Lagrangian Code Research Articles

Shock wave propagation in prompt supernova explosion and the MSW effect of neutrino

The MSW effect of supernova neutrino is one of foci of recent neutrino astrophysics. It is still an open question how the shock wave propagation affects the neutrino oscillation. Using an implicit Lagrangian code for general relativistic spherical hydrodynamics, we succeeded in numerical simulations of breakout of shock wave propagation through the stellar envelope. We first discuss our successful result of shock wave propagation which is generated by adiabatic collapse of iron core, and compare with non-adiabatic models. Secondly, we apply our model to the neutrino oscillation and calculate survival probabilities of three light-neutrino families. We discuss how the flux and energy spectrum of each neutrino species can change due to the MSW effect.

Simulations of the output of the Ne-like Ti x-ray laser at32.6nmdriven by the grazing-incidence pumping scheme

Numerical simulations are presented for the output of a Ne-like Ti soft x-ray laser at $32.6\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ driven by the grazing-incidence pumping (GRIP) scheme, using a modified one-dimensional (1D) lagrangian hydrodynamic code MED103 coupled with an atomic physics data package and a 2D RAYTRACE code as a postprocessor. We predict that saturation can be achieved at a plasma length of $0.2\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ by an optimized driving configuration with an energy of only $200\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}$. We also compare the output beam quality of the Ne-like Ti soft x-ray laser generated by an optimized pump energy of $200\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}$ and a higher energy of $385\phantom{\rule{0.3em}{0ex}}\mathrm{mJ}$. Better spatial coherence is expected for a higher pump energy.

A comparative study of numerical models for Eulerian–Lagrangian simulations of turbulent evaporating sprays

The paper comparatively assesses the computational performance of a selected number of theoretical and semi-empirical liquid droplet evaporation models that focus on thermodynamic non-equilibrium effects, physical property estimation methods and convective and blowing effect corrections for the calculation of the heat and mass transfer rates. Three different test cases are examined in order to establish the most appropriate model, in terms of both physical accuracy and numerical efficiency for implementation in two-phase CFD spray simulations. The considered cases span from a single, isolated droplet, evaporating in a convective environment, to a fully turbulent, evaporating, hollow cone spray; for the latter case, an in-house Eulerian–Lagrangian CFD code is used. Predictions are validated against experimental data for all test cases and the most promising model is established on the basis of accuracy and CPU time requirements. As a result, the “infinite conductivity” equilibrium droplet evaporation model, combined with an analytic expression for the convective and blowing effect correction can be proposed as most appropriate for CFD spray applications.

“Curl- q”: A vorticity damping artificial viscosity for essentially irrotational Lagrangian hydrodynamics calculations

The bane of Lagrangian hydrodynamics calculations in multi-dimensions is the appearance of vorticity that causes tangling of the mesh and consequent run termination. This vorticity may be numerical or physical in origin, and is in addition to the spurious ‘‘hourglass’’ modes associated with quadrilateral or hexahedral zones that in pure form have both zero curl and divergence associated with their velocity field. The purpose of this note is to introduce a form of vorticity damping, based on a previously published edgecentered artificial viscosity [1], that extends the runtime and range of calculations over which a pure Lagrangian code can compute. Since the explicit inclusion of an artificial viscosity into the fluid equations is often referred to as the ‘‘q term’’, we denote this new term as the ‘‘curl-q’’, because it is a function of the curl of the velocity field in a zone. It is formulated in the context of the ‘‘discrete, compatible formulation of Lagrangian hydrodynamics’’ [2,3]. This employs a staggered placement of variables in space (velocity and position at nodes, with density and stresses in zones), but a predictor/corrector time integration scheme so that all variables are known at the same time level, allowing total energy to be exactly conserved [2]. This new ‘‘curl-q’’ does not resolve shock waves and is always to be utilized with an artificial viscosity that performs this task. In order to set the stage for the introduction of the new curl-q force, the edge-centered artificial viscosity given in [1] is briefly reviewed in a slightly simplified form; after this the curl-q force is formulated as an analogy to this edge-centered artificial viscosity. Numerical results are given in both 2D and 3D that display its effectiveness. In particular, results are contrasted between this new term and a recently published tensor artificial viscosity [4]. It is shown that these two forms give quite similar results in 2D. We end with a brief discussion concerning the validity of the use of this type of numerical device. In Fig. 1 is shown a quadrilateral zone with its defining points, i = 1–4, and associated median mesh vectors ~Si. These vectors point in the indicated direction and have a magnitude of the surface area that lies between their defining points in 2D or in 3D [2,5]. In terms of the median mesh vector~S1, the force exerted by the edgecentered artificial viscosity between points ‘‘1’’ and ‘‘2’’ from zone ‘‘z’’ is given by

Collisionless expansion of a Gaussian plasma into a vacuum

The collisionless expansion into a vacuum of a one-dimensional Gaussian plasma bunch of size much larger than the Debye length is studied both analytically and with a hybrid Lagrangian code. Both the case of an isothermal plasma fed by an external energy source and the case of an energy-conserving plasma are studied. It is shown that charge-separation effects are responsible for deviations with respect to known self-similar solutions. In particular, wave breaking occurs because the inner parts of the expanding plasma acquire faster velocities than the outer parts. As a result an ion front appears at the edge of the plasma. In the isothermal case the self-similar solution progressively applies to the whole plasma, while the fastest ion’s velocity asymptotically reaches the same values as in the sharp boundary case. In the energy-conserving case the self-similar description only applies to a part of the plasma, which approximately corresponds to the part which was initially quasineutral, and the final velocity of the fastest ions is significantly smaller than in the sharp boundary case.

The repair paradigm: New algorithms and applications to compressible flow

The repair paradigm leads to several algorithms for redistributing mass, momentum and energy, while adhering to local maximum principles, as an adjunct to the remapping step in certain compressible flow codes that use remapping, such as Arbitrary–Lagrangian–Eulerian codes, or for just redistributing mass in advection codes. In the case of advection of a concentration, repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells, thus guaranteeing at least that the new concentration is between zero and one. For compressible flow, density, velocity and internal energy are similarly constrained while maintaining conservation of mass, momentum and total energy. In this way, positive density and internal energy are achieved as a side effect. We propose a new algorithm, combining both local and global repair, that maintains causality and is efficient in a parallel computational setting. The local/global algorithm is independent of the order in which the distribution is performed, and it maintains 1D symmetry. This is applied to advection in two dimensions, and to, among others, the LeBlanc problem, the Sedov problem, and an interacting 2D blast wave problem. The latter is done with a Lagrangian code for which rezoning, remapping and repair are essential.

Current singularities at finitely compressible three-dimensional magnetic null points

The formation of current singularities at line-tied two- and three-dimensional (2D and 3D, respectively) magnetic null points in a nonresistive magnetohydrodynamic environment is explored. It is shown that, despite the different separatrix structures of 2D and 3D null points, current singularities may be initiated in a formally equivalent manner. This is true no matter whether the collapse is triggered by flux imbalance within closed, line-tied null points or driven by externally imposed velocity fields in open, incompressible geometries. A Lagrangian numerical code is used to investigate the finite amplitude perturbations that lead to singular current sheets in collapsing 2D and 3D null points. The form of the singular current distribution is analyzed as a function of the spatial anisotropy of the null point, and the effects of finite gas pressure are quantified. It is pointed out that the pressure force, while never stopping the formation of the singularity, significantly alters the morphology of the current distribution as well as dramatically weakening its strength. The impact of these findings on 2D and 3D magnetic reconnection models is discussed.

Hydrodynamic instabilities in axisymmetric geometry self-similar models and numerical simulations

Hydrodynamic instabilities play an important role in the target compression for inertial confinement fusion (ICF). We present the analytical solution of a perturbed isentropic implosion. We compare the analytical solution to the results obtained with perturbation and bi-dimensional Lagrangian hydrodynamic codes. We also compare results from bi-dimensional code and perturbation code on an ICF like test case.

Fluid and kinetic simulation of inertial confinement fusion plasmas

The main features of codes for inertial confinement fusion studies are outlined, and a few recent simulation results are presented. The two-dimensional Lagrangian fluid code DUED is used to study target evolution, including beam-driven compression, hydrodynamic stability, hot spot formation, ignition and burn. An electro-magnetic particle-in-cell (PIC) code is applied to the study of ultraintense laser–plasma interaction and generation of fast electron jets. A relativistic 3D collisionless fluid model addresses relativistic electron beam propagation in a dense plasma.

Computations of primordial black-hole formation

Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe. Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear. We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium. For super-critical perturbations, we confirm the results of previous work concerning scaling laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes. For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold. We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law.

An optimization of a multi-layered plate under ballistic impact

This paper may be the first trial regarding the optimal design of a multi-layered plate under ballistic impact. An optimal design of a multi-layered plate to endure ballistic impact is suggested by using size optimization based on numerical simulations. The NET2D, a Lagrangian explicit time-integration finite element code for impact analyses, is used to find the optimal parameter values. Three different materials such as mild steel and aluminum for a multi-layered plate structure and die steel for the pellet are assumed. In order to consider the effects of strain rate hardening, strain hardening and thermal softening, the Johnson–Cook model is used as the constitutive models for the simulation. Several mesh types of different size and aspect ratio are tried to check the effect of mesh on the solution and to obtain the appropriate mesh density. The measuring domain is selected to reduce the analyzing time without affecting the sensitivity. The response surface method based on the design of experiments is used to obtain the optimal design. The average temperature or the equivalent plastic strain is introduced as a response for the optimization of the impact problem. Furthermore, the perforation criteria with the equivalent plastic strain to determine whether the plate structure is perforated or not is suggested. The optimized thickness of each layer in which perforation does not occur and the strength of multi-layer is maximized is obtained at a constant velocity of a pellet with a designated total thickness.

One-dimensional numerical simulation of laser-driven flyer plates

A one-dimensional (1D) bulk absorption model to simulate the accelerating process of laser-driven flyer plate is presented, in which the laser energy deposition and acceleration of flyer plates are included. In the laser intensity range of 108–1010W∕cm2, the gasdynamical behavior of the ablated foil plasma is described with a modified 1D Lagrangian hydrodynamic code. The histories and profiles of physical parameters including pressure, temperature, velocity, opacity, and average ionization charge number are given by this model. According to the calculation, an aluminum foil of original thickness of 10μm may be accelerated to a velocity of about 2km∕s, the calculated average ionized charge number of plasma Z is >3 peak temperature and plasma pressure are in the order of 2.1×105K and 1.5×103GPa, respectively, and the maximum Rosseland opacity is about 1.9×105cm−1. The calculated particle velocity distribution in plasma is linear, which confirms the assumption employed in the Gurney model.

A New Approach to the Reactive Burn Modeling of Heterogeneous Explosives

This paper discusses a new approach to building mechanistic models for shock ignition and other reactive phenomena in heterogeneous solid explosives. In the proposed approach, hot spots are treated on a unifying framework (a hot-spot cell) to integrate the three, principal response behaviors of heterogeneous energetic materials: energy localization (hot-spot formation), temperature dependent chemical reactions, and heat flow. Hot spots are treated statistically through use of a size distribution and its evolution equation. Chemical reactions start at the surface of hot spots, and the products in turn become reactants for the subsequent bulk reactions. A heat conduction equation is solved for the temperature states in the hot-spot cell. Ignition and growth of the reactive burn depend on the complex interactions of the three processes: localized energy deposition, chemical reactions (surface as well as bulk), and heat flow. Coupling of the hot-spot model to hydrodynamic flow equations is based at present on a single cell, using mass averaged mixture equations of state and a common particle velocity for the constituents. The overall model is implemented on a two-dimensional Lagrangian code called CASH. To demonstrate the predictive capability of the model, we show several exploratory calculations using RDX as a model material.more » They include (1) shock ignition and growth-to-detonation, (2) quenching, and (3) curved detonation in a cylindrical specimen.« less

Explosive disruption of polytropes: a one-dimensional hydrodynamic calculation

We study explosions of stellar models using a one-dimensional Lagrangian hydrodynamics code. We calculate how much mass is liberated as a function of the energy of explosion for a variety of pre-explosion polytropic structures and for equations of state with a range of radiation-to-gas pressure ratios. The results show that simple assumptions about the amount of mass lost in an explosion can be quite inaccurate, and that even one-dimensional stellar models exhibit a rich phenomenology. The mass loss fraction rises from about 50 to 100 per cent as a function of the explosion energy in an approximately discontinuous manner. Combining our results with those of other, more realistic models, we suggest that Nova Scorpii (J1655 - 40) may have experienced significant mass fallback because the explosion energy was less than the critical value. We infer that the original progenitor was less than twice the mass of today's remnant.

A Lagrangian multibody code for deriving the symbolic state-space equations of motion for open-loop systems containing flexible beams

In this study, the development of a symbolic multibody code is described. This code uses Lagrange's equations to derive automatically the state-space equations of motion, in relative coordinates, for open-loop systems made up of rigid bodies and flexible Timoshenko beams. The algorithm is then encoded in Mathematica as the package MultiFlex.m and the straightforward application of MultiFlex to two examples with 1 degree of freedom and 14 degrees of freedom, respectively, is presented. The MultiFlex package represents one part of a suite of programs which are being developed by the authors in order to provide entirely symbolic analysis of multi-degree of freedom problems in engineering dynamics.

Simulation of high-intensity laser–plasma interactions by use of the 2D Lagrangian code “ATLANT-HE”

Hot electrons may significantly influence interaction of ultrashort laser pulses with solids. Accurate consideration of resonant absorption of laser energy and hot electron generation at a critical surface was achieved through the developed physical and mathematical models. A two-dimensional (2D) ray-tracing algorithm has been developed to simulate laser beam refraction and Bremsstrahlung absorption with allowance for nonlinear influence of a strong electromagnetic field. Hot electron transport was considered as a straight-line flow weakening by a friction force calculated in the approximation of the average state of ionization. Developed models were coupled with the 2D Lagrangian gas dynamic code “ATLANT” that takes into account nonlinear heat transport. The developed program has been applied to simulate irradiation of Al foils by picosecond laser double pulses. Hot electron transport and heating resulted in thin foil explosions. The transition from the exploding foil regime to the ablative one with foil thickening has been simulated and analyzed at various values of laser light intensity. In second series of calculations we have modeled the interaction of a nanosecond iodine laser with a two-layered target.

Mergers of irrotational neutron star binaries in conformally flat gravity

We present the first results from our new general relativistic, Lagrangian hydrodynamics code, which treats gravity in the conformally flat (CF) limit. The evolution of fluid configurations is described using smoothed particle hydrodynamics (SPH), and the elliptic field equations of the CF formalism are solved using spectral methodsin spherical coordinates. The code was tested on models for which the CF limit is exact, finding good agreement with the classical Oppenheimer-Volkov solution for a relativistic static spherical star as well as the exact semi-analytic solution for a collapsing spherical dust cloud. By computing the evolution of quasi-equilibrium neutron star binary configurations in the absence of gravitational radiation backreaction, we have confirmed that these configurations can remain dynamically stable all the way to the development of a cusp. With an approximate treatment of radiation reaction, we have calculated the complete merger of an irrotational binary configuration from the innermost point on an equilibrium sequence through merger and remnant formation and ringdown, finding good agreement withprevious relativistic calculations. In particular, we find that mass loss is highly suppressed by relativistic effects, but that, for a reasonably stiff neutron star equation of state, the remnant is initially stable against gravitational collapse because of its strong differential rotation. The gravity wave signal derived from our numerical calculation has an energy spectrum which matches extremely well with estimates based solely on quasi-equilibrium results, deviating from the Newtonian power-law form at frequencies below 1 kHz, i.e., within the reach of advanced interferometric detectors.

Formation of molten metal films during metal-on-metal slip under extreme interfacial conditions

The present paper describes results of plate-impact pressure-shear friction experiments conducted to study time-resolved growth of molten metal films during dry metal-on-metal slip under extreme interfacial conditions. By employing tribo-pairs comprising hard tool-steel against relatively low melt-point metals such as 7075-T6 aluminum alloys, interfacial friction stress ranging from 100 to 400 MPa and slip speeds of approximately 100 m/ s have been generated. These relatively high levels of friction stress combined with high slip-speeds generate conditions conducive for interfacial temperatures to approach the melting point of the lower melt point metal (Al alloy) comprising the tribo-pair. A Lagrangian finite element code is developed to understand the evolution of the thermo-mechanical fields and their relationship to the observed slip response. The code accounts for dynamic effects, heat conduction, contact with friction, and full thermo-mechanical coupling. At temperatures below the melting point the material is described as an isotropic thermally softening elastic-viscoplastic solid. For material elements with temperatures in excess of the melt point a purely Newtonian fluid constitutive model is employed. The results of the hybrid experimental-computational study provides new insights into the thermoelastic–plastic interactions during high speed metal-on-metal slip under extreme interfacial conditions. During the early part of frictional slip the coefficient of kinetic friction is observed to decrease with increasing slip velocity. During the later part transition in interfacial slip occurs from dry metal-on-metal sliding to the formation of molten Al films at the tribo-pair interface. Under these conditions the interfacial resistance approaches the shear strength of the molten aluminum alloy under normal pressures of approximately 1– 3 GPa and shear strain rates of ∼10 7 s −1 . The results of the study indicate that under these extreme conditions molten aluminum films maintain a shearing resistance as high as 100 MPa . Scanning electron microscopy of the slip surfaces reveal molten aluminum to be smeared on the tribo-pair interface. Knoop hardness measurements in 7075-T6 Al alloy at various depths from the slip interface indicate that the hardness increases approximately linearly with depth and reaches a plateau at approximately 40 μm from the surface.

Similarity solutions for magnetic bubble expansion

A model for an expanding magnetic bubble or plasmoid is introduced, corresponding to a large aspect ratio torus, having one-dimensional (cylindrical) symmetry but with three dimensional expansion, with the length of the cylinder expanding in time in the same manner as the radius. This model has a general class of similarity equations in ideal magnetohydrodynamics (MHD) for spherical expansion. There are two parameters c, d characterizing the similarity solutions, depending on boundary conditions and conservation relations. These solutions exhibit either tangential discontinuities or shocks at the boundary, depending on the values of the constants c and d. Some of the solutions have magnetic fluxes within the bubble increasing with time, but with smaller or zero magnetic fields outside the bubble, requiring a shock and a dynamo in the shock region. The results of simulations of one class of solutions with a Lagrangian MHD code show good agreement. Some of the properties of fully toroidal solutions of the similarity equations are derived. This model has applications to a magnetic bubble from an accretion disk around an active galactic nucleus (AGN), appropriate to the phase in which the bubble has expanded to a size much greater than the disk field length scales but much smaller than any exterior scales. At this stage the magnetic reconnection and flux conversion stage associated with setting up the expanding bubble is completed. The model may also apply to a plasmoid formed in the solar corona.

A 3D Collision Scheme For CompressibleMedia In A General Connectivity LagrangianFormulation

In multi-material hydrodynamic problems described using the Lagrangian formulation, the interaction between different material meshes is carried out using collision schemes. The basic schemes simulate interactions using momentum transfer equations, which are solved for every pair of colliding meshes. In this paper we present a novel collision scheme, which extends the basic ones. After completing the momentum transfer stage, our scheme prevents penetration between meshes gradually, while detecting future collisions, ensuring that the penetration depth is kept under a specified limit. Collisions of more than two meshes at the same point are treated specifically, by solving vectorized momentum transfer equations. This approach, which suppresses sliding for these points, prevents numerical penetrations of mesh vertices into the interface between two other meshes. Since this special treatment is invoked only when it is necessary, it does not significantly affect the efficiency of the scheme. We demonstrate the generality of our scheme in a problem in which one mesh is calculated using a Lagrangian code, while the other mesh is calculated concurrently using an ALE code. The two meshes are coupled using our collision scheme, by passing information between them (and the codes), and by updating their boundaries’