Spatially-varying meshless approximation method for enhanced computational efficiency

TL;DR

This paper introduces a spatially varying stencil strategy for meshless methods by combining regular-node MON-based discretization with scattered-node RBF-FD near domain irregularities to reduce stencil size and computational cost. Regular nodes employ a small MON stencil with $n=5$, while scattered nodes use Polyharmonic Splines augmented with monomials to yield $n=12$, enabling faster weight computation and evaluation. The approach is validated on the de Vahl Davis natural convection benchmark and on 2D and 3D irregular domains, achieving substantial speedups (about 35–50%) with comparable accuracy, and demonstrating the importance of the scattered-layer width $\delta_h$ and $h$-refinement toward irregular boundaries. These results suggest a practical pathway for efficient, accurate meshless simulations of complex geometries in natural convection and related PDE problems, with future work focusing on transition-layer effects and mixed convection scenarios.

Abstract

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatial variation of 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 discretised with regular nodes. Consequently, in regions covered with regular nodes the approximation using solely the monomial basis can be performed, effectively reducing the required stencil size compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour. The performance of the proposed hybrid scattered-regular approximation approach, in terms of computational efficiency and accuracy of the numerical solution, is studied on natural convection driven fluid flow problems. We start with the solution of the de Vahl Davis benchmark case, defined on square domain, and continue with two- and three-dimensional irregularly shaped domains. We show that the spatial variation of the two approximation methods can significantly reduce the computational complexity, with only a minor impact on the solution accuracy.

Figures (11)

• Figure 1: Irregular domain discretization example (left) and spatial distribution of approximation methods along with corresponding example stencils (right).
• Figure 2: Condition numbers $\kappa(\mathbf{M})$ for the Laplacian operator: entire computational domain (left) and a zoomed-in section around the irregularly shaped obstacle (right).
• Figure 3: The de Vahl Davis sketch (left) and example hybrid scattered-regular domain discretization (right).
• Figure 4: Example solution at the stationary state. Temperature field (left) and velocity magnitude (right).
• Figure 5: Time evolution of the average Nusselt number along the cold edge calculated with the densest considered discretization. Three reference results Kosec and Šarler kosec2008solution, Sadat and Couturier sadat and Wan et. al. wan are also added.