Lagrangian Code Research Articles

A modern concept of Lagrangian hydrodynamics

AbstractWe offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite‐volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.

Characterization of similar Marshak waves observed at the LMJ

We detail results of two experiments performed at the Laser Mégajoule (LMJ) facility aimed at studying similar supersonic Marshak waves propagating in a low-density SiO2 aerogel enclosed in metallic tubes. Similar means here that these two experiments, driven by the same input radiation temperature history, use purposely very different tubes in terms of length (L = 1200 or 2000 μm), diameter (2R = 1000 or 2000 μm), nature of the wall (gold or copper), and aerogel densities (ρ = 30 or 20 mg/cm3), yet the transit time and the radiation temperature of the fronts at the tube exit are the same for both shots. Marshak waves are characterized at the exit using simultaneously for the first time to our knowledge, a one dimensional soft x-ray imager from which the radiation front transit time and curvature are measured and also a broadband x-ray spectrometer to infer its temperature history. These constraining results are then successfully compared to those from simple analytical models [Cohen et al., Phys. Rev. Res. 2, 023007 (2020) and Hurricane et al., Phys. Plasmas 13, 113303 (2006)] and from the three dimensional Lagrangian radiation-hydrodynamics code TROLL to get information on x-ray energy losses. Controlled compensation effects between the length, diameter, and nature of the tubes (governing these losses) are such that the radiation temperature drop along the tubes is eventually the same for these two similar shots.

Impulse coupling measurement of metallic and carbon targets during laser ablation through ballistic pendulum experiments and simulations

Laser ablation propulsion and orbit cleaning are developing areas of research. The general aim of laser-based techniques applied to this field is to maximize the momentum transfer produced by a laser shot. This work presents results from ballistic pendulum experiments under vacuum on aluminum, copper, tin, gold, and porous graphite targets. The work has focused on the metrology of the laser experiments to ensure good stability over a wide range of laser parameters (laser intensity ranging from 4 GW/cm2 to 8.7 TW/cm2, pulse duration from 80 ps to 15 ns, and wavelengths of 528 or 1057 nm). The results presented compile data from three experimental campaigns spanning from 2018 to 2021 on two different laser platforms and using different pulse durations, energies, and wavelengths. The study is complemented by the simulation of the momentum from the mono-dimensional Lagrangian code ESTHER. The first part of this work gives a detailed description of the experimental setup used, the ESTHER code, and the treatment of the simulations. The second part focuses on the experimental results. The third part describes the simulation results and provides a comparison with the experimental data. The last part presents possible improvements for future work on the subject.

Interpretative 3D MHD modelling of deuterium SPI into a JET H-mode plasma

The pre-thermal quench (pre-TQ) dynamics of a pure deuterium ( D2 ) shattered pellet injection (SPI) into a 3MA / 7MJ JET H-mode plasma is studied via 3D non-linear MHD modelling with the JOREK code. The interpretative modelling captures the overall evolution of the measured density and radiated power. The simulations also identify the importance of the drifts of ablation plasmoids towards the tokamak low field side (LFS) and the impurities in the background plasma in fragment penetration, assimilation, radiative cooling and MHD activity in D2 SPI experiments. It is found that plasmoid drifts lead to an about 70% reduction of the central line-integrated density (compared to a simulation without drifts) in the JET D2 SPI discharge considered. Impurities that pre-exist before the SPI as well as those from possible impurity influxes related to the SPI are shown to dominate the radiation in the considered discharge. With inputs from JOREK simulations, modelling with the Lagrangian particle-based pellet code PELOTON reproduces the deviation of the SPI fragments in the direction of the major radius as observed by the fast camera. This confirms the role of rocket effects and plasmoid drifts in the considered discharge and reinforces the validity of the JOREK modelling. The limited core density rise due to plasmoid drifts and the strong radiative cooling and MHD activity with impurities (depending on their species and concentration) could limit the effectiveness of LFS D2 SPI in runaway electron avoidance and are worth considering in the design of the ITER disruption mitigation system.

A novel V-cut method for explosive-free breakage of biaxially loaded rock using soundless chemical demolition agents

Interest in soundless chemical demolition agents (SCDAs), also known as expansive cements, as potentially viable alternatives to explosives for rock fragmentation, has been growing in recent years. Consequently, there is an increasing amount of literature on the use of SCDA for the breakage of rock blocks and boulders. Limited research has been conducted so far on the breakage of excavation fronts, such as tunnel or drift faces, using SCDA. This is due to the perception that the planar compressive in-situ stresses in the face would inhibit the creation and propagation of fracturing due to expansive pressure. This study proposes a novel V-cut method for demolishing rock panels under biaxial stress using SCDA. This method was examined through large-scale tests and numerical modelling. The rock panels were subjected to high biaxial confinements of 26 and 40 MPa. Such a level of confinement corresponds to an in-situ stress state 1000 m below the surface in the Canadian shield. The V-cut drillhole pattern employs two sets of three SCDA holes angled at 45° from the face of a Stanstead granite panel. The drillhole arrangement aims to create a V-shaped wedge in the plane of major principal stress. When angled drillholes are subjected to expansive pressure, they tend to cast out of the panel face, causing fragmentation. Two panels of 1 m × 1 m × 0.25 m were successfully demolished using the proposed method. The three-dimensional fast Lagrangian analysis code FLAC3D modelling was used to reconstruct the panel failure mechanism owing to the V-cut. This study demonstrates the feasibility of fragmenting an excavation front, such as a rock excavation face, with SCDA using a V-cut drill hole pattern while subjected to high biaxial confinement.

Mergers of double NSs with one high-spin component: brighter kilonovae and fallback accretion, weaker gravitational waves

ABSTRACT Neutron star (NS) mergers where both stars have negligible spins are commonly considered as the most likely ‘standard’ case. In globular clusters, however, the majority of NSs have been spun up to millisecond (ms) periods and, based on observed systems, we estimate that a non-negligible fraction of all double NS mergers ($\sim 4\pm 2\, {{\ \rm per\ cent}}$) contains one component with a spin of a (few) ms. We use the Lagrangian numerical relativity code SPHINCS_BSSN to simulate mergers where one star has no spin and the other has a dimensionless spin parameter of χ = 0.5. Such mergers exhibit several distinct signatures compared to irrotational cases. They form only one, very pronounced spiral arm and they dynamically eject an order of magnitude more mass of unshocked material at the original, very low electron fraction. One can therefore expect particularly bright, red kilonovae. Overall, the spinning case collisions are substantially less violent and they eject smaller amounts of shock-generated semirelativistic material. Therefore, the ejecta produce a weaker blue/ultraviolet kilonova precursor signal, but – since the total amount is larger – brighter kilonova afterglows months after the merger. The spinning cases also have significantly more fallback accretion and thus could power late-time X-ray flares. Since the post-merger remnant loses energy and angular momentum significantly less efficiently to gravitational waves, such systems can delay a potential collapse to a black hole and are therefore candidates for merger-triggered gamma-ray bursts with longer emission time-scales.

Resolution criteria to avoid artificial clumping in Lagrangian hydrodynamic simulations with a multiphase interstellar medium

ABSTRACT Large-scale cosmological galaxy formation simulations typically prevent gas in the interstellar medium (ISM) from cooling below $\approx 10^4\, \mathrm{K}$. This has been motivated by the inability to resolve the Jeans mass in molecular gas ($\ll 10^5\, \mathrm{M}_{\odot }$) which would result in undesired artificial clumping. We show that the classical Jeans criteria derived for Newtonian gravity are not applicable in the simulated ISM if the spacing of resolution elements representing the dense ISM is below the gravitational force softening length and gravity is therefore softened and not Newtonian. We re-derive the Jeans criteria for softened gravity in Lagrangian codes and use them to analyse gravitational instabilities at and below the hydrodynamical resolution limit for simulations with adaptive and constant gravitational softening lengths. In addition, we define criteria for which a numerical runaway collapse of dense gas clumps can occur caused by oversmoothing of the hydrodynamical properties relative to the gravitational force resolution. This effect is illustrated using simulations of isolated disc galaxies with the smoothed particle hydrodynamics code swift. We also demonstrate how to avoid the formation of artificial clumps in gas and stars by adjusting the gravitational and hydrodynamical force resolutions.

QES-Plume v1.0: a Lagrangian dispersion model

Abstract. Low-cost simulations providing accurate predictions of transport of airborne material in urban areas, vegetative canopies, and complex terrain are demanding because of the small-scale heterogeneity of the features influencing the mean flow and turbulence fields. Common models used to predict turbulent transport of passive scalars are based on the Lagrangian stochastic dispersion model. The Quick Environmental Simulation (QES) tool is a low-computational-cost framework developed to provide high-resolution wind and concentration fields in a variety of complex atmospheric-boundary-layer environments. Part of the framework, QES-Plume, is a Lagrangian dispersion code that uses a time-implicit integration scheme to solve the generalized Langevin equations which require mean flow and turbulence fields. Here, QES-Plume is driven by QES-Winds, a 3D fast-response model that computes mass-consistent wind fields around buildings, vegetation, and hills using empirical parameterizations, and QES-Turb, a local-mixing-length turbulence model. In this paper, the particle dispersion model is presented and validated against analytical solutions to examine QES-Plume’s performance under idealized conditions. In particular, QES-Plume is evaluated against a classical Gaussian plume model for an elevated continuous point-source release in uniform flow, the Lagrangian scaling of dispersion in isotropic turbulence, and a non-Gaussian plume model for an elevated continuous point-source release in a power-law boundary-layer flow. In these cases, QES-Plume yields a maximum relative error below 6 % when compared with analytical solutions. In addition, the model is tested against wind-tunnel data for a uniform array of cubical buildings. QES-Plume exhibits good agreement with the experiment with 99 % of matched zeros and 59 % of the predicted concentrations falling within a factor of 2 of the experimental concentrations. Furthermore, results also emphasize the importance of using high-quality turbulence models for particle dispersion in complex environments. Finally, QES-Plume demonstrates excellent computational performance.

Arkenstone – I. A novel method for robustly capturing high specific energy outflows in cosmological simulations

ABSTRACT Arkenstone is a new model for multiphase, stellar feedback-driven galactic winds designed for inclusion in coarse resolution cosmological simulations. In this first paper of a series, we describe the features that allow Arkenstone to properly treat high specific energy wind components and demonstrate them using idealized non-cosmological simulations of a galaxy with a realistic circumgalactic medium (CGM), using the arepo code. Hot, fast gas phases with low mass loadings are predicted to dominate the energy content of multiphase outflows. In order to treat the huge dynamic range of spatial scales involved in cosmological galaxy formation at feasible computational expense, cosmological volume simulations typically employ a Lagrangian code or else use adaptive mesh refinement with a quasi-Lagrangian refinement strategy. However, it is difficult to inject a high specific energy wind in a Lagrangian scheme without incurring artificial burstiness. Additionally, the low densities inherent to this type of flow result in poor spatial resolution. Arkenstone addresses these issues with a novel scheme for coupling energy into the transition region between the interstellar medium (ISM) and the CGM, while also providing refinement at the base of the wind. Without our improvements, we show that poor spatial resolution near the sonic point of a hot, fast outflow leads to an underestimation of gas acceleration as the wind propagates. We explore the different mechanisms by which low and high specific energy winds can regulate the star formation rate of galaxies. In future work, we will demonstrate other aspects of the Arkenstone model.

The Lagrangian numerical relativity code SPHINCS_BSSN_v1.0

We present version 1.0 of our Lagrangian numerical relativity code SPHINCS_BSSN. This code evolves the full set of Einstein equations, but contrary to other numerical relativity codes, it evolves the matter fluid via Lagrangian particles in the framework of a high-accuracy version of smooth particle hydrodynamics (SPH). The major new elements introduced here are: (i) a new method to map the stress–energy tensor (known at the particles) to the spacetime mesh, based on a local regression estimate; (ii) additional measures that ensure the robust evolution of a neutron star through its collapse to a black hole; and (iii) further refinements in how we place the SPH particles for our initial data. The latter are implemented in our code SPHINCS_ID which now, in addition to LORENE, can also couple to initial data produced by the initial data library FUKA. We discuss several simulations of neutron star mergers performed with SPHINCS_BSSN_v1.0, including irrotational cases with and without prompt collapse and a system where only one of the stars has a large spin (χ = 0.5).

Quantitative proton radiography and shadowgraphy for arbitrary intensities

Charged-particle radiography and shadowgraphy data can be directly inverted to obtain a line-integrated transverse Lorentz force or a line-integrated transverse refractive index gradient if intensity modulations due to scattering and absorption are negligible, and angular deflections are small. We develop a new direct-inversion algorithm based on plasma physics and compare it to a new Monge–Ampère code and an existing power diagram code (Kasim et al., 2017). The measured or source intensity is represented by electrons subject to drag, and the other intensity by fixed ions. The decrease in kinetic plus electrostatic energy determines convergence. The displacement of the electrons from their initial to their equilibrium positions determines the line-integrated force or refractive index gradient. We have implemented two approaches: PIC (particle in cell) and Lagrangian fluid, in 1-D and 2-D. The PIC code works for arbitrary intensities, can work efficiently in parallel, and can make use of existing codes. The Lagrangian code requires less memory and is faster than the PIC code without massively parallel processing, but fails in 2-D for large intensity modulations. The Monge–Ampère code is by far the fastest in 2-D, without massively parallel processing, but fails for intensities with large voids, high contrast ratios and large deflections across the boundaries, and could not obtain the degree of convergence possible with the PIC code. The power diagram code was by far the slowest and most memory intensive, and failed for large peaks in the measured intensity.

In-depth investigation of natural convection thermal characteristics of BALI experiment through Eulerian computational fluid dynamics code and comparison with Lagrangian code

In-vessel retention through external reactor vessel cooling (IVR-ERVC) is a severe accident management (SAM) strategy that has been adopted and used in many nuclear reactors such as AP1000, APR1400, and light water reactor etc. Some reactor accidents have raised concerns about nuclear reactors among residents, leading to a decrease in residents' acceptability and many studies on SAM are being conducted. Experiments on IVR-ERVC are almost impossible due to its specificity, so fluid characteristics are analyzed through BALI experiments with similar condition. In this study, computational fluid dynamics (CFD) via Reynolds-averaged Navier-Stokes (RANS) and large eddy simulation (LES) for BALI experiments were performed. Steady-state CFD analysis was performed on three turbulence models, and SST k-ω model was in good agreement with the experimental measurement temperature within the maximum error range of 1.9%. LES CFD analysis was performed based on the RANS analysis results and it was confirmed that the temperature and wall heat flux for depth was consistent within an error range of 1.0% with BALI experiment. The LES CFD analysis results were compared with those of the Lagrangian-based solver. LES matched the temperature distribution better than SOPHIA, but SOPHIA calculated the position of boundary between stratified layer and convective layer more accurately. On the other hand, Lagrangian-based solver predicted several small eddy behaviors of the convective layer and LES predicted large vortex behavior. The vibration characteristics near the cooling part of the BALI experimental device were confirmed through Fast Fourier Transform (FFT) investigation. It was found that the power spectral density for pressure at least 10 times higher near the side cooling than near the top cooling.

Code Comparison in Galaxy-scale Simulations with Resolved Supernova Feedback: Lagrangian versus Eulerian Methods

We present a suite of high-resolution simulations of an isolated dwarf galaxy using four different hydrodynamical codes: Gizmo, Arepo, Gadget, and Ramses. All codes adopt the same physical model, which includes radiative cooling, photoelectric heating, star formation, and supernova (SN) feedback. Individual SN explosions are directly resolved without resorting to subgrid models, eliminating one of the major uncertainties in cosmological simulations. We find reasonable agreement on the time-averaged star formation rates as well as the joint density–temperature distributions between all codes. However, the Lagrangian codes show significantly burstier star formation, larger SN-driven bubbles, and stronger galactic outflows compared to the Eulerian code. This is caused by the behavior in the dense, collapsing gas clouds when the Jeans length becomes unresolved: Gas in Lagrangian codes collapses to much higher densities than that in Eulerian codes, as the latter is stabilized by the minimal cell size. Therefore, more of the gas cloud is converted to stars and SNe are much more clustered in the Lagrangian models, amplifying their dynamical impact. The differences between Lagrangian and Eulerian codes can be reduced by adopting a higher star formation efficiency in Eulerian codes, which significantly enhances SN clustering in the latter. Adopting a zero SN delay time reduces burstiness in all codes, resulting in vanishing outflows as SN clustering is suppressed.

Eulerian–Lagrangian modeling of phase transition for application to cavitation-driven chemical processes

Hydrodynamic cavitation is a promising technology for several applications, like disinfection, sludge treatment, biodiesel production, degradation of organic emerging pollutants as pharmaceutical, and dye degradation. Due to local saturation conditions, cavitating liquid exhibits generation, growth, and subsequent collapse of vapor-filled cavities. The cavities' collapse brings very high pressure and temperature; this last aspect is essential in some chemical processes because it induces the decomposition of water molecules into species with a high oxidation potential, which can react with organic substances. Properly exploiting this process requires a highly accurate prediction of pressure peak values. To this purpose, we implemented a multi-phase Eulerian–Lagrangian code to solve the fluid-dynamic problem, coupled with the Rayleigh–Plesset equation, to capture the evolution of bubbles with the required accuracy. The algorithm was validated against experimental data acquired with optical techniques for different cavitation-shedding mechanisms. Then, we used the developed tool to investigate the decoloration of organic substances from a cavitation Venturi tube operating at different pressure. We compared the obtained results with the experimental observation to assess the reliability of the developed code as a predictive tool for cavitation and the possibility of using the code itself to assess scale-up criteria for possible industrial applications.

Investigation of Nanoscale Scratching on Copper with Conical Tools Using Particle-Based Simulation

In this study, a modeling approach based on smooth particle hydrodynamics (SPH) was implemented to simulate the nanoscale scratching process using conical tools with different negative rake angles. The implemented model enables the study of the topography of groove profiles, scratching forces, and the residual plastic strain beneath the groove. An elastoplastic material model was employed for the workpiece, and the tool–workpiece interaction was defined by a contact model adopted from the Hertz theory. An in-house Lagrangian SPH code was implemented to perform nano-scratching simulations. The SPH simulation results were compared with nanoscale scratching experimental data available in the literature. The simulation results revealed that the normal force was more dominant compared to the cutting force, in agreement with experimental results reported for a conical tip tool with a 60° negative rake angle. In addition, the simulated groove profile was in good agreement with the groove profile produced in the aforementioned experiment. The numerical simulations also showed that the normal and cutting forces increased with the increase in the scratching depth and rake angle. Although the cutting and ploughing mechanisms were noticed in nano-scratching, the ploughing mechanism was more dominant for increased negative rake angles. It was also observed that residual plastic strain exists below the groove surface, and that the plastically deformed layer thickness beneath a scratched groove is larger for more negative values of the tool rake angle and higher scratching depths.

Massively-parallel Lagrangian particle code and applications

Massively-parallel, distributed-memory algorithms for the Lagrangian particle hydrodynamic method (Samulyak et al., 2018) have been developed, verified, and implemented. The key component of parallel algorithms is a particle management module that includes a parallel construction of octree databases, dynamic adaptation and refinement of octrees, and particle migration between parallel subdomains. The particle management module is based on the p4est (parallel forest of k-trees) library. The massively-parallel Lagrangian particle code has been applied to a variety of fundamental science and applied problems. A summary of Lagrangian particle code applications to the injection of impurities into thermonuclear fusion devices and to the simulation of supersonic hydrogen jets in support of laser-plasma wakefield acceleration research has also been presented.

Parametric instability in warped astrophysical discs: growth, saturation, and feedback

ABSTRACT Attempts to understand the dynamics of warped astrophysical discs have garnered significant attention, largely motivated by the growing catalogue of observed distorted systems. Previous studies have shown that the evolution of the warp is crucially regulated by the internal flow fields established by the undulating geometry. These are typically modelled as laminar horizontal, shearing flows which oscillate back and forth at approximately the orbital frequency. However this shearing motion is known to be susceptible to a hydrodynamic, parametric instability of inertial waves which might modify the warped dynamics. Whilst the linear growth phase is well understood, the subsequent non-linear saturation combined with the self-consistent feedback onto the warp has not been studied. In this work, we implement a novel numerical setup using the recent ring model framework of Fairbairn and Ogilvie, within the Lagrangian code gizmo. We formally identify several locally growing modes in the simulation, as predicted by a three-mode coupling analysis of the instability, and find decent agreement with the theoretical growth rates. We understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength parametric couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wave-like turbulence. The Reynolds stresses, transporting energy from the warp to the small scales, can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Consequently, this model might help inform future global studies which are commonplace but typically do not resolve the parametric instability.

Regularized Interior Point Methods for Constrained Optimization and Control

Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an algorithm that synergistically combines them. Building a sequence of closely related subproblems and approximately solving each of them, this approach inherently exploits warm-starting, early termination, and the possibility to adopt subsolvers tailored to specific problem structures. Moreover, by relaxing the equality constraints with a proximal penalty, the regularized subproblems are feasible and satisfy a strong constraint qualification by construction, allowing the safe use of efficient solvers. We show how regularization benefits the underlying linear algebra and a detailed convergence analysis indicates that limit points tend to minimize constraint violation and satisfy suitable optimality conditions. Finally, numerical results indicate that the combined approach compares favorably, in terms of robustness, against both interior point and augmented Lagrangian codes.

Impact of temporal modulations on laser-induced damage of fused silica at 351 nm

Abstract Laser-induced damage (LID) on high-power laser facilities is one of the limiting factors for the increase in power and energy. Inertial confinement fusion (ICF) facilities such as Laser Mégajoule or the National Ignition Facility use spectral broadening of the laser pulse that may induce power modulations because of frequency modulation to amplitude modulation conversion. In this paper, we study the impact of low and fast power modulations of laser pulses both experimentally and numerically. The MELBA experimental testbed was used to shape a wide variety of laser pulses and to study their impact on LID. A 1D Lagrangian hydrodynamic code was used to understand the impact of different power profiles on LID.

Rapid simulations of halo and subhalo clustering

The analysis of cosmological galaxy surveys requires realistic simulations for their interpretation. Forward modelling is a powerful method to simulate galaxy clustering without the need for an underlying complex model. This approach requires fast cosmological simulations with a high resolution and large volume, to resolve small dark matter halos associated to single galaxies. In this work, we present fast halo and subhalo clustering simulations based on the Lagrangian perturbation theory code PINOCCHIO, which generates halos and merger trees. The subhalo progenitors are extracted from the merger history and the survival of subhalos is modelled. We introduce a new fitting function for the subhalo merger time, which includes a redshift dependence of the fitting parameters. The spatial distribution of subhalos within their hosts is modelled using a number density profile. We compare our simulations with the halo finder ROCKSTAR applied to the full N-body code GADGET-2. The subhalo velocity function and the correlation function of halos and subhalos are in good agreement. We investigate the effect of the chosen number density profile on the resulting subhalo clustering. Our simulation is approximate yet realistic and significantly faster compared to a full N-body simulation combined with a halo finder. The fast halo and subhalo clustering simulations offer good prospects for galaxy forward models using subhalo abundance matching.