Meshless Approximation and Helmholtz-Hodge Decomposition of Vector Fields.

Giuseppe Patane

doi:10.1109/tvcg.2020.3016588

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.

Highlights

Vector fields are commonly used to model several phenomena, such as natural events, diffusive processes, electric and electromagnetic fields; for instance, vector fields represent the velocity and direction of an object or the magnitude and direction of a force
Experimental results For the selection of the centres of the meshless approximation of a 2D vector field defined on a regular (Fig. 12, wind field) and irregular sampling (Fig. 13, a fluid flow), we apply the kernel-based sampling, a uniform sampling, and a random sampling
The error is measured as the angle between the input and the reconstruction vector fields; the white colour corresponds to a null angle and black identifies the maximum angle, which is lower or equal to π in our experiments

Summary

Introduction

Vector fields are commonly used to model several phenomena, such as natural events, diffusive processes, electric and electromagnetic fields; for instance, vector fields represent the velocity and direction of an object or the magnitude and direction of a force. Measuring and simulating a flow field typically generate a large amount of unstructured, sparse, and noisy vector data, which are a challenging input for the reconstruction of a global representation. The components of the reconstructed vector field must satisfy additional properties such as being conservative (e.g., incompressible fluid flow), irrotational (e.g., magnetic field), harmonic, and invariant with respect to a set of transformations. In these examples, we typically have heterogeneous data (e.g., scalar values, vectors) in terms of data structures (e.g., vector fields, scalar functions), spatial distribution or resolution (e.g., regular grids, meshes, sparse samples), and data values of any dimension. The HHD is applied to simplify or edit the structure of

Methods

Results

Conclusion