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At least 1,063 records · Page 59

Numerical studies of the thermal design sensitivity calculation for a reaction-diffusion system with discontinuous derivatives

The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.

Hou, Jean W.↗

Analysis of operator splitting errors for near-limit flame simulations

High-fidelity simulations of ignition, extinction and oscillatory combustion processes are of practical interest in a broad range of combustion applications. Splitting schemes, widely employed in reactive flow simulations, could fail for stiff reaction–diffusion systems exhibiting near-limit flame phenomena. The present work first employs a model perfectly stirred reactor (PSR) problem with an Arrhenius reaction term and a linear mixing term to study the effects of splitting errors on the near-limit combustion phenomena. Analysis shows that the errors induced by decoupling of the fractional steps may result in unphysical extinction or ignition. The analysis is then extended to the prediction of ignition, extinction and oscillatory combustion in unsteady PSRs of various fuel/air mixtures with a 9-species detailed mechanism for hydrogen oxidation and an 88-species skeletal mechanism for n-heptane oxidation, together with a Jacobian-based analysis for the time scales. The tested schemes include the Strang splitting, the balanced splitting, and a newly developed semi-implicit midpoint method. Results show that the semi-implicit midpoint method can accurately reproduce the dynamics of the near-limit flame phenomena and it is second-order accurate over a wide range of time step size. For the extinction and ignition processes, both the balanced splitting and midpoint method can yield accurate predictions, whereas the Strang splitting can lead to significant shifts on the ignition/extinction processes or even unphysical results. With an enriched H radical source in the inflow stream, a delay of the ignition process and the deviation on the equilibrium temperature are observed for the Strang splitting. On the contrary, the midpoint method that solves reaction and diffusion together matches the fully implicit accurate solution. The balanced splitting predicts the temperature rise correctly but with an over-predicted peak. For the sustainable and decaying oscillatory combustion from cool flames, both the Strang splitting and the midpoint method can successfully capture the dynamic behavior, whereas the balanced splitting scheme results in significant errors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Error Estimates for the Optimal Control of a Parabolic Fractional PDE

In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

97 MATHEMATICS AND COMPUTING↗

Unified Models of Turbulence and Nonlinear Wave Evolution in the Extended Solar Corona and Solar Wind

The PI (Cranmer) and Co-I (A. van Ballegooijen) made significant progress toward the goal of building a "unified model" of the dominant physical processes responsible for the acceleration of the solar wind. The approach outlined in the original proposal comprised two complementary pieces: (1) to further investigate individual physical processes under realistic coronal and solar wind conditions, and (2) to extract the dominant physical effects from simulations and apply them to a one-dimensional and time-independent model of plasma heating and acceleration. The accomplishments in the report period are thus divided into these two categories: 1a. Focused Study of Kinetic MHD Turbulence. We have developed a model of magnetohydrodynamic (MHD) turbulence in the extended solar corona that contains the effects of collisionless dissipation and anisotropic particle heating. A turbulent cascade is one possible way of generating small-scale fluctuations (easy to dissipate/heat) from a pre-existing population of low-frequency Alfven waves (difficult to dissipate/heat). We modeled the cascade as a combination of advection and diffusion in wavenumber space. The dominant spectral transfer occurs in the direction perpendicular to the background magnetic field. As expected from earlier models, this leads to a highly anisotropic fluctuation spectrum with a rapidly decaying tail in the parallel wavenumber direction. The wave power that decays to high enough frequencies to become ion cyclotron resonant depends on the relative strengths of advection and diffusion in the cascade. For the most realistic values of these parameters, though, there is insufficient power to heat protons and heavy ions. The dominant oblique waves undergo Landau damping, which implies strong parallel electron heating. We thus investigated the nonlinear evolution of the electron velocity distributions (VDFs) into parallel beams and discrete phase-space holes (similar to those seen in the terrestrial magnetosphere) which are an alternate means of heating protons via stochastic interactions similar to particle-particle collisions. 1b. Focused Study of the Multi-Mode Detailed Balance Formalism. The PI began to explore the feasibility of using the "weak turbulence," or detailed-balance theory of Tsytovich, Melrose, and others to encompass the relevant physics of the solar wind. This study did not go far, however, because if the "strong" MHD turbulence discussed above is a dominant player in the wind's acceleration region, this formalism is inherently not applicable to the corona. We will continue to study the various published approaches to the weak turbulence formalism, especially with an eye on ways to parameterize nonlinear wave reflection rates. 2. Building the Unified Model Code Architecture. We have begun developing the computational model of a time-steady open flux tube in the extended corona. The model will be "unified" in the sense that it will include (simultaneously for the first time) as many of the various proposed physical processes as possible, all on equal footing. To retain this generality, we have formulated the problem in two interconnected parts: a completely kinetic model for the particles, using the Monte Carlo approach, and a finite-difference approach for the self-consistent fluctuation spectra. The two codes are run sequentially and iteratively until complete consistency is achieved. The current version of the Monte Carlo code incorporates gravity, the zero-current electric field, magnetic mirroring, and collisions. The fluctuation code incorporates WKJ3 wave action conservation and the cascade/dissipation processes discussed above. The codes are being run for various test problems with known solutions. Planned additions to the codes include prescriptions for nonlinear wave steepening, kinetic velocity-space diffusion, and multi-mode coupling (including reflection and refraction).

Cranmer, Steven R.↗

An extension of the localized artificial diffusivity method for immiscible and high density ratio flows

The localized artificial diffusivity (LAD) method is widely regarded as the preferred multi-material regularization scheme for the compact finite difference method, because it is conservative, easy to implement, and generally robust for a wide range of multi-material problems. However, traditional LAD methods face significant challenges when applied to flows with large density ratios and when maintaining thermodynamic equilibrium across material interfaces. These limitations arise from the formulation of the artificial diffusivity flux and the reliance on enthalpy diffusion for interface regularization. Additionally, traditional LAD methods struggle to ensure stability under large density ratio conditions, fail to maintain a finite interface thickness, and are therefore unsuitable for modeling immiscible interfaces. Here, in this work, we discuss the origins of these issues in traditional LAD methods and propose modifications which enable the simulation of large density ratio and immiscible flows. The proposed method targets the artificial diffusion fluxes at gradients and ringing in the volume fraction, rather than the mass fraction in traditional methods, to consistently regularize large density ratio interfaces. Furthermore, the proposed method introduces an artificial bulk density diffusion term to enforce equilibrium conditions across interfaces. To address the challenge of modeling immiscible flows, a conservative diffuse interface term is incorporated into the formulation to ensure a finite interface thickness. Specific consideration is taken in the design of the method to ensure that these crucial properties are maintained for N -material flows. The effectiveness of the proposed method is demonstrated through a series of canonical test cases, and its accuracy is validated by comparison with experimental data on micro-bubble collapse in water. These results highlight the method’s robustness and its ability to overcome the limitations of traditional LAD approaches.

Artificial diffusivity↗

Vibration characteristics of the HPOTP (High Pressure Oxygen Turbopump) of the SSME (Space Shuttle Main Engine)

A review is presented of various rotordynamic problems which have been encountered and eliminated in developing the current flight engines, and continuing subsynchronous problems which are being encountered in developing a 109% power level engine. The basic model for the High Pressure Oxygen Turbopump (HPOTP) of the SSME including the structural dynamic model for the rotor and housing and component models for the liquid and gas seals, turbine-clearance excitation forces, and impeller-diffuser forces are discussed. Results from a linear model are used to examine the synchronous response and stability characteristics of the HPOTP, examining bearing load and stability problems associated with the second critical speed. Various seal modifications are examined and shown to have favorable consequences with respect to bearing reactions and stability.

Childs, D. W.↗

Galvanostatic Intermittent Titration Technique Reinvented: Part I. A Critical Review

The galvanostatic intermittent titration technique (GITT), introduced in 1977 by Weppner and Huggins, provided a readily accessible means to measuring the chemical diffusion coefficient of electrochemical electrode materials. The method continues to be widely used today, but the reported diffusivity values are highly inconsistent, ranging as much as four orders of magnitude for some Li layered oxide compositions. Even qualitative trends of diffusivity are inconsistent, suggesting significant flaws in the implementation of the method. Other variants of the GITT method also suffer from similar inconsistency problems. Here we identify numerous sources of significant error including composition-dependent reaction overpotentials, mathematical flaws in the relaxation analysis methods, finite-size and non-planar geometry effects, inter-particle inhomogeneity issues, early transient effects, and surface area uncertainties. We propose a simple relaxation analysis scheme using the time variable t relax + τ - t relax , where t relax is relaxation time and τ is the galvanostatic pulse duration. We also propose to use dense diffusion-limited samples to isolate the bulk-diffusion process in the time domain. Chemical diffusivity can be extracted much more reliably with this improved implementation of the GITT method.

25 ENERGY STORAGE↗

On the acceleration of charged particles at relativistic shock fronts

The diffusive acceleration of highly relativistic particles at a shock is reconsidered. Using the same physical assumptions as Blandford and Ostriker (1978), but dropping the restriction to nonrelativistic shock velocities, the authors find approximate solutions of the particle kinetic equation by generalizing the diffusion approximation to higher order terms in the anisotropy of the particle distribution. The general solution of the transport equation on either side of the shock is constructed, which involves the solution of an eigenvalue problem. By matching the two solutions at the shock, the spectral index of the resulting power law is found by taking into account a sufficiently large number of eigenfunctions. Low-order truncation corresponds to the standard diffusion approximation and to a somewhat more general method described by Peacock (1981). In addition to the energy spectrum, the method yields the angular distribution of the particles and its spatial dependence.

Kirk, J. G.↗

A Finite-Element Approach for the Partitioning of Carbon in Q&P Steel

This paper reports a Galerkin finite-element analysis of carbon partitioning from martensite into austenite during the quenching and partitioning (Q&P) processes in steels. In contrast to classical or sophisticated diffusion field models, an alternative nonlinear governing equation based on chemical potential and composition is considered. The model is applied to simulate the carbon partitioning of modified 22MnB5 alloys assuming an immobile austenite–martensite phase boundary and, in turn, is compared with experimental measurements of the volume fraction and carbon content of retained austenite. The simulations show outstanding results on the influence of Si and C content on the partitioning parameters.

36 MATERIALS SCIENCE↗

The effects of phase decorrelation on the dynamics of the bump-on-tail instability

The dynamics of the bump-on-tail instability has been studied. The novelty of the work is the analysis of how the bump-on-tail dynamics is affected by an extrinsic stochastisation of the phase of the wave-particle interaction; here referred to as phase decorrelation. For this purpose, a nonlinear Monte Carlo model has been developed. When the characteristic time scale for macroscopic phase decorrelation becomes shorter than time scales of nonlinear wave-particle dynamics, the system may be described quasilinearly, with the phase decorrelation being replaced by a quasilinear diffusion coefficient in particle energy. A purely quasilinear Monte Carlo model, which is typically less computationally demanding than the fully nonlinear description due to the reduced dimensionality of phase space, has been developed for comparison. In this paper, parameter regimes, where the nonlinear and the quasilinear descriptions quantitatively agree on a macroscopic level, have been investigated, using combined theoretical and numerical analyses. Qualitative effects on the macroscopic dynamics by the presence of phase decorrelation and/or by structures of the energy distribution function in the proximity of the wave-particle resonance are also studied.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Velocity space evolution of a minority energetic electron population undergoing the anomalous Doppler instability

The kinetic evolution in velocity space of a minority suprathermal electron population that is undergoing the anomalous Doppler instability (ADI) is investigated using the results from fully nonlinear numerical simulations that self-consistently evolve particles and fields in a plasma. Electron trajectories in phase space during different stages of the ADI are captured, and are analysed in relation to the characteristics of the excited electric fields and of the overall distribution of particles. For some electrons, trapping and mirroring effects are observed during the saturation phase. A relationship between the second order moments of the perpendicular electron distribution function and time is established, and is used to investigate the range of applicability of analytical approximations drawn from classical theory, that involve a quasilinear wave-driven diffusion operator.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

COSMIC-RAY PITCH-ANGLE SCATTERING IN IMBALANCED MHD TURBULENCE SIMULATIONS

Pitch-angle scattering rates for cosmic-ray particles in MHD simulations with imbalanced turbulence are calculated for fully evolving electromagnetic turbulence. We compare with theoretical predictions derived from the quasilinear theory of cosmic-ray diffusion for an idealized slab spectrum and demonstrate how cross helicity affects the shape of the pitch-angle diffusion coefficient. Additional simulations in evolving magnetic fields or static field configurations provide evidence that the scattering anisotropy in imbalanced turbulence is not primarily due to coherence with propagating Alfvén waves, but an effect of the spatial structure of electric fields in cross-helical MHD turbulence.

79 ASTRONOMY AND ASTROPHYSICS↗

Modulating effect of evanescent waves on thin film growth

Atomic-scale smooth thin films are keys to successful integration and proper function of many multilayer-structured devices. However, the intrinsic islandlike growth mode prevents human beings from realizing such ultrasmooth films of many important functional materials. To solve this problem, we propose a negative entropy-infusing method that employs evanescent waves to enhance the downward interlayer diffusion of adatoms and thus transform the islandlike growth mode to the layer-by-layer growth mode. In this work, the formulas of the optical force and the lowered diffusion barrier were derived, and the application of this theory on a simplified example demonstrates significantly improved surface morphologies through numerical simulations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Random Walks With Tweedie: A Unified View of Score-Based Diffusion Models [In the Spotlight]

We present a concise derivation for several influential score-based diffusion models that relies on only a few textbook results. Diffusion models have recently emerged as powerful tools for generating realistic, synthetic signals—particularly natural images—and often play a role in state-of-the-art algorithms for inverse problems in image processing. While these algorithms are often surprisingly simple, the theory behind them is not, and multiple complex theoretical justifications exist in the literature. Here, in this study, we provide a simple and largely self-contained theoretical justification for score-based diffusion models that is targeted towards the signal processing community. This approach leads to generic algorithmic templates for training and generating samples with diffusion models. We show that several influential diffusion models correspond to particular choices within these templates and demonstrate that alternative, more straightforward algorithmic choices can provide comparable results. This approach has the added benefit of enabling conditional sampling without any likelihood approximation.

97 MATHEMATICS AND COMPUTING↗

Grazing-incidence X-ray diffraction from a crystal with subsurface defects

The diffraction of X rays incident on a crystal surface under grazing angles under conditions of total external reflection has been investigated. An approach is proposed in which exact solutions to the dynamic problem of grazing-incidence diffraction in an ideal crystal are used as initial functions to calculate the diffuse component of diffraction in a crystal with defects. The diffuse component of diffraction is calculated for a crystal with surface defects of a dilatation-center type. Exact formulas of the continuum theory which take into account the mirror-image forces are used for defect-induced atomic displacements. Scattering intensity maps near Bragg peaks are constructed for different scan modes, and the conditions for detecting primarily the diffuse component are determined. The results of dynamic calculations of grazing-incidence diffraction in defect-containing crystals are compared with calculations in the kinematic approximation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

X-ray diffraction study of BaTiO{sub 3} single crystals before and after fast-neutron irradiation

The neutron irradiation of ferroelectrics is efficiently used to form structural states that cannot be obtained by conventional technologies. To date, the effect of neutron irradiation on the structure and properties of BaTiO{sub 3} has been studied for only ceramic materials. We have considered the influence of fast-neutron irradiation (F = 1 × 10{sup 17} cm{sup −2}) on the structure and properties of BaTiO{sub 3} single crystals for the first time. The structural changes occurring in irradiated BaTiO{sub 3} and their correlation with the behavior of dielectric and nonlinear optical characteristics are analyzed with the aid of a specially developed method for taking into account the experimental correction to diffuse scattering. Neutron irradiation to the aforementioned dose retains the polar structure of the material and only slightly changes atomic displacements. The radiationinduced structural changes occur according to the high-temperature type to form a structure similar to the cubic modification of unirradiated BaTiO{sub 3} crystal.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

LOCAL SIMULATIONS OF INSTABILITIES DRIVEN BY COMPOSITION GRADIENTS IN THE ICM

The distribution of helium in the intracluster medium (ICM) permeating galaxy clusters is not well constrained due to the very high plasma temperature. Therefore, the plasma is often assumed to be homogeneous. A nonuniform helium distribution can, however, lead to biases when measuring key cluster parameters. This has motivated one-dimensional models that evolve the ICM composition assuming that the effects of magnetic fields can be parameterized or ignored. Such models for nonisothermal clusters show that helium can sediment in the cluster core, leading to a peak in concentration offset from the cluster center. The resulting profiles have recently been shown to be linearly unstable when the weakly collisional character of the magnetized plasma is considered. In this paper, we present a modified version of the MHD code Athena, which makes it possible to evolve a weakly collisional plasma subject to a gravitational field and stratified in both temperature and composition. We thoroughly test our implementation and confirm excellent agreement against several analytical results. In order to isolate the effects of composition, in this initial study we focus our attention on isothermal plasmas. We show that plasma instabilities, feeding off gradients in composition, can induce turbulent mixing and saturate by rearranging magnetic field lines and alleviating the composition gradient. Composition profiles that increase with radius lead to instabilities that saturate by driving the average magnetic field inclination to roughly 45°. We speculate that this effect may alleviate the core insulation observed in homogeneous settings, with potential consequences for the associated cooling flow problem.

79 ASTRONOMY AND ASTROPHYSICS↗

Turbulent dynamo in a conducting fluid and a partially ionized gas

By following the Kazantsev theory and taking into account both microscopic and turbulent diffusion of magnetic fields, we develop a unified treatment of the kinematic and nonlinear stages of a turbulent dynamo process, and we study the dynamo process for a full range of magnetic Prandtl number P {sub m} and ionization fractions. We find a striking similarity between the dependence of dynamo behavior on P {sub m} in a conducting fluid and R (a function of ionization fraction) in a partially ionized gas. In a weakly ionized medium, the kinematic stage is largely extended, including not only exponential growth but a new regime of dynamo characterized by a linear-in-time growth of magnetic field strength, and the resulting magnetic energy is much higher than the kinetic energy carried by viscous-scale eddies. Unlike the kinematic stage, the subsequent nonlinear stage is unaffected by microscopic diffusion processes and has a universal linear-in-time growth of magnetic energy with the growth rate as a constant fraction 3/38 of the turbulent energy transfer rate, showing good agreement with earlier numerical results. Applying the analysis to the first stars and galaxies, we find that the kinematic stage is able to generate a field strength only an order of magnitude smaller than the final saturation value. But the generation of large-scale magnetic fields can only be accounted for by the relatively inefficient nonlinear stage and requires longer time than the free-fall time. It suggests that magnetic fields may not have played a dynamically important role during the formation of the first stars.

79 ASTRONOMY AND ASTROPHYSICS↗