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We present a new algorithm for the automatic one-shot generation of scattered node sets on irregular two dimensional (2D) and three dimensional (3D) domains using Poisson disk sampling coupled to novel parameter-free, high-order parametric spherical-radial-basis-function-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 2 and 3 dimensions 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 a radial-basis-function-based--finite-difference 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 2 and 3 dimensions.

Robust Node Generation for Mesh-free Discretizations on Irregular Domains and Surfaces