# On generation of node distributions for meshless PDE discretizations

## TL;DR

This paper introduces an efficient algorithm for generating locally regular node distributions with variable density for meshless PDE discretizations, suitable for complex domains and validated through thermo-fluid simulations.

## Contribution

The paper presents a novel algorithm capable of generating high-quality, variable-density node layouts with proven minimal spacing guarantees, applicable in multiple dimensions.

## Key findings

- Algorithm achieves $O(N)$ complexity for uniform spacing and $O(N \,\log N)$ for variable spacing.
- Generated nodes are suitable for RBF-FD method in 2D and 3D thermo-fluid problems.
- Comparison shows improved node quality and solution accuracy over existing methods.

## Abstract

In this paper we present an algorithm that is able to generate locally regular node layouts with spatially variable nodal density for interiors of arbitrary domains in two, three and higher dimensions. It is demonstrated that the generated node distributions are suitable to use in the RBF-FD method, which is demonstrated by solving thermo-fluid problem in 2D and 3D. Additionally, local minimal spacing guarantees are proven for both uniform and variable nodal densities. The presented algorithm has time complexity $O(N)$ to generate $N$ nodes with constant nodal spacing and $O(N \log N)$ to generate variably spaced nodes. Comparison with existing algorithms is performed in terms of node quality, time complexity, execution time and PDE solution accuracy.