Addressing the checkerboard problem in an Eulerian meshless method for incompressible flows

TL;DR

This paper presents a novel adaptive point cloud algorithm to effectively address the pressure checkerboard problem in Eulerian meshless methods for incompressible flows, improving accuracy without extensive mesh refinement.

Contribution

The paper introduces an adaptive point addition algorithm based on an error indicator to solve the checkerboard problem in meshless methods, avoiding re-meshing and extensive refinement.

Findings

The algorithm successfully reduces pressure oscillations in simulations.

It requires fewer additional points compared to uniform refinement.

Results match well with existing literature.

Abstract

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.