Helmholtz Decomposition Remains Opened for Future Researchers

IEEE Transactions on Visualization and Computer Graphics, 2013

The Helmholtz-Hodge Decomposition (HHD) describes the decomposition of a flow field into its divergence-free and curlfree components. Many researchers in various communities like weather modeling, oceanology, geophysics, and computer graphics are interested in understanding the properties of flow representing physical phenomena such as incompressibility and vorticity. The HHD has proven to be an important tool in the analysis of fluids, making it one of the fundamental theorems in fluid dynamics. The recent advances in the area of flow analysis have led to the application of the HHD in a number of research communities such as flow visualization, topological analysis, imaging, and robotics. However, because the initial body of work, primarily in the physics communities, research on the topic has become fragmented with different communities working largely in isolation often repeating and sometimes contradicting each others results. Additionally, different nomenclature has evolved which further obscures the fundamental connections between fields making the transfer of knowledge difficult. This survey attempts to address these problems by collecting a comprehensive list of relevant references and examining them using a common terminology. A particular focus is the discussion of boundary conditions when computing the HHD. The goal is to promote further research in the field by creating a common repository of techniques to compute the HHD as well as a large collection of example applications in a broad range of areas.

Journal of Computational Physics, 2007

In 1999, Jean-Paul Caltagirone and Jérô me Breil have developed in their paper [Caltagirone, J. Breil, Sur une méthode de projection vectorielle pour la résolution des équations de Navier-Stokes, C.R. Acad. Sci. Paris 327(Série II b) (1999) 1179-1184] a new method to compute a divergence-free velocity. They have used the grad(div) operator to extract the solenoidal part of a given vector field. In this contribution we explain how this method can be considered as a real Helmholtz decomposition and we present a stable approximation in the framework of spectral methods. Numerical results are presented to illustrate the efficiency of this approach.

Mathematics and Visualization, 2017

In our paper, we discuss generalized vector field decompositions that mainly have been derived from the classical Helmholtz-Hodge-decomposition. The ability to decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics.

2015

American Journal of Physics, 2006

Journal of Mathematical Analysis and Applications, 2023

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in R n , we replace the vector potential in R 3 by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n−1)/2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka-Volterra equations with n species.

arXiv (Cornell University), 2020

This paper introduces a simplified method to extend the Helmholtz Decomposition to n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described by a superposition of n(n − 1)/2 rotations within the coordinate planes. The source potential and the rotation potential are obtained by convolving the source and rotation densities with the fundamental solutions of the Laplace equation. The rotation-free gradient of the source potential and the divergence-free rotation of the rotation potential sum to the original vector field. The approach relies on partial derivatives and a Newton potential operator and allows for a simple application of this standard method to highdimensional vector fields, without using concepts from differential geometry and tensor calculus.

2015

2020

To understand the interaction of a fluid and a rigid body, we use the concept of B-evolution. Then in a similar way to the usual Navier-Stokes system, we obtain a Helmholtz type decomposition. Using B-evolution theory and the decomposition, we work on the semigroup to analyze the linear part of the system.