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In this study, a method is proposed to systematically generate solutions of the two-dimensional Toda lattice equation in terms of previously known solutions ϕ(x,y) of the two-dimensional Laplace's equation. The solitonic solution of Nakamura's [J. Phys. Soc. Jpn. 52 (1983) 380] is shown to correspond to one particular choice of ϕ(x,y).
In this paper we report two distributed and communication-efficient algorithms based on the multi-agent system are proposed to solve a system of linear equations with the Laplacian sparse system matrix. One algorithm is based on the gradient descent method in optimization. In this algorithm, the agents only share partial information instead of all of their collective state vectors to save significant communication. The other algorithm is obtained by approximating Newton’s method for a faster convergence rate. Although it requires twice as much communication as the first one, it is still communication-efficient given the low dimension of the information shared among agents. The convergence at a linear rate is proved for both algorithms, and a comprehensive comparison of their convergence rate, communication burden, and computation costs is also performed. The proposed algorithms can be applied to various systems to solve those problems that can be modeled as a system of linear equations with a Laplacian sparse system matrix. Simulation results with the electric power system illustrate their effectiveness.
Nitrogen, the most abundant element in Earth's atmosphere, is also a primary component of solid nitride minerals found in meteorites and on Earth's surface. If they remain stable to high pressures and temperatures, these nitrides may also be important reservoirs of nitrogen in planetary interiors. We used synchrotron X-ray diffraction to measure the thermal equation of state and phase stability of titanium nitride (TiN) in a laser-heated diamond anvil cell at pressures up to ~70 GPa and temperatures up to ~2500 K. TiN maintains the cubic B1 (NaCl-type) crystal structure over the entire pressure and temperature range explored. It has K 0 = 274(4) GPa, K 0 ' = 3.9(2), and γ 0 = 1.39(4) for a fixed V 0 = 76.516(30) Å 3 (based on experimental measurements), q = 1, and θ 0 = 579 K. Additionally, we collected Raman spectra of TiN up to ~60 GPa, where we find that the transverse acoustic (TA), longitudinal acoustic (LA), and transverse optical (TO) phonon modes exhibit mode Grüneisen parameters of 1.66(17), 0.54(15), and 0.93(4), respectively. Based on our equation of state, TiN has a density of ~5.6–6.4 g/cm 3 at Earth's lower mantle conditions, significantly more dense than both the mantle of the Earth and the estimated densities of the mantles of other terrestrial planets, but less dense than planetary cores. Furthermore, we find that TiN remains stable against physical decomposition at the pressures and temperatures found within Earth's mantle, making it a plausible reservoir for deep planetary nitrogen if chemical conditions allow its formation.
By using gravity/gauge correspondence, we construct a holographic model, constrained to mimic the lattice QCD equation of state at zero density, to investigate the temperature and baryon chemical potential dependence of the equation of state. We also obtained the energy loss of light and heavy partons within the hot and dense plasma represented by the heavy quark drag force, Langevin diffusion coefficients and jet quenching parameter at the critical point and across the first-order transition line predicted by the model.
In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.
In this study, shock initiation experiments have been carried out on the polymer-bonded explosive PBX 9012 [nominally 90% octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazine (HMX), 10% vinylidene–hexafluoropropylene copolymer (Viton A) by weight] in order to provide calibration data for the explosive’s unreacted equation-of-state (EOS) and shock-to-detonation transition for reactive burn rate calibration. The input pressures covered the range of 1.86–4.43 GPa. This provided run-to-detonation depths ranging from > 22.3 mm for the lowest pressure shot to 4.91 mm at the highest pressure. The relative shock sensitivity of PBX 9012 is compared to other HMX-explosives in terms of the Pop-plot, showing that the studied explosive is more sensitive than other similar HMX-based counterparts (with notable exceptions). The uncertainty in the shock velocity determinations from the shock tracker measurements are also investigated, yielding new uncertainty measures in the generated Hugoniot data and run-to-detonation coordinates. Finally, the unreacted equation-of-state is determined using a linear U s – u p Mie–Grüneisen relation and the Davis reactants EOS analytical form, the latter being more suited for reactive burn modeling.
We introduce the anti-symmetric representation of the extended magnetohydrodynamic (MHD) equations. In this representation, the use of the anti-symmetric flux operator (∇·ν+ν·∇) results in conservation theorems with discrete analogs. Inherently robust numerical applications are achieved with little effort, and conservation to machine precision is possible with simple numerical schemes. Starting from the two-fluid equations, we construct a single-fluid MHD model based on generalized center-of-mass variables for the mass (√ρ), momentum (√ρν), and pressure (√ p ). This model is shown to possess identical conservation properties to the two-fluid system, with the only restriction being the use of a single temperature. Common approximations to the Braginskii heat fluxes and to the gyroviscous stress tensor are cast into our representation for convenience. The discrete conservation properties are verified using the classic Orszag–Tang vortex problem. In addition to the favorable mass, momentum, and energy conservation properties, the time reversibility of the simulations is demonstrated.
Through an extensive data analysis of powder X-ray diffraction data obtained at pressures up to at least 78 GPa, we report the experimental equations of state for all lanthanide nitrides between NdN and LuN, excluding the radioactive Pm. By fitting the obtained unit cell volumes as a function of pressure with the third order Birch–Murnaghan equation of state, we find that the bulk modulus increases with an increasing lanthanide number from K 0 = 146(12) GPa for NdN to 182(7) GPa in EuN. Hereafter, the bulk modulus reaches a plateau for the rest of the series except for TmN which has a lower bulk modulus. We find that the first derivative of the bulk modulus is around 4 for all compounds except TbN, which displays a significantly different compression behavior. In addition, we find a B1 to B10 pressure-induced phase transition in NdN, SmN, EuN, and GdN at increasingly higher pressures. In fact, we observe that the onset pressure of the phase transition increases linearly with Ln atomic number. From TbN and onwards, we do not observe any sign of a B1 to B10 transition indicating that the transition pressure exceeds the studied pressure range. Therefore, we believe that, for the heavier lanthanides, the linear relationship between the onset pressure and the lanthanide number does not hold and even higher pressures are needed to observe the transition. This coherent study of the series of lanthanide nitrides offers a unique opportunity for benchmark studies of computational methods applied to compounds with 4f electrons.
Mechanical equation-of-state data of initially liquid and solid CO2 shock-compressed to terapascal conditions are reported. Diamond-sapphire anvil cells were used to vary the initial density and state of CO2 samples that were then further compressed with laser-driven shock waves, resulting in a data set from which precise derivative quantities, including Grüneisen parameter and sound speed, are determined. Reshock states are measured to 800 GPa and map the same pressure-density conditions as the single shock using different thermodynamic paths. The compressibility data reported here do not support current density-functional-theory calculations, but are better represented by tabular equation-of-state models.
A simple self-consistent prototype equation of state (EOS) based on the Generalized van der Waals (vdW) partition function has been demonstrated to describe the phase transition of simple fluids in nanopores with uniform size. Different from those commonly presented in the literature, the new EOS does not need an auxiliary equation that is conventionally applied to provide the capillary pressure derived from surface tension. Lastly, the encouraging performance of the EOS calls for further extension to applications with more complex fluids and porous media.
In this work, the partially averaged Navier–Stokes (PANS) equations are used to predict the variable-density Rayleigh–Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number 500. This is a prototypical problem of material mixing, featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS Besnard–Harlow–Rauenzahn-linear eddy viscosity model closure. The numerical simulations are conducted at different levels of physical resolution and test three approaches to set the parameters $f_\phi$ defining the range of physically resolved scales. The computations demonstrate the efficiency (accuracy vs cost) of the PANS model predicting the spatiotemporal development of the RT flow. Results comparable to large-eddy simulations and direct numerical simulations are obtained at significantly lower physical resolution without the limitations of the Reynolds-averaged Navier–Stokes equations in these transitional flows. The data also illustrate the importance of appropriate selection of the physical resolution and the resolved fraction of each dependent quantity $\phi$ of the turbulent closure, $f_\phi$. These two aspects determine the ability of the model to resolve the flow phenomena not amenable to modeling by the closure and, as such, the computations’ fidelity.
Although drift-ordered fluid models are widely applied in tokamak edge turbulence simulations, the models used are acknowledged not to conserve energy or even electrical charge. The present paper aims to remove many of the existing pitfalls in drift-fluid models, however, with the objective of finding a solution simple enough to be implemented in numerical applications. Our main result is an improved version of the drift-Braginskii equations involving a generalized vorticity function. In the new drift-Braginskii system, the quasi-neutrality condition translates into a transport equation for a generalized vorticity, expressed in conservation form, and related to the total mass-weighted circulation. It is found that kinetic energy conservation can be achieved if the polarization flow is defined recursively. In conclusion, the resulting model conserves the kinetic energy associated with E×B and diamagnetic flows and retains the associated perpendicular kinetic energy flux.
Here, the pressure–volume equations of state of palladium and rhodium statically compressed in neon are presented. Vinet fits give parameters for palladium: V0=58.678(73) Å 3 , B 0 =189.3(30) GPa, B' 0 =5.473(63), and rhodium: V 0 =55.062(63) Å3, B 0 =241.3(65) GPa, B' 0 =5.34(24). Both metals are observed to react with hydrocarbons under pressure to form hydrides. Existing equations of state are discussed with regard to potential inadvertent hydrogen contamination as a source for discrepancies and anomalous fitted parameters.
Traditionally, hydrodynamics simulations are performed with a single equation-of-state (EOS) to describe each material. These EOSs typically have a physics-informed functional form with adjustable parameters that are calibrated in order to replicate small-scale data. However, because the calibration data have uncertainty and there are typically inherent degeneracies in fitting the EOS, there are actually multiple EOSs that might be consistent with calibration data. In this work, we perform uncertainty quantification (UQ) for the reactant and product equations of state for the high explosive PBX 9501 to yield an ensemble of EOSs that match the uncertain small-scale calibration data. We then simulate an experiment of an explosively formed penetrator repeatedly with different EOSs to both validate the UQ analysis and determine the effects of EOS uncertainty on the prediction of quantities of interest in the experiment. In general, we find good agreement between the simulation predictions and the experimental measurements, and we identify an EOS variable that contributes most directly to the spread in the predictions as the EOSs are varied.
Abstract Tantalum pentoxide (Ta 2 O 5 ) is among the most technologically useful heavy transition metal oxides. Unfortunately, its crystal structure is the subject of long-standing and unresolved disagreement. Among other consequences, this uncertainty has made it impossible to formulate a robust high pressure equation of state for Ta 2 O 5 . Here, we report the results of high pressure x-ray diffraction experiments indexed against a comprehensive list of proposed Ta 2 O 5 phases. Five of the proposed phases produce good matches to experimental observations, and the compressibility parameters for these phases are all consistent within uncertainty. This means that regardless of the particular phase chosen, the Ta 2 O 5 equation of state can be described with bulk modulus K 0 = 138 ± 3.68 GPa and pressure derivative K 0 ′ = 1.82 ± 0.45 . Combining these experimental results with new density-functional theory calculations allows us to identify the λ phase as the best structural model of Ta 2 O 5 at ambient conditions.
We report the P-V-T equation of state measurements of B 4 C to 50 GPa and approximately 2500 K in laser-heated diamond anvil cells. We obtain an ambient temperature, third-order Birch–Murnaghan fit to the P-V data that yields a bulk modulus K 0 of 221(2) GPa and derivative, (dK/dP) 0 of 3.3(1). These were used in fits with both a Mie–Grüneisen–Debye model and a temperature-dependent, Birch–Murnaghan equation of state that includes thermal pressure estimated by thermal expansion (α) and a temperature-dependent bulk modulus (dK 0 /dT). The ambient pressure thermal expansion coefficient (α 0 + α 1 T), Grüneisen γ(V) = γ 0 (V/V 0 ) q and volume-dependent Debye temperature, were used as input parameters for these fits and found to be sufficient to describe the data in the whole P-T range of this study.
We utilize the now substantial amount of astrophysical observations of neutron stars (NSs), along with perturbative quantum chromodynamics (pQCD) calculations at high density, to directly constrain the NS equation of state (EOS). To this end, we construct nonparametric EOS priors by using Gaussian processes trained on 75 EOSs, which include models with either hadrons, hyperons, or quarks at high densities. We create a prior using the full EOS sample (model agnostic), and one prior for each EOS family to test model discrimination. We introduce a novel inference approach, which allows the simultaneous sampling of intrinsic and extrinsic parameters of binary NS mergers, as well as a nonparametric equation of state. We showcase this method in a Bayesian updating scheme by first performing a complete analysis of the binary NS merger event GW170817 with minimal assumptions, and sequentially adding information from x-ray and radio NS observations, along with pQCD calculations. Besides providing standard constraints, such as the pressure at twice nuclear saturation density 𝑝(2𝜌 sat ) = 4.3$^{+0.6}_{−0.6}$ × 10 34 dyne/cm 2 , at 95% confidence level, for the model agnostic prior, our methodology shows how the choice of EOS families used in conditioning changes the inferred astrophysical properties of the EOS, namely tidal deformability and maximum supported NS mass. We find hyperonic priors predicting higher tidal deformabilities for a 1.4𝑀 ⊙ NS, and hadronic priors being preferred by the considered astrophysical data.
We report measurements of the compressibility of ramp compressed tantalum to a final stress of 2.3 TPa corresponding to threefold volumetric compression. Using these data, we extended the experimental constraint on the Ta cold compression curve by an order of magnitude in pressure. Furthermore, by combining the resulting data with previous measurements of shock compression and ambient pressure heating, we construct an experimentally bounded and thermodynamically consistent equation of state model for Ta which has 2% uncertainty in pressure at 1 TPa. We therefore propose Ta as an in situ pressure scale for laser-heated static compression experiments which were recently able to reach terapascal pressures and thousands of degrees Kelvin. Our new equation of state of Ta is experimentally constrained at extreme pressures and temperatures relevant to a wide range of planetary interiors and will allow for more accurate comparison between experimental measurements and theory at extreme conditions.