# Fast variable density 3-D node generation

## TL;DR

This paper introduces a fast, scalable method for generating variable density nodes in 2-D and 3-D, enhancing mesh-free PDE solvers by efficiently creating quasi-uniform node distributions with variable spacing.

## Contribution

The paper presents a novel advancing front method for rapid generation of variable density nodes in higher dimensions, with linear computational complexity.

## Key findings

- Generation cost is O(N) in 2-D and 3-D.
- Method generalizes to higher dimensions.
- Efficient node distribution for mesh-free solvers.

## Abstract

Mesh-free solvers for partial differential equations perform best on scattered quasi-uniform nodes. Computational efficiency can be improved by using nodes with greater spacing in regions of less activity. We present an advancing front type method to generate variable density nodes in 2-D and 3-D with clear generalization to higher dimensions. The exhibited cost of generating a node set of size $N$ in 2-D and 3-D with the present method is O(N).