## 1 Introduction  
Recently, computer-aided engineering, such as computational fluid dynamics (CFD), finite element analysis, and multiphysics simulations using computers, has been getting a great deal of attention in various fields, such as the academic, research, and industrial sectors, owing to the rapid advancement of computational techniques. For CFD analysis, grid systems are generally used in the Eulerian frame for both structured and unstructured grids and solvers. Currently, it is common to adopt unstructured grids and solvers that discretize the governing equations using the finite volume method because of the ease of expressing complex objects. The accuracy of the unstructured grids and solvers is relatively lower than that of the structured ones despite its high computational cost. However, the stability and accuracy of CFD simulations using these methods vary depending on the quality of the grid, and it is inefficient to carry out grid-based simulations for flows such as in free surfaces with strong nonlinearity and/or movement of a body, where grids need to be deformed, deleted, or regenerated during the simulation. To enhance its efficiency, various numerical methods such as overset grid method [1], immersed boundary method [2], level set method [3], and so forth have been developed for the grid-based simulations. In contrast to the grid-based method, in which the connection information with neighboring grids must be maintained, the gridless method, also called the meshless or mesh-free method, requires only the location information and physical quantities of the computational points when approximating the spatial derivatives at the points. There are lots of gridless methods in the literature. Among them, smoothed particle hydrodynamics (SPH) [4], element-free Galerkin (EFG) method [5], reproducing kernel particle method (RPKM) [6], partition of unity [7], hp-clouds [8], finite point method (FPM) [9], generalized finite difference method (GFDM) [10], meshless local Petrov-Galerkin (MLPG) method [11], radial basis function (RBF) method [12; 13], and moving particle simulation (MPS) [14] are the popular ones. These methods have been developed until recently and are being applied in various fields. The primary areas of advancement in meshfree methods are to address issues with essential boundary enforcement, numerical quadrature, and contact and large deformations [15]. The smoothed point interpolation method [16], smoothed finite element method [17] and gradient smoothing method [18] can be applied for CFD problems. Gridless methods can be classified in several ways. According to the form of the partial differential equations, that is, the corresponding governing equations, the methods based on the weak form of the governing equations, such as EFG, RKPM, and MLPG, are generally stable and accurate but computationally expensive and require global or local meshes in the computational domain. In contrast, SPH, FPM, RBF, and MPS, which adopt strong forms of governing equations, are generally less stable and precise than the weak-form-based methods when Neumann boundary conditions are imposed [19]. Another classification may be based on the adopting frame, that is, the way in which the flow is described. MPS and SPH, which are Lagrangian gridless methods, have been widely used for estimating the impact force in free surface flows with strong nonlinearity and for approximating the motion of a floating body [20; 21], especially in the field of naval architecture and ocean engineering. The method introduced by [PERSON] [22] is a Eulerian method that has a higher accuracy for flows near a body than Lagrangian methods because the computational points do not move, and the distance between the fluid and body boundary can easily be controlled. For fluid flows, one classification may be whether compressibility is to be considered, which depends on the flow of interest and significantly affects the numerical algorithms for CFD.