Modeling and adjoints for continuous systems.
Trajectory differential equations in state- variable form, presenting compatible difference equation adjoint scheme for trajectory integration
Denham, W. F.↗
SEARCH · Engineering Papers
Results for “equations of state”
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An alternative approach to the numerical simulation of steady inviscid flow
A numerical procedure for the efficient simulation of steady inviscid flow is described and its utility is demonstrated. The method is uniformly valid for application in the subsonic, transonic and supersonic flow regimes. It does not rely on the introduction of additional assumptions beyond those necessary to obtain the Euler equations from the Navier-Stokes equations, nor does it make use of a time-asymptotic solution of the unsteady equations of motion. Application of the herein-defined surrogate equation technique allows the formulation of stable, fully-conservative, type-dependent finite difference equations for use in obtaining numerical solutions to systems of first-order partial differential equations, such as the steady-state Euler equations or their various approximations. Computational results are presented for the full Euler equations used to simulate rotational subsonic flow and for the transonic small disturbance equations. For the latter case, a computational efficiency greater than that obtained by means of the standard perturbation potential approach is indicated.
Johnson, G. M.↗
The origin of the moon and the single-impact hypothesis. I
One of the newer ideas regarding the origin of the moon is concerned with a single-impact hypothesis. It is pointed out that this theory has the advantage of overcoming most of the difficulties with the classical theories. The angular momentum of the earth-moon system can easily be obtained by varying the initial conditions of the impact. A series of three-dimensional numerical simulations of the collision between the earth and an object of about 1/10 its mass is presented. Different impact velocities, impact parameters, and initial internal energies are considered. Attention is given to assumptions, the equation of state, numerical techniques utilizing the momentum equation and the energy conservation equation, tests, and initial conditions and units.
Benz, W.↗
Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique
The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.
Nordmann, R.↗
One-way nesting for a primitive equation ocean model
Prognostic numerical models for atmospheric and oceanic circulations require initial fields, boundary conditions, and forcing functions in addition to a consistent set of partial differential equations, including a state relation and equations expressing conservation of mass, momentum, and energy. Depending on the horizontal domain to be modeled, the horizontal boundary conditions are either physically obvious or extremely difficult to specify consistently. If the entire atmosphere is modeled, periodic horizontal boundary conditions are appropriate. On the other hand, the physical horizontal boundaries on the entire ocean are solid walls. Obviously, the normal velocity at a solid wall is zero while the specification of the tangential velocity depends on the mathematical treatment of the horizontal viscous terms. Limitations imposed by computer capacity and cost, as well as research interests, have led to the use of limited area models to study flows in the atmosphere and ocean. The limited area models do not have physical horizontal boundaries, merely numerical ones. Correctly determining these open boundary conditions for limited-area numerical models has both intrigued and frustrated numerical modelers for decades. One common approach is to use the closed or solid wall boundary conditions for a limited-area model. The argument given for this approach is that the boundary conditions affect flow near the walls but that none of these effects are propagated into the interior. Therefore, one chooses a big enough domain that the central region of interest is not corrupted by the boundary flow. Research in progress to model the North Atlantic circulation vividly illustrates the pitfalls of this approach. Two model runs are compared: (1) the southern boundary at 20S between latitudes 0 and 40W is artificially closed; and (2) the same boundary is specified as open with an inward transport of 15 Sv (determined from a global model with the same physics) uniformly spread across the boundary. A comparison of both runs is presented.
Blake, D. W.↗
Memory-efficient emulation of physical tabular data using quadtree decomposition
Computationally expensive functions are sometimes replaced in simulations with an emulator that approx-imates the true function (e.g., equations of state, wavelength-dependent opacity, or composition-dependent materials properties). For functions that have a constrained domain of interest, this can be done by discretizing the domain and performing a local interpolation on the tabulated function values of each local domain. For these so-called tabular data methods, the method of discretizing the domain and mapping the input space to each subdomain can drastically influence the memory and computational costs of the emulator. This is especially true for functions that vary drastically in different regions. We present a method for domain discretization and mapping that utilizes quadtrees, which results in significant reductions in the size of the emulator with minimal increases to computational costs or loss of global accuracy. We apply our method to the electron-positron Helmholtz free energy equation of state and show over an order of magnitude reduction in memory costs for reasonable levels of numerical accuracy.
97 MATHEMATICS AND COMPUTING↗
Application of machine learning in the determination of impact parameter in the 132 Sn+ 124 Sn system
Here, 132 Sn + 124 Sn collisions at a beam energy of 270 MeV/nucleon were performed at the Radioactive Isotope Beam Factory (RIBF) in RIKEN to investigate the nuclear equation of state. Reconstructing the impact parameter is one of the important tasks in the experiment as it relates to many observable. In this work, we employ three commonly used algorithms in machine learning, the artificial neural network (ANN), the convolutional neural network (CNN), and the light gradient boosting machine (LightGBM), to determine the impact parameter by analyzing either the charged particle spectra or several features simulated with events from the ultrarelativistic quantum molecular dynamics (UrQMD) model. To closely imitate experimental data and investigate the generalizability of the trained machine learning algorithms, incompressibility of nuclear equation of state and the in-medium nucleon-nucleon cross sections are varied in the UrQMD model to generate the training data. The mean absolute error Δb between the true and the predicted impact parameter is smaller than 0.45 fm if training and testing sets are sampled from the UrQMD model with the same parameter set. However, if training and testing sets are sampled with different parameter sets, Δb would increase to 0.8 fm. The generalizability of the trained machine learning algorithms suggests that these machine learning algorithms can be used reliably to reconstruct the impact parameter in experiment.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Characterization of the inner edge of the neutron star crust
The poorly known crustal equation of state plays a critical role in many observational phenomena associated with a neutron star. Here using semiclassical Monte Carlo simulations, we explore the possible configurations of the inner edge of the neutron star crust for a variety of baryon densities and proton fractions. Applying the Kirkwood-Buff theory to these two-component systems, we observe how the isothermal compressibility reaches a maximum when isolated nonsymmetric (or quasispherical) clusters are formed in an extremely dilute neutron gas. To determine the neutron fraction, we suggest a geometrical model based on the behavior of the proton-neutron pair correlation function. Accordingly, the equation of state of the inner crust is calculated, illustrating that the nuclear energy in β equilibrium follows a power-law behavior with baryon density. As a possible astrophysical outcome of this study, our results could help refine the mass-radius relation. Finally, our results pave the way towards further investigations of the impact of the proton-neutron pair correlation function on transport properties within the neutron star crust.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
A compact x-ray diffraction system for dynamic compression experiments on pulsed-power generators
Pulsed-power generators can produce well-controlled continuous ramp compression of condensed matter for high-pressure equation-of-state studies using the magnetic loading technique. X-ray diffraction (XRD) data from dynamically compressed samples provide direct measurements of the elastic compression of the crystal lattice, onset of plastic flow, strength–strain rate dependence, structural phase transitions, and density of crystal defects, such as dislocations. Here, we present a cost-effective, compact, pulsed x-ray source for XRD measurements on pulsed-power-driven ramp-loaded samples. This combination of magnetically driven ramp compression of materials with a single, short-pulse XRD diagnostic will be a powerful capability for the dynamic materials’ community to investigate in situ dynamic phase transitions critical to equation of states. Finally, we present results using this new diagnostic to evaluate lattice compression in Zr and Al and to capture signatures of phase transitions in CdS.
47 OTHER INSTRUMENTATION↗
Polyimide dynamically compressed to decomposition pressures: Two-wave structures captured by velocimetry and modeling
We performed a series of six plate impact experiments on polyimide and modeled them using new reactant and products equations of state combined with an Arrhenius rate model. The first experiment was diagnosed with embedded electromagnetic velocity gauges through which we directly observed attenuation of the lead shock to an approximately constant state over a propagation distance of roughly 4 mm. Simulated gauge profiles were in excellent qualitative agreement with experiment and suggested a sluggish chemical reaction that did not proceed to completion. The remaining five experiments were conducted in a transmission geometry and diagnosed velocimetrically at the sample/window interface. All five of these yielded profiles with a sharp shock followed by a more gradual approach to maximum interface velocity that was “rounded” to varying degree. These profiles proved difficult to interpret unambiguously due to the convolution of the reactive wave upon first shock with reflection of the lead wave and reshock or release by the window. Comparison with thermochemical calculations strongly suggests that the point of maximum interface velocity corresponds to the equilibrium reshock or release locus. We discuss the implications of this point for the practice of impedance matching based on the reflected Hugoniot of reactive materials such as polymers. The reactant and thermochemical products equations of state are developmental SESAME tables 97710 and 97720, respectively.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Dynamic experiments to study the α - ε phase transition in cerium
The ability to understand and predict the response of matter at extreme conditions requires knowledge of a material’s equation-of-state including the location of phase boundaries, transition kinetics, and the evolution of material strength. Cerium is a material with a complex phase diagram that continues to attract significant scientific interest. Recent dynamic experiments have provided information on the low-pressure γ–α phase transition, sound speed, and Hugoniot data for the higher-pressure α phase, as well as the incipient shock melt transition. Despite these efforts, there are still regions of the phase diagram that are largely unexplored dynamically, including the high-pressure region below the melt boundary. Along a room temperature isotherm, diamond anvil cell data report a transition to the ϵ phase between 13 and 17 GPa. At higher temperatures, similar diamond anvil cell data show significant disagreement regarding the existence, location, and slope of the ϵ-phase boundary. In this work, double-shock loading was used to access the α–ϵ region of the phase diagram to obtain equation-of-state information and to determine the location of the ϵ-phase boundary for shock loading.
74 ATOMIC AND MOLECULAR PHYSICS↗
TOUGH3/EOS7MG (EOS7MG) v1.1
EOS7MG is a new TOUGH3 module for modeling water, brine, and multiple gas mixtures in the subsurface multiphase flow system. The module allows mixture of as more as three gas components selecting from a list of predefined gases (CH4, C2H6, C3H8, H2S, CO2, N2, NO2, H2, and AIR). EOS7MG was developed based on EOS7C and inherited the general modeling capabilities of the TOUGH3 platform. It uses cubic equation of states for gas density, viscosity, and enthalpy calculation. The module provides the selection options for Peng-Robinson, Redlich-Kwong, or Soave-Redlich-Kwong equations of state. The subsurface flow and transport of gas and aqueous phase mixtures over a wide range of pressures and temperatures is modeled with a multiphase Darcy's Law and an accurate chemical equilibrium approach as originally implemented in EOS7C is used for calculation of the solubilities of different mass components. EOS7MG was designed to allow flexible number of components based on the number of gases included in the model. The potential application of this module may include subsurface gas storage, energy storage, CCS, enhanced gas recovery, and more.
Zhang, Keni↗
Hydro-Code Implementation and Testing of a Kinetic Phase Transition Framework
In this report we describe the Kinetic Phase Transition (KPT) framework that has been worked out over the last 10 years (from around 2014) and the implementation of it into three different codes, the one-dimensional hydro- LASLO and the three-dimensional magneto-hydro- ALEGRA, Sandia codes, via subroutines in the LAMBDA Equations of State and constitutive models package, and Flag, an arbitrary Lagrangian-Eulerian multiphysics code developed within the Lagrangian Applications project (LAP) at LANL. We discuss the introduction of phase mass (and/or volume) fractions that are needed in a code for it to be ‘phase aware’, that is, not only the thermodynamic state is known in each point but also the mixture of the materials’ phases in that point. Further we point to the need of a full Equations of State for each phase in a material to achieve phase awareness and we review the equilibrium phase model, where a phase mixture is at its lowest Gibbs free energy state, to make this point clear. Contrasting the kinetic phase transition to this equilibrium model seamlessly introduce us to the KPT framework that is subsequently thoroughly discussed. While the determination of the total state and the states and mass fractions of phases in each point is a problem that can borrow many of its numerical details from Eulerian codes and mixture of materials (not phases), the update of mass fractions with time in a KPT framework needs a new set of considerations. General for any update model is that we need to prevent mass fractions from becoming unphysical (negative or their sum to be larger than one). We have solved this problem by implementing a subdivision of the hydro time step that prevents the phase from being fully present to not present at all in one subdivided time step by limiting the size of the subdivided time step. This scheme also corrects numerical problems from abrupt changes in parameter values, the so called Gibbs phenomena, that gives rise to slushing between phases in the KPT framework. Interspersed throughout the report are discussions on different thermodynamics considerations. EOS validity windows, limitations on the EOS phase space, are needed for the KPT framework and are discussed separately and exemplified. The KPT framework described in this report has been verified by code comparison, but validation is still an active area of research. There is room for improvement in the update model, both in the model for determination of rates and in how to prevent the mass fractions from becoming unphysical. In addition, the parameters in the KPT update model and the placement of the phase boundary in the EOS phase space, and interactions with other constitutive models, are closely related and interfering with each other. One possible way forward is to simultaneously develop KPT parameters, EOS, and constitutive models for each material.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Sources of uncertainty in direct seismological measurements of the solar helium abundance
The methods by which Dappen et al. (1988, 1990, 1991) and Dziembowski et al. (1990, 1991, 1992) recently obtained discrepant estimates of the helium abundance in the solar convection zone are compared. The aim of the investigation reported in this paper is to identify the main source of the discrepancy. Using as proxy data eigenfrequencies of a set of modes of a theoretical solar model, computed with the same physics as were the frequencies of a reference model with which these data are compared, the two methods yield similar results. Thus we ascertain that the principal source of the discrepancy is not in the inversions themselves, which yield essentially a measure of the variation of the adiabatic exponent gamma of the material in the He II ionization zone. Instead it is in the approximations adopted in the treatment of heavy elements in the equation of state used to relate the variation of gamma to chemical composition. We obtain acceptably consistent results when inverting solar data by two methods using the same equation of state. We attempt to identify the likely residual sources of uncertainty.
Kosovichev, A. G.↗
Path integral molecular dynamics: A high-fidelity approach to quantum dynamics of electrons
We investigate electron transport in the uniform electron gas using ring-polymer molecular dynamics (RPMD). Working in the weakly coupled, non-degenerate regime, we use RPMD to probe how the onset of quantum diffraction effects at high temperature reshapes electron–electron collisions and leads to a classical-to-quantum crossover in macroscopic transport properties. Static thermodynamics obtained with RPMD are consistent with the weak-coupling equation of state, confirming correct quantum Boltzmann sampling. Real-time transport extracted from mean square displacements exhibits the expected ballistic-to-diffusive transition and a systematic reduction of the electronic self-diffusivity as quantum effects strengthen, due to quantum diffraction modifying electron–electron collisions. Direct ring-polymer scattering simulations reveal diffractive “softening” of binary deflections, providing a micro-to-macro link between collision physics and diffusion. The present study establishes RPMD as a quantitative, trajectory-based tool for electron transport across the classical–quantum crossover and furnishes benchmarks for improving Coulomb-log interpolation models. We outline extensions to multi-component plasmas and a path to incorporate Fermi–Dirac statistics within path-integral dynamics.
Electronic transport↗
Phase stability of a eutectic high entropy alloy under extremes of pressures and temperatures
Additively manufactured high-entropy alloys are of interest because of their unique combination of high yield strength and large ductility achieved with far-from-equilibrium crystalline phases and micro/nanostructure morphology. We report on the phase transformation and thermal equation of state of the eutectic high-entropy alloy (EHEA) Al 18 Co 20 Cr 10 Fe 10 Ni 40 W 2 , produced by laser powder-bed fusion (L-PBF). The EHEA was studied in a large-volume Paris–Edinburgh cell using energy-dispersive x-ray diffraction to a pressure of 5.5 GPa and a temperature of 1723 K. Static compression studies in diamond anvil cells using angle-dispersive x-ray diffraction extended the high-pressure structural data to 317 GPa at ambient temperature. The initial dual-phase nanolamellar face-centered cubic (FCC) and body-centered cubic (BCC) structure of Al 18 Co 20 Cr 10 Fe 10 Ni 40 W 2 transforms into a single FCC phase under high pressure, with the BCC-to-FCC phase transformation completing at 9 ± 2 GPa. The FCC phase remained stable up to the highest pressure of 317 GPa. The measured thermal equation of state for the FCC phase of Al 18 Co 20 Cr 10 Fe 10 Ni 40 W 2 is presented up to 5.5 GPa and 1473 K. We observed melting of the EHEA at 1698 ± 25 K at a pressure of 5.5 GPa, and the recrystallized sample shows an increased fraction of the CsCl-type (B2) phase at ambient conditions following release from the high-pressure high-temperature state. The BCC-to-FCC phase transition completion pressure is correlated with the nanolamellae thickness of the BCC layer in this diffusion-less transformation at ambient temperature.
36 MATERIALS SCIENCE↗
Ab Initio Phase Diagram of Chromium to 2.5 TPa
Chromium possesses remarkable physical properties such as hardness and corrosion resistance. Chromium is also a very important geophysical material as it is assumed that lighter Cr isotopes were dissolved in the Earth’s molten core during the planet’s formation, which makes Cr one of the main constituents of the Earth’s core. Unfortunately, Cr has remained one of the least studied 3d transition metals. In a very recent combined experimental and theoretical study (Anzellini et al., Scientific Reports, 2022), the equation of state and melting curve of chromium were studied to 150 GPa, and it was determined that the ambient body-centered cubic (bcc) phase of crystalline Cr remains stable in the whole pressure range considered. However, the importance of the knowledge of the physical properties of Cr, specifically its phase diagram, necessitates further study of Cr to higher pressure. In this work, using a suite of ab initio quantum molecular dynamics (QMD) simulations based on the Z methodology which combines both direct Z method for the simulation of melting curves and inverse Z method for the calculation of solid–solid phase transition boundaries, we obtain the theoretical phase diagram of Cr to 2.5 TPa. We calculate the melting curves of the two solid phases that are present on its phase diagram, namely, the lower-pressure bcc and the higher-pressure hexagonal close-packed (hcp) ones, and obtain the equation for the bcc-hcp solid–solid phase transition boundary. We also obtain the thermal equations of state of both bcc-Cr and hcp-Cr, which are in excellent agreement with both experimental data and QMD simulations. We argue that 2180 K as the value of the ambient melting point of Cr which is offered by several public web resources (“Wikipedia,” “WebElements,” “It’s Elemental,” etc.) is most likely incorrect and should be replaced with 2135 K, found in most experimental studies as well as in the present theoretical work.
equation of state↗
Cosmological evolution of ultralight axionlike scalar fields
In this paper, we examine the cosmological evolution of ultralight axionlike (ULA) scalar fields with potentials of the form V(Φ)=m 2 f 2 [1–cos(Φ/f)] n , with particular emphasis on the deviation in their behavior from the corresponding small–Φ power-law approximations to these potentials: V(Φ)∝Φ 2n . We show that in the slow-roll regime, when Φ ˙2 /2 $\ll$ V(Φ), the full ULA potentials yield a more interesting range of possibilities for quintessence than do the corresponding power law approximations. For rapidly oscillating scalar fields, we derive the equation of state parameter and oscillation frequency for the ULA potentials and show how they deviate from the corresponding power-law values. We derive an analytic expression for the equation of state parameter that better approximates the ULA value than does the pure power-law approximation.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗