  Particle-based methods are a practical tool in computational fluid dynamics,
and novel types of methods have been proposed. However, widely developed
Lagrangian-type formulations suffer from the nonuniform distribution of
particles, which is enhanced over time and result in problems in computational
efficiency and parallel computations. To mitigate these problems, a
mesh-constrained discrete point (MCD) method was developed for stationary
boundary problems (Matsuda et al., 2022). Although the MCD method is a meshless
method that uses moving least-squares approximation, the arrangement of
particles (or discrete points (DPs)) is specialized so that their positions are
constrained in background meshes to obtain a closely uniform distribution. This
achieves a reasonable approximation for spatial derivatives with compact
stencils without encountering any ill-posed condition and leads to good
performance in terms of computational efficiency. In this study, a novel
meshless method based on the MCD method for incompressible flows with moving
boundaries is proposed. To ensure the mesh constraint of each DP in moving
boundary problems, a novel updating algorithm for the DP arrangement is
developed so that the position of DPs is not only rearranged but the DPs are
also reassigned the role of being on the boundary or not. The proposed method
achieved reasonable results in numerical experiments for well-known moving
boundary problems.
