Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “general relativity equations and solutions”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Existence of frozen-in coordinate systems

The 'frozen-in' coordinate systems were first introduced in the works on 'reconnection' and 'magnetic barrier' theories (see review by M.l.Pudovkin and V.S.Semenov, Space Sci. Rev. 41,1 1985). The idea was to utilize the mathematical apparatus developed for 'general relativity' theory to simplify obtaining solutions to the ideal MHD equations set. Magnetic field (B), plasma velocity (v), and their vector product were used as coordinate vectors. But there exist no stationary solutions of ideal MHD set that satisfies the required boundary conditions at infinity (A.D.Chertkov, Solar Wind Seven Conf.,Pergamon Press,1992,165) having non-zero vector product of v and B where v and B originate from the same sphere. The existence of a solution is the hidden mine of the mentioned theories. The solution is constructed in the coordinate system, which is unknown and indeterminate before obtaining this solution. A substitution of the final solution must be done directly into the initial MHD set in order to check the method. One can demonstrate that 'solutions' of Petschek's problem, obtained by 'frozen-in' coordinate systems, does not satisfy just the 'frozen-in' equation, i.e. induction equation. It stems from the fact that Petschek's 're-connection' model, treated as a boundary problem, is over determined. This problem was incorrectly formulated.

Chertkov, A. D.↗
DOE OSTI2024

Amplitudes, supersymmetric black hole scattering at $\mathcal{O}\left({G}^5\right)$, and loop integration

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in $\mathcal{N}$ = 8 supergravity at the fifth post-Minkowskian order and to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present the solution to the differential equations for all master integrals necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in $\mathcal{N}$ = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in $\mathcal{N}$ = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

Black Holes↗
DOE OSTI2022

Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

Herein we formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results [1]-[4], indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Magnetic frame-dragging correction to the electromagnetic solution of a compact neutron star

ABSTRACT Neutron stars are usually modelled as spherical, rotating perfect conductors with a predominant intrinsic dipolar magnetic field anchored to their stellar crust. Due to their compactness, General Relativity corrections must be accounted for in Maxwell’s equations, leading to modified interior and exterior electromagnetic solutions. We present analytical solutions for slowly rotating magnetized neutron stars, taking into account the magnetic frame-dragging correction. For typical compactness values, i.e. Rs ∼ 0.5 [R*], we show that the new terms lead to a per cent order correction in the magnetic field orientation and strength compared to the case, with no magnetic frame-dragging correction. Also, we obtain a self-consistent redistribution of the surface azimuthal current. We verify the validity of the derived solution through two-dimensional particle-in-cell simulations of an isolated neutron star. Defining the azimuthal electric and magnetic field amplitudes during the transient phase as observables, we prove that the magnetic frame-dragging correction reduces the transient wave amplitude, as expected from the analytical solution. We show that simulations are more accurate and stable, when we include all first-order terms. The increased accuracy at lower spatiotemporal resolutions translates into a reduction in simulation runtimes.

Torres, R. (ORCID:0000000291820228)↗
DOE OSTI2025

Lectures on statistical mechanics

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

plasma dynamics↗
NASA NTRS1971

Linearization of Schwarzschild's line element - Application to the clock paradox.

This article studies the relativistic theory of the motion of a particle in the presence of a uniform acceleration field. The problem is introduced as a linearization of the fundamental line element of general relativity. The linearized line element is a solution of Einstein's field equations. The equations of geodesics corresponding to this line element are solved and applied to the clock paradox problem.-

Broucke, R.↗
NASA NTRS1995

Electric fields and field-aligned currents in polar regions of the solar corona: 3-D MHD consideration

The calculation of the solar rotation electro-dynamical effects in the near-the-Sun solar wind seems more convenient from the non-inertial corotating reference frame. This implies some modification of the 3-D MHD equations generally on the base of the General Theory of Relativity. The paper deals with the search of stationary (in corotating non-inertial reference frame) solutions of the modified 3-D MHD equations for the in near-the-Sun high latitude sub-alfvenic solar wind. The solution is obtained requiring electric fields and field-aligned electric currents in the high latitude near-the-Sun solar wind. Various scenario are explored self-consistently via a number of numerical experiments. The analogy with the high latitude Earth's magnetosphere is used for the interpretation of the results. Possible observational manifestations are discussed.

Pisanko, Yu. V.↗
NASA NTRS1978

An integral solution to a nonlinear diffusion problem

The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.

Hussaini, M. Y.↗
NASA NTRS1988

Ion-ion waves in the auroral region - Wave excitation and ion heating

The properties of the ion-ion mode which is excited in plasmas when two or more cold ion beams are streaming relative to one another are investigated assuming a warm electron distribution at rest (the model consistent with particle distributions observed in auroral plasma cavities). Numerical solutions are derived for the generalized electrostatic dispersion equation for parallel propagation and for oblique propagation of ion waves. It is shown that the relative ion temperature and concentration have significant effects on the stability of the accelerated plasma and the expected ion heating. Finally, a relationship between ion drift and thermal speed is derived using the marginal stability of ion-ion waves for nonzero ion temperature; the relationship was found to compare favorably with DE-1 ion observations at the high-altitude boundary of the auroral cavity.

Dusenbery, P. B.↗
NASA NTRS1972

A general relaxation method for inverse solution of the full radiative transfer equation.

The relaxation method for the inverse solution of the full radiative transfer equation is generalized to solve for all atmospheric parameters which appear in the integrand as functions or functionals, without any a priori information related to the expected solution. The method is illustrated by examples in the earth's atmosphere for the determination of water vapor mixing ratio profiles from observations in the 6.3-micron band.

Chahine, M. T.↗
NASA NTRS1977

Unsteady supersonic aerodynamic theory by the method of potential gradient

A generalized solution of the hyperbolic wave equation has been derived. The method used has been further developed to relate the velocity components at a field point to the potential gradient distribution in the dependence domain. Singular integrals have been evaluated in closed form, while numerical integration methods are suggested for treating more complex but analytic functions. Idealization of the lifting surfaces by trapezoidal elements with two sides parallel to the streamlines is computationally efficient because line integrations along the other two sides need only be considered. Furthermore, all the integrands vanish on the Mach cone and the need for determining the hyperbolic curves of intersection of the cone with the lifting surface is avoided. Generalized aerodynamic coefficients for three AGARD planforms have been calculated and compared with the available results.

Jones, W. P.↗
DOE OSTI2020

Covariant phase space with boundaries

The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

36 MATERIALS SCIENCE↗
NASA NTRS1978

Computing viscous flows

Although the goals and techniques of computational aerodynamics and computational fluid dynamics differ, advancement in the physical and mathematical aspects of the latter are required for progress in aerodynamic computation. The most attractive approach is the use of hybrid methods where both the equations treated and the solution algorithms reflect the local character of the flow. A working general turbulence model that is only peripherally related to the availability of large fast computers would provide a significant breakthrough in computational aerodynamics. There is no unanimity of opinion as to what may be the optimum algorithm or family of algorithms in the next decade. While it is premature to develop an optimum processor, such a machine dedicated to study the structure of solutions to the three-dimensional time-dependent Navier-Stokes equations and to the computability of turbulence would be very valuable.

Murphy, J. D.↗
NASA NTRS1960

Solution of Nonlinear Systems

The problem of solving systems of nonlinear equations has been relatively neglected in the mathematical literature, especially in the textbooks, in comparison to the corresponding linear problem. Moreover, treatments that have an appearance of generality fail to discuss the nature of the solutions and the possible pitfalls of the methods suggested. Probably it is unrealistic to expect that a unified and comprehensive treatment of the subject will evolve, owing to the great variety of situations possible, especially in the applied field where some requirement of human or mechanical efficiency is always present. Therefore we attempt here simply to pose the problem and to describe and partially appraise the methods of solution currently in favor.

Turner, L. R.↗
NASA NTRS1975

A characterization of transionospheric fading communication channel

Using a multiple-scatter propagation theory, the characteristics of the transionospheric satellite-earth communication links are related to the parameters of the ionosphere. General equations for the frequency correlation functions are derived. Some approximate solutions are obtained and numerical examples are given. Also discussed are the degree of coherence and potential applications of the procedure to characterize the intense fading channel.

Liu, C. H.↗
NASA NTRS1991

Use of non-spherical gravity harmonics for relative motion GN/C (category 2)

Detailed analysis of the Automatic Rendezvous and Capture problem indicate a need for three different regions of mathematical description for the GN&C algorithms: (1) multi-vehicle orbital mechanics to the rendezvous interface point, i.e., within 100 nm; (2) relative motion solutions (such as Clohessy-Wiltshire type) from the far-field to the near-field interface, i.e., within 1 nm and; (3) close proximity motion - the near-field motion where the relative differences in the gravitational and orbit inertial accelerations can be neglected from the equations of motion. Limit boundaries to these regions can be precisely defined by further analysis and will be functions of the tracking measurement accuracies and the computer resources available for the solution of the algorithms. This paper analyzes the relative motion in Regions 2 and 3 above and present the derivation and discussion of the general case of non-spherical gravitational perturbed relative motion. Mathematical deviations from the numerically integrated spherical gravity case and solutions from the Clohessy-Wiltshire equations are presented in the analysis. Based upon this preliminary analysis, it is recommended that further efforts be used to assess the relative position and velocity differences in Region 2 due to non-spherical gravity harmonics and that viable GN&C algorithms be developed to include these gravity perturbations (especially the effects of the first gravity harmonic, J2).

Henderson, David M.↗
NASA NTRS1976

Integration-free interval doubling for Riccati equation solutions

Various algorithms are given for the case of constant coefficients. The algorithms are based on two ideas: first, relate the Re solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration free superlinearly convergent iterative solution to the algebraic RE. The algorithm, arranged in square-root form, is thought to be numerically stable and competitive with other methods of solving the algebraic RE.

Bierman, G. J.↗