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Results for “variational principle”

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.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Ellipsoidal figures of equilibrium - Compressible models

The results of Chandrasekhar (1969) are generalized to polytropes, using a formalism based on ellipsoidal energy variational principle to construct approximate stellar equilibrium solutions and study their stability. After reviewing the energy variational method and describing the approach, several equivalent stability conditions are established and secular vs. dynamical instabilities are discussed. Then, the equilibrium structure equations are derived for isolated, rotating polytropes, and axisymmetric configurations (compressible Maclaurin spheroids) are considered. Particular attention is given to triaxial configurations, either in a state of uniform rotation (generalizing the classical Jacobi ellipsoids) or with internal fluid motions of uniform vorticity (the compressible analogues of Riemann-S ellipsoids) and to the stability of these single star configurations. The compressible generalizations of the Roche and Roche-Riemann problems for a polytrope in orbit about a point-mass companion are solved, and the generalized Darwin problem for two identical polytropes in a binary is considered.

Lai, Dong↗
NASA NTRS1986

Theory of beat-resonant coupling of electrostatic modes

A general expression is derived for the beat-resonant coupling electrostatic modes in a Vlasov plasma. The result for the coupling of two modes has a simple structure: the appropriate momentum gradient of the equilibrium particle distribution is weighted by a positive coupling coefficient and averaged over the resonance surface in momentum space. The contributions of all the resonance surfaces are then summed. This basic structure had been previously exhibited only for specific homogeneous plasma models. The present theory, which unifies and greatly simplifies these individual treatments, is based on a variational formulation of the Vlasov-Poisson equations. Using Lie transforms, the variational principle is reexpressed in oscillation-center variables, and then the nonlinear wave dynamics are obtained from the independent variations of the wave phase and the wave amplitude. The power of the method is then applied to a strongly magnetized, strongly inhomogeneous, non-neutral plasma model.

Crawford, John David↗
NASA NTRS1991

Variational formulation of hybrid problems for fully 3-D transonic flow with shocks in rotor

Based on previous research, the unified variable domain variational theory of hybrid problems for rotor flow is extended to fully 3-D transonic rotor flow with shocks, unifying and generalizing the direct and inverse problems. Three variational principles (VP) families were established. All unknown boundaries and flow discontinuities (such as shocks, free trailing vortex sheets) are successfully handled via functional variations with variable domain, converting almost all boundary and interface conditions, including the Rankine Hugoniot shock relations, into natural ones. This theory provides a series of novel ways for blade design or modification and a rigorous theoretical basis for finite element applications and also constitutes an important part of the optimal design theory of rotor bladings. Numerical solutions to subsonic flow by finite elements with self-adapting nodes given in Refs., show good agreement with experimental results.

Liu, Gao-Lian↗
NASA NTRS1992

Importance of parametrizing constraints in quantum-mechanical variational calculations

In variational calculations of quantum mechanics, constraints are sometimes imposed explicitly on the wave function. These constraints, which are deduced by physical arguments, are often not uniquely defined. In this work, the advantage of parametrizing constraints and letting the variational principle determine the best possible constraint for the problem is pointed out. Examples are carried out to show the surprising effectiveness of the variational method if constraints are parameterized. It is also shown that misleading results may be obtained if a constraint is not parameterized.

Chung, Kwong T.↗
NASA NTRS1984

Hybrid stress finite elements for large deformations of inelastic solids

A new hybrid stress finite element algorithm, based on a generalization of Fraeijs de Veubeke's complementary energy principle is presented. Analyses of large quasistatic deformation of inelastic solids (hypoelastic, plastic, viscoplastic) are within its capability. Principle variables in the formulation are the nominal stress rate and spin. A brief account is given of the boundary value problem in these variables, and the 'equivalent' variational principle. The finite element equation, along with initial positions and stresses, comprise an initial value problem. Factors affecting the choice of time integration schemes are discussed. Results found by application of the new algorithm are compared to those obtained by a velocity based finite element algorithm.

Reed, K. W.↗
DOE OSTI2023

A Three-Dimensional, Thermodynamically and Variationally Consistent, Fully Coupled, Electro-Chemo-Thermo-Mechanical Model of Solid-State Batteries

Here we present a theoretical framework for high fidelity modeling of three-dimensional electro-chemo-thermo-mechanical coupled systems such as solid-state batteries. The framework provides a useful means of utilizing molecular level information to predict the overall performance of the system at the continuum level. The central statement of the theoretical framework is a thermodynamically consistent inf-sup problem. The total potential of the coupled system consists of the Helmholtz free energy, the electrical kinetic potential, the chemical potential, the Fourier potential, the chemical reaction potential, and the external power expenditure. All governing equations derive from variational principles. We also show that the resulting initial-boundary-value problem (IBVP) is consistent with existing, widely adopted equations for modeling batteries. Moreover, the IBVP can be naturally discretized using the (variational) finite element method to simulate practical solid-state batteries with arbitrary geometries. Several examples illustrate multiphysics coupling between pairs of the physical field equations, culminating in a comprehensive three-dimensional model of charge-discharge in a solid state battery.

25 ENERGY STORAGE↗
NASA NTRS1977

Scale-covariant theory of gravitation and astrophysical applications

A scale-covariant theory of gravitation is presented which is characterized by a set of equations that are complete only after a choice of the scale function is made. Special attention is given to gauge conditions and units which allow gravitational phenomena to be described in atomic units. The generalized gravitational-field equations are derived by performing a direct scale transformation, by extending Riemannian geometry to Weyl geometry through the introduction of the notion of cotensors, and from a variation principle. Modified conservation laws are provided, a set of dynamical equations is obtained, and astrophysical consequences are considered. The theory is applied to examine certain homogeneous cosmological solutions, perihelion shifts, light deflections, secular variations of planetary orbital elements, stellar structure equations for a star in quasi-static equilibrium, and the past thermal history of earth. The possible relation of the scale-covariant theory to gauge field theories and their predictions of cosmological constants is discussed.

Canuto, V.↗
DOE OSTI2023

An extended three-field principle to scale-bridge the granular micromechanics of polymer-bonded particulate materials

Observed phenomena of damage, viscoelasticity, and viscoplasticity in polymer-bonded particulate materials (PBPM), like polymer-bonded explosive (PBX), are accommodated in a scale-bridging constitutive model based on granular micromechanics. An extended three-field variational principle with min - max optimization problem is introduced to homogenize the bonded aggregates’ grain-scale contact mechanics. Similarly to prior applications of a three-field principle, volume constraints are invoked to embed microscale-to-macroscale correspondences for kinematic and static volumetric measures, like mean strain and its energy-conjugate stress invariant. Unlike prior applications, our three-field principle realizes a strain-driven model for effective constitutive behavior due to bonded intergranular contact. In conclusion, the resultant micro-sphere integration-based homogenization is demonstrated in comparison to existing publicly-available measurements to capture the extreme tension/compression constitutive asymmetry exhibited by PBPMs (specifically PBX-9501) during material softening.

42 ENGINEERING↗
NASA NTRS1984

Theoretical study of miscibility and glass-forming trends in mixtures of polystyrene spheres

A theoretical study of glass-forming trends and miscibility in mixtures of polystyrene spheres (polyballs) of different diameters, suspended in an aqueous solution, is presented. The polyballs are assumed to be charged and to interact via a Debye-Hueckel screened Coulomb potential. The Helmholtz free energy is calculated from a variational principle based on the Gibbs-Bogoliubov inequality, in which a mixture of hard spheres of different diameters is chosen as the reference system. It is found that when the charges of the two types of polyballs are sufficiently different, the variationally determined ratio of hard-sphere diameters differs substantially, leading to packing difficulties characteristic of glass formation. The experimentally observed range of glass formation corresponds to a ratio of hard-sphere diameters of 0.8 or less. Calculations of the free energy as a function of concentration indicate that the liquid polyball mixture is stable against the phase separation, even for widely different polyball charges.

Shih, W.-H.↗
NASA NTRS1995

Variational Calculations of Ro-Vibrational Energy Levels and Transition Intensities for Tetratomic Molecules

A description is given of an algorithm for computing ro-vibrational energy levels for tetratomic molecules. The expressions required for evaluating transition intensities are also given. The variational principle is used to determine the energy levels and the kinetic energy operator is simple and evaluated exactly. The computational procedure is split up into the determination of one dimensional radial basis functions, the computation of a contracted rotational-bending basis, followed by a final variational step coupling all degrees of freedom. An angular basis is proposed whereby the rotational-bending contraction takes place in three steps. Angular matrix elements of the potential are evaluated by expansion in terms of a suitable basis and the angular integrals are given in a factorized form which simplifies their evaluation. The basis functions in the final variational step have the full permutation symmetries of the identical particles. Sample results are given for HCCH and BH3.

Schwenke, David W.↗
NASA NTRS1974

Conservation of wave action

It is pointed out that two basic principles appear in the theory of wave propagation, including the existence of a phase variable and a law governing the intensity, in terms of a conservation law. The concepts underlying such a conservation law are explored. The waves treated are conservative in the sense that they obey equations derivable from a variational principle applied to a Lagrangian functional. A discrete oscillating system is considered. The approach employed also permits in a natural way the definition of a local action density and flux in problems in which the waves are modal or general.

Hayes, W. D.↗
NASA NTRS1987

Relaxation of magnetotail plasmas

A quasi-thermodynamic model is presented for the relaxation of magnetotail plasmas during substorms, followed by quiet times. It is proposed that the plasma relaxes to a state of low-potential energy subject to a small number of global constraints. The constraints are exactly preserved by all ideal motions and, approximately, by a wide class of motions of the plasma undergoing magnetic reconnection. A variational principle which minimizes the free energy predicts the relaxed state. Exact, two-dimensional solutions of the relaxed state are obtained. A universal feature of the exact solutions is a chain of magnetic islands along the tail axis. Sufficient conditions for the stability of relaxed states are obtained from the second variation of the free-energy functional.

Bhattacharjee, A.↗
NASA NTRS1985

Analysis of all-frequency variational behavior of the Kirchhoff approximation for a classic surface-scattering model

In testing a stochastic variational principle at high frequencies by using a Kirchhoffean trial function in an idealized model for surface scattering - a randomly embossed plane - we have found not only the predicted high-frequency improvement but also an unexpected low-frequency improvement in the calculated scattering amplitudes. To investigate systematically the all-frequency variational behavior, we consider here the deterministic one-boss case - Rayleigh's classic model whose exact solution is available for comparison - over all wavelengths, polarizations, and configurations of incidence and scattering. We examine analytically in particular the long-wave limit of the variational-Kirchhoff amplitudes; the results demonstrate improvements in both wavelength and angle depedence for horizontal (TM) polarization and some variational improvements for vertical (TE) polarization. This low-frequency behavior in tandem with the foreseen high-frequency improvement leads to good variational-Kirchhoff results through the intermediate resonance-frequency regime for this model.

Bird, J. F.↗
NASA NTRS1986

Variational divergence in wave scattering theory with Kirchhoffean trial functions

In a recent study of variational improvement of the Kirchhoff approximation for electromagnetic scattering by rough surfaces, a key ingredient in the variational principle was found to diverge for important configurations (e.g., backscatter) if the polarization had any vertical component. The cause and a cure of this divergence are discussed here. The divergence is demonstrated to occur for arbitrary perfectly conducting scatterers and its universal characterstics are determined, by means of a general divergence criterion that is derived. A variational cure for the divergence is prescribed, and it is tested successfully on a standard scattering model.

Bird, J. F.↗
NASA NTRS1984

A note on the variational method of Stokes and DeMarcus for radiative transfer in planetary atmospheres

A generalization of the variational principle of Stokes and DeMarcus (1971) by the construction of two bifunctionals is proposed. The equation of radiative transfer is obtained by imposing the stationary condition on such functionals, and the reflectivity and transmissivity at arbitrary angle are given by their extremum values. Further simplification is possible when trail functions are used that are linear in the variational parameters, in which case the source function derived from the method of Stokes and DeMarcus can give the reflectivity at arbitrary emergent angle to second order. The usefulness and limitations of these variational priinciples are explored with extensive numerical results.

Chow, K. S. K.↗
NASA NTRS1974

On numerically accurate finite element

A general criterion for testing a mesh with topologically similar repeat units is given, and the analysis shows that only a few conventional element types and arrangements are, or can be made suitable for computations in the fully plastic range. Further, a new variational principle, which can easily and simply be incorporated into an existing finite element program, is presented. This allows accurate computations to be made even for element designs that would not normally be suitable. Numerical results are given for three plane strain problems, namely pure bending of a beam, a thick-walled tube under pressure, and a deep double edge cracked tensile specimen. The effects of various element designs and of the new variational procedure are illustrated. Elastic-plastic computation at finite strain are discussed.

Nagtegaal, J. C.↗
NASA NTRS1990

A 4-node assumed-stress hybrid shell element with rotational degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or drilling degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element. This process is accomplished by assuming quadratic variations for both in-plane and out-of-plane displacement fields and linear variations for both in-plane and out-of-plane rotation fields along the edges of the element. In addition, the degrees of freedom at midside nodes are approximated in terms of the degrees of freedom at corner nodes. During this process the rotational degrees of freedom at the corner nodes enter into the formulation of the element. The stress field are expressed in the element natural-coordinate system such that the element remains invariant with respect to node numbering.

Aminpour, Mohammad A.↗
NASA NTRS1984

A variational formulation for time-dependent climate models

A variational principle for time dependent diffusion problems is presented and is demonstrated by applying it to simple seasonal climate models. Two cases are treated. The first, a North-Coakley-type model with constant properties, is used as a tutorial example for the application of the technique. For the second case, heat capacity and thermal conductivity are considered to be latitude dependent in order to treat the effects of land/ocean distribution on the seasonal temperature distribution over the earth. The variational equations are derived and approximate analytical solutions are developed which delineate the influences of the physical asymmetries of the hemispheres in producing an asymmetric annual cycle.

Smith, G. L.↗