A particle-based method using the mesh-constrained discrete point approach for two-dimensional Stokes flows

TL;DR

This paper introduces a mesh-constrained discrete point (MCD) approach that enhances meshless methods for 2D Stokes flows by using compact stencils and a background Cartesian mesh, improving computational efficiency and accuracy.

Contribution

The study proposes a novel MCD approach that constrains point distributions with a Cartesian mesh, enabling compact stencils and efficient, accurate meshless simulations of Stokes flows.

Findings

DP distribution is independent of spatial resolution

Flow solutions match theoretical and reference results

Velocity and pressure converge with expected order

Abstract

Meshless methods inherently do not require mesh topologies and are practically used for solving continuum equations. However, these methods generally tend to have a higher computational load than conventional mesh-based methods because calculation stencils for spatial discretization become large. In this study, a novel approach for the use of compact stencils in meshless methods is proposed, called the mesh-constrained discrete point (MCD) approach. The MCD approach introduces a Cartesian mesh system to the background of a domain. And the approach rigorously constrains the distribution of discrete points (DPs) in each mesh by solving a dynamic problem with nonlinear constraints. This can avoid the heterogeneity of the DP distribution at the mesh-size level and impose compact stencils with a fixed degree of freedom for derivative evaluations. A fundamental formulation for arrangements of DPs and an application to unsteady Stokes flows are presented in this paper. Numerical tests were performed for the distribution of DPs and flow problems in co-axial and eccentric circular channels. The proposed MCD approach achieved a reasonable distribution of DPs independently of the spatial resolution with a few iterations in pre-processing. Additionally, solutions using the obtained DP distributions in Stokes flow problems were in good agreement with theoretical and reference solutions. The results also confirmed that the numerical accuracies of velocity and pressure achieved the expected convergence order, even when compact stencils were used.