broken ice were generated when the head of the vehicle hit the lower surface of the ice. The upper surface of ice at the contact area arched upwards and cracks appeared in the ice. As the vehicle continued to move, the cracks in the ice expanded and a large amount of broken ice was generated. The broken ice splashed upwards along with the vehicle, and eventually the ice was bent and broke off.

As shown in

Table 3. Velocity of the vehicle breaking ice under different ejection pressures.

Ejection Pressure (MPa)	Ice Thickness (mm)	Velocity Before Breaking Ice (m/s)	Residual Velocity After Breaking Ice (m/s)	Velocity Loss Rate
3	200	13.08	7.80	40.37%
4	200	15.44	10.62	31.22%
5	200	16.63	11.53	30.67%

, the initial velocity of the vehicle increased with the ejection pressure. The higher the initial velocity of the vehicle, the larger the residual velocity after breaking ice and the lower the velocity loss rate.

Based on the experiment model, a numerical simulation model for the vehicle breaking ice at a high velocity was established to further examine the accuracy of the JH-2 sea ice constitutive model, the FEM-SPH adaptive algorithm, and the critical strain value used in this paper. The initial velocity was 13.08 m/s, which was consistent with the initial velocity of the vehicle when the ejection pressure was 3 MPa in the experiment. The experimental result was compared with the numerical simulation results, and the critical strain values (${\epsilon }_c$) used in the numerical simulation were 0.15, 0.25, 0.35, and 0.45.

As revealed in Figure 18, the value of the critical strain has an impact on the results. When the critical strain value is 0.35, the numerical simulation result shows a high degree of similarity with the experimental result. As shown in Figure 19, the crack growth and fracture phenomena of ice obtained from numerical simulation are highly similar to the experimental results. This could prove that the sea ice constitutive model, FEM-SPH adaptive algorithm and the critical strain value adopted in this paper are reasonable and reliable.

3.4. Convergence Test of the Water-Exiting Vehicle Breaking Ice Computation Model

We adopted hexahedral meshing for the vehicle and local mesh refinement for ice. The mesh size of ice in the middle area near the vehicle was refined, while the outer mesh size was larger. The local mesh refinement was also employed for air and water with the refined mesh size in the middle and larger mesh size on both sides.

Mesh size has a high influence on the numerical computation results. In order to test the convergence of mesh, three sets of meshes were used with the mesh size gradually refined. The mesh quantity was 2.31 × 10⁶, 4.93 × 10⁶, and 9.55 × 10⁶, which were denominated as scheme 1, scheme 2, and scheme 3. The three sets of meshes were used in the numerical computation, and the initial velocity of the vehicle was set to 40 m/s. This paper used a 128-core simulation server for numerical simulation and adopted an MPP parallelization solution strategy.

The results of the mesh convergence test are given in Figure 20. It is evidently revealed that the calculation results for velocity, acceleration, and stress gradually converge with the increasing mesh quantity, and the computation results of schemes 2 and 3 are highly consistent. It can be regarded that the computation results converge when the computation model adopts the set of meshes in scheme 2. The model for the water-exiting vehicle breaking ice after meshing in scheme 2 is shown in Figure 21.

4. Numerical Simulation

4.1. Influence of Ice Thickness

The ice thickness was set to 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm, while the initial velocity of the vehicle was set to 40 m/s. On this basis, we explored the influence of ice thickness on the vehicle breaking ice.

4.1.1. Influence on the Motion Characteristics of the Vehicle

As illustrated in Figure 22 and Figure 23, the velocity of the vehicle keeps decreasing in the process of breaking ice, and the ice thickness plays a significant role in the variation of the vehicle’s velocity. When the ice thickness increases from 10 mm to 50 mm, the residual velocity of the vehicle after breaking ice is 35.9 m/s, 31.6 m/s, 28.0 m/s, 25.6 m/s, and 21.6 m/s, respectively, with the velocity loss rate of 10.3%, 21.0%, 30.0%, 36.0%, and 46.0%. With the increase in the ice thickness, the residual velocity of the vehicle after breaking ice decreases and the velocity loss rate becomes larger, which is attributed to the continuous conversion of the vehicle’s kinetic energy into the internal energy of the ice during the ice-breaking process of the vehicle. With the identical initial velocity of the vehicle, the thicker the ice, the longer the time for the kinetic energy to convert into internal energy. Meanwhile, the residual kinetic energy of the vehicle after breaking ice becomes lower, so the residual velocity is smaller.

As shown in Figure 24, after the flow field of the vehicle moving in the water is stabilized, the acceleration of the vehicle is approximately 5000~20,000 m/s². When the vehicle passes through the ice, its acceleration reaches a peak and then gradually decreases to 0. The acceleration peak of the vehicle is little affected by the ice thickness. When the acceleration of the vehicle is 0, it could be regarded that the vehicle has completely passed through the ice. When the ice thickness is 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm, it takes the vehicle 0.61 ms, 0.99 ms, 1.20 ms, 1.69 ms, and 2.02 ms, respectively, to completely pass through the ice. The time taken by the vehicle to pass through the ice increases with the ice thickness.

Figure 25 shows the dynamic characteristic curves of the vehicle breaking ice with thicknesses of 10 mm and 50 mm. It can be seen that the process of the vehicle breaking through the ice is divided into three stages: (1) Movement in water: At this time, the vehicle has not come into contact with the ice, the speed decreases relatively slowly, and the acceleration value is small, corresponding to 0 ms to 0.1 ms in (a) and 0 ms to 0.1 ms in (b). (2) Ice-breaking: At this stage, the vehicle collides with the ice and is subjected to the impact load of the ice, causing the acceleration to increase rapidly and the speed to decrease significantly, corresponding to 0.1 ms to 0.4 ms in (a) and 0.1 ms to 1.75 ms in (b); (3) Separation of vehicle and ice: At this point, the head of the vehicle has broken through the ice, and the force between the vehicle and the ice is mainly friction. The speed change is small, and the acceleration gradually approaches 0, corresponding to 0.4 ms to 1.0 ms in (a) and 1.75 ms to 2.0 ms in (b).

4.1.2. Influence of Ice Thickness on the Load Characteristics of the Vehicle

The variation of the stress at the head of the vehicle at different ice thicknesses is presented in Figure 26. Evidently, the stress at the head of the vehicle constantly fluctuates with time. When ice is 10~50 mm thick, the maximum stress at the head is 9.48 × 10⁷ Pa, 1.09 × 10⁸ Pa, 1.14 × 10⁸ Pa, 1.46 × 10⁸ Pa, and 1.56 × 10⁸ Pa, respectively. The maximum stress on the vehicle goes up with the increasing ice thickness. After reaching the maximum stress, the stress drops quickly.

Figure 27 shows the damage process of ice at different thicknesses when the vehicle breaks ice. Water and air are hidden to indicate the variation of the cloud diagrams clearly. Evidently, the amount of broken ice increases with the ice thickness. When the ice is 10 mm thick, a low amount of broken ice is generated when the vehicle breaks ice. However, a large amount of broken ice appears when the ice thickness increases to 50 mm. In the ice 10 mm or 20 mm thick, the hole made by the vehicle after breaking the ice is approximately rectangular. When the ice is above 30 mm thick, the hole is nearly circular. The above phenomenon can be explained from two aspects: the nonlinear mechanical properties of ice and the types of stress waves. (1) Since ice transforms from a ductile material into a brittle material in the process of being broken, the thicker the ice, the slower the transformation. Therefore, it takes a longer time to reach the critical point of the transformation in the process of breaking the ice. When the vehicle breaks ice, the ice is in two states at the same time, that is, ductility and brittleness. The thicker the ice, the more uneven the distribution of the two states and the more noticeable the nonlinear effect is. In practice, the thinner ice is subject to a lower nonlinear effect, so the hole made on it is more perfectly rectangular. Differently, the thicker ice is under a higher nonlinear effect, which causes the irregular edges of the hole and makes it appear circular. (2) When the ice layer is thin, the impact energy of the vehicle on the ice layer is mainly transmitted through bending stress waves. For thicker ice layers, the impact energy is mainly transmitted through compression waves around the collision point, causing high compressive stress on the lower surface of the ice layer and triggering conical radial cracks. The compression waves are reflected as tensile waves on the upper surface of the ice layer, causing layer cracks. The two types of cracks expand and merge to form an approximately circular hole.

In the process of breaking ice, the vehicle is exposed to a very high impact load, which transfers axially along it. To fully reflect the mechanical state of the vehicle in the process of breaking ice, an axial distance is defined to be from a point on the vehicle to the vertex of the vehicle head. Six points are selected with their axial distance of 0 mm, 20 mm, 40 mm, 60 mm, 80 mm, and 100 mm as shown in Figure 28.

The vehicle is subject to different stresses at different times, but there is always the maximum stress. The relationship of the maximum stress with axial distance and ice thickness is illustrated in Figure 29. It is noticed that the maximum stress occurs at the vertex of the vehicle head, and then declines axially, but goes up near the middle of the vehicle (with an axial distance of 60 mm). This is attributed to the transfer of impact load and the inertial effect: (1) The vehicle is exposed to a large impact load during its collision with ice, and stress first appears at its head. Subsequently, the stress is transferred axially along the vehicle in the form of wave. However, the wave of stress continuously attenuates during the transfer process due to the thermal loss and internal friction of the material. (2) In the process of breaking ice, the vehicle starts to slow down under the effect of ice resistance, which results in an inertial effect. The inertial effect imposes an inertial force on the vehicle in the direction from the tail to the head. The stress on the vehicle is jointly caused by the impact load and the inertial force. The highest impact load exists at the head of the vehicle, which is accompanied by the maximum stress there. A small impact load happens in the middle of the vehicle, but a large inertial force occurs there. For this reason, stress will go up near the middle of the vehicle.

4.2. Influence of the Initial Velocity of the Vehicle

In this section, the velocity of the vehicle was set to 20 m/s, 30 m/s, 40 m/s, 50 m/s, 60 m/s, 70 m/s, and 80 m/s in order to explore the influence of the initial velocity of the vehicle in the process of breaking ice.

4.2.1. Influence on the Motion Characteristics of the Vehicle

As illustrated in Figure 30, when the initial velocity of the vehicle is 20 m/s, 30 m/s, 40 m/s, 50 m/s, 60 m/s, 70 m/s, and 80 m/s, the velocity loss rate of the vehicle after breaking ice is 20.01%, 14.42%, 10.65%, 9.24%, 8.37%, 7.79%, and 7.45%, respectively. The velocity loss rate of the vehicle decreases with the increase in its initial velocity, which agrees with the conclusion drawn from the experiment of the vehicle breaking ice. When the initial velocity of the vehicle exceeds 70 m/s, the velocity loss rate has a little change, so it is believed that the velocity loss rate has converged at that time. This means that when designing a water-exiting vehicle, its actual maximum speed should not exceed 70 m/s, as exceeding this value will have little effect on enhancing the vehicle’s icebreaking capability.

As revealed in Figure 31 and Figure 32, the acceleration of the vehicle in the process of breaking ice increases with the initial velocity. When the vehicle begins to touch the ice, its acceleration soars. At the initial velocity of 20 m/s, the vehicle has a maximum acceleration of 8470 m/s² in the process of breaking ice. At the initial velocity of 80 m/s, the maximum acceleration reaches 47,320 m/s². Ice is a material highly sensitive to strain rate, and it becomes stronger when the strain rate is larger, which makes the vehicle face stronger resistance in the process of breaking ice. Therefore, the faster the vehicle moves, the greater the resistance it faces and the greater the acceleration.

4.2.2. Influence on the Load Characteristics of the Vehicle

As shown in Figure 33, the maximum stress at the vehicle head increases with the initial velocity of the vehicle. When the initial velocity is 20~80 m/s, the maximum stress at the vehicle head is 4.93 × 10⁷ Pa, 5.61 × 10⁷ Pa, 9.48 × 10⁷ Pa, 1.09 × 10⁸ Pa, 1.20 × 10⁸ Pa, 1.55 × 10⁸ Pa, and 1.70 × 10⁸ Pa, respectively. This phenomenon is caused by the high strain rate when the ice collides with the vehicle moving at a high velocity. In this case, the strength of the ice increases, which causes higher resistance to the vehicle. In the process of breaking ice, the vehicle generates a great amount of broken ice, which might splash and hit the vehicle head. For this reason, a number of small peaks still occur after the maximum stress occurred.

In Figure 34, the cloud diagrams of ice damage are presented for the vehicle’s initial velocities of 20 m/s, 40 m/s, 60 m/s, and 80 m/s in the process of breaking ice. Evidently, in the early stage, the vehicle head intrudes into the ice and causes a small amount of broken ice in the vicinity of the collision point. When the vehicle constantly moves upwards, the damaged area of the ice expands, and the central part of the ice is “pushed upwards”. After the vehicle head passes through the ice, the cylindrical section of the vehicle has friction against the ice, and a large amount of broken ice is scattered around the vehicle head. The time taken by the vehicle to pass through the ice at different initial velocities varies significantly. When the initial velocity is 20 m/s, 40 m/s, 60 m/s, and 80 m/s, it takes 0.76 ms, 0.44 ms, 0.27 ms, and 0.20 ms for the vehicle to pass through the ice, respectively.

The relationship of the vehicle’s maximum stress with axial distance and velocity is illustrated in Figure 35. When the initial velocity of the vehicle is 20 m/s, the stress varies slightly with the axial distance. When the velocity of the vehicle exceeds 30 m/s, the stress first decreases axially along the vehicle but goes up at the axial distance of 60 mm and then drops again. When the axial distance is 0~60 mm, the stress increases with the initial velocity if the axial distance does not change.

4.3. Evolution of the Fluid Field in the Process of the Vehicle Breaking Ice

A vehicle may experience cavitation when it moves in water. In the presence of ice, the cavitation will be more complicated as the evolution of the fluid field when the vehicle exits water. Figure 36 gives the cloud diagrams for the flow field of the vehicle when it passes through the ice of 10 mm thick at the initial velocities of 40 m/s and 80 m/s, respectively. As illustrated, cavitation occurs when the vehicle exits water. In the process, the air cavity at the vehicle head squeezes against the ice and then collapses. In the meantime, the air cavity wraps the shoulders of the vehicle. As the vehicle continues to move forward, the cavity along the column section continuously comes into contact with the ice, repeatedly undergoing a process of “growth and collapse,” which intensifies the cavitation phenomenon.

4.4. Sensitivity Analysis of the Main Parameters

In order to study the relative importance of ice thickness and initial velocity on the icebreaking results, 20 working conditions were simulated by changing the ice thickness and initial velocity (specific parameters are shown in

Table 4. Parameters for different working conditions.

Parameters	Value
initial velocity (m/s)	20, 40, 60, 80
ice thickness (mm)	10, 20, 30, 40, 50

), and a sensitivity analysis of the results (velocity loss rate, stress) was performed based on the Sobol method. The larger the Sobol index, the more sensitive the results are to the parameter. As can be seen from Figure 37, compared to ice thickness, velocity loss rate and stress are more sensitive to changes in initial velocity.

5. Conclusions

Aiming at the problem of the multi-physics field coupling effect in the process of a water-exiting vehicle breaking ice, this paper presented an FEM-SPH-SALE coupling algorithm, which combines the high computational efficiency of the FEM with the applicability of the SPH to resolve the problem of deformation and damage. An experiment of the vehicle breaking ice was carried out with different ejection pressures. The accuracy of the proposed FEM-SPH-SALE coupling algorithm was verified by virtue of the four-point ice bending simulation, wedge-body water-entry simulation, and vehicle breaking ice experiment. In this way, we explored the influence of ice thickness and the vehicle’s initial velocity on the motion characteristics, load characteristics, and flow field evolution of the vehicle. Some conclusions are mainly drawn as follows:

(1)

When the ice thickness is 200 mm, and the ejection pressure is 3 MPa, 4 MPa, and 5 MPa, after breaking ice, the velocity of the vehicle decreases from 13.08 m/s, 15.44 m/s, and 16.63 m/s to 7.80 m/s, 10.62 m/s, and 11.53 m/s, respectively. The velocity loss rate of the vehicle in the process of breaking ice decreases with the increasing initial velocity of the vehicle.

(2)

When the ice thickness increases from 10 mm to 50 mm, the residual velocity of the vehicle after breaking ice drops from 35.9 m/s to 21.6 m/s, the velocity loss rate goes up from 10.3% to 46.0%, and the maximum stress at the head rises from 9.48 × 10⁷ Pa to 1.56 × 10⁸ Pa. The velocity loss rate of the vehicle and the maximum stress at the head become larger with the increase in ice thickness. The influence of ice thickness on the acceleration of the vehicle is mainly reflected in the duration of acceleration but not the peak of acceleration.

(3)

When the vehicle passes through ice of 10 mm thickness at an initial velocity of more than 70 m/s, the velocity loss rate reaches convergence. The acceleration of the vehicle and the maximum stress at the head increase with the rise of the initial velocity. When the initial velocity is 80 m/s, the maximum acceleration of the vehicle in the process of breaking ice is 47,320 m/s², and the maximum stress at the head is 1.70 × 10⁸ Pa.

(4)

At the early stage of ice-breaking, the vehicle head intrudes into the ice and causes a small amount of broken ice in the vicinity of the collision point. With the motion of the vehicle, the central part of the ice is “pushed upwards”, and a large amount of broken ice moves upwards with the vehicle. The amount of broken ice increases with the thickness of the ice.

(5)

Due to the transmission of stress wave and inertial effect, the stress on the vehicle in the process of breaking ice tends to “decrease, increase, and then decrease” axially from the vertex of the vehicle head. Therefore, when designing a water-exiting vehicle, it is necessary to reinforce the structure of the head and middle sections of the vehicle.

(6)

When the vehicle breaks ice, the air cavities at the head collapse under the squeeze of the ice, and those in the cylindrical section of the vehicle repeat the process of “generation/collapse”, which results in the phenomenon of intensive cavitation.

(7)

When ice reaches the critical strain and transforms into SPH particles, the interaction between a large number of high-speed SPH particles and the fluid will cause pressure oscillations, energy dissipation, and tensile instability phenomena, which will become an important issue in future research.