## 1 Introduction  
The water entry of a torpedo is a typical example of water entries of structures, relating to large deformation of the free liquid surface. The impact load and the corresponding moving status induced by the water entry of the torpedo may result in strength damage to the torpedo structure or internal device failure. Thus, they should be carefully considered in the design and analysis of a torpedo. Three types of research methods are widely employed in investigating water entry processes of torpedoes, including the experimental method [1], theoretical method [2], and numerical method [3]. With the rapid development of computer technology, the numerical method has become a mainstream research method owing to its essential advantages of low cost and fast calculating speed compared to the other two methods. The traditional mesh-based numerical methods, such as the volume of fluid (VOF) [4, 5], the finite volume method (FVM) [6], the boundary element method (BEM) [7], and the constrained interpolation profile (CIP) [8], are widely applied in simulating phenomena of fluid-structure interactions. However, meshes generated by these methods are prone to distortions, leading to inaccurate calculation results when applied to problems involving large deformations of free interfaces, such as the water entry of a torpedo. In recent years, the meshless numerical method has been extensively developed owing to its unique natural advantages in overcoming the problems above [9]. Rather than using meshes as the computational basis, it discretizes the computational domain into particles and then solves various integral equations or partial differential equations (PDEs) on the particles. The smoothed particle hydrodynamics (SPH) method is a typical meshless numerical method that uses a collection of particles with specific physical properties, such as density, viscosity, mass, velocity, and temperature, to represent fluids [10]. The movement and the changes in the properties of particles are controlled via mass, momentum, and energy conservation equations. The numerical solution to these equations is obtained by employing the kernel approximation and the weighted integration of particle interpolation [11]. The SPH method has an inherent advantage in solving strongly nonlinear problems of the water entry of the torpedo, induced by complex free surface splashing and large deformation. Lots of work has been conducted to improve the numerical accuracy and stability of the SPH method in capturing the moving boundary, the large deformation of the fluid, and the free surface breakage in water entry processes [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. However, there remain certain shortcomings in the SPH method. First, the SPH method performs calculation processes based on a series of discrete particles. Most studies have, first, commonly applied meshes over the computational domain and then extracted particles from the meshes to discretize the computational domain [26]. This existing particle generation method has low surface discretization accuracy and poor discretization flexibility. Second, the SPH method widely adopts a uniform particle distribution over the entire computational domain. To accurately capture the dramatic variations in the flow field around a moving three-dimensional (3D) torpedo, the total number of particles allocated in the computational domain is often very large. It consumes large amounts of computer memory and processing power and thus makes the SPH method less efficient. This work thoroughly investigates the complex water entry of a 3D torpedo by applying an improved adaptive SPH algorithm. First, a meshless and efficient point distribution method is established for the 3D computational domain of the water entry of the 3D torpedo. Second, based on the adaptive SPH technique reported by [PERSON] et al. [27], a dynamic refinement and coarsening scheme of particles within the target region is proposed.