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

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.

On Generation of Node Distributions for Meshless PDE Discretizations

“…From the discussion presented in section 3 it is clear that although state of the art placing algorithms provide a solid spatial discretization methodology for strong form meshless methods, there is still room for improvements, especially in the generalization to higher dimensions, flexibility regarding variable nodal density, and treatment of irregular domains. Improving upon the work of Fornberg and Flyer [11] and Shankar et al [32], we propose a new algorithm that overcomes some of limitations of FF and SKF algorithms. We will refer to the proposed node placing algorithm as PNP in the rest of the paper.…”

“…An often observed property of numerical discretization methods is the spectrum of discretized partial differential operator. For example, the spectrum of discretized Laplace operator should have only eigenvalues with negative real part, and a relatively small spread along the imaginary axis [32]. Figure 14 shows the spectrum of Laplace operator discretized with 2nd order RBF-FD PHS on PNP nodes shown in Figure 9.…”

“…To the best of our knowledge, algorithms presented in [11,32] are currently among the best available. However, they have some shortcomings, namely [11] only works in two dimensions and [32] does not support variable nodal spacing. In this paper, we present an algorithm that overcomes these shortcomings.…”

“…For algorithms that can fill domains with varying density, conformal mappings can be used to generate nodes on curved surfaces by appropriately modifying the node density [11]. The paper by Shankar et al [32] also includes an algorithm for generation of an appropriate boundary discretization, based on RBF geometric model and super-sampling. This paper does not deal with the task of generating a boundary discretization and focuses on discretizations of domain interiors, assuming that the boundary discretization already exists when required.…”

“…State of the art algorithms. To the best of our knowledge, recently published algorithms by Fornberg and Flyer [11] and Shankar et al [32] best satisfy the requirements described in section 2 and are hence used as a base for further development. Both algorithms are first briefly described in the following sections.…”