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At least 145 records · Page 8

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NASA NTRS1973

Mixed finite-difference scheme for analysis of simply supported thick plates.

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.

Noor, A. K.↗

NASA NTRS1990

The effect of random Alfven waves on the propagation of hydromagnetic waves in a finite-beta plasma

Using first-order smoothing theory, Fourier analysis and perturbation methods, the evolution equation of the wave spectrum as well as the nonlinear forces generated by random Alfven waves in a finite-beta plasma with phenomenological Landau-damping effects are obtained. The effect of microscale random Alfven waves on the propagation of large-scale hydromagnetic waves is also investigated by solving the mean-field equations. It is shown that parallel-propagating random Alfven waves are modulationally stable and that obliquely propagating random Alfven waves can be modulationally unstable when the energy of random waves is converted to slow magnetoacoustic waves that can be Landau-damped, providing a dissipation mechanism for the Alfven waves.

Hamabata, Hiromitsu↗

NASA NTRS1989

Side-wall gas 'creep' and 'thermal stress convection' in microgravity experiments on film growth by vapor transport

While 'no-slip' boundary conditions and the Navier-Stokes equations of continuum fluid mechanics have served the vapor transport community well until now, it is pointed out that transport conditions within highly nonisothermal ampoules are such that the nonisothermal side walls 'drive' the dominant convective flow, and the familiar Stokes-Fourier-Fick laws governing the molecular fluxes of momentum, energy, and (species) mass in the 'continuum' field equations will often prove to be inadequate, even at Knudsen numbers as small as 0.001. The implications of these interesting gas kinetic phenomena under microgravity conditions, and even under 'earth-bound' experimental conditions, are outlined here, along with a tractable approach to their systematic treatment.

Rosner, Daniel E.↗

NASA NTRS1990

A numerical method for phase-change problems

A highly accurate and efficient finite-difference method for phase-change problems with multiple moving boundaries of irregular shape is developed by employing a coordinate transformation that immobilizes moving boundaries and preserves the conservative forms of the original governing equations. The numerical method is first presented for one-dimensional phase-change problems (involving large density variation between phases, heat generation, and multiple moving boundaries) and then extended to solve two-dimensional problems (without change of densities between phases). Numerical solutions are obtained non-iteratively using an explicit treatment of the interfacial mass and energy balances and an implicit treatment of the temperature field equations. The accuracy and flexibility of the present numerical method are verified by solving some phase-change problems and comparing the results with existing analytical, semi-analytical and numerical solutions. Results indicate that one- and two-dimensional phase-change problems can be handled easily with excellent accuracies.

Kim, Charn-Jung↗

DOE OSTI2023

Mean-field limit of a hybrid system for multi-lane multi-class traffic

This article aims to study coupled mean-field equation and ODEs with discrete events motivated by vehicular traffic flow. Precisely, multi-lane traffic flow in presence of human-driven and autonomous vehicles is considered, with autonomous vehicles possibly influenced by external policymakers. First, a finite-dimensional hybrid system is developed based on the continuous Bando-Follow-the-Leader dynamics coupled with discrete events due to lane-change maneuvers. Then the mean-field limit of the finite-dimensional hybrid system is rigorously derived for the dynamics of the human-driven vehicles. The microscopic lane-change maneuvers of the human-driven vehicles generate a source term for the mean-field PDE. This leads to an infinite-dimensional hybrid system described by coupled Vlasov-type PDE, ODEs, and discrete events.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Semiclassical dynamics of Hawking radiation

Here, we consider gravity in 3+1 spacetime dimensions coupled to N scalar matter fields in a semiclassical limit where $N\to\infty$. The dynamical evolution of a black hole including the back-reaction of the Hawking radiation on the metric is formulated as an initial-value problem. The quantum stress-energy tensor is evaluated using a point-splitting regularization along spacelike geodesics. To account for the quantum entanglement of the matter fields, they are treated as a set of bilocal collective fields defined on spacelike hypersurfaces. The resulting semiclassical field equations include terms up to fourth order in derivatives that can be treated in a perturbative $\hbar$ expansion. The formulation we arrive at should be amenable to numerical simulation of time-dependent semiclassical spacetime.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1993

On the spring and mass of the Dirac oscillator

The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the Hamiltonian for the Dirac oscillator yields the Klein-Gordon equation with a potential of the form: (ar(sub 2) + b(L x S)), where a and b are constants. To obtain the Dirac oscillator, a 'minimal substitution' is made in the Dirac equation, where the ordinary derivative is replaced with a covariant derivative. However, an unusual feature of the covariant derivative in this case is that the potential is a non-trivial element of the Clifford algebra. A theory which naturally gives rise to gage potentials which are non-trivial elements of the Clifford algebra is that based on local automorphism invariance. An exact solution of the automorphism gage field equations which reproduces both the potential term and the mass term of the Dirac oscillator is presented.

Crawford, James P.↗

NASA NTRS1984

Structural optimization with constraints on transients response

An exceptionally elegant method for structural optimization with constraints on the static response presented by Shield and Prager is discussed. Their derivation of the optimality condition was facilitated by a reformulation of the structural elasticity equations in terms of what was then a new variational principle, the principle of stationary mutual potential energy. Their optimality condition relates the design variable to an appropriately defined mutual strain energy. An alternative but related approach, based upon the principle of stationary mutual complementary energy, presented by N. C. Haung, is also discussed. The simplicity of these principles lies in the facts that the energy functionals are stationary at the solution to the field equations and that their stationary value is proportional to the quantity to be optimized.

Reiss, R.↗

DOE OSTI2022

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.

42 ENGINEERING↗

NASA NTRS1973

Theory of a ring laser

Development of a systematic formulation of the theory of a ring laser which is based on first principles and uses a well-known model for laser operation. A simple physical derivation of the electromagnetic field equations for a noninertial reference frame in uniform rotation is presented, and an attempt is made to clarify the nature of the Fox-Li modes for an open polygonal resonator. The polarization of the active medium is obtained by using a Fourier-series method which permits the formulation of a strong-signal theory, and solutions are given in terms of continued fractions. It is shown that when such a continued fraction is expanded to third order in the fields, the familiar small-signal ring-laser theory is obtained.

Menegozzi, L. N.↗

NASA NTRS1994

Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation

In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.

Tam, Christopher K. W.↗

NASA NTRS1974

Do massive Brans-Dicke theories of gravitation imitate Brans-Dicke theories with nonzero divergence of energy-momentum tensor

Acharya and Hogan (1973) have introduced a massive scalar field into the usual Brans-Dicke (1961) theory of gravitation. Formally they obtain certain field equations. The assumption of a nonzero divergence for T mu nu (or equivalently the introduction of sources) formally imitates a massive Brans-Dicke and satisfies the condition of Acharya and Hogan that the theory be indistinguishable with the classical test of the Einstein theory. Although it was shown elsewhere that the modified Brans-Dicke theory agrees with the classical test under certain conditions, there were no specified limits on omega, a similar circumstance discovered by Acharya and Hogan for the massive scalar field.

Smalley, L. L.↗

DOE OSTI2023

Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler–Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to “physics-informed” gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Gravitational production of heavy particles during and after inflation

Abstract We investigate the gravitational production of a scalar fieldχwith a mass exceeding the Hubble scale during inflationm χ ≳H I , employing both analytical and numerical approaches. We demonstrate that the steepest descent method effectively captures the epochs and yields of gravitational production in a compact and simple analytical framework. These analytical results align with the numerical solutions of the field equation. Our study covers three spacetime backgrounds: de Sitter, power-law inflation, and the Starobinsky inflation model. Within these models, we identify two distinct phases of particle production: during and after inflation. During inflation, we derive an accurate analytic expression for the particle production rate, accounting for a varying Hubble rate. After inflation, the additional burst of particle production depends on the inflaton mass around its minimum. When this mass is smaller than the Hubble scale during inflation,H I , there is no significant extra production. However, if the inflaton mass is larger, post-inflation production becomes the dominant contribution. Furthermore, we explore the implications of gravitationally produced heavy fields for dark matter abundance, assuming their cosmological stability.

Physics↗

NASA NTRS1996

A Three-Dimensional Linearized Unsteady Euler Analysis for Turbomachinery Blade Rows

A three-dimensional, linearized, Euler analysis is being developed to provide an efficient unsteady aerodynamic analysis that can be used to predict the aeroelastic and aeroacoustic response characteristics of axial-flow turbomachinery blading. The field equations and boundary conditions needed to describe nonlinear and linearized inviscid unsteady flows through a blade row operating within a cylindrical annular duct are presented. In addition, a numerical model for linearized inviscid unsteady flow, which is based upon an existing nonlinear, implicit, wave-split, finite volume analysis, is described. These aerodynamic and numerical models have been implemented into an unsteady flow code, called LINFLUX. A preliminary version of the LINFLUX code is applied herein to selected, benchmark three-dimensional, subsonic, unsteady flows, to illustrate its current capabilities and to uncover existing problems and deficiencies. The numerical results indicate that good progress has been made toward developing a reliable and useful three-dimensional prediction capability. However, some problems, associated with the implementation of an unsteady displacement field and numerical errors near solid boundaries, still exist. Also, accurate far-field conditions must be incorporated into the FINFLUX analysis, so that this analysis can be applied to unsteady flows driven be external aerodynamic excitations.

Montgomery, Matthew D.↗

NASA NTRS2003

Resource Letter GrW-1: Gravitational Waves

The phenomenon of gravitational radiation was one of the first predictions of Einstein's general theory of relativity. Progress in understanding this radiation theoretically was slow at first, owing to the difficulty of the nonlinear field equations and the subtleties of their physical effects. The experimental side of this subject also has taken a long time to develop, with efforts at detection severely challenged by the extreme weakness of the waves impinging on the Earth. However, as the 21st century begins, observations of the gravitational waves from astrophysical sources such as black holes, neutron stars, and stellar collapse are expected to open a new window on the universe. Vigorous experimental programs centered on ground-based detectors are being carried out worldwide, and a space-based detector is in the planning stages. On the theoretical side, much effort is being expended to produce robust models of the astrophysical sources and accurate calculations of the waveforms they produce. In this Resource Letter, a set of basic references will be presented first, to provide a general introduction to and overview of the literature in this field. The focus then will shift to highlighting key resources in more specialized areas at the forefront of current research.

White, Nicholas E.↗

NASA NTRS1993

Ducted fan acoustic radiation including the effects of nonuniform mean flow and acoustic treatment

Forward and aft acoustic propagation and radiation from a ducted fan is modeled using a finite element discretization of the acoustic field equations. The fan noise source is introduced as equivalent body forces representing distributed blade loading. The flow in and around the nacelle is assumed to be nonuniform, reflecting the effects of forward flight and flow into the inlet. Refraction due to the fan exit jet shear layer is not represented. Acoustic treatment on the inlet and exhaust duct surfaces provides a mechanism for attenuation. In a region enclosing the fan a pressure formulation is used with the assumption of locally uniform flow. Away from the fan a velocity potential formulation is used and the flow is assumed nonuniform but irrotational. A procedure is developed for matching the two regions by making use of local duct modal amplitudes as transition state variables and determining the amplitudes by enforcing natural boundary conditions at the interface between adjacent regions in which pressure and velocity potential are used. Simple models of rotor alone and rotor/exit guide vane generated noise are used to demonstrate the calculation of the radiated acoustic field and to show the effect of acoustic treatment. The model has been used to assess the success of four techniques for acoustic lining optimization in reducing far field noise.

Eversman, Walter↗

DOE OSTI2022

Black hole scattering and partition functions

When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by “renormalizing” the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, we discover that the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler-like region, and the renormalized DOS captures the quasinormal modes for the scalar. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We comment on possible connections with recent works from an algebraic point of view.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗