Meshless Approximation and Helmholtz-Hodge Decomposition of Vector Fields

TL;DR

This paper presents a meshless method for Helmholtz-Hodge decomposition of vector fields using radial basis functions, applicable to arbitrary domains and dimensions, demonstrated on 2D and 3D data.

Contribution

It introduces a novel meshless approach for vector field decomposition using radial basis functions, applicable to any domain dimension and without mesh assumptions.

Findings

Effective decomposition of 2D and 3D vector fields

Conditions on kernels ensure derivative existence

Validated on sensor data and simulations

Abstract

The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.