When dealing with the complex deformation of free surface such as wave breaking, traditional mesh-based Computational Fluid Dynamics (CFD) methods often face problems arising alongside grid distortion and re-meshing. Therefore, the meshless method became robust for treating large displaced free surface and other boundaries caused by moving structures. The particle method, which is an important branch of meshless method, is mainly divided into the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods. Different from the SPH method, which involves continuity and treat density as a variable when building kernel functions, the kernel function in the MPS method is a weight function which treats density as a constant, and the spatial derivatives are discretized by establishing the gradient operator and Laplace operator separately. In other words, the first- or second-order continuity of the kernel functions in the MPS method is not a necessity as in SPH, though it might be desirable. At present, the MPS method has been successfully applied to various violent-free surface flow problems in ocean engineering and diverse applications have been comprehensively demonstrated in a number of review papers. This work will focus on algorithm developments of the MPS method and to provide all perspectives in terms of numerical algorithms along with their pros and cons.

To improve the accuracy of solving the Poisson equation and the efficiency of handling complex boundary shapes in the particle method, this paper proposes a Local Regular-distributed Background Particles (LRBP) as an alternative to traditional boundary handling methods. This method avoids the trouble of arranging virtual particles by introducing background particles, making it suitable for problems with complex boundary shapes. In addition, based on the framework of the weak form Poisson equation, the boundary conditions are easily applied, and the calculations are more accurate. Furthermore, this method allows for different interpolation methods inside and outside the boundary, providing flexibility and versatility. These characteristics are well demonstrated in the validation examples, which indicate its potential to solve complex flow problems.