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Basic differential equations governing compressible turbulent boundary layer flow are reviewed, including conservation of mass and energy, momentum equations derived from Navier-Stokes equations, and equations of state. Closure procedures were broken down into: (1) simple or zeroth-order methods, (2) first-order or mean field closure methods, and (3) second-order or mean turbulence field methods.
A mathematical method for determining the flow field about power-law bodies in hypersonic flow conditions is developed. The second-order solutions, which reflect the effects of the second-order terms in the equations, are obtained by applying the method of small perturbations in terms of body slenderness parameter to the zeroth-order solutions. The method is applied by writing each flow variable as the sum of a zeroth-order and a perturbation function, each multiplied by the axial variable raised to a power. The similarity solutions are developed for infinite Mach number. All results obtained are for no flow through the body surface (as a boundary condition), but the derivation indicates that small amounts of blowing or suction through the wall can be accommodated.
This final technical report summarizes the outcomes of a DOE-funded project on federated scientific machine learning (FL) under nonsmooth, nonconvex, and hierarchical optimization settings. The project develops new mathematical models, algorithms, and theoretical guarantees for decentralized stochastic, bilevel, and minimax optimization problems arising in DOE mission-relevant applications. A unified framework of randomized and zeroth-order federated optimization methods is introduced, providing provable convergence, communication efficiency, and sample-complexity guarantees. The report documents algorithmic design, theoretical analysis, and empirical validation of the proposed federated learning methods. The project also contributes to workforce development through graduate training and dissemination of results via publications and seminars.
Discussion of various zeroth-order approximations to s-wave scattering of electrons by hydrogen atoms below the first excitation threshold. The formalism previously developed by the author (1967, 1968) is applied to Feshbach operators to derive integro-differential equations, with the optical-potential set equal to zero, for the singlet and triplet cases. Phase shifts of s-wave scattering are computed in the zeroth-order approximation of the Feshbach operator method and in the static-exchange approximation. It is found that the convergence of numerical computations is faster in the former approximation than in the latter.
A theory is proposed for analyzing the inviscid interpretation of two streams in the case when the difference in total pressure between the streams is relatively small. A stream is considered which discharges from a nozzle or reservoir into a partially moving and partially stationary environment in such a way that the flows leave the solid boundaries in a tangential direction where the two streams first interact. The problem is solved by expanding in a small parameter related to the difference in total pressure between the streams, the zeroth-order solution is obtained by classical methods, and a technique similar to that employed in thin-airfoil theory is used to transfer the first-order boundary conditions to the zeroth-order boundary. A procedure is developed to transform the problem into one that can be solved by standard techniques of the theory of sectionally analytic functions. Solutions are obtained for flows with and without free streamlines, and the general theory is applied to several specific flow configurations.
A new analysis procedure is presented which solves for the flow variables of an incompressible-flow annular pressure seal in which the rotor has a large static displacement from the centered position. The analysis begins with a set of governing equations based on a turbulent bulk-flow model and Moody's friction equation. No simplification of these bulk-flow equations is required for the solution procedure. Perturbation of the flow variables yields a set of zeroth and first-order equations. The zeroth-order equations (which model the large static displacement) are integrated by means of an efficient new method which employs fast Fourier transforms. Further integration of the zeroth-order pressures yields the hydrodynamic reactive force. Predictions for the hydrodynamic forces from this analysis procedure are in excellent agreement with available experimental results.
The paper considers a thin elastic bidirectional symmetric composite laminate subjected to a uniform axial strain. All interlaminar stresses are determined by a zeroth-order perturbation analysis of the governing equations. The method of solution considers the laminate to be composed of an interior region (removed from the free edge) and a boundary layer region. The interior region is analyzed by dropping terms related to powers of the thickness-to-width ratio in the dimensionless elasticity equations. The boundary layer region is studied by introducing a stretching transformation to the governing equations. A uniform solution is then formed by satisfying Prandtl's matching principle of perturbation theory. Results for a four-layer bidirectional graphite-epoxy laminate are compared with existing finite-difference results. It is shown that the perturbation analysis provides more insight into the stress distribution near the free edge.
A modified coupled pair functional (CPF) method is presented for the configuration interaction problem that dramatically improves properties for cases where the Hartree-Fock reference configuration is not a good zeroth-order wave function description. It is shown that the tendency for CPF to overestimate the effect of higher excitations arises from the choice of the geometric mean for the partial normalization denominator. The modified method is demonstrated for ground state dipole moment calculations of the NiH, CuH, and ZnH transition metal hydrides, and compared to singles-plus-doubles configuration interaction and the Ahlrichs et al. (1984) CPF method.
A stochastic feedback control law for a space vehicle performing an aeroassisted plane-change maneuver is developed. The stochastic control law is designed to minimize the energy loss while taking into consideration the uncertainty in the atmospheric density. The solution is based on expansion of the stochastic Hamilton-Jacobi-Bellman equation (or dynamic programming) about a zeroth-order known integrable solution. The resulting guidance law is expressed as a series expansion in the noise power spectral densities. A numerical example indicates the potential improvement of this method.
The state of contaminant molecules, the deposition rate on key surfaces, and the heat transfer rate were estimated by the use of a zeroth-order approximation. Optical surfaces of infrared telescopes cooled to about 20 K should be considered to be covered with at least several deposition layers of condensible molecules without any contamination controls. The effectiveness of the purge gas method of contamination controls was discussed. This method attempts to drive condensible molecules from the telescope tube by impacts with a purge gas in the telescope tube. For this technique to be sufficiently effective, the pressure of the purge gas must be more than 2 x .000001 torr. The influence caused by interactions of the purged gas with the particulate contaminants was found to slightly increase the resident times of the particulate contaminants within the telescope field of view.
Artificial boundary conditions (BCs) play a ubiquitous role in numerical simulations of transport phenomena in several diverse fields, such as fluid dynamics, electromagnetism, acoustics, geophysics, and many more. They are essential for accurately capturing the behavior of physical systems whenever the simulation domain is truncated for computational efficiency purposes. Ideally, an artificial BC would allow relevant information to enter or leave the computational domain without introducing artifacts or unphysical effects. Boundary conditions designed to control spurious wave reflections are referred to as nonreflective boundary conditions (NRBCs). Another approach is given by the perfectly matched layers (PMLs), in which the computational domain is extended with multiple dampening layers, where outgoing waves are absorbed exponentially in time. Here, in this work, the definition of PML is revised in the context of the lattice Boltzmann method. The impact of adopting different types of BCs at the edge of the dampening zone is evaluated and compared, in terms of both accuracy and computational costs. It is shown that for sufficiently large buffer zones, PMLs allow stable and accurate simulations even when using a simple zeroth-order extrapolation BC. Moreover, employing PMLs in combination with NRBCs potentially offers significant gains in accuracy at a modest computational overhead, provided the parameters of the BC are properly tuned to match the properties of the underlying fluid flow.
In this paper, we present a novel derivative-free framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning to quantum computing involve settings where only stochastic function values are obtained via a zeroth-order oracle, which has no available gradient information and necessitates the usage of derivative-free optimization methodologies. Our approach includes estimating gradients using stochastic function evaluations and integrating adaptive sampling techniques to control the accuracy in these stochastic approximations. Our framework encapsulates several gradient estimation techniques, including standard finite-difference, Gaussian smoothing, sphere smoothing, randomized coordinate finite-difference, and randomized subspace finite-difference methods. We provide theoretical convergence guarantees for our framework and analyze the worst-case iteration and sample complexities associated with each gradient estimation method. Finally, we demonstrate the empirical performance of the methods on logistic regression and nonlinear least squares problems.
The Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST) presents an unprecedented opportunity to advance the search for astrobiologically relevant data across diverse astronomical environments. Through its extensive imaging capabilities, LSST will enhance our inventory of Solar System objects and assess their potential astrobiological conditions. Beyond the Solar System, LSST will survey billions of stars, contributing to the detection of exoplanets, characterizing planetary atmospheres at a zeroth-order level, identifying transient phenomena, and exploring conditions within habitable zones. In this paper, we highlight LSST’s unique contribution to astrobiology, complementing other missions and expanding the search for life beyond Earth in the coming decades.
Two global-chemistry models for oxidative deposition of jet fuels are evaluated by integrating them into a Computational Fluid Dynamics with Chemistry (CFDC) code. A previously developed two-step global-chemistry model was found to be insufficient to describe the thermal-oxidation and -deposition rates associated with a Jet-A fuel. A new global-chemistry model has been developed systematically based on observations from flowing and static experiments. The global-autoxidation reaction is modified such that the reaction rate becomes zeroth-order with respect to the dissolved oxygen concentration. The generation of deposit-forming precursor is coupled with the autoxidation reaction by introducing a radical species ROO. A formulation for the sticking probability has also been developed. Deposition profiles are well represented by this new model under a variety of temperature and flow conditions. The model correctly predicts the changes in magnitude and spatial location of the deposition peak due to changes in flow. The CFDC model, which is designed for flowing systems, has been extended to static experiments. The model incorporates a non-depleting species F(sub s) representing all non-oxygen compounds responsible for deposition. Static experiments were found to provide a useful and inexpensive method for estimating the concentration of F(sub s) in the fuel.
The second harmonic (SH) of an axicon generated Bessel-Gauss beam is created through the nonlinear interaction of photons with a crystal, resulting in the energy doubling of the output photons. In this work, we show experimentally that in addition to frequency doubling, the SH of Bessel-Gauss beams under asymmetric aberrations from an elliptic axicon exhibit intriguing beam formation. Particularly, the central region of the SH beam profile is composed of two central spots of various geometries surrounded by nested ellipses; one of which is the configuration of two central gamma dots with similar radius knotted by nested ellipses for a zeroth-order Bessel-Gauss pump. These SH beams consistently maintain their spatial profile throughout propagation, reminiscing the behavior of screw dislocations in wave patterns. Our numerical simulations produce beam dynamics consistent with that of experiments and further implicate the remarkable interweaving of bright spots with dark vortices. This is especially noticeable when the beams dynamically oscillate along the optical axis, resulting in the genesis of spatially polarized beams with a knotted framework. The insights gained from our study establish a novel paradigm for exploring interactions of Bessel-like beams with vortex dynamics. This, in turn, has the potential to spark innovations in optical applications, fostering new methodologies to harness and manipulate complex light structures. Our experimental findings could spur new methods of generating logical states of light and new opportunities for material processing control. Published by the American Physical Society 2025
Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.
Electron binding energies in neutral atoms have been calculated relativistically, with the requirement of complete relaxation. Hartree-Fock-Slater wave functions served as zeroth-order eigenfunctions to compute the expectation of the total Hamiltonian. A first-order correction to the local approximation was thus included. Quantum-electrodynamic corrections were made. For all elements with atomic numbers ranging from 2 to 106, the following quantities are listed: total energies, electron kinetic energies, electron-nucleus potential energies, electron-electron potential energies consisting of electrostatic and Breit interaction (magnetic and retardation) terms, and vacuum polarization energies. Binding energies including relaxation are listed for all electrons in all atoms over the indicated range of atomic numbers. A self-energy correction is included for the 1s, 2s, and 2p(1/2) levels. Results for selected atoms are compared with energies calculated by other methods and with experimental values.
Asymptotic solutions for the reflectivity of a very rough surface are obtained in closed form up to second order in variance of slope. The zeroth-order solution gives the reflectivity of a specular surface and the sum of the higher-order terms yields the net effect of the rough surface. Within the common regions of validity of small rms slope and incident angle not close to grazing, the result agrees very well with those obtained with numerical methods.