Helmholtz-Hodge Decomposition Research Articles

Turbulence and vorticity in Galaxy clusters generated by structure formation

Turbulence is a key ingredient for the evolution of the intracluster medium, whose properties can be predicted with high-resolution numerical simulations. We present initial results on the generation of solenoidal and compressive turbulence in the intracluster medium during the formation of a small-size cluster using highly resolved, non-radiative cosmological simulations, with a refined monitoring in time. In this first of a series of papers, we closely look at one simulated cluster whose formation was distinguished by a merger around z ∼ 0.3. We separate laminar gas motions, turbulence and shocks with dedicated filtering strategies and distinguish the solenoidal and compressive components of the gas flows using Hodge–Helmholtz decomposition. Solenoidal turbulence dominates the dissipation of turbulent motions (∼95 per cent) in the central cluster volume at all epochs. The dissipation via compressive modes is found to be more important (∼30 per cent of the total) only at large radii (≥0.5rvir) and close to merger events. We show that enstrophy (vorticity squared) is good proxy of solenoidal turbulence. All terms ruling the evolution of enstrophy (i.e. baroclinic, compressive, stretching and advective terms) are found to be significant, but in amounts that vary with time and location. Two important trends for the growth of enstrophy in our simulation are identified: first, enstrophy is continuously accreted into the cluster from the outside, and most of that accreted enstrophy is generated near the outer accretion shocks by baroclinic and compressive processes. Secondly, in the cluster interior vortex, stretching is dominant, although the other terms also contribute substantially.

Subdivision exterior calculus for geometry processing

This paper introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design.

A radial basis function method for computing Helmholtz–Hodge decompositions

A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is specified on the boundary, and one where the tangential component of the curl-free part of the field specified. These two decompositions can then be combined to obtain a full Helmholtz-Hodge decomposition of the field, i.e. the sum of divergence-free, curl-free, and harmonic fields. All decompositions are computed from samples of the field at (possibly scattered) nodes over the domain, and all boundary conditions are imposed on the vector fields, not their potentials, distinguishing this technique from many current methods. Sobolev-type error estimates for the various decompositions are provided and demonstrated with numerical examples.

Dynamic Response of Swirl Stabilized Turbulent Premixed Flames Based on the Helmholtz-Hodge Velocity Decomposition

The dynamic response of fully premixed flames stabilized in strongly swirled flows undergoing vortex breakdown is investigated with axisymmetric unsteady RANS simulations. The analysis relies on the well known Helmholtz-Hodge decomposition of the velocity field into its irrotational and rotational components. A novel methodology based on the linearization of the progress variable transport equation is developed to determine the separate contribution of these velocity components to the Flame Transfer Function (FTF). Due to the phase delay between the convected tangential velocity and instantaneously propagating axial velocity perturbations, a non-monotonic frequency dependence of the swirl number amplitude downstream the swirl generator is detected. In line with experimental observations, such non-monotonic frequency dependence is found also for the amplitude and phase of the FTF. This behaviour is associated here with rotational velocity perturbations generated by the Central Recirculation Zone (CRZ) generated by the phenomenon of vortex breakdown which, responding in a fashion totally similar to the swirl number perturbation, produces flame surface area fluctuations with the same distribution versus frequency.

Helmholtz–Hodge decomposition and the analysis of 2D vector field ensembles

The Helmholtz–Hodge decomposition is a very useful instrument for vector field interpretation because it traditionally distinguishes conservative (curl-free) components from mass-preserving (divergence-free) components. In this work, we propose to use the Helmholtz–Hodge decomposition as a basic tool for the analysis of an ensemble of 2D vector fields. Given a vector field ensemble E, our approach firstly generates the corresponding curl-free, divergence-free, and harmonic component ensembles by using the Natural Helmholtz–Hodge decomposition to each vector field in E. With these ensembles in hand, our proposal not only quantifies, via a statistical analysis, how much each component ensemble is point-wisely correlated to the original vector field ensemble, but also allows to investigate the uncertainty of curl-free, divergence-free, and harmonic components separately. We test the proposed method with a synthetic vector field ensemble and with a simulated vector field ensemble.

Energy circulations in singular beams diffracted through an isosceles right triangular aperture

The Poynting vector in a singular optical field follows a helical path. Hence, it is the presence of a phase singularity that provides the circulating energy flow component to an optical field. These circulations in turn are responsible for the orbital angular momentum of light. This field when diffracted through an isosceles right triangular aperture generates straight line fringes in the far-field diffraction. We have addressed the modifications that happen to the energy circulations and the orbital angular momentum of light due to diffraction. Transverse components of the energy circulations in the diffracted field are studied by the Helmholtz-Hodge decomposition method. Interference experiments are presented to reveal the presence of phase singularities in the diffraction. This paper presents a complete analysis of the phase, phase gradient, and the intensity distributions in the diffraction pattern.

Detection and characterization of an optical vortex by the branch point potential method: analytical and simulation results.

Singularities are discontinuities in optical wavefronts that can be produced by turbulence effects. Since the presence of singularities in a wavefront severely degrades the adaptive optics correction performance, their detection is very important. The gradient of the wavefront phase, as measured by the Shack-Hartmann wavefront sensor in the presence of singularities, can be considered as the sum of the rotational and irrotational parts. The rotational part of the phase gradient originating from the phase singularities can be considered as a potential based on Helmholtz-Hodge decomposition. The potential at the singularities positions appears as peaks and valleys of the potential depending on the positive or negative charges of singularities. In this article, the detection of phase singularities based on the branch point potential (BPP) method is investigated. The irrotational part of the gradient produces a background potential where singularities positions appear as local extremum of the potential. With our method, the irrotational part of the gradient is eliminated and the value of peaks and valleys is increased. In addition, in this method, the potential value characterizes the optical singularities. Here, analytical and simulation results for the detection of general forms of the singularity are presented. Our simulations show the performance of singularities detection in noisy conditions.

Internal energy flows of coma-affected singular beams in low-numerical-aperture systems.

The circulating phase gradient component of a singular beam gets modified when focused by a low-numerical-aperature system suffering from coma aberration. The gradient due to this coma aberration splits the higher charge vortex into elementary vortices and distributes them spatially. This splitting depends on the charge and polarity of the incident singular beam as well as the sign and magnitude of the aberration coefficient. The transverse component of the Poynting vector field distribution at the focal plane is decomposed into the curl or solenoidal component and divergence or irrotational component using the Helmholtz-Hodge decomposition technique. The solenoidal component that relates to the orbital angular momentum carries the circulating energy, while the irrotational component shows the sources and sinks of the energy. Intriguing results of the study of energy flow around the edge dislocations apart from the point phase defects in the irrotational components are also presented.

A Kinematics Scalar Projection Method (KSP) for Incompressible Flows with Variable Density

A new scalar projection method presented for simulating incompressible flows with variable density is proposed. It reverses conventional projection algorithm by computing first the irrotational component of the velocity and then the pressure. The first phase of the projection is purely kinematics. The predicted velocity field is subjected to a discrete Hodge-Helmholtz decomposition. The second phase of upgrade of pressure from the density uses Stokes’ theorem to explicitly compute the pressure. If all or part of the boundary conditions is then fixed on the divergence free physical field, the system required to be solved for the scalar potential of velocity becomes a Poisson equation with constant coefficients fitted with Dirichlet conditions.

CONVERGENCE ANALYSIS ON GIBOU-MIN METHOD FOR THE SCALAR FIELD IN HODGE-HELMHOLTZ DECOMPOSITION

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. Using the L 2 -orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be O(h 1.5 ) with step size h: In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

The Natural Helmholtz-Hodge Decomposition for Open-Boundary Flow Analysis.

The Helmholtz-Hodge decomposition (HHD), which describes a flow as the sum of an incompressible, an irrotational, and a harmonic flow, is a fundamental tool for simulation and analysis. Unfortunately, for bounded domains, the HHD is not uniquely defined, traditionally, boundary conditions are imposed to obtain a unique solution. However, in general, the boundary conditions used during the simulation may not be known known, or the simulation may use open boundary conditions. In these cases, the flow imposed by traditional boundary conditions may not be compatible with the given data, which leads to sometimes drastic artifacts and distortions in all three components, hence producing unphysical results. This paper proposes the natural HHD, which is defined by separating the flow into internal and external components. Using a completely data-driven approach, the proposed technique obtains uniqueness without assuming boundary conditions a priori. As a result, it enables a reliable and artifact-free analysis for flows with open boundaries or unknown boundary conditions. Furthermore, our approach computes the HHD on a point-wise basis in contrast to the existing global techniques, and thus supports computing inexpensive local approximations for any subset of the domain. Finally, the technique is easy to implement for a variety of spatial discretizations and interpolated fields in both two and three dimensions.

Encoding cortical dynamics in sparse features.

Distributed cortical solutions of magnetoencephalography (MEG) and electroencephalography (EEG) exhibit complex spatial and temporal dynamics. The extraction of patterns of interest and dynamic features from these cortical signals has so far relied on the expertise of investigators. There is a definite need in both clinical and neuroscience research for a method that will extract critical features from high-dimensional neuroimaging data in an automatic fashion. We have previously demonstrated the use of optical flow techniques for evaluating the kinematic properties of motion field projected on non-flat manifolds like in a cortical surface. We have further extended this framework to automatically detect features in the optical flow vector field by using the modified and extended 2-Riemannian Helmholtz–Hodge decomposition (HHD). Here, we applied these mathematical models on simulation and MEG data recorded from a healthy individual during a somatosensory experiment and an epilepsy pediatric patient during sleep. We tested whether our technique can automatically extract salient dynamical features of cortical activity. Simulation results indicated that we can precisely reproduce the simulated cortical dynamics with HHD; encode them in sparse features and represent the propagation of brain activity between distinct cortical areas. Using HHD, we decoded the somatosensory N20 component into two HHD features and represented the dynamics of brain activity as a traveling source between two primary somatosensory regions. In the epilepsy patient, we displayed the propagation of the epileptic activity around the margins of a brain lesion. Our findings indicate that HHD measures computed from cortical dynamics can: (i) quantitatively access the cortical dynamics in both healthy and disease brain in terms of sparse features and dynamic brain activity propagation between distinct cortical areas, and (ii) facilitate a reproducible, automated analysis of experimental and clinical MEG/EEG source imaging data.

The Helmholtz-Hodge Decomposition—A Survey

The Helmholtz-Hodge Decomposition (HHD) describes the decomposition of a flow field into its divergence-free and curl-free 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.

Comments on the "Meshless Helmholtz-Hodge Decomposition"

The Helmholtz-Hodge decomposition (HHD) is one of the fundamental theorems of fluids describing the decomposition of a flow field into its divergence-free, curl-free, and harmonic components. Solving for the HHD is intimately connected to the choice of boundary conditions which determine the uniqueness and orthogonality of the decomposition. This article points out that one of the boundary conditions used in a recent paper "Meshless Helmholtz-Hodge Decomposition" is, in general, invalid and provides an analytical example demonstrating the problem. We hope that this clarification on the theory will foster further research in this area and prevent undue problems in applying and extending the original approach.

Fast discrete Helmholtz–Hodge decompositions in bounded domains

We present new fast discrete Helmholtz–Hodge decomposition (DHHD) methods for efficiently computing at the order O(ε) the divergence-free (solenoidal) or curl-free (irrotational) components and their associated potentials for a given L2(Ω) vector field in a bounded domain. The solution algorithms solve suitable penalized boundary-value elliptic problems involving either the grad(div) operator in the vector penalty-projection (VPP) or the rot(rot) operator in the rotational penalty-projection (RPP) with adapted right-hand sides of the same form. Therefore, they are extremely well-conditioned, fast and cheap, avoiding having to solve the usual Poisson problems for the scalar or vector potentials. Indeed, each (VPP) or (RPP) problem only requires two conjugate-gradient iterations whatever the mesh size, when the penalty parameter ε is sufficiently small. We state optimal error estimates vanishing as O(ε) with a penalty parameter ε as small as desired up to machine precision, e.g. ε=10−14. Some numerical results confirm the efficiency of the proposed (DHHD) methods, very useful for solving problems in electromagnetism or fluid dynamics.

Helmholtz Hodge decomposition of scalar optical fields

It is shown that the vector field decomposition method, namely, the Helmholtz Hodge decomposition, can also be applied to analyze scalar optical fields that are ubiquitously present in interference and diffraction optics. A phase gradient field that depicts the propagation and Poynting vector directions can hence be separated into solenoidal and irrotational components.

Split Bregman iteration and infinity Laplacian for image decomposition

In this paper, we address the issue of decomposing a given real-textured image into a cartoon/geometric part and an oscillatory/texture part. The cartoon component is modeled by a function of bounded variation, while, motivated by the works of Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, AMS, 2001], we propose to model the oscillating component v by a function of the space G of oscillating functions, which is, in some sense, the dual space of BV(Ω). To overcome the issue related to the definition of the G-norm, we introduce auxiliary variables that naturally emerge from the Helmholtz–Hodge decomposition for smooth fields, which yields to the minimization of the L∞-norm of the gradients of the new unknowns. This constrained minimization problem is transformed into a series of unconstrained problems by means of Bregman Iteration. We prove the existence of minimizers for the involved subproblems. Then a gradient descent method is selected to solve each subproblem, becoming related, in the case of the auxiliary functions, to the infinity Laplacian. Existence/Uniqueness as well as regularity results of the viscosity solutions of the PDE introduced are proved.

Characterization of Interictal Epileptiform Discharges with Time-Resolved Cortical Current Maps Using the Helmholtz–Hodge Decomposition

Source estimates performed using a single equivalent current dipole (ECD) model for interictal epileptiform discharges (IEDs) which appear unifocal have proven highly accurate in neocortical epilepsies, falling within millimeters of that demonstrated by electrocorticography. Despite this success, the single ECD solution is limited, best describing sources which are temporally stable. Adapted from the field of optics, optical flow analysis of distributed source models of MEG or EEG data has been proposed as a means to estimate the current motion field of cortical activity, or “cortical flow.” The motion field so defined can be used to identify dynamic features of interest such as patterns of directional flow, current sources, and sinks. The Helmholtz–Hodge Decomposition (HHD) is a technique frequently applied in fluid dynamics to separate a flow pattern into three components: (1) a non-rotational scalar potential U describing sinks and sources, (2) a non-diverging scalar potential A accounting for vortices, and (3) an harmonic vector field H. As IEDs seem likely to represent periods of highly correlated directional flow of cortical currents, the U component of the HHD suggests itself as a way to characterize spikes in terms of current sources and sinks. In a series of patients with refractory epilepsy who were studied with magnetoencephalography as part of their evaluation for possible resective surgery, spike localization with ECD was compared to HHD applied to an optical flow analysis of the same spike. Reasonable anatomic correlation between the two techniques was seen in the majority of patients, suggesting that this method may offer an additional means of characterization of epileptic discharges.

Feature detection and tracking in optical flow on non-flat manifolds

Optical flow is a classical approach to estimating the velocity vector fields associated to illuminated objects traveling onto manifolds. The extraction of rotational (vortices) or curl-free (sources or sinks) features of interest from these vector fields can be obtained from their Helmholtz–Hodge decomposition (HHD). However, the applications of existing HHD techniques are limited to flat, 2D domains. Here we demonstrate the extension of the HHD to vector fields defined over arbitrary surface manifolds. We propose a Riemannian variational formalism, and illustrate the proposed methodology with synthetic and empirical examples of optical-flow vector field decompositions obtained on a variety of surface objects.

A McLean Theorem for the moduli space of Lie solutions to mass transport equations

On compact manifolds which are not simply connected, we prove the existence of “fake” solutions to the optimal transportation problem. These maps preserve volume and arise as the exponential of a closed 1-form, hence appear geometrically like optimal transport maps. The set of such solutions forms a manifold with dimension given by the first Betti number of the manifold. In the process, we prove a Hodge–Helmholtz decomposition for vector fields. The ideas are motivated by the analogies between special Lagrangian submanifolds and solutions to optimal transport problems.