Search NASA⌕ Search

SEARCH · Search NASA

Results for “Closed form solution”

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 793 records · Page 44
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Periodic motions (close to stationary) of an axisymmetric satellite with magnetic damping

Close to stationary periodic motions of an axisymmetric satellite in a circular orbit are considered. The satellite was equipped with a spherical magnetic damper. The investigation was conducted on the assumption that a strong magnet was installed on the damper float. Stationary rotations of the satellite around the axis of symmetry are selected as the generating solutions. The solutions are constructed in the form of power series of the smaller parameter, and they are extended numerically to the region of random values of the damping coefficient. The stability of the resulting solutions was investigated.

Ovchinnikov, M. Y.↗
NASA NTRS2008

Comparing Wild 2 Particles to Chondrites and IDPS

We compare the observed composition ranges of olivine, pyroxene and Fe-Ni sulfides in Wild 2 grains, comparing these with chondritic IDPs and chondrite classes to explore whether these data suggest affinities to known hydrous materials in particular. Wild 2 olivine has an extremely wide composition range, from Fo4-100 with a pronounced frequency peak at Fo99. The composition range displayed by the low-calcium pyroxene is also very extensive, from En52 to En100, with a significant frequency peak centered at En95. These ranges are as broad or broader than those reported for any other extraterrestrial material. Wild 2 Fe-Ni sulfides mainly have compositions close to that of FeS, with less than 2 atom % Ni - to date, only two pentlandite grains have been found among the Wild-grains suggesting that this mineral is not abundant. The complete lack of compositions between FeS and pentlandite (with intermediate solid solution compositions) suggests (but does not require) that FeS and pentlandite condensed as crystalline species, i.e. did not form as amorphous phases, which later became annealed. While we have not yet observed any direct evidence of water-bearing minerals, the presence of Ni-bearing sulfides, and magnesium-dominated olivine and low-Ca pyroxene does not rule out their presence at low abundance. We do conclude that modern major and minor element compositions of chondrite matrix and IDPs are needed.

Zolensky, Michael↗
DOE OSTI2021

An Analytical Solution for the Initiation and Early Progression of Fretting Wear in Spherical Contacts

Abstract This article derives analytical solutions to calculate the wear volume at the initiation of fretting motion and its early progression over the first few oscillation cycles. The Archard-based model considers a deformable hemisphere that is contact with a deformable flat block. The material pairs investigated are special alloys, the Inconel 617/Incoloy 800H, and Inconel 617/Inconel 617. The analytical study begins with a unidirectional frictional sliding contact, where the local interfacial sliding distance and the nominal sliding distance at the initiation of gross slip are derived. The obtained analytical expressions for unidirectional sliding are then used to derive the corresponding wear volume for the initiation and early progression of gross slip and the wear volume for a general fretting cycle under elastic conditions. These analytical derivations are all verified by the finite element analysis (FEA). The FEA method and the analytical solutions render virtually identical results for both similar and dissimilar material pairs. The effects of plasticity on the wear volume under elastic–plastic conditions are also investigated. It is found that the fretting wear volumes obtained from the FEA simulations, which include plasticity, are close to those obtained from the analytical expressions for purely elastic regimes. All the results are presented in normalized forms, which can be easily generalized and applied to three-dimensional fretting wear of other material pairs.

Engineering↗
NASA NTRS1973

Atmospheric and cloud structures of the Jovian planets

Wet adiabatic atmospheric models are computed for the four Jovian planets. Possible cloud-forming condensates considered are H2O ice, aqueous NH3 solution, NH4SH, and solid NH3, CH4, and Ar. Some techniques for computer calculation of condensation in a multiphase system of variable composition are discussed. Nominal models are computed for all four planets, and the effects of variations in composition and different pressure-temperature regimes are investigated in detail for Jupiter. The meaning of the calculated closed densities and their relation to the actual densities are discussed. The computed atmospheric profiles represent average conditions, subject to considerable local variations.

Weidenschilling, S. J.↗
NASA NTRS1987

Chemistry-split techniques for viscous reactive blunt body flow computations

The weak-coupling structure between the fluid and species equations has been exploited and resulted in three, closely related, time-iterative implicit techniques. While the primitive variables are solved in two separated groups and each by an Alternating Direction Implicit (ADI) factorization scheme, the rate-species Jacobian can be treated in either full or diagonal matrix form, or simply ignored. The latter two versions render the split technique to solving for species as scalar rather than vector variables. The solution is completed at the end of each iteration after determining temperature and pressure from the flow density, energy and species concentrations. Numerical experimentation has shown that the split scalar technique, using partial rate Jacobian, yields the best overall stability and consistency. Satisfactory viscous solutions were obtained for an ellipsoidal body of axis ratio 3:1 at Mach 35 and an angle of attack of 20 degrees.

Li, C. P.↗
DOE OSTI2021

Solidification dynamics in metal additive manufacturing: analysis of model assumptions

Solidification characteristics in the meltpool drive the process-microstructure relationship which helps dictate the material properties of as-built parts in additive manufacturing; therefore, being able to accurately and quickly simulate the size, shape, and solidification characteristics in the melt pool is of great interest to the field. In this study, we investigate various important physical phenomena (dynamic material properties, fluid-flow, radiation and vaporization) which can either be included or neglected in a continuum finite volume model (FVM) and their effect on the solidification conditions. Additionally, since the simplest form of such a model (conduction only) has an analytic solution which is much faster, its viability is also considered. Since the inclusion of some of these physical phenomena will inherently change the net energy input as well as the amount of energy needed to achieve melting of a control volume, each set of included phenomena had an effective absorption efficiency which was calibrated to closely match the dimensions of the melt pool to that of the ground truth data. The ground truth data for this study was defined to be the output of the FVM which included all the physical phenomena (OF). This study then goes on to compare the effects on solidification conditions each of these calibrated models has. It was found that most of the change in solidification conditions comes from the inclusion of latent heat. A posterior correlation factor (PCF) is then introduced to enable an analytic model to predict similar solidification conditions to OF model.

36 MATERIALS SCIENCE↗
NASA NTRS1938

Compressible flow about symmetrical Joukowski profiles

The method of Poggi is employed for the determination of the effects of compressibility upon the flow past an obstacle. A general expression for the velocity increment due to compressibility is obtained. The general result holds whatever the shape of the obstacle; but, in order to obtain the complete solution, it is necessary to know a certain Fourier expansion of the square of the velocity of flow past the obstacle. An application is made to the case flow of a symmetrical Joukowski profile with a sharp trailing edge, fixed in a stream of an arbitrary angle of attack and with the circulation determined by the Kutta condition. The results are obtained in a closed form and are exact insofar as the second approximation to the compressible flow is concerned, the first approximation being the result for the corresponding incompressible flow. Formulas for lift and moment analogous to the Blasius formulas in incompressible flow are developed and are applied to thin symmetrical Joukowski profiles for small angles of attack.

Kaplan, Carl↗
NASA NTRS1985

Interactive aircraft flight control and aeroelastic stabilization

Aeroservoelastic optimization techniques were studied to determine a methodology for maximization of the stable flight envelope of an idealized, actively controlled, flexible airfoil. The equations of motion for the airfoil were developed in state-space form to include time-domain representations of aerodynamic forces and active control. The development of an optimization scheme to stabilize the aeroelastic system over a range of airspeeds, including the design airspeed is outlined. The solution approach was divided in two levels: (1) the airfoil structure, with a design variable represented by the shear center position; and (2) the control system. An objective was stated in mathematical form and a search was conducted with the restriction that each subsystem be constrained to be optimal in some sense. Analytical expressions are developed to compute the changes in the eigenvalues of the closed-loop, actively controlled system. A stability index is constructed to ensure that stability is present at the design speed and at other airspeeds away from the design speed.

Weisshaar, T. A.↗
DOE OSTI2024

Two-loop master integrals for leading-color $$ pp\to t\overline{t}H $$ amplitudes with a light-quark loop

Abstract We compute the two-loop master integrals for leading-color QCD scattering amplitudes including a closed light-quark loop in$$ t\overline{t}H $$ t t ¯ H production at hadron colliders. Exploiting numerical evaluations in modular arithmetic, we construct a basis of master integrals satisfying a system of differential equations inϵ-factorized form. We present the analytic form of the differential equations in terms of a minimal set of differential one-forms. We explore properties of the function space of analytic solutions to the differential equations in terms of iterative integrals which can be exploited for studying the analytic form of related scattering amplitudes. Finally, we solve the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space. As the first computation of a set of two-loop seven-scale master integrals, our results provide valuable input for analytic studies of scattering amplitudes in processes involving massive particles and a large number of kinematic scales.

Physics↗
DOE OSTI2020

Assembly of Disordered Cocontinuous Morphologies by Multiblock Copolymers with Random Block Sequence and Length Dispersity

Among the wide variety of nanostructures accessible by self-assembly of block copolymers, cocontinuous morphologies are of great interest for separations, catalysis, sensing, and energy storage due to their ability to integrate distinct properties of the constituent polymers. Furthermore, we investigate the tendency to form disordered cocontinuous phases of multiblock copolymers (MBCs) with random block sequence and length dispersity prepared by step-growth polymerization of pre-made telechelic polymers, specifically using thiol-Michael addition between diacrylate functionalized polystyrene (PS) and poly (D,L - lactide) (PLA) chains and a small molecule dithiol. In the phase diagram so established, disordered cocontinuous nanostructures occupy a wide window (≈ 25 % vol) at a modest segregation strength of χN = 15, close to the critical value for microphase separation. At a higher segregation strength of χN = 30, only a moderately wide (≈ 10 vol %) cocontinuous window is found, with MBCs instead tending to form ordered nanostructures. Although the breadth of the cocontinuous widow is less than that found in prior work on analogous randomly end-linked copolymer networks (RECNs), the MBC architecture offers offsetting advantages in terms of solution and melt processability.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Machine learning of superconducting critical temperature from Eliashberg theory

Abstract The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional superconductors, including the retardation of the interaction and the Coulomb pseudopotential, to predict the critical temperature T c . McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature within this theory, which depends on the electron–phonon spectral function α 2 F ( ω ). Here we show that modern machine-learning techniques can substantially improve these formulae, accounting for more general shapes of the α 2 F function. Using symbolic regression and the SISSO framework, together with a database of artificially generated α 2 F functions and numerical solutions of the Eliashberg equations, we derive a formula for T c that performs as well as Allen–Dynes for low- T c superconductors and substantially better for higher- T c ones. This corrects the systematic underestimation of T c while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen–Dynes formula for the prediction of higher-temperature superconductors.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Metal–bicarbonate ion pairing in alkaline aqueous solutions from multilevel embedded correlated wavefunction theory and molecular dynamics

In this work, we examine ion-pairing mechanisms of bicarbonates in alkaline aqueous solutions with the divalent metal ions most abundantly present in seawater, namely, Ca 2+ and Mg 2+ . We employ a rare-event enhanced sampling approach within first-principles molecular dynamics to explore regions of phase space spanning solvent-shared to contact ion pairs. Second-order Møller–Plesset perturbation theory (MP2) corrections are subsequently applied in an embedding framework (EMB) to refine the electronic structure of stationary states and associated reaction barriers along the free-energy profiles while retaining the extended solvent effects at the density functional theory (DFT) level. Ca 2+ –HCO 3 − was previously hypothesized to exist in a solvent-shared ion pair (SSHIP) by DFT studies with an endergonic contact ion pair (CIP) formation; however, our EMB-MP2 refinement of the DFT ion-pairing pathways reveals that Ca 2+ and HCO 3 − form a virtually barrier-free CIP in alkaline solutions, with even more energetic ease than the widely studied Ca–CO 3 ion pair. We find qualitative agreement between DFT and EMB-MP2 for Mg 2+ —unlike Ca 2+ , Mg 2+ refuses to shed its strong hydration shell, thereby preferring a SSHIP state with a significant activation barrier to crossover to the CIP forms—a trait reminiscent of ion pairing in Mg–CO 3 and closely related to the kinetic limitations underlying the famous subject of the dolomite problem. Our study highlights the importance of improved electronic structure descriptions of liquids, modeled as a condensed phase of matter lacking in long-range crystalline order. It also strongly suggests that Ca 2+ –HCO 3 − CIPs are likely precursors involved in prenucleation of CaCO 3 mineral formation in seawater.

Sharma, Vidushi [Princeton Plasma Physics Laborato↗
NASA NTRS2003

On the Applicability of High-frequency Approximations to Lilley's Equation

Three forms of the high-frequency asymptotic Green's function for Lilley's equation are reviewed and compared to the exact solution over wide range of Strouhal numbers. The asymmetric approximation, which applies to sources away form the jet axis, and the quasi-symmetric approximation, which is arrived at by making a near-axis source assumption, are both obtained for parallel round jets from a formal Fourier-transform solution. The ray-theory solution, which is the only high-frequency approximation that can be applied to more general mean flows, follows from a WKB ansatz and is shown to be closely related to the asymmetric approximation. The comparisons show that the best overall prediction of the exact Green's function is given by the asymmetric approximation which remains accurate down to a Strouhal number of 1/2. The close relationship between the asymmetric and ray-theory approximations suggests that the high-frequency asymptotic Green's function for more general mean flows would be similarly successful.

Wundrow, David W.↗
NASA NTRS1989

OPTICON: Pro-Matlab software for large order controlled structure design

A software package for large order controlled structure design is described and demonstrated. The primary program, called OPTICAN, uses both Pro-Matlab M-file routines and selected compiled FORTRAN routines linked into the Pro-Matlab structure. The program accepts structural model information in the form of state-space matrices and performs three basic design functions on the model: (1) open loop analyses; (2) closed loop reduced order controller synthesis; and (3) closed loop stability and performance assessment. The current controller synthesis methods which were implemented in this software are based on the Generalized Linear Quadratic Gaussian theory of Bernstein. In particular, a reduced order Optimal Projection synthesis algorithm based on a homotopy solution method was successfully applied to an experimental truss structure using a 58-state dynamic model. These results are presented and discussed. Current plans to expand the practical size of the design model to several hundred states and the intention to interface Pro-Matlab to a supercomputing environment are discussed.

Peterson, Lee D.↗
NASA NTRS1996

Lattice Truss Structural Response Using Energy Methods

A deterministic methodology is presented for developing closed-form deflection equations for two-dimensional and three-dimensional lattice structures. Four types of lattice structures are studied: beams, plates, shells and soft lattices. Castigliano's second theorem, which entails the total strain energy of a structure, is utilized to generate highly accurate results. Derived deflection equations provide new insight into the bending and shear behavior of the four types of lattices, in contrast to classic solutions of similar structures. Lattice derivations utilizing kinetic energy are also presented, and used to examine the free vibration response of simple lattice structures. Derivations utilizing finite element theory for unique lattice behavior are also presented and validated using the finite element analysis code EAL.

Kenner, Winfred Scottson↗
DOE OSTI2025

In situ monitoring of lanthanide reactions with oxide species via combined absorption spectroscopy and electrochemical methods

Molten salts for engineering scale applications of spent nuclear fuel pyrochemical processing will inevitably have some level of oxygen impurities which can form insoluble oxide and oxychloride species with fission products. This work demonstrates real-time concentration monitoring of two lanthanide (Ln3+) fission products, Nd3+ and Pr3+, and their reactions with oxygen (O2-) impurities to form insoluble products in LiCl-NaCl-KCl eutectic salt. Combined absorption spectroscopy and electrochemical testing were used to track lanthanide concentrations. O2- impurity levels were controlled by adding Li2O to lanthanide-salt solutions. After the introduction of O2- impurities, Ln3+ concentrations were monitored via time-resolved absorption spectroscopy. Concentrations of both Ln3+ species in solution decreased with time as insoluble products formed. The initial impurity concentration controlled whether insoluble products were predominantly oxychlorides or mixtures of oxychlorides and oxides. However, absorption spectroscopy is limited for weakly absorbing species, such as Pr3+, and under conditions of high impurity concentrations where solutions can be turbid. To circumvent this limitation, the concentrations of Pr3+ were monitored with cyclic voltammetry (CV). Reaction rates and extent of Pr3+ removal from solution as monitored by CV agreed closely with results found from spectroscopic monitoring. This demonstrates that simultaneous electrochemical testing complements the capabilities of absorption spectroscopy to monitor reactions of fission products in molten salts.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Silicon Anodes with Improved Calendar Life Enabled By Multivalent Additives

Silicon is widely recognized as the most promising upgrade for graphite anodes due to its much higher capacity, natural abundance, and ability to be directly applied in the slurry-based, roll-to-roll production lines. However, in addition to the fast capacity decay, silicon anodes also suffer from inferior calendar life in practical applications due to the unstable solid-electrode interface (SEI). Until now, strategies to effectively improve the calendar life by tailored SEIs remain largely unclear, especially in high-Si content, zero-graphite anodes. Here, silicon anodes with superior calendar life are developed by adding small concentrations of multivalent salts into the baseline electrolyte. The Ca additive reacts with the F ions in the electrolyte, forming a layer of nanocrystalline CaF 2 that is closely coated around the silicon particles. The CaF 2 -enabled new SEI is strong and dense, which effectively protects the silicon core from side reactions, leading to lower capacity decay after calendar aging at high voltage. More importantly, the Ca additive is effective universally for all available commercial silicon or SiO sources. This study provides a feasible and low-cost solution for developing silicon anodes with long calendar life, paving the way towards commercially viable silicon anodes.

25 ENERGY STORAGE↗
NASA NTRS1938

Two-dimensional subsonic compressible flow past elliptic cylinders

The method of Poggi is used to calculate, for perfect fluids, the effect of compressibility upon the flow on the surface of an elliptic cylinder at zero angle of attack and with no circulation. The result is expressed in a closed form and represents a rigorous determination of the velocity of the fluid at the surface of the obstacle insofar as the second approximation is concerned. Comparison is made with Hooker's treatment of the same problem according to the method of Janzen and Rayleight and it is found that, for thick elliptic cylinders, the two methods agree very well. The labor of computation is considerably reduced by the present solution.

Kaplan, Carl↗