In this paper, we look at the pressure checkerboard problem that arises in an Eulerian meshless method that solves the incompressible Navier-Stokes equations using the generalized finite difference method (GFDM). Although, the checkerboard problem has been dealt with extensively in mesh-based methods, the literature in connection with meshless methods is comparatively scarce. In this paper, we explore the occurrence of the checkerboard problem in a meshless method. A few unsuccessful attempts to resolve the checkerboard problem are reported. The successful fix for the problem entails an algorithm that adapts the point cloud by adding points in the regions of pressure oscillations. The algorithm uses an error indicator that detects the presence of the checkerboard oscillations in the solution. The algorithm minimizes the computational effort since it ensures the use of additional points only in regions of concern, as directed by the error indicator, in contrast to an approach of using a highly refined set of points throughout the domain. It also requires no a priori estimates of the regions where the oscillations occur and integrates conveniently in the framework of the meshless method since no re-meshing strategies are involved. The results are compared with literature and a good match is observed.

翻译：本文研究了采用广义有限差分法（GFDM）求解不可压缩纳维-斯托克斯方程时，在欧拉无网格方法中出现的压力棋盘格问题。尽管棋盘格问题在基于网格的方法中已得到广泛研究，但针对无网格方法的相关文献相对匮乏。本文探讨了无网格方法中棋盘格问题的产生机制，并报告了若干未成功的解决尝试。问题的有效解决手段在于提出一种算法，该算法通过向压力振荡区域添加点来自适应调整点云分布。算法采用误差指示器检测解中是否存在棋盘格振荡，从而最小化计算工作量——仅根据误差指示器的指示在需要关注的区域添加额外点，而非在整个计算域采用高度加密的点集。该方法无需预先估计振荡发生区域，且由于不涉及网格重构策略，能便捷地集成到无框架方法体系中。将计算结果与文献进行对比，观察到良好的一致性。