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.

SourceResearch connectionFrom yearThrough year

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

At least 73 records · Page 4

All sources NASA NTRS DOE OSTI

DOE OSTI2023

Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma

A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field. Expanding a distribution function in general moments, a set of ordinary differential equations is obtained. Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the first-order density, temperature, and flow velocity in terms of radial gradients of the zeroth-order pressure and temperature. Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity are also obtained by solving the non-Maxwellian moment equations. The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations. Furthermore, the method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field.

neoclassical transport↗

DOE OSTI2023

Evolution of coupled weakly driven waves in a dissipative plasma

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1990

Numerical Methods For Chemically Reacting Flows

Issues related to numerical stability, accuracy, and resolution discussed. Technical memorandum presents issues in numerical solution of hyperbolic conservation laws containing "stiff" (relatively large and rapidly changing) source terms. Such equations often used to represent chemically reacting flows. Usually solved by finite-difference numerical methods. Source terms generally necessitate use of small time and/or space steps to obtain sufficient resolution, especially at discontinuities, where incorrect mathematical modeling results in unphysical solutions.

Leveque, R. J.↗

DOE OSTI2025

Poisson Equation for a (General) Homogeneous d-Dimensional Ellipsoid with Applications to Beam Envelope Tracking

This note describes the solution of the free-space Poisson equation in the interior of a $d$-dimensional homogeneous ellipsoid, and the associated space charge fields. An explicit formula (\ref{Sformula}) is provided that relates the $d\times d$ matrix describing the space charge (quadratic) potential to the $d\times d$ covariance matrix of the ellipsoid. For the cases $d=2$ and $d=3$, this result is used to determine the linear map corresponding to a space charge kick, that may be used to push the beam $6\times 6$ covariance matrix during envelope tracking. The treatment of upright ellipsoids for $d=2$ and $d=3$ is well-represented in the literature. However, the approach taken here emphasizes a general ellipsoid with arbitrary correlations in any dimension. The Appendix provides a general solution of the free-space Poisson equation in dimension $d$ for a source distribution with ellipsoidal symmetry.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

Continuous dependence of fixed points of condensing maps

Many problems in analysis are concerned with the dependence upon parameters of fixed points of maps. For contraction mappings, criteria are relatively easy to obtain and have been known for some time. In the study of solutions of functional differential equations, more general results were needed. It is the purpose of this paper to give a rather general fixed-point theorem for condensing maps depending on a parameter, to prove continuous dependence and to indicate how many of the previous results are special cases.

Hale, J. K.↗

NASA NTRS1991

Closed-form solutions for linear regulator design of mechanical systems including optimal weighting matrix selection

Vibration in modern structural and mechanical systems can be reduced in amplitude by increasing stiffness, redistributing stiffness and mass, and/or adding damping if design techniques are available to do so. Linear Quadratic Regulator (LQR) theory in modern multivariable control design, attacks the general dissipative elastic system design problem in a global formulation. The optimal design, however, allows electronic connections and phase relations which are not physically practical or possible in passive structural-mechanical devices. The restriction of LQR solutions (to the Algebraic Riccati Equation) to design spaces which can be implemented as passive structural members and/or dampers is addressed. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical system. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist.

Hanks, Brantley R.↗

DOE OSTI2020

Resurgent trans-series for generalized Hastings–McLeod solutions

Here, we show that the physical Hastings–McLeod solution of the integrable Painlevé II equation generalizes in a natural way to a class of non-integrable equations, in a way that preserves many of the significant qualitative properties. The Hastings–McLeod solution of Painlevé II is an important and universal example of resurgent relations between perturbative and non-perturbative physics. We derive the trans-series structure of the generalized Hastings–McLeod solutions, demonstrating that integrability is not essential for the resurgent asymptotic properties of the solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1973

Reduction of truncation errors in modal analysis

A condensation method and computer program are described for large discrete parameter vibration analysis of complex structures that greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range. A dynamic transformation is obtained from the partitioned equations of motion that relates modes not explicitly in the condensed solution to the retained modes at a selected system frequency. The generalized mass and stiffness matrices, obtained with existing modal synthesis methods, are reduced using this transformation and solved. Revised solutions are then obtained using new transformations at the calculated eigenvalues and are also used to assess the accuracy of the results. Computations are made tractable by simplified forms of the transformation that result with various modal synthesis methods. Three examples using the dynamic transformation in conjunction with a General Electric stiffness coupling method and the method of Craig and Bampton indicate large reductions in truncation errors and demonstrate the method for sequential groups of modes.

Kuhar, E. J.↗

NASA NTRS1973

A dynamic transformation method for modal synthesis.

This paper presents a condensation method for large discrete parameter vibration analysis of complex structures that greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range. A dynamic transformation is obtained from the partitioned equations of motion that relates modes not explicity in the condensed solution to the retained modes at a selected system frequency. The generalized mass and stiffness matrices, obtained with existing modal synthesis methods, are reduced using this transformation and solved. Revised solutions are then obtained using new transformations at the calculated eigenvalues and are also used to assess the accuracy of the results. If all the modes of interest have not been obtained, the results are used to select a new set of retained coordinates and a new transformation frequency, and the procedure is repeated for another group of modes.

Kuhar, E. J.↗

DOE OSTI2025

An eigenvalue-based method for computing the relaxed pressure in compressible multiphase flow with N phases

The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N + 1 equations and N + 1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. Furthermore, this thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.

Eigenvalue algorithm↗

NASA NTRS1979

Investigation of aerodynamic characteristics of subsonic wings

An analytical strake design procedure is investigated. A numerical solution to the governing strake design equation is used to generate a series of strakes which are tested in a water tunnel to study their vortex breakdown characteristics. The strakes are scaled for use on a half-scale model of the NASA-LaRC general research fuselage with a 44 degrees trapezoidal wing. An analytical solution to the governing design equation is obtained. The strake design procedure relates the potential-flow leading-edge suction and pressure distributions to vortex stability. Several suction distributions are studied and those which are more triangular and peak near the tip generate strakes that reach higher angles of attack before vortex breakdown occurs at the wing trailing edge. For the same suction distribution, a conical rather than three dimensional pressure specification results in a better strake shape as judged from its vortex breakdown characteristics.

Dejarnette, F. R.↗

NASA NTRS1982

Magnetohydrodynamic equilibrium. III - Helically symmetric fields. IV - Nonequilibrium of nonsymmetric hydrodynamic topologies

It is pointed out that plasma confinement in stable equilibrium states constitutes a fundamental and still unresolved question in plasma astrophysics and thermonuclear fusion research. The problem has two parts related to the equilibrium states themselves and their mechanical stability. The question of the existence of general solutions of the field and fluid equations for the steady dynamical interaction of inviscid compressible fluids of high electrical conductivity with magnetic and gravity fields is considered. In the absence of fluid motions, the presented equations become the familiar equations of magnetostatics supplemented by an equation of state. Starting from this simplest case of magnetostatic equilibrium, the investigation proceeds to the more complex case of magnetohydrodynamic equilibrium. Examples of helically symmetric fields are presented to illustrate the use of the formulation for treating the dynamics of helically symmetric hydromagnetic flows.

Tsinganos, K. C.↗

NASA NTRS1999

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Lorenzo, Carl F.↗

NASA NTRS1979

Coupled motion of rigid bodies about their center of mass

Nontrivial analytical solutions for the coupled motion of two rigid bodies about their center of mass are obtained on the assumptions that the rigid bodies are coupled by a massless rigid boom and that no external forces are acting on the system. Both relative rotational and translational motions of the two bodies are considered. General equations of motion are derived by regarding the two bodies as consisting of two distinct systems of particles and by applying the principle of conservation of angular momentum. It is shown that a basic nontrivial solution can be obtained for the translational problem if an assumption is made concerning the relative orientation of one principal axis of inertia of each body and that fundamental nontrivial solutions are readily obtained for the rotational problem if an additional assumption is made with respect to the symmetry of one body. Certain stability criteria are found for some of these motions by defining regions of constraint for the relative translational and rotational elements.

Jezewski, D. J.↗

DOE OSTI2022

Extension of the PINN diffusion model to k-eigenvalue problems

This paper extends our recent work on the Physics-Informed Neural Networks (PINN) approach for the fixed source diffusion models and applies it to the diffusion theory based k-eigenvalue problems. To make the PINN equitable for the eigenvalue problems, we introduce a novel integral regularization term to the loss function in the framework, and allow the direct inference of the principal eigenvalue and the associated eigenfunction. The regularization term enforces a pre-defined value on the integration of the model predictions, and this value can be directly related to a physical property of the system. We also introduce an additional learnable parameter to approximate the principal eigenvalue. As a proof of principle, we solve the one-group two-dimensional k-eigenvalue neutron diffusion equation in this work. We then provide two numerical examples to demonstrate the applicability of the PINN approach. In each example, we solve the k-eigenvalue diffusion equation in a multi-region configuration constrained with a set of Robin boundary conditions for generality. We use a FEM solution based on the power-iteration method to verify the results of the PINN solution. The results showed relative percentage error in the predicted eigenvalue of about 0.77% and about 1.2% for example 1 and example 2, respectively. The mean absolute error in the predicted flux for example 1 is ∼ 0.002 and for example 2 is ∼ 0.0024. These results indicate some preliminary successes of the PINN application to k-eigenvalue problems. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS2018

Green's Function Applicable to Turbofan Exhaust Noise in Jets with an External Center-Body

The problem of propagation of sound across the shear layer in a turbofan jet exhaust with an external center-body is discussed. The wave equation of interest is compressible Rayleigh equation. Two forms of the equation are considered, and the Green's function solutions subject to appropriate surface conditions on the center-body and flight condition in the ambient are presented. Directivity studied in a heated exhaust at temperature ratio of 2.0 and Mach number 0.90 indicate that a rigid center-body tends to increase the sound propagation at forward angles relative to an exhaust without a center-body, while application of suitable surface liner may significantly reduce this enhancement. A general form of the far-field solution to the propagation equation in a parallel flow under supersonic conditions and in the neighborhood of the shear layer singularity is also discussed.

Noise↗

NASA NTRS1982

A cylindrical shell with an arbitrarily oriented crack

The general problem of a shallow shell with constant curvatures is considered. It is assumed that the shell contains an arbitrarily oriented through crack and the material is specially orthotropic. The nonsymmetric problem is solved for arbitrary self equilibrating crack surface tractions, which, added to an appropriate solution for an uncracked shell, would give the result for a cracked shell under most general loading conditions. The problem is reduced to a system of five singular integral equations in a set of unknown functions representing relative displacements and rotations on the crack surfaces. The stress state around the crack tip is asymptotically analyzed and it is shown that the results are identical to those obtained from the two dimensional in plane and antiplane elasticity solutions. The numerical results are given for a cylindrical shell containing an arbitrarily oriented through crack. Some sample results showing the effect of the Poisson's ratio and the material orthotropy are also presented.

Yahsi, O. S.↗

NASA NTRS1983

A cylindrical shell with an arbitrarily oriented crack

The general problem of a shallow shell with constant curvatures is considered. It is assumed that the shell contains an arbitrarily oriented through crack and the material is specially orthotropic. The nonsymmetric problem is solved for arbitrary self equilibrating crack surface tractions, which, added to an appropriate solution for an uncracked shell, would give the result for a cracked shell under most general loading conditions. The problem is reduced to a system to five singular integral equations in a set of unknown functions representing relative displacements and rotations on the crack surfaces. The stress state around the crack tip is asymptotically analyzed and it is shown that the results are identical to those obtained from the two dimensional in plane and antiplane elasticity solutions. The numerical results are given for a cylindrical shell containing an arbitrarily oriented through crack. Some sample results showing the effect of the Poisson's ratio and the material orthotropy are also presented. Previously annunced in STAR as N83-16783

Yahsi, O. S.↗