A stable moving particle semi-implicit method with renormalized Laplacian model improved for incompressible free-surface flows

Imposing accurate wall boundary conditions in corrective‐matrix‐based moving particle semi‐implicit method for free surface flow

“…First, the simulated results are compared with experiment ones for validation. The simulated histories of front positions are compared with literature results 37 in Figure 11, where good agreements are observed. The water depth histories are contrasted in Figure 12, where satisfactory agreements between simulations and experiment are observed.…”

“…where n * i is the temporary PND; N i is the number of neighbor particles, and N 0 is the constant number of neighbor particles based on the initial regular particle distribution. If Equations (37) and (38) are not satisfied simultaneously, the reference particle will be regarded as an internal particle; otherwise, the free surface detection will be further considered based on the geometry conditions in Step (2), similar to Reference 27. Namely, if the following condition:…”

“…The original treatment method in MPS are as follows 2 : (1) free surface particles provide zero pressure in solving the pressure Poisson equation (PPE); (2) the same Laplacian, divergence and gradient models are applied to both free surface and internal particles; and (3) the particle collision model is utilized to guarantee the stability 33 . However, it is difficult to directly apply the high‐order accurate schemes with corrective matrix to the free surface particles due to the incomplete neighbor support, as mentioned by Wang et al 34 and Duan et al 31 Meanwhile, the virtual particle methods are proved to be helpful to improve accuracy and stability (eg, References 3,14,32,35‐38), because virtual particles can compensate the neighbor particles outside free surface. Shibata et al 3 and Chen et al 35 derived the discretization models for free surface particles considering virtual particles.…”

“…Incidentally, the positions of virtual particles are not necessary according to References 3, 14, 35. More advanced methods to specify the positions of virtual particles were developed by Liu et al 37,38 Besides, various methods by modifying and adjusting PS to enhance the stability at free surface were also developed by Lind et al, 39 Khayyer et al (optimized particle shifting, OPS), 40 Duan et al, 32 Oger et al, 41 Wang et al, 42 and Jandaghiana and Shakibaeinia 43 . Nevertheless, the general boundary condition treatment method of exact first‐order accuracy is still missing for free surface in literature.…”

Machine‐learning‐based surface tension model for multiphase flow simulation using particle method

“…This work is based on MPS method. During the past two decades, significant improvements have been achieved to improve the accuracy and stability of the MPS method, including corrective gradient model, 21,22 high‐order Laplacian model, 8,23,24 least‐square MPS, 25 and conservative pairwise‐relaxing method 26 . Extensive study is conducted in multiphase flow simulation using MPS 27‐32 .…”