Robust Node Generation for Meshfree Discretizations on Irregular Domains and Surfaces

TL;DR

This paper introduces a scalable, parameter-free algorithm for generating scattered nodes on irregular domains using Poisson disk sampling and SBF-based boundary modeling, suitable for meshfree discretizations.

Contribution

It presents a novel, high-order geometric modeling approach combined with Poisson disk sampling for automatic node generation on irregular domains, including dynamic boundary adjustments.

Findings

Algorithm exhibits O(N) complexity in 2D and 3D.

Effective in producing quasi-uniform node sets with minimal boundary seed nodes.

Demonstrates accuracy and scalability for meshfree discretizations of PDEs.

Abstract

We present a new algorithm for the automatic one-shot generation of scattered node sets on irregular 2D and 3D domains using Poisson disk sampling coupled to novel parameter-free, high-order parametric Spherical Radial Basis Function (SBF)-based geometric modeling of irregular domain boundaries. Our algorithm also automatically modifies the scattered node sets locally for time-varying embedded boundaries in the domain interior. We derive complexity estimates for our node generator in 2D and 3D that establish its scalability, and verify these estimates with timing experiments. We explore the influence of Poisson disk sampling parameters on both quasi-uniformity in the node sets and errors in an RBF-FD discretization of the heat equation. In all cases, our framework requires only a small number of "seed" nodes on domain boundaries. The entire framework exhibits O(N) complexity in both 2D and 3D.