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Results for “divergence theorem”

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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Applying an Oriented Divergence Theorem to Swept Face Remap

Here we present a novel oriented divergence theorem and apply the results to a swept face remap method (conservative data transfer between two meshes) in arbitrary Langrangian–Eulerian hydrodynamics. In our setting, we compute the material flux along swept regions between corresponding faces in the source and target meshes. Since the swept region may add material, subtract material, or do both when it intersects itself, we cannot apply the conventional divergence theorem without accounting for orientation and self-overlaps. In this work, we encode the swept region orientation and geometry with a map from the unit n -dimensional cube, and then apply an oriented analog of divergence theorem to compute the material flux. We present efficient implementation strategies for the presented method. We also provide numerical evidence supporting our results and discuss extensions to more general mesh topologies.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2019

Quantification of CO2 and CH4 emissions over Sacramento, California, based on divergence theorem using aircraft measurements

Emission estimates of carbon dioxide (CO2) and methane (CH4) and the meteorological factors affecting them are investigated over Sacramento, California, using an aircraft equipped with a cavity ring-down greenhouse gas sensor as part of the Alpha Jet Atmospheric eXperiment (AJAX) project. To better constrain the emission fluxes, we designed flights in a cylindrical pattern and computed the emission fluxes from two flights using a kriging method and Gauss’s divergence theorem. Differences in wind treatment and assumptions about background concentrations affect the emission estimates by a factor of 1.5 to 7. The uncertainty is also impacted by meteorological conditions and distance from the emission sources. The vertical layer averaging affects the flux estimate, but the choice of raw wind or mass-balanced wind is more important than the thickness of the vertical averaging for massbalanced wind for both urban and local scales. The importance of vertical mass transfer for flux estimates is examined, and the difference in the total emission estimate with and without vertical mass transfer is found to be small, especially at the local scale. The total flux estimates accounting for the entire circumference are larger than those based solely on measurements made in the downwind region. This indicates that a closed-shape flight profile can better contain total emissions relative to a one-sided curtain flight because most cities have more than one point source and wind direction can change with time and altitude. To reduce the uncertainty of the emission estimate, it is important that the sampling strategy account not only for known source locations but also possible unidentified sources around the city. Our results highlight that aircraft-based measurements using a closed-shape flight pattern are an efficient and useful strategy for identifying emission sources and estimating local- and city-scale greenhouse gas emission fluxes.

Ju-Mee Ryoo↗
NASA NTRS1994

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗
NASA NTRS1989

Advanced development of BEM for elastic and inelastic dynamic analysis of solids

Direct Boundary Element formulations and their numerical implementation for periodic and transient elastic as well as inelastic transient dynamic analyses of two-dimensional, axisymmetric and three-dimensional solids are presented. The inelastic formulation is based on an initial stress approach and is the first of its kind in the field of Boundary Element Methods. This formulation employs the Navier-Cauchy equation of motion, Graffi's dynamic reciprocal theorem, Stokes' fundamental solution, and the divergence theorem, together with kinematical and constitutive equations to obtain the pertinent integral equations of the problem in the time domain within the context of the small displacement theory of elastoplasticity. The dynamic (periodic, transient as well as nonlinear transient) formulations have been applied to a range of problems. The numerical formulations presented here are included in the BEST3D and GPBEST systems.

Banerjee, P. K.↗
NASA NTRS1972

A reciprocal theorem for a mixture theory

A dynamic reciprocal theorem for a linearized theory of interacting media is developed. The constituents of the mixture are a linear elastic solid and a linearly viscous fluid. In addition to Steel's field equations, boundary conditions and inequalities on the material constants that have been shown by Atkin, Chadwick and Steel to be sufficient to guarantee uniqueness of solution to initial-boundary value problems are used. The elements of the theory are given and two different boundary value problems are considered. The reciprocal theorem is derived with the aid of the Laplace transform and the divergence theorem and this section is concluded with a discussion of the special cases which arise when one of the constituents of the mixture is absent.

Martin, C. J.↗
NASA NTRS1976

A tensor formulation of the equation of transfer for spherically symmetric flows

A tensor formulation of the equation of radiative transfer is derived in a seven-dimensional Riemannian space such that the resulting equation constitutes a divergence in any coordinate system. After being transformed to a spherically symmetric comoving coordinate system, the transfer equation contains partial derivatives in angle and frequency, as well as optical depth due to the effects of aberration and the Doppler shift. However, by virtue of the divergence form of this equation, the divergence theorem may be applied to yield a numerical differencing scheme which is expected to be stable and to conserve luminosity. It is shown that the equation of transfer derived by this method in a Lagrangian coordinate system may be reduced to that given by Castor (1972), although it is, of course, desirable to leave the equation in divergence form.

Haisch, B. M.↗
DOE OSTI2026

Resilience Measurement Framework For Post-deployment Artificial Intelligence (ai) Integrated Systems

Resilience is largely defined as the ability to adapt or recover from adverse conditions, stresses, attacks, or compromises on systems that use or are enabled by digital resources. In Artificial Intelligence Management and Research for Advanced Networked Testbed Hub (AMARANTH), resilience is measured in the amount of time it took from the beginning of a testing period for the model to reach predictions outside of the original 95% confidence interval or using the Kullback-Leibler (KL) divergence theorem, the Population Stability Index (PSI), and traditional methods such as root mean squared error (RMSE) threshold. Artificial Intelligence (AI) model drift is of significant concern when deploying AI-integrated systems into critical and/or secure environments. Drift can impact resilience of the AI-integrated system post-deployment and requires consistent maintenance and upkeep to ensure the model is accurate and precise. To quantify model drift and predict the point when a model's drift becomes unacceptable, we describe using Kullback-Leibler (KL) divergence, Population Stability Index (PSI) and/or confidence interval width estimations to determine the point of failure and time to failure of a model post-deployment. Through simple code functions, the KL-divergence, PSI, confidence interval, and root mean squared (RMSE) point of failures can be used to derive when a model needs to be maintained as well as the impact of adversarial action through statistical means.

Yockey, Patience [Idaho National Laboratory (INL),↗
DOE OSTI2023

First Moments of a Polyhedron Clipped by a Paraboloid

We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron’s faces with the paraboloid. Here, we provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Towards and FVE-FAC Method for Determining Thermocapillary Effects on Weld Pool Shape

Several practical materials processes, e.g., welding, float-zone purification, and Czochralski crystal growth, involve a pool of molten metal with a free surface, with strong temperature gradients along the surface. In some cases, the resulting thermocapillary flow is vigorous enough to convect heat toward the edges of the pool, increasing the driving force in a sort of positive feedback. In this work we examine this mechanism and its effect on the solid-liquid interface through a model problem: a half space of pure substance with concentrated axisymmetric surface heating, where surface tension is strong enough to keep the liquid free surface flat. The numerical method proposed for this problem utilizes a finite volume element (FVE) discretization in cylindrical coordinates. Because of the axisymmetric nature of the model problem, the control volumes used are torroidal prisms, formed by taking a polygonal cross-section in the (r, z) plane and sweeping it completely around the z-axis. Conservation of energy (in the solid), and conservation of energy, momentum, and mass (in the liquid) are enforced globally by integrating these quantities and enforcing conservation over each control volume. Judicious application of the Divergence Theorem and Stokes' Theorem, combined with a Crank-Nicolson time-stepping scheme leads to an implicit algebraic system to be solved at each time step. It is known that near the boundary of the pool, that is, near the solid-liquid interface, the full conduction-convection solution will require extremely fine length scales to resolve the physical behavior of the system. Furthermore, this boundary moves as a function of time. Accordingly, we develop the foundation of an adaptive refinement scheme based on the principles of Fast Adaptive Composite Grid methods (FAC). Implementation of the method and numerical results will appear in a later report.

Canright, David↗
NASA NTRS2016

New Approach to Characterize CO2 and CH4 Emissions over Sacramento, California Using an Airborne Aircraft Measurement

Emission fluxes of carbon dioxide (CO2) and methane (CH4) over Sacramento, California, USA, are estimated using an aircraft equipped with a cavity ring-down greenhouse gas sensor through the Alpha Jet Atmospheric eXperiment (AJAX). To better quantify the emission fluxes over the entire city and two small point sources within the area, we designed flights in a cylindrical pattern and computed the emission fluxes from 9 flights over all seasons using the Kriging method based on Gausss divergence theorem. The concentrations at the downwind side of Sacramento show somewhat persistent patterns among the 9 flights, but the magnitudes and locations of the fluxes vary depending on the individual weather conditions and seasonality on a given flight day. The net emission fluxes tend to be smaller in summer (July) than winter (November). Both trace gas mixing ratios as well as the wind speed and direction show high spatial variability both horizontally and vertically. The local maximum emission fluxes over a landfill and a rice field measured on July 29, 2015, were 2-3 times lower than the maximum measured over the entire city. These low emission fluxes suggest that not only the local source but also the transport of greenhouse gases due to diurnal and seasonal variation in winds have a strong effect on the CO2 and CH4 flux estimation of the both local emissions and city-size emissions. Our results highlight that the aircraft-based approach is effective and useful for capturing city-size emission fluxes and estimating the greenhouse gas emissions and its sources.

airborne↗
NASA NTRS1971

Kinetic theory of a two-dimensional magnetized plasma.

Several features of the equilibrium and nonequilibrium statistical mechanics of a two-dimensional plasma in a uniform dc magnetic field are investigated. The charges are assumed to interact only through electrostatic potentials. The problem is considered both with and without the guiding-center approximation. With the guiding-center approximation, an appropriate Liouville equation and BBGKY hierarchy predict no approach to thermal equilibrium for the spatially uniform case. For the spatially nonuniform situation, a guiding-center Vlasov equation is discussed and solved in special cases. For the nonequilibrium, nonguiding-center case, a Boltzmann equation, and a Fokker-Planck equation are derived in the appropriate limits. The latter is more tractable than the former, and can be shown to obey conservation laws and an H-theorem, but contains a divergent integral which must be cut off on physical grounds. Several unsolved problems are posed.

Vahala, G.↗
DOE OSTI2024

Flavor fragmentation function factorization

A definition of partonic jet flavor that is both theoretically well-defined and experimentally robust would have profound implications for measurements and predictions especially for heavy flavor applications. Recently, a definition of jet flavor was introduced as the net flavor flowing along the direction of the Winner-Take-All axis of a jet which is soft safe to all orders, but not collinear safe. Here, we exploit the lack of collinear safety and propose a factorization theorem of perturbative flavor fragmentation functions that resum collinear divergences and describe the evolution of flavor from the short distance of jet production to the long distance at which hadronization occurs. Collinear flavor evolution is governed by a small modification of the DGLAP equations. We present a detailed all-orders analysis and identify exact relations that must hold amongst the various anomalous dimensions by probability conservation and the existence of fixed points of the renormalization group flow. We explicitly validate the factorization theorem at one-loop order, and demonstrate its consistency at two loops in particular flavor channels. Starting at two-loops, constraints on phase space imposed by flavor measurements potentially allow for non-trivial soft contributions, but we demonstrate that they are scaleless and so explicitly vanish, ensuring that soft particles are summed inclusively and all divergences are exclusively collinear in nature. This factorization theorem opens the door to precision calculations with identified flavor in the infrared.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Flavor Fragmentation Function Factorization

A definition of partonic jet flavor that is both theoretically well-defined and experimentally robust would have profound implications for measurements and predictions especially for heavy flavor applications. Recently, a definition of jet flavor was introduced as the net flavor flowing along the direction of the Winner-Take-All axis of a jet which is soft safe to all orders, but not collinear safe. Here, we exploit the lack of collinear safety and propose a factorization theorem of perturbative flavor fragmentation functions that resum collinear divergences and describe the evolution of flavor from the short distance of jet production to the long distance at which hadronization occurs. Collinear flavor evolution is governed by a small modification of the DGLAP equations. We present a detailed all-orders analysis and identify exact relations that must hold amongst the various anomalous dimensions by probability conservation and the existence of fixed points of the renormalization group flow. We explicitly validate the factorization theorem at one-loop order, and demonstrate its consistency at two loops in particular flavor channels. Starting at two-loops, constraints on phase space imposed by flavor measurements potentially allow for non-trivial soft contributions, but we demonstrate that they are scaleless and so explicitly vanish, ensuring that soft particles are summed inclusively and all divergences are exclusively collinear in nature. This factorization theorem opens the door to precision calculations with identified flavor in the infrared.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1986

Nongeostrophic theory of zonally averaged circulation. I - Formulation

A nongeostrophic theory of zonally averaged circulation is formulated using the nonlinear primitive equations (mass conservation, thermodynamics, and zonal momentum) on a sphere. The relationship between the mean meridional circulation and diabatic heating rate is studied. Differences between results of nongeostropic theory and the geostrophic formulation concerning the role of eddy forcing of the diabatic circulation and the nonlinear nearly inviscid limit versus the geostrophic limit are discussed. Consideration is given to the Eliassen-Palm flux divergence, the Eliassen-Palm pseudodivergence, the nonacceleration theorem, and the nonlinear nongeostrophic Taylor relationship.

Tung, Ka Kit↗
NASA NTRS1984

Generalization of the quasi-geostrophic Eliassen-Palm flux to include eddy forcing of condensation heating

A modified Eulerian form of the Eliassen-Palm flux which includes the effect of eddy forcing on condensation heating is defined. With the two-dimensional vector flux in the meridional plane which is a function of the zonal mean eddy fluxes replaced by the modified flux, both the Eliassen-Palm theorem and a modified but more general form of the nonacceleration theorem for quasi-geostrophic motion still hold. Calculations of the divergence of the modified flux and of the eddy forcing of the moisture field are presented.

Stone, P. H.↗