Lagrangian Representation Research Articles

Compatible Pairs of Dubrovin–Novikov Poisson Brackets and Lagrangian Representations of Integrable Hierarchies

Compatible Pairs of Dubrovin–Novikov Poisson Brackets and Lagrangian Representations of Integrable Hierarchies

The maintenance of coherent vortex topology by Lagrangian chaos in drift-Rossby wave turbulence

This work introduces the “potential vorticity bucket brigade,” a mechanism for explaining the resilience of vortex structures in magnetically confined fusion plasmas and geophysical flows. Drawing parallels with zonal jet formation, we show how inhomogeneous patterns of mixing can reinforce, rather than destroy non-zonal flow structure. We accomplish this through an exact stochastic Lagrangian representation of vorticity transport, together with a near-integrability property, which relates coherent flow topology to fluid relabeling symmetries. We demonstrate these ideas in the context of gradient-driven magnetized plasma turbulence, though the tools we develop here are model-agnostic and applicable beyond the system studied here.

Symmetry harmonization: exploring deformed oscillators and dissipative dynamics through the glass of Newton–Hooke algebra

Motivated by the symmetry in the non-relativistic limit of anti-de Sitter geometry, we employ planar dynamical models featuring exotic (deformed) harmonic oscillators, presented through direct and indirect Lagrangian representations. The latter introduces Bateman dissipative oscillator system. Analyzing these dynamic systems with a first-order Lagrangian scheme, our phase-space-based approach utilizes the moment map components to reveal the underlying symmetry algebra. This obtained algebra, interpreted as an extended version of Newton–Hooke (NH) cosmological symmetry algebras, has the potential to cast an augmented non-relativistic shadow over the expanding universe, offering an insightful perspective on extended NH spacetime in 2+1 dimensions through our dynamical realizations.

Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

<p>Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.</p>

Bridging Eulerian and Lagrangian Poisson-Boltzmann solvers by ESES.

The Poisson-Boltzmann (PB) model is a widely used electrostatic model for biomolecular solvation analysis. Formulated as an elliptic interface problem, the PB model can be numerically solved on either Eulerian meshes using finite difference/finite element methods or Lagrangian meshes using boundary element methods. Molecular surface generators, which produce the discretized dielectric interfaces between solutes and solvents, are critical factors in determining the accuracy and efficiency of the PB solvers. In this work, we investigate the utility of the Eulerian Solvent Excluded Surface (ESES) software for rendering conjugated Eulerian and Lagrangian surface representations, which enables us to numerically validate and compare the quality of Eulerian PB solvers, such as the MIBPB solver, and the Lagrangian PB solvers, such as the TABI-PB solver. Furthermore, with the ESES software and its associated PB solvers, we are able to numerically validate an interesting and useful but often neglected source-target symmetric property associated with the linearized PB model.

Self-similar solutions for fuzzy dark matter

Self-similar solutions for fuzzy dark matter are very different from their counterparts in the standard cold dark matter model. In contrast to the familiar hierarchical collapse of the current model for structure formation, they correspond to an inverse hierarchical blow-up. This fact highlights the gravitational cooling process, which in the absence of dissipation, allows the system to eject excess energy through the intermittent expulsion of clumps of matter. These surprising behaviours are due to the wave properties of the Schrödinger equation and to the quantum pressure. These features are printed in the Eulerian density-velocity representation of the nonrelativistic scalar field, or in the Lagrangian representation in the mass-shell trajectories.

ModalNeRF: Neural Modal Analysis and Synthesis for Free‐Viewpoint Navigation in Dynamically Vibrating Scenes

AbstractRecent advances in Neural Radiance Fields enable the capture of scenes with motion. However, editing the motion is hard; no existing method allows editing beyond the space of motion existing in the original video, nor editing based on physics. We present the first approach that allows physically‐based editing of motion in a scene captured with a single hand‐held video camera, containing vibrating or periodic motion. We first introduce a Lagrangian representation, representing motion as the displacement of particles, which is learned while training a radiance field. We use these particles to create a continuous representation of motion over the sequence, which is then used to perform a modal analysis of the motion thanks to a Fourier transform on the particle displacement over time. The resulting extracted modes allow motion synthesis, and easy editing of the motion, while inheriting the ability for free‐viewpoint synthesis in the captured 3D scene from the radiance field. We demonstrate our new method on synthetic and real captured scenes.

Coupling Phase-Field LB–MP Method for Multiphase Fluid–Deformable Solid Interaction Problems Involving Large Density and Viscosity Contrasts

A novel coupling phase-field lattice Boltzmann–material point (phase-field LB–MP) method is proposed in this work for multiphase fluid–solid bidirectional interaction problems. In this method, the phase-field lattice Boltzmann (LB) scheme is used to describe the multiphase incompressible fluid behaviors for its capability of dealing with binary-fluid system with large contrasts in both density and viscosity. The convected particle-domain interpolation-based material point method (MPM) is further adopted to capture the soft solid responses for its advantage in large deformation simulation and solid boundary representation. To bridge the solid and fluid phases, the Lagrangian marker points-based representation is proposed to explicitly depict the multiphase fluid–solid interface. Moreover, the hybrid bounce-back model is employed to effectively evaluate the unknown distribution functions and the contact angle specifying strategy for wetting boundary condition is developed for curved interface. The Galilean-invariant momentum exchange scheme and the distribution model are respectively taken into account to deal with hydrodynamic and capillary forces, and the interface-tracking force imposing strategy is developed for large deformation of solid. Additionally, the refilling algorithm is also considered for moving or deformable fluid–solid interface. There is no re-meshing operation needed for both fluid and solid parts in our phase-field LB–MP method. Three numerical examples are presented for progressively rigorous validations, which demonstrate that our phase-field LB–MP method has the capability of solving the multiphase fluid–deformable solid bidirectional interaction problems involving solid large deformation and binary-fluid behavior with large contrasts in both density and viscosity.

Lagrangian description of the atmospheric flow from Eulerian tracer advection with relaxation

AbstractThe Lagrangian representation of fluid flows offers a natural perspective to study many kinds of physical mechanisms. By contrast, the Eulerian representation is more convenient from a diagnostic point of view. This article attempts to combine elements of both worlds by proposing an Eulerian method that allows one to extract Lagrangian information about the atmospheric flow. The method is based on the offline advection of passive tracer fields and includes a relaxation term. The latter device allows one to run the integration in a continuous fashion without the need for reinitialization. As a result one obtains accumulated Lagrangian information, for example, about the recent parcel displacement or the recent parcel‐based diabatic heating, at each point of an Eulerian grid at any time step. The method is implemented with a pseudospectral algorithm suitable for gridded global atmospheric data and compared with the more traditional trajectory method. The method's utility is demonstrated on the basis of a few examples, which relate to cloud formation and the development of temperature anomalies. The examples highlight that the method provides a convenient diagnostic of parcel‐based changes, paving an intuitive way to explore the physical processes involved. Due to its gridpoint‐based nature, the proposed method can be applied to large data sets in a straightforward and computationally efficient manner, suggesting that the method is particularly useful for climatological analyses.

Initial stages of gravity-driven flow of two fluids of equal depth

Short-time behavior of gravity-driven free surface flow of two fluids of equal depth and different densities is studied. Initially, the fluids are at rest and separated with a vertical rigid plate of negligible thickness. Then, the plate disappears suddenly and a gravity-driven flow of the fluids starts. The flow in an early stage is described by the potential theory. The initial flow in the leading order is described by a linear problem, which is solved by the Fourier series method. The motions of the interface between the fluids and their free surfaces are investigated. The singular behaviors of the velocity field at the bottom point, where the interface meets the rigid bottom, and the top point, where the interface meets both free surfaces, are analyzed in detail. The flow velocity is shown to be log-singular at the bottom point. The leading-order inner asymptotic solution is constructed in a small vicinity of this point. It is shown that the flow close to the bottom point is self-similar. The motion of the interface is independent of any parameters, including the density ratio, of the problem in specially stretched variables. In the limiting case of negligible density of one of the fluids, the results of the classical dam break problem are recovered. The Lagrangian representation is employed to capture the behavior of the interface and the free surfaces at the top, where the fluid interface meets the free surfaces. The shapes of the free surfaces and the interface in the leading order computed by using the Lagrangian variables show a jump discontinuity of the free surface near the top point where the free surfaces and the interface meet. Inner region formulation is derived near the top point.

Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

We study the regularity of the flow {varvec{X}}(t,y), which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution rho in L^infty ({mathbb {R}}^{d+1}) of the continuity equation ∂tρ+div(ρb)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t \\rho + {{\\,\ extrm{div}\\,}}(\\rho {\\varvec{b}}) = 0, \\end{aligned}$$\\end{document}with {varvec{b}}in L^1_t {{,textrm{BV},}}_x. We prove that {varvec{X}} is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{{\\varvec{X}}(t,y+rz) - {\\varvec{X}}(t,y)}{r} \\underset{r \\rightarrow 0}{\\rightarrow }\\ W(t,y) z \\quad \ ext {in measure}, \\end{aligned}$$\\end{document}where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}}{\ extrm{d}t} W(t, y) = \\frac{(D {\\varvec{b}})_y(\ extrm{d}t)}{J(t-,y)} W(t-, y), \\end{aligned}$$\\end{document}where (D{varvec{b}})_y(textrm{d}t) is the disintegration of the measure int D {varvec{b}}(t,cdot ) , textrm{d}t with respect to the partition given by the trajectories {varvec{X}}(t, y) and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}}{\ extrm{d}t} J(t,y) = ({{\\,\ extrm{div}\\,}}{\\varvec{b}})_y(\ extrm{d}t) = \ extrm{Tr}(D{\\varvec{b}})_y(\ extrm{d}t). \\end{aligned}$$\\end{document}The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a {{,textrm{BV},}} vector field.

Modelling of solidification structure and macro-segregation in an industrial DC cast slab

A cellular automaton (CA)-finite element (FE) model was implemented for multi-scale modelling of micro-segregation, mesoscopic grain structure and macroscopic segregation during direct chill (DC) casting of industrial billets or slabs. The macroscopic transport of mass, momentum, energy and solutes was solved on an FE grid, while the mesoscopic grain structure governed by nucleation, growth kinetics and grain evolution was calculated on a CA grid. The solidification path was determined using a modified micro-segregation model for multi-component aluminium alloys. An Euler representation was used for pre-processing and post-processing, and a Lagrangian representation was used for expanding the calculation domain and for resolving the CAFE model. By simulating a DC casting experiment of the 2024 aluminium alloy, a typical grain structure was reproduced, and the composition map showed a reasonable deviation. This model was applied to an industrial-scale DC cast slab of an Al-3.5Cu-1.5 Mg (wt. %) alloy, and three simulations with different nucleation undercoolings were performed for a grain-unrefined slab, a grain-refined slab and an equilibrium solidified slab, respectively. The slabs tended to solidify at equilibrium with the decreasing nucleation undercooling. The earlier release of latent heat yielded a smaller liquid undercooling region ahead of the solidification front, and a finer grain structure. A typical grain structure with coarse equiaxial grains at the centre and fine columnar grains near the bottom surface as well as sidewall was observed for the grain-unrefined slab. By contrast, the grain structure of the grain-refined slab was fully equiaxial. Furthermore, the grain structure, temperature field, melt flow and macro-segregation were quantitatively investigated.

Filament based plasma

Simulation of stellar atmospheres, such as that of our own sun, is a common task in CGI for scientific visualization, movies and games. A fibrous volumetric texture is a visually dominant feature of the solar corona---the plasma that extends from the solar surface into space. These coronal fibers can be modeled as magnetic filaments whose shape is governed by the magnetohydrostatic equation. The magnetic filaments provide a Lagrangian curve representation and their initial configuration can be prescribed by an artist or generated from magnetic flux given as a scalar texture on the sun's surface. Subsequently, the shape of the filaments is determined based on a variational formulation. The output is a visual rendering of the whole sun. We demonstrate the fidelity of our method by comparing the resulting renderings with actual images of our sun's corona.

The rectifiability of the entropy defect measure for Burgers equation

We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting.

LBcuda: A high-performance CUDA port of LBsoft for simulation of colloidal systems

We present LBcuda, a GPU accelerated version of LBsoft, our open-source MPI-based software for the simulation of multi-component colloidal flows. We describe the design principles, the optimization and the resulting performance as compared to the CPU version, using both an average cost GPU and high-end NVidia GPU cards (V100 and the latest A100). The results show a substantial acceleration for the fluid solver reaching up to 200 GLUPS (Giga Lattice Updates Per Second) on a cluster made of 512 A100 NVIDIA cards simulating a grid of eight billion lattice points. These results open attractive prospects for the computational design of new materials based on colloidal particles. Program summaryProgram Title: LBcudaCPC Library link to program files:https://doi.org/10.17632/v6fvmzpcrn.1Developer's repository link:https://github.com/copmat/LBcudaLicensing provisions: 3-Clause BSD LicenseProgramming language: CUDA FortranNature of problem: Hydro-dynamics of colloidal multi-component systems and Pickering emulsions.Solution method: Lattice-Boltzmann method solving the Navier-Stokes equations for the fluid dynamics within an Eulerian description. Particle solver describing colloidal particles within a Lagrangian representation coupled to the fluid solver. The numerical solution of the coupling algorithm includes the back reaction effects for each force terms according to a fluid-particle multi-scale paradigm.

Characterisation of the Tyre Spray Ejected Downstream of a Bluff Automotive Body

<div class="section abstract"><div class="htmlview paragraph">Considerations of surface contamination and airborne spray are becoming increasingly significant throughout the automotive design process. Advanced driver assistance systems, such as autonomous cruise control, are growing in popularity. These systems rely on external sensors, the performance of which may be impaired by both direct obstruction and spray. Existing experimental methods of assessing front-end surface contamination and wiper performance have typically utilised fixed spray-grids positioned upstream of the vehicle. The resulting spray is largely steady in nature, in contrast to the unsteady flow-field and tyre spray that would be produced by preceding vehicles. This paper presents the numerical analysis of the spray ejected downstream of a simplified automotive body. The continuous phase (air) is solved using a DDES-based approach coupled with a Lagrangian representation of the dispersed phase (water). Two configurations are examined, a square-back configuration and a variation employing 20° rear-end side tapers. The inclusion of side tapering results in a significant change in wake topology and the resulting spray cloud. Good agreement is achieved between initial single-phase predictions of the continuous phase and existing experimental data. The spray cloud of both configurations is found to be highly unsteady, driven by vortical structures in the near- and far-wake, and is altered significantly by what are relatively minor changes to the geometry. Proper Orthogonal Decomposition reveals comparable spatial modes in the mass flux field of the dispersed phase and the continuous phase velocity field downstream of the vehicle. However, the correlation between the temporal coefficients of these modes is relatively weak. This highlights the presence of slip effects between the two phases, coming as a result of particle inertia, and the need to consider both phases simultaneously in future studies of spray dynamics.</div></div>

Demonstrating the viability of Lagrangian in situ reduction on supercomputers

Performing exploratory analysis and visualization of large-scale time-varying computational science applications is challenging due to inaccuracies that arise from under-resolved data. In recent years, Lagrangian representations of the vector field computed using in situ processing are being increasingly researched and have emerged as a potential solution to enable exploration. However, prior works have offered limited estimates of the encumbrance on the simulation code as they consider “theoretical” in situ environments. Further, the effectiveness of this approach varies based on the nature of the vector field, benefitting from an in-depth investigation for each application area. With this study, an extended version of Sane et al. (2021), we contribute an evaluation of Lagrangian analysis viability and efficacy for simulation codes executing at scale on a supercomputer. We investigated previously unexplored cosmology and seismology applications as well as conducted a performance benchmarking study by using a hydrodynamics mini-application targeting exascale computing. To inform encumbrance, we integrated in situ infrastructure with simulation codes, and evaluated Lagrangian in situ reduction in representative homogeneous and heterogeneous HPC environments. To inform post hoc accuracy, we conducted a statistical analysis across a range of spatiotemporal configurations as well as a qualitative evaluation. Additionally, our study contributes cost estimates for distributed-memory post hoc reconstruction. In all, we demonstrate viability for each application — data reduction to less than 1% of the total data via Lagrangian representations, while maintaining accurate reconstruction and requiring under 10% of total execution time in over 90% of our experiments.

Graph neural network-accelerated Lagrangian fluid simulation

We present a data-driven model for fluid simulation under Lagrangian representation. Our model, Fluid Graph Networks (FGN), uses graphs to represent the fluid field. In FGN, fluid particles are represented as nodes and their interactions are represented as edges. Instead of directly predicting the acceleration or position correction given the current state, FGN decomposes the simulation scheme into separate parts — advection, collision, and pressure projection. For these different predictions tasks, we propose two kinds of graph neural network structures, node-focused networks and edge-focused networks. We show that the learned model can produce accurate results and remain stable in scenarios with different geometries. In addition, FGN is able to retain many important physical properties of incompressible fluids, such as low velocity divergence, and adapt to time step sizes beyond the one used in the training set. FGN is also computationally efficient compared to classical simulation methods as it operates on a smaller neighborhood and does not require iteration at each timestep during the inference.

Generalized characteristics for finite entropy solutions of Burgers’ equation

We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers’ equation ∂tu+∂xu22=0,whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution’s hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.

On the structure of weak solutions to scalar conservation laws with finite entropy production

We consider weak solutions with finite entropy production to the scalar conservation law $$\begin{aligned} \partial _t u+\mathrm {div}_x F(u)=0 \quad \text{ in } (0,T)\times \mathbb {R}^d. \end{aligned}$$ Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.