## 1 Introduction  
Smoothed particle hydrodynamics (SPH) is a fully Lagrangian mesh-free method initially developed for astrophysics in 1970s [1; 2]. It has been widely used in large deformation problems such as fluid motions where the continuum hydrodynamics equations are solved with a set of interacting fluid particles. For incompressible flows, there are generally two ways to impose incompressibility. One is to run the simulations in the quasi-incompressible limit by assuming a small Mach number to ensure the density fluctuation within 1% [3; 4; 5], which is known as weakly compressible smoothed particle hydrodynamics (WCSPH). This method is conceptually easy as pressure values are computed straightforwardly and are independent of other quantities. However, very small time steps are required, and the pressure values obtained are usually noisy since it is sensitive with density variations [6; 7]. The other one is called truly incompressible SPH (ISPH) in which incompressibility is enforced by setting the velocity divergence to be zero, and the pressure is calculated by solving a Poisson's equation at every time step [8; 9; 10]. Comparison of these two methods can be found in many works [11] and the ISPH method is selected for the current research. Since SPH methods are based on integral approximation and further approximated by averaging the neighboring particles within the kernel domain, which will be presented in Section 3, the computations are all based on summation of neighboring particles inside the kernel domain. For the particles without enough neighboring particles, i.e., when the kernel domain is truncated by the boundary, the simulation would not be successful. Hence, it is one of the most challenging problems to find an ideal boundary treatment in the SPH method. According to the existing work, the boundary treatment can be mainly classified into three categories: repulsive force, ghost particles, and semi-analytical boundaries. In repulsive force boundary treatment, a line of denser particles is introduced along the wall boundaries where forces were generated to push away the approaching fluid particles. The repulsive force treatment is easy to implement and is effective in the prevention of inner fluid particles from penetrating. However, it often leads to spurious effects and other quantities calculated may not be accurate since the kernel domain is still truncated for fluid particles near the near wall [3; 12]. In the ghost particle boundary treatment, several lines of virtual particles are placed outside the solid boundary to make a complete smoothing domain for particles near the wall. One type of ghost particles, called mirror particles, possess the same physical properties as those of real fluid particles. They are generated at every time step along the normal direction of the inner fluid particles treating the boundary as a mirror [13; 14]. With these mirror particles the spurious effect is eliminated, and the physical values of interior particles could be calculated more accurately. However, special consideration is required to calculate the exact positions of the mirror particles, especially for the angled points, which can be time consuming. Furthermore, fluid particles can move through the wall boundary due to the relatively weak fence effect [15]. To solve this problem, another type of ghost particles called dynamic boundary particles are suggested to be distributed outside the boundary with fixed positions [16]. However, they are not suitable for deforming boundaries. Furthermore, it is not always easy to ensure enough dynamic particles in the kernel domain, and inner particle penetration could thus be found in some cases [17]. Moreover, ghost particles may take up too much space, and sometimes they even overlap with the real fluid particles. In conclusion, ghost particles are useful to keep a complete kernel domain for the near-wall inner particles, so that the physical properties such as density could be calculated correctly.