Spatially dependent node regularity in meshless approximation of partial differential equations

TL;DR

This work introduces HyNP, a dimension‑independent, hybrid regular–scattered meshless node placement algorithm that places scattered nodes near geometric details and regular nodes elsewhere to reduce stencil size and computational cost. It presents two local operator-approximation schemes (RBF-FD with PHS augmentation and LRBFCM with Gaussians) and analyzes stability via condition numbers, demonstrating that the hybrid discretization maintains accuracy while significantly reducing runtime in 2D/3D natural convection and Boussinesq problems. The results show that discretization quality, measured by separation distance and empty-space metrics, remains robust across the hybrid interface, and the approach yields notable speedups (e.g., ~50–57%) without sacrificing solution fidelity. The work suggests that spatially varying node regularity is a practical path toward scalable meshless simulations on irregular domains and complex geometries, with potential extensions to more challenging flow and elasticity problems.

Abstract

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatially varying computational node regularity. Rather than covering the entire domain with scattered nodes, only regions with geometric details are covered with scattered nodes, while the rest of the domain is discretized with regular nodes. A simpler approximation can be used in regions covered by regular nodes, effectively reducing the required stencil size and computational cost compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour. This paper is an extended version of conference paper entitled "Spatially-varying meshless approximation method for enhanced computational efficiency" [arXiv:2303.01760] presented at "International Conference on Computational Science (ICCS) 2023". The paper is extended with discussion on development and implementation of a hybrid regular-scattered node positioning algorithm (HyNP). The performance of the proposed HyNP algorithm is analysed in terms of separation distance and maximal empty sphere radius. Furthermore, it is demonstrated that HyNP nodes can be used for solving problems from fluid flow and linear elasticity, both in 2D and 3D, using meshless methods. The extension also provides additional analyses of computational efficiency and accuracy of the numerical solution obtained on the spatially-variable regularity of discretization nodes. In particular, different levels of refinement aggressiveness and scattered layer widths are considered to exploit the computational efficiency gains offered by such solution procedure.

Figures (23)

• Figure 1: A visualisation of hybrid fill algorithm progression on a domain with variable node density and irregularly shaped areas with regular node positioning.
• Figure 2: Visualisation of two and three dimensional irregularly shaped clover domains.
• Figure 3: Visualisation of the test domain discretized with different regularity functions $g$ and comparison of fill quality measure distributions between scattered and hybrid fill algorithms on a 2D domain.
• Figure 4: Comparison of fill quality measure distributions between scattered and hybrid fill algorithms on a 3D domain.
• Figure 5: Irregular domain discretization example (left) and spatial distribution of approximation methods along with corresponding example stencils (right).