  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.
