Helmholtz-Hodge Decomposition Research Articles

Evolution of CFD numerical methods and physical models towards a full discrete approach

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.

Vortex-p: A Helmholtz-Hodge and Reynolds decomposition algorithm for particle-based simulations

Astrophysical turbulent flows display an intrinsically multi-scale nature, making their numerical simulation and the subsequent analyses of simulated data a complex problem. In particular, two fundamental steps in the study of turbulent velocity fields are the Helmholtz-Hodge decomposition (compressive+solenoidal; HHD) and the Reynolds decomposition (bulk+turbulent; RD). These problems are relatively simple to perform numerically for uniformly-sampled data, such as the one emerging from Eulerian, fix-grid simulations; but their computation is remarkably more complex in the case of non-uniformly sampled data, such as the one stemming from particle-based or meshless simulations. In this paper, we describe, implement and test vortex-p, a publicly available tool evolved from the vortex code, to perform both these decompositions upon the velocity fields of particle-based simulations, either from smoothed particle hydrodynamics (SPH), moving-mesh or meshless codes. The algorithm relies on the creation of an ad-hoc adaptive mesh refinement (AMR) set of grids, on which the input velocity field is represented. HHD is then addressed by means of elliptic solvers, while for the RD we adapt an iterative, multi-scale filter. We perform a series of idealised tests to assess the accuracy, convergence and scaling of the code. Finally, we present some applications of the code to various SPH and meshless finite-mass (MFM) simulations of galaxy clusters performed with OpenGadget3, with different resolutions and physics, to showcase the capabilities of the code.

Isotropic random tangential vector fields on spheres

In this paper we characterized isotropic random tangential vector fields on d-spheres for d≥1 by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a d-sphere for d≥3. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.

Decomposition of the mechanical stress tensor: from the compressible Navier–Stokes equation to a turbulent potential flow model

We frame the mechanical stress tensor decomposition in a general procedure which involves the Helmholtz–Hodge decomposition. We highlight the impact of the mechanical stress tensor decomposition on the Navier–Stokes equation, with emphasis on the dissipation function. For fluids with low compressibility, we draw some insights on the Reynolds Averaged Navier–Stokes equations, and on the Reynolds stress tensor decomposition. We derive a turbulent potential flow model, and investigate the transition from viscous potential flow to turbulent potential flow. Under low Mach number approximation, we apply the turbulent potential flow model to one-dimensional propagation of large amplitude pressure waves in liquid-filled pipe.

Feasibility of Estimating Sea Surface Height Anomalies from Surface Ocean Currents and Winds

Abstract We propose a method to reconstruct sea surface height anomalies (SSHA) from vector surface currents and winds. This analysis is motivated by the proposed satellite ODYSEA, which is a Doppler scatterometer that measures coincident surface vector winds and currents. If it is feasible to estimate SSHA from these measurements, then ODYSEA could provide collocated fields of SSHA, currents, and winds over a projected wide swath of ∼1700 km. The reconstruction also yields estimates of the low-frequency surface geostrophic, Ekman, irrotational, and nondivergent current components and a framework for separation of balanced and unbalanced motions. The reconstruction is based on a steady-state surface momentum budget including the Ekman drift, Coriolis acceleration, and horizontal advection. The horizontal SSHA gradient is obtained as a residual of these terms, and the unknown SSHA is solved for using a Helmholtz–Hodge decomposition given an imposed SSHA boundary condition. We develop the reconstruction using surface currents, winds, and SSHA off the U.S. West Coast from a 43-day coupled ROMS–WRF simulation. We also consider how simulated ODYSEA measurement and sampling errors and boundary condition uncertainties impact reconstruction accuracy. We find that temporal smoothing of the currents for periods of 150 h is necessary to mitigate large reconstruction errors associated with unbalanced near-inertial motions. For the most realistic case of projected ODYSEA measurement noise and temporal sampling, the reconstructed SSHA fields have an RMS error of 2.1 cm and a model skill (squared correlation) of 0.958 with 150-h resolution. We conclude that an accurate SSHA reconstruction is feasible using information measured by ODYSEA and external SSHA boundary conditions.

An alternative to the Navier–Stokes equation based on the conservation of acceleration

The derivation of the Navier–Stokes equation in continuum mechanics leads to a number of consequences which are discussed in depth. In spite of its very high representativity of real flows, this equation presents some artefacts due to the whole notion of the continuous medium. An alternative to the Navier–Stokes equation is proposed, based on the conservation of energy per unit mass instead of momentum. The classical inertial frame of reference is replaced by a set of local frames of reference where interactions are treated as cause and effect. Invoking the principle of equivalence between energy and mass, the latter is eliminated from the quantities used in this new formalism. All quantities, variables and physical properties are thus expressed in units of mass. The law of motion is established in the form of the conservation of acceleration, an energy per unit of mass and length. The acceleration is thus written in the form of a Helmholtz–Hodge decomposition, in two terms, the first curl-free and the second divergence-free as a function of two potentials, scalar and vector. Maxwell's idea of federating the laws of electrodynamics and magnetism to establish electromagnetism is taken up here to establish the new law of motion as a nonlinear wave equation. This approach makes it possible to demonstrate that this law is relativistic from the start. The form of the equation of motion in two Lagrangians gives access to symmetries related to the conservation of certain quantities according to Noether's theorem.

Opposite effects of flame-generated potential and solenoidal velocity fluctuations on flame surface area in moderately intense turbulence

To explore the influence of dilatational and rotational motions on iso-scalar surface area within a premixed turbulent flame, three-dimensional compressible direct numerical simulation data obtained by Dave et al. (2018) from a lean, complex-chemistry, hydrogen-air flame propagating in intense small-scale turbulence (a ratio of laminar flame thickness to Kolmogorov length scale is about 20) in a box are analyzed using Helmholtz–Hodge decomposition of velocity field into solenoidal and potential components. The obtained results indicate that the flame surface area is created by potential velocity fluctuations generated due to combustion-induced thermal expansion, whereas the rotational motion acts to smooth wrinkles of iso-scalar surfaces within local preheat and reaction zones and, consequently, to reduce the flame surface area. The latter finding challenges the classical concept of flame-generated turbulence and is attributed to flame-generated solenoidal velocity fluctuations. Specifically, the incoming tangential (to the flame) vorticity is suppressed by baroclinic torque, which also generates vorticity in another tangential direction, with the latter (flame-generated) solenoidal velocity fluctuations working to smooth the flame surface. Thus, even under conditions of moderately intense turbulence, flame surface area can primarily be created by potential velocity perturbations caused by combustion-induced thermal expansion.

Information geometry of dynamics on graphs and hypergraphs

We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz–Hodge decomposition and the Otto structure in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{2}$$\\end{document}-Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can broaden the applicability of information geometry to various problems of linear and nonlinear dynamics.

Modeling capillary flows by conservation of acceleration and surface energy

Modeling of capillary flows on a surface or at the interface of two fluids is approached from a formulation with two contributions, one described by the divergence of the normal to the surface and the second represented by its curl. These are the two curl-free and divergence-free components of a Helmholtz–Hodge decomposition of the capillary acceleration; the first is the gradient of the scalar potential of the acceleration and the second the dual curl of the vector potential. Proposed formulation is characterized by (i) definition of a directional curvature based on the dihedral angle instead of an average curvature, (ii) an intrinsic anisotropic surface tension per unit mass excluding the presence of density in the capillary terms of the equation of motion and (iii) introduction of a capillary potential, an energy per unit mass. This energetic approach leads to an equation of capillary motion of a surface without thickness; the same formalism can be integrated as a source in an equation of motion of immiscible fluids.

A method to investigate mid and high-frequency vibration energy transfer paths using the irrotational intensity

Structural intensity fields provide a clear understanding of the noise, vibration and harshness (NVH) behavior of complex structures by showing the vibration energy flow between zones of excitation and dissipation. The use of models with fine meshes for automotive structures has allowed structural intensity calculations to be performed at high frequencies. However, fields are difficult to understand at medium and high frequencies: the intensity field has a vortical character that hides the vibration energy flow in these frequency ranges. In this work, a methodology is presented to filter out vortices in the structural intensity field. The approach uses the Hodge–Helmholtz decomposition: the structural intensity field is decomposed into rotational, irrotational, and harmonic components. Among them, only the irrotational component is needed to describe the path of energy flow from sources to sinks. The proposed methodology involves calculating the dynamic response and solving a diffusion equation similar to the thermal conduction equation to obtain the irrotational field of interest. Plate examples are used to illustrate the efficiency of the method, which provides a better understanding of the mid- and high-frequency energy transmission paths.

A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

In this paper, we propose and analyse a numerical method to solve 2D Dirichlet time-harmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure (P-) and Shear (S-) waves via a suitable Helmholtz–Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the L2-norm for the displacement field, in which the contributions to the error associated with the P- and S- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the P- and S- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented.

Cosmic-Ray Drag and Damping of Compressive Turbulence

While it is well known that cosmic rays (CRs) can gain energy from turbulence via second-order Fermi acceleration, how this energy transfer affects the turbulent cascade remains largely unexplored. Here, we show that damping and steepening of the compressive turbulent power spectrum are expected once the damping time becomes comparable to the turbulent cascade time. Magnetohydrodynamic simulations of stirred compressive turbulence in a gas-CR fluid with diffusive CR transport show clear imprints of CR-induced damping, saturating at , where is the turbulent energy input rate. In that case, almost all of the energy in large-scale motions is absorbed by CRs and does not cascade down to grid scale. Through a Hodge–Helmholtz decomposition, we confirm that purely compressive forcing can generate significant solenoidal motions, and we find preferential CR damping of the compressive component in simulations with diffusion and streaming, rendering small-scale turbulence largely solenoidal, with implications for thermal instability and proposed resonant scattering of E ≳ 300 GeV CRs by fast modes. When CR transport is streaming dominated, CRs also damp large-scale motions, with kinetic energy reduced by up to 1 order of magnitude in realistic E CR ∼ E g scenarios, but turbulence (with a reduced amplitude) still cascades down to small scales with the same power spectrum. Such large-scale damping implies that turbulent velocities obtained from the observed velocity dispersion may significantly underestimate turbulent forcing rates, i.e., .

No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution.The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution.We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour.Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.

Integration of operator-validated contours in deformable image registration for dose accumulation in radiotherapy.

Deformable image registration (DIR) is a core element of adaptive radiotherapy workflows, integrating daily contour propagation and/or dose accumulation in their design. Propagated contours are usually manually validated and may be edited, thereby locally invalidating the registration result. This means the registration cannot be used for dose accumulation. In this study we proposed and evaluated a novel multi-modal DIR algorithm that incorporated contour information to guide the registration. This integrates operator-validated contours with the estimated deformation vector field and warped dose. The proposed algorithm consisted of both a normalized gradient field-based data-fidelity term on the images and an optical flow data-fidelity term on the contours. The Helmholtz-Hodge decomposition was incorporated to ensure anatomically plausible deformations. The algorithm was validated for same- and cross-contrast Magnetic Resonance (MR) image registrations, Computed Tomography (CT) registrations, and CT-to-MR registrations for different anatomies, all based on challenging clinical situations. The contour-correspondence, anatomical fidelity, registration error, and dose warping error were evaluated. The proposed contour-guided algorithm considerably and significantly increased contour overlap, decreasing the mean distance to agreement by a factor of 1.3 to 13.7, compared to the best algorithm without contour-guidance. Importantly, the registration error and dose warping error decreased significantly, by a factor of 1.2 to 2.0. Our contour-guided algorithm ensured that the deformation vector field and warped quantitative information were consistent with the operator-validated contours. This provides a feasible semi-automatic strategy for spatially correct warping of quantitative information even in difficult and artefacted cases.

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Backscatter of scalar variance in turbulent premixed flames

To explore the direction of inter-scale transfer of scalar variance between subgrid scale (SGS) and resolved scalar fields, direct numerical simulation data obtained earlier from two complex-chemistry lean hydrogen–air flames are analysed by applying Helmholtz–Hodge decomposition (HHD) to the simulated velocity fields. Computed results show backscatter of scalar (combustion progress variable$c$) variance, i.e. its transfer from SGS to resolved scales, even in a highly turbulent flame characterized by a unity-order Damköhler number and a ratio of Kolmogorov length scale to thermal laminar flame thickness as low as 0.05. Analysis of scalar fluxes associated with the solenoidal and potential velocity fields yielded by HHD shows that the documented backscatter stems primarily from the potential velocity perturbations generated due to dilatation in instantaneous local flames, with the backscatter being substantially promoted by a close alignment of the spatial gradient of mean scalar progress variable and the potential-velocity contribution to the local SGS scalar flux. The alignment is associated with the fact that combustion-induced thermal expansion increases local velocity in the direction of$\boldsymbol {\nabla } c$. These results call for development of SGS models capable of predicting backscatter of scalar variance in turbulent flames in large eddy simulations.

Reconstruction of the Instantaneous Images Distorted by Surface Waves via Helmholtz–Hodge Decomposition

Imaging through water waves will cause complex geometric distortions and motion blur, which seriously affect the correct identification of an airborne scene. The current methods main rely on high-resolution video streams or a template image, which limits their applicability in real-time observation scenarios. In this paper, a novel recovery method for the instantaneous images distorted by surface waves is proposed. The method first actively projects an adaptive and adjustable structured light pattern onto the water surface for which random fluctuation will cause the image to degrade. Then, the displacement field of the feature points in the structured light image is used to estimate the motion vector field of the corresponding sampling points in the scene image. Finally, from the perspective of fluid mechanics, the distortion-free scene image is reconstructed based on the Helmholtz-Hodge Decomposition (HHD) theory. Experimental results show that our method not only effectively reduces the distortion to the image, but also significantly outperforms state-of-the-art methods in terms of computational efficiency. Moreover, we tested the real-scene sequences of a certain length to verify the stability of the algorithm.

Macroscopic Network Circulation for Planar Graphs

The analysis of networks, aimed at suitably defined functionality, often focuses on partitions that capture desired features. Chief among the relevant concepts is a 2-partition; this underlies the Cheeger inequality and highlights a ‘`constriction’' that limits accessibility between the respective parts of the network. In a similar spirit, we explore a notion of global circulation which necessitates a concept of a 3-partition that exposes this macroscopic feature of network flows. Graph circulation is often present in transportation networks as well as in certain biological networks. We introduce a notion of circulation for general graphs and then focus on planar graphs. For the latter we explain that a scalar potential characterizes circulation in complete analogy with the curl of planar vector fields and we present an algorithm for determining values of the potential and, hence, quantify circulation. We then discuss notions of circulation, explain how these may depend on graph embedding, draw parallels with Helmholtz-Hodge decomposition of vector fields, and conclude with an application in detecting abnormalities in cardiac circulatory physiology.

High-Order Directional Fields

We introduce a framework for representing face-based directional fields of an arbitrary piecewise-polynomial order. Our framework is based on a primal-dual decomposition of fields, where the exact component of a field is the gradient of piecewise-polynomial conforming function, and the coexact component is defined as the adjoint of a dimensionally-consistent discrete curl operator. Our novel formulation sidesteps the difficult problem of constructing high-order non-conforming function spaces, and makes it simple to harness the flexibility of higher-order finite elements for directional-field processing. Our representation is structure-preserving, and draws on principles from finite-element exterior calculus. We demonstrate its benefits for applications such as Helmholtz-Hodge decomposition, smooth PolyVector fields, the vector heat method, and seamless parameterization.

Conditioned structure functions in turbulent hydrogen/air flames

Direct numerical simulation data obtained from two turbulent, lean hydrogen–air flames propagating in a box are analyzed to explore the influence of combustion-induced thermal expansion on turbulence in unburned gas. For this purpose, Helmholtz–Hodge decomposition is applied to the computed velocity fields. Subsequently, the second-order structure functions conditioned to unburned reactants are sampled from divergence-free solenoidal velocity field or irrotational potential velocity field, yielded by the decomposition. Results show that thermal expansion significantly affects the conditioned potential structure functions not only inside the mean flame brushes but also upstream of them. Upstream of the flames, first, transverse structure functions for transverse potential velocities grow with distance r between sampling points more slowly when compared to the counterpart structure functions sampled from the entire or solenoidal velocity field. Second, the former growth rate depends substantially on the distance from the flame-brush leading edge, even at small r. Third, potential root mean square (rms) velocities increase with the decrease in distance from the flame-brush leading edge and are comparable with solenoidal rms velocities near the leading edge. Fourth, although the conditioned axial and transverse potential rms velocities are always close to one another, thus implying isotropy of the potential velocity field in unburned reactants, the potential structure functions exhibit a high degree of anisotropy. Fifth, thermal expansion effects are substantial even for the solenoidal structure functions and even upstream of a highly turbulent flame. These findings call for development of advanced models of turbulence in flames, which allow for the discussed thermal expansion effects.